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Calculated Shard: 152 (from laksa118)

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curl -X POST \
  'http://laksa152.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical, ml_categories_json AS ml_categories_json, ml_types_json AS ml_types_json, ml_intent_types_json AS ml_intent_types_json, meta_language AS meta_language, attrs_author AS attrs_author, ifNull(toUnixTimestamp(attrs_publish_time), 0) AS attrs_publish_time, ifNull(toUnixTimestamp(attrs_original_publish_time), 0) AS attrs_original_publish_time, ifNull(attrs_is_republished, 0) AS attrs_is_republished, ifNull(attrs_nr_words, 0) AS attrs_nr_words, ifNull(attrs_boilerpipe_nr_words, 0) AS attrs_boilerpipe_nr_words, ifNull(body_ext_links_number, 0) AS body_ext_links_number, ifNull(body_int_links_number, 0) AS body_int_links_number, ifNull(meta_nofollow, 0) AS meta_nofollow, ifNull(meta_noarchive, 0) AS meta_noarchive, ifNull(props_was_rendered, 0) AS props_was_rendered, ifNull(src_redirect, \'\') AS src_redirect, ifNull(download_time_msec, 0) AS download_time_msec, ifNull(download_ttfb_msec, 0) AS download_ttfb_msec, ifNull(download_size, 0) AS download_size FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Specific_heat_capacity\')), getAhrefsUnparsedNoserviceFromURL(\'https://en.wikipedia.org/wiki/Specific_heat_capacity\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity","crawl_time":1776314929,"first_indexed_time":1375989344,"http_code":200,"src_unparsed":"org,wikipedia!en,\/wiki\/Specific_heat_capacity s443","src_root_hash":"17790707453426894952","history_drop_reason":null,"meta_title":"Specific heat capacity - Wikipedia","meta_descriptions":[],"attrs_boilerpipe_text":"Specific heat capacity\nOther names\nSpecific heat\nCommon symbols\nc\nSI unit\nJ⋅kg\n−1\n⋅K\n−1\nIn\nSI base units\nm\n2\n⋅K\n−1\n⋅s\n−2\nIntensive\n?\nYes\nDimension\nL\n2\n⋅T\n−2\n⋅Θ\n−1\nIn\nthermodynamics\n, the\nspecific heat capacity\n(symbol\nc\n) of a substance is the amount of\nheat\nthat must be added to one unit of mass of the substance in order to cause an increase of one unit in\ntemperature\n. It is also referred to as\nmassic heat capacity\nor as the\nspecific heat.\nMore formally it is the\nheat capacity\nof a sample of the substance divided by the\nmass\nof the sample.\n[\n1\n]\nThe\nSI\nunit of specific heat capacity is\njoule\nper\nkelvin\nper\nkilogram\n, J⋅kg\n−1\n⋅K\n−1\n.\n[\n2\n]\nFor example, the heat required to raise the temperature of\n1 kg\nof\nwater\nby\n1 K\nis\n4184 joules\n, so the specific heat capacity of water is\n4184 J⋅kg\n−1\n⋅K\n−1\n.\n[\n3\n]\nSpecific heat capacity often varies with temperature, and is different for each\nstate of matter\n. Liquid water has one of the highest specific heat capacities among common substances, about\n4184 J⋅kg\n−1\n⋅K\n−1\nat 20 °C, but that of ice, just below 0 °C, is only\n2093 J⋅kg\n−1\n⋅K\n−1\n. The specific heat capacities of\niron\n,\ngranite\n, and\nhydrogen\ngas are about 449 J⋅kg\n−1\n⋅K\n−1\n, 790 J⋅kg\n−1\n⋅K\n−1\n, and 14300 J⋅kg\n−1\n⋅K\n−1\n, respectively.\n[\n4\n]\nWhile the substance is undergoing a\nphase transition\n, such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature.\nThe specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity\nat constant pressure\n) than when it is heated in a closed vessel that prevents expansion (specific heat capacity\nat constant volume\n). These two values are usually denoted by\nand\n, respectively; their quotient\nis the\nheat capacity ratio\n.\nThe term\nspecific heat\nmay also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C;\n[\n5\n]\nmuch in the fashion of\nspecific gravity\n. Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of\nmoles\n, one gets the\nmolar heat capacity\ninstead, whose SI unit is joule per kelvin per mole, J⋅mol\n−1\n⋅K\n−1\n. If the amount is taken to be the\nvolume\nof the sample (as is sometimes done in engineering), one gets the\nvolumetric heat capacity\n, whose SI unit is joule per kelvin per\ncubic meter\n, J⋅m\n−3\n⋅K\n−1\n.\nDiscovery of specific heat\n[\nedit\n]\nJoseph Black\nOne of the first scientists to use the concept was\nJoseph Black\n, an 18th-century medical doctor and professor of medicine at\nGlasgow University\n. He measured the specific heat capacities of many substances, using the term\ncapacity for heat\n.\n[\n6\n]\nIn 1756 or soon thereafter, Black began an extensive study of heat.\n[\n7\n]\nIn 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by\nDaniel Gabriel Fahrenheit\non behalf of Dutch physician\nHerman Boerhaave\n. For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: \"Quicksilver [mercury] ... has less capacity for the matter of heat than water.\"\n[\n8\n]\n[\n9\n]\nThe specific heat capacity of a substance, usually denoted by\nor\n, is the heat capacity\nof a sample of the substance, divided by the mass\nof the sample:\n[\n10\n]\nwhere\nrepresents\nthe amount of heat needed to uniformly raise the temperature of the sample by a small increment\n.\nLike the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature\nof the sample and the\npressure\napplied to it. Therefore, it should be considered a function\nof those two variables.\nThese parameters are usually specified when giving the specific heat capacity of a substance. For example, \"Water (liquid):\n= 4187 J⋅kg\n−1\n⋅K\n−1\n(15 °C).\"\n[\n11\n]\nWhen not specified, published values of the specific heat capacity\ngenerally are valid for some\nstandard conditions for temperature and pressure\n.\nHowever, the dependency of\non starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier\nand approximates the specific heat capacity by a constant\nsuitable for those ranges.\nSpecific heat capacity is an\nintensive property\nof a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier \"specific\" in front of an extensive property often indicates an intensive property derived from it.\n[\n12\n]\n)\nThe injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and\/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure\nand starting temperature\n. Two particular choices are widely used:\nThe value of\nis always less than the value of\nfor all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the\nheat capacity ratio\nof gases is typically between 1.3 and 1.67.\n[\n13\n]\nThe specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale.\nThe specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops.\nThe specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a\nphase change\n, such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement.\nThe specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a\ncalorimeter\n, and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as\ndifferential scanning calorimetry\n.\n[\n14\n]\n[\n15\n]\nGraph of temperature of phases of water heated from\n−100 °C\nto\n200 °C\n– the dashed line example shows that melting and heating\n1 kg\nof ice at\n−50 °C\nto water at\n40 °C\nneeds\n600 kJ\nThe specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container.  On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the\ncoefficient of thermal expansion\nand the\ncompressibility\nof the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.\n[\ncitation needed\n]\nInternational system\n[\nedit\n]\nThe SI unit for specific heat capacity is joule per kelvin per kilogram\n⁠\nJ\n\/\nkg⋅K\n⁠\n, J⋅K\n−1\n⋅kg\n−1\n.  Since an increment of temperature of one\ndegree Celsius\nis the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J\/(kg⋅°C).  Sometimes the\ngram\nis used instead of kilogram for the unit of mass: 1 J⋅g\n−1\n⋅K\n−1\n= 1000 J⋅kg\n−1\n⋅K\n−1\n.\nThe specific heat capacity of a substance (per unit of mass) has\ndimension\nL\n2\n⋅Θ\n−1\n⋅T\n−2\n, or (L\/T)\n2\n\/Θ. Therefore, the SI unit J⋅kg\n−1\n⋅K\n−1\nis equivalent to\nmetre\nsquared per\nsecond\nsquared per\nkelvin\n(m\n2\n⋅K\n−1\n⋅s\n−2\n).\nImperial engineering units\n[\nedit\n]\nProfessionals in\nconstruction\n,\ncivil engineering\n,\nchemical engineering\n, and other technical disciplines, especially in the\nUnited States\n, may use\nEnglish Engineering units\nincluding the\npound\n(lb = 0.45359237 kg) as the unit of mass, the\ndegree Fahrenheit\nor\nRankine\n(°R =\n⁠\n5\n\/\n9\n⁠\nK, about 0.555556 K) as the unit of temperature increment, and the\nBritish thermal unit\n(BTU ≈ 1055.056 J),\n[\n16\n]\n[\n17\n]\nas the unit of heat.\nIn those contexts, the unit of specific heat capacity is BTU\/lb⋅°R, or 1\n⁠\nBTU\n\/\nlb⋅°R\n⁠\n= 4186.68\n⁠\nJ\n\/\nkg⋅K\n⁠\n.\n[\n18\n]\nThe BTU was originally defined so that the average specific heat capacity of water would be 1 BTU\/lb⋅°F.\n[\n19\n]\nNote the value's similarity to that of the calorie - 4187 J\/kg⋅°C ≈ 4184 J\/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C.\nIn chemistry, heat amounts were often measured in\ncalories\n. Confusingly, there are two common units with that name, respectively denoted\ncal\nand\nCal\n:\nthe\nsmall calorie\n(\ngram-calorie, cal\n) is 4.184 J exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal\/(°C⋅g).\nThe\ngrand calorie\n(\nkilocalorie, kilogram-calorie, food calorie, kcal, Cal\n) is 1000 small calories, 4184 J exactly. It was defined so that the specific heat capacity of water would be 1 Cal\/(°C⋅kg).\nWhile these units are still used in some contexts (such as kilogram calorie in\nnutrition\n), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually:\n1 \n⁠\ncal\n\/\n°C⋅g\n⁠\n= 1 \n⁠\nCal\n\/\n°C⋅kg\n⁠\n= 1 \n⁠\nkcal\n\/\n°C⋅kg\n⁠\n= 4184 \n⁠\nJ\n\/\nkg⋅K\n⁠\n[\n20\n]\n= 4.184 \n⁠\nkJ\n\/\nkg⋅K\n⁠\n.\nNote that while cal is\n1\n⁄\n1000\nof a Cal or kcal, it is also per\ngram\ninstead of\nkilo\ngram\n: ergo, in either unit, the specific heat capacity of water is approximately 1.\nThe temperature of a sample of a substance reflects the average\nkinetic energy\nof its constituent particles (atoms or molecules) relative to its center of mass.  However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the\nequipartition theorem\n.\nStatistical mechanics\npredicts that at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy, unless multiple electronic states are accessible at room temperature (such is the case for atomic fluorine).\n[\n21\n]\nThus, the\nheat capacity per mole\nat room temperature is the same for all of the noble gases as well as for many other atomic vapors.  More precisely,\nand\n, where\nis the\nideal gas unit\n(which is the product of\nBoltzmann conversion constant\nfrom\nkelvin\nmicroscopic energy unit to the macroscopic energy unit\njoule\n, and the\nAvogadro number\n).\nTherefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional)\natomic weight\n. That is, approximately,\nFor the noble gases, from helium to xenon, these computed values are\nGas\nHe\nNe\nAr\nKr\nXe\n4.00\n20.17\n39.95\n83.80\n131.29\n(J⋅K\n−1\n⋅kg\n−1\n)\n\n3118\n618.3\n312.2\n148.8\n94.99\n(J⋅K\n−1\n⋅kg\n−1\n)\n\n5197\n1031\n520.3\n248.0\n158.3\nA polyatomic gas molecule can store energy in additional degrees of freedom. Its kinetic energy contributes to the heat capacity in the same way as monatomic gases, but there are also contributions from the\nrotations\nof the molecule and vibration of the atoms relative to each other (including internal\npotential energy\n).\nThe heat capacity may also have contribution from\nexcited electronic states\nfor molecules with a sufficiently small energy gap between the ground state and the excited state, such as in\nNO\n.\n[\n22\n]\nFor a few systems, quantum spin statistics can also be important contributions to the heat capacity, even at room temperature. The analysis of the heat capacity of\nH\n2\ndue to ortho\/para separation,\n[\n23\n]\nwhich arises from\nnuclear spin\nstatistics, has been referred to as \"one of the great triumphs of post-quantum mechanical statistical mechanics.\"\n[\n24\n]\nThese extra\ndegrees of freedom\nor \"modes\" contribute to the specific heat capacity of the substance. Namely, when energy is introduced into a gas with polyatomic molecules, only part of it will increase their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number of degrees of freedom of the molecules.\n[\n25\n]\n[\n26\n]\n[\n27\n]\nQuantum statistical mechanics\npredicts that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (quanta), and that this affects the system's thermodynamic properties.  Depending on the temperature, the average energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom.  Those modes are said to be \"frozen out\".  In that case, the specific heat capacity of the substance increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat.\nFor example, the molar heat capacity of\nnitrogen\nN\n2\nat constant volume is\n(at 15 °C, 1 atm), which is\n.\n[\n28\n]\nThat is the value expected from the\nEquipartition Theorem\nif each molecule had 5 kinetic degrees of freedom.  These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms.  Because of those two extra degrees of freedom, the specific heat capacity\nof\nN\n2\n(736 J⋅K\n−1\n⋅kg\n−1\n) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K\n−1\n⋅kg\n−1\n), by a factor of\n⁠\n5\n\/\n3\n⁠\n. The vibrational and electronic degrees of freedom do not contribute significantly to the heat capacity in this case, due to the relatively large energy level gaps for both vibrational and electronic excitation in this molecule.\nThis value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C.  In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still \"frozen out\".  At about that temperature, those modes begin to \"un-freeze\" as vibrationally excited states become accessible. As a result\nstarts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K\n−1\n⋅mol\n−1\nat 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.\n[\n29\n]\nThe last value corresponds almost exactly to the value predicted by the Equipartition Theorem, since in the high-temperature limit the theorem predicts that the vibrational degree of freedom contributes twice as much to the heat capacity as any one of the translational or rotational degrees of freedom.\nDerivations of heat capacity\n[\nedit\n]\nRelation between specific heat capacities\n[\nedit\n]\nStarting from the\nfundamental thermodynamic relation\none can show,\nwhere\nA derivation is discussed in the article\nRelations between specific heats\n.\nFor an\nideal gas\n, if\nis expressed as\nmolar\ndensity in the above equation, this equation reduces simply to\nMayer\n's relation,\nwhere\nand\nare\nintensive property\nheat capacities expressed on a per mole basis at constant pressure and constant volume, respectively.\nSpecific heat capacity\n[\nedit\n]\nThe specific heat capacity of a material on a per mass basis is\nwhich in the absence of phase transitions is equivalent to\nwhere\nis the heat capacity of a body made of the material in question,\nis the mass of the body,\nis the volume of the body, and\nis the density of the material.\nFor gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include\nisobaric\n(constant pressure,\n) or\nisochoric\n(constant volume,\n) processes. The corresponding specific heat capacities are expressed as\nA related parameter to\nis\n, the\nvolumetric heat capacity\n. In engineering practice,\nfor solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript\n, as\n. Of course, from the above relationships, for solids one writes\nFor pure homogeneous\nchemical compounds\nwith established\nmolecular or molar mass\nor a\nmolar quantity\nis established, heat capacity as an\nintensive property\ncan be expressed on a per\nmole\nbasis instead of a per mass basis by the following equations analogous to the per mass equations:\nwhere\nn\n= number of moles in the body or\nthermodynamic system\n. One may refer to such a\nper mole\nquantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis.\nPolytropic heat capacity\n[\nedit\n]\nThe\npolytropic\nheat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change\nThe most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between\n1\nand the adiabatic exponent (\nγ\nor\nκ\n)\nDimensionless heat capacity\n[\nedit\n]\nThe\ndimensionless\nheat capacity of a material is\nwhere\nC\nis the heat capacity of a body made of the material in question (J\/K)\nn\nis the\namount of substance\nin the body (\nmol\n)\nR\nis the\ngas constant\n(J⋅K\n−1\n⋅mol\n−1\n)\nN\nis the number of molecules in the body. (dimensionless)\nk\nB\nis the\nBoltzmann constant\n(J⋅K\n−1\n)\nAgain,\nSI\nunits shown for example.\nIn the\nIdeal gas\narticle, dimensionless heat capacity\nis expressed as\n.\nHeat capacity at absolute zero\n[\nedit\n]\nFrom the definition of\nentropy\nthe absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature\nT\nf\nThe heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the\nthird law of thermodynamics\n. One of the strengths of the\nDebye model\nis that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached.\nThe theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3\nR\n, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas.\nThe Dulong–Petit limit results from the\nequipartition theorem\n, and as such is only valid in the classical limit of a\nmicrostate continuum\n, which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at\nstandard ambient temperature\n, quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3\nR\nper mole of\natoms\nin the solid, although in molecular solids, heat capacities calculated\nper mole of molecules\nin molecular solids may be more than 3\nR\n. For example, the heat capacity of water ice at the melting point is about 4.6\nR\nper mole of molecules, but only 1.5\nR\nper mole of atoms. The lower than 3\nR\nnumber \"per atom\" (as is the case with diamond and beryllium) results from the \"freezing out\" of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3\nR\nper mole of atoms of the Dulong–Petit theoretical maximum.\nFor a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of\nphonons\n. See\nDebye model\n.\nTheoretical estimation\n[\nedit\n]\nThe path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below.\nWater (liquid): CP = 4185.5 J⋅K\n−1\n⋅kg\n−1\n(15 °C, 101.325 kPa)\nWater (liquid): CVH = 74.539 J⋅K\n−1\n⋅mol\n−1\n(25 °C)\nFor liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However,  different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr).\n[\nnotes 1\n]\nRelation between heat capacities\n[\nedit\n]\nMeasuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see\ncoefficient of thermal expansion\nand\ncompressibility\n). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws.\nThe\nheat capacity ratio\n, or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.\nFor an\nideal gas\n, evaluating the partial derivatives above according to the\nequation of state\n, where\nR\nis the\ngas constant\n, for an ideal gas\n[\n30\n]\nSubstituting\nthis equation reduces simply to\nMayer\n's relation:\nThe differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.\nSpecific heat capacity\n[\nedit\n]\nThe specific heat capacity of a material on a per mass basis is\nwhich in the absence of phase transitions is equivalent to\nwhere\nis the heat capacity of a body made of the material in question,\nis the mass of the body,\nis the volume of the body,\nis the density of the material.\nFor gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include\nisobaric\n(constant pressure,\n) or\nisochoric\n(constant volume,\n) processes. The corresponding specific heat capacities are expressed as\nFrom the results of the previous section, dividing through by the mass gives the relation\nA related parameter to\nis\n, the\nvolumetric heat capacity\n. In engineering practice,\nfor solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript\n, as\n. Of course, from the above relationships, for solids one writes\nFor pure\nhomogeneous\nchemical compounds\nwith established\nmolecular or molar mass\n, or a\nmolar quantity\n, heat capacity as an\nintensive property\ncan be expressed on a per-\nmole\nbasis instead of a per-mass basis by the following equations analogous to the per mass equations:\nwhere\nn\nis the number of moles in the body or\nthermodynamic system\n. One may refer to such a per-mole quantity as\nmolar heat capacity\nto distinguish it from specific heat capacity on a per-mass basis.\nPolytropic heat capacity\n[\nedit\n]\nThe\npolytropic\nheat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change:\nThe most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (\nγ\nor\nκ\n).\nDimensionless heat capacity\n[\nedit\n]\nThe\ndimensionless\nheat capacity of a material is\nwhere\nis the heat capacity of a body made of the material in question (J\/K),\nn\nis the\namount of substance\nin the body (\nmol\n),\nR\nis the\ngas constant\n(J\/(K⋅mol)),\nN\nis the number of molecules in the body (dimensionless),\nk\nB\nis the\nBoltzmann constant\n(J\/(K⋅molecule)).\nIn the\nideal gas\narticle, dimensionless heat capacity\nis expressed as\nand is related there directly to half the number of degrees of freedom per particle.  This holds true for quadratic degrees of freedom, a consequence of the\nequipartition theorem\n.\nMore generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the\ndimensionless entropy\nper particle\n, measured in\nnats\n.\nAlternatively, using base-2 logarithms,\nrelates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in\nbits\n.\n[\n31\n]\nHeat capacity at absolute zero\n[\nedit\n]\nFrom the definition of\nentropy\nthe absolute entropy can be calculated by integrating from zero to the final temperature\nT\nf\n:\nThermodynamic derivation\n[\nedit\n]\nIn theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an\nequation of state\nand an\ninternal energy function\n.\nState of matter in a homogeneous sample\n[\nedit\n]\nTo apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass\n.  Assume that the evolution of the system is always slow enough for the internal pressure\nand temperature\nbe considered uniform throughout.  The pressure\nwould be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air.\nThe state of the material can then be specified by three parameters: its temperature\n, the pressure\n, and its\nspecific volume\n, where\nis the volume of the sample.  (This quantity is the reciprocal\nof the material's\ndensity\n.)  Like\nand\n, the specific volume\nis an intensive property of the material and its state, that does not depend on the amount of substance in the sample.\nThose variables are not independent. The allowed states are defined by an\nequation of state\nrelating those three variables:\nThe function\ndepends on the material under consideration. The\nspecific internal energy\nstored internally in the sample, per unit of mass, will then be another function\nof these state variables, that is also specific of the material. The total internal energy in the sample then will be\n.\nFor some simple materials, like an\nideal gas\n, one can derive from basic theory the equation of state\nand even the specific internal energy\nIn general, these functions must be determined experimentally for each substance.\nConservation of energy\n[\nedit\n]\nThe absolute value of this quantity\nis undefined, and (for the purposes of thermodynamics) the state of \"zero internal energy\" can be chosen arbitrarily.  However, by the\nlaw of conservation of energy\n, any infinitesimal increase\nin the total internal energy\nmust be matched by the net flow of heat energy\ninto the sample, plus any net mechanical energy provided to it by  enclosure or surrounding medium on it.  The latter is\n, where\nis the change in the sample's volume in that infinitesimal step.\n[\n32\n]\nTherefore\nhence\nIf the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount\n, then the term\nis zero (no mechanical work is done).  Then, dividing by\n,\nwhere\nis the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume\nof the material.\nFor the heat capacity at constant pressure, it is useful to define the\nspecific enthalpy\nof the system as the sum\n. An infinitesimal change in the specific enthalpy will then be\ntherefore\nIf the pressure is kept constant, the second term on the left-hand side is zero, and\nThe left-hand side is the specific heat capacity at constant pressure\nof the material.\nConnection to equation of state\n[\nedit\n]\nIn general, the infinitesimal quantities\nare constrained by the equation of state and the specific internal energy function.  Namely,\nHere\ndenotes the (partial) derivative of the state equation\nwith respect to its\nargument, keeping the other two arguments fixed, evaluated at the state\nin question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state.  The constant-volume and constant-pressure changes are only two particular directions in this space.\nThis analysis also holds no matter how the energy increment\nis injected into the sample, namely by\nheat conduction\n, irradiation,\nelectromagnetic induction\n,\nradioactive decay\n, etc.\nRelation between heat capacities\n[\nedit\n]\nFor any specific volume\n, denote\nthe function that describes how the pressure varies with the temperature\n, as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at\n. Analogously, for any pressure\n, let\nbe the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at\n. Namely, those functions are such that\nand\nfor any values of\n.  In other words, the graphs of\nand\nare slices of the surface defined by the state equation, cut by planes of constant\nand constant\n, respectively.\nThen, from the\nfundamental thermodynamic relation\nit follows that\nThis equation can be rewritten as\nwhere\nboth depending on the state\n.\nThe\nheat capacity ratio\n, or adiabatic index, is the ratio\nof the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.\nCalculation from first principles\n[\nedit\n]\nThe\npath integral Monte Carlo\nmethod is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3\nR\n= 24.94 joules per kelvin per mole of atoms (\nDulong–Petit law\n,\nR\nis the\ngas constant\n). Low temperature approximations for both gases and solids at temperatures less than their characteristic\nEinstein temperatures\nor\nDebye temperatures\ncan be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material.\n[\n33\n]\nFor an\nideal gas\n, evaluating the partial derivatives above according to the\nequation of state\n, where\nR\nis the\ngas constant\n, for an ideal gas\n[\n34\n]\nSubstituting\nthis equation reduces simply to\nMayer\n's relation:\nThe differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.\nPhysics portal\nEnthalpy of fusion\n(latent heat of melting)\nEnthalpy of vaporization\n(latent heat of vaporization)\nFrenkel line\nHeat capacity ratio\nHeat equation\nHeat transfer coefficient\nHistory of thermodynamics\nJoback method\n(Estimation of heat capacities)\nLatent heat\nMaterial properties (thermodynamics)\nQuantum statistical mechanics\nR-value (insulation)\nStatistical mechanics\nTable of specific heat capacities\nThermal mass\nThermodynamic databases for pure substances\nThermodynamic equations\nVolumetric heat capacity\n^\nIUPAC\n,\nCompendium of Chemical Terminology\n, 5th ed. (the \"Gold Book\") (2025). Online version: (2006–) \"\nStandard Pressure\n\".\ndoi\n:\n10.1351\/goldbook.S05921\n.\n^\nHalliday, David; Resnick, Robert; Walker, Jearl (2001).\nFundamentals of Physics\n(6th ed.). New York, NY US:\nJohn Wiley & Sons\n.\n^\nOpen University (2008).\nS104 Book 3 Energy and Light\n, p. 59.\nThe Open University\n.\nISBN\n \n9781848731646\n.\n^\nOpen University (2008).\nS104 Book 3 Energy and Light\n, p. 179.\nThe Open University\n.\nISBN\n \n9781848731646\n.\n^\nEngineering ToolBox (2003).\n\"Specific Heat of some common Substances\"\n.\n^\n(2001):\nColumbia Encyclopedia\n, 6th ed.; as quoted by\nEncyclopedia.com\n. Columbia University Press. Accessed on 2019-04-11.\n^\nLaidler, Keith J. (1993).\nThe World of Physical Chemistry\n. Oxford University Press.\nISBN\n \n0-19-855919-4\n.\n^\nRamsay, William\n(1918).\nThe life and letters of Joseph Black, M.D\n. Constable. pp. \n38–\n39.\n^\nBlack, Joseph (1807). Robison, John (ed.).\nLectures on the Elements of Chemistry: Delivered in the University of Edinburgh\n. Vol. 1. Mathew Carey. pp. \n76–\n77.\n^\nWest, John B. (2014-06-15).\n\"Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases\"\n.\nAmerican Journal of Physiology. Lung Cellular and Molecular Physiology\n.\n306\n(12):\nL1057–\nL1063.\ndoi\n:\n10.1152\/ajplung.00020.2014\n.\nISSN\n \n1040-0605\n.\nPMID\n \n24682452\n.\n^\nInternational Bureau of Weights and Measures\n(2006),\nThe International System of Units (SI)\n(PDF)\n(8th ed.),\nISBN\n \n92-822-2213-6\n,\narchived\n(PDF)\nfrom the original on 2021-06-04\n, retrieved\n2021-12-16\n^\n\"Water – Thermal Properties\"\n. Engineeringtoolbox.com\n. Retrieved\n2021-03-29\n.\n^\nInternational Union of Pure and Applied Chemistry, Physical Chemistry Division.\n\"Quantities, Units and Symbols in Physical Chemistry\"\n(PDF)\n. Blackwell Sciences. p. 7.\nThe adjective specific before the name of an extensive quantity is often used to mean divided by mass.\n^\nLange's Handbook of Chemistry, 10th ed., page 1524.\n^\nQuick, C. R.; Schawe, J. E. K.; Uggowitzer, P. J.; Pogatscher, S. (2019-07-01).\n\"Measurement of specific heat capacity via fast scanning calorimetry—Accuracy and loss corrections\"\n.\nThermochimica Acta\n. Special Issue on occasion of the 65th birthday of Christoph Schick.\n677\n:\n12–\n20.\nBibcode\n:\n2019TcAc..677...12Q\n.\ndoi\n:\n10.1016\/j.tca.2019.03.021\n.\nISSN\n \n0040-6031\n.\n^\nPogatscher, S.; Leutenegger, D.; Schawe, J. E. K.; Uggowitzer, P. J.; Löffler, J. F. (September 2016).\n\"Solid–solid phase transitions via melting in metals\"\n.\nNature Communications\n.\n7\n(1) 11113.\nBibcode\n:\n2016NatCo...711113P\n.\ndoi\n:\n10.1038\/ncomms11113\n.\nISSN\n \n2041-1723\n.\nPMC\n \n4844691\n.\nPMID\n \n27103085\n.\n^\nKoch, Werner (2013).\nVDI Steam Tables\n(4 ed.). Springer. p. 8.\nISBN\n \n978-3-642-52941-2\n.\nPublished under the auspices of the\nVerein Deutscher Ingenieure\n(VDI).\n^\nCardarelli, Francois (2012).\nScientific Unit Conversion: A Practical Guide to Metrication\n. M.J. Shields (translation) (2 ed.). Springer. p. 19.\nISBN\n \n978-1-4471-0805-4\n.\n^\nFrom direct values: 1\n⁠\nBTU\n\/\nlb⋅°R\n⁠\n× 1055.06\n⁠\nJ\n\/\nBTU\n⁠\n× (\n⁠\n1\n\/\n0.45359237\n⁠\n)\n⁠\nlb\n\/\nkg\n⁠\nx\n⁠\n9\n\/\n5\n⁠\n⁠\n°R\n\/\nK\n⁠\n= 4186.82\n⁠\nJ\n\/\nkg⋅K\n⁠\n^\n°F=°R\n^\n°C=K\n^\nMcQuarrie, Donald A. (1973).\nStatistical Thermodynamics\n. New York, NY:\nUniversity Science Books\n. pp. \n83–\n85.\n^\n\"6.6: Electronic Partition Function\"\n.\nChemistry LibreTexts\n. 2020-11-26\n. Retrieved\n2024-12-16\n.\n^\nBonhoeffer, K.F.; Harteck, P. (1926).\n\"Über Para- und Orthowasserstoff\"\n.\nZ. Phys. Chem\n.\n4B\n:\n113–\n141.\ndoi\n:\n10.1515\/zpch-1929-0408\n.\n^\nMcQuarrie, Donald A. (1973).\nStatistical Thermodynamics\n. New York, NY: University Science Books. p. 107.\n^\nFeynman, R.,\nThe Feynman Lectures on Physics\n, Vol. 1, ch. 40, pp. 7–8\n^\nReif, F. (1965).\nFundamentals of statistical and thermal physics\n. McGraw-Hill. pp. \n253–254\n.\n^\nKittel, Charles; Kroemer, Herbert (2000).\nThermal physics\n. W. H. Freeman. p. 78.\nISBN\n \n978-0-7167-1088-2\n.\n^\nThornton, Steven T. and Rex, Andrew (1993)\nModern Physics for Scientists and Engineers\n, Saunders College Publishing\n^\nChase, M.W. Jr. (1998)\nNIST-JANAF Themochemical Tables, Fourth Edition\n, In\nJournal of Physical and Chemical Reference Data\n, Monograph 9, pages 1–1951.\n^\nYunus A. Cengel and Michael A. Boles,\nThermodynamics: An Engineering Approach\n, 7th Edition, McGraw-Hill, 2010,\nISBN\n \n007-352932-X\n.\n^\nFraundorf, P. (2003). \"Heat capacity in bits\".\nAmerican Journal of Physics\n.\n71\n(11): 1142.\narXiv\n:\ncond-mat\/9711074\n.\nBibcode\n:\n2003AmJPh..71.1142F\n.\ndoi\n:\n10.1119\/1.1593658\n.\nS2CID\n \n18742525\n.\n^\nFeynman, Richard,\nThe Feynman Lectures on Physics\n, Vol. 1, Ch. 45\n^\nS. Benjelloun, \"Thermodynamic identities and thermodynamic consistency of Equation of States\",\nLink to Archiv e-print\nLink to Hal e-print\n^\nCengel, Yunus A. and Boles, Michael A. (2010)\nThermodynamics: An Engineering Approach\n, 7th Edition, McGraw-Hill\nISBN\n \n007-352932-X\n.\nEmmerich Wilhelm & Trevor M. Letcher, Eds., 2010,\nHeat Capacities: Liquids, Solutions and Vapours\n, Cambridge, U.K.:Royal Society of Chemistry,\nISBN\n \n0-85404-176-1\n. A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, \"Heat Capacities: Introduction, Concepts, and Selected Applications\" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as\nphotoacoustic\nmethods, e.g., Jan Thoen & Christ Glorieux, \"Photothermal Techniques for Heat Capacities,\" and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc.\n(2012-05may-24)\nPhonon theory sheds light on liquid thermodynamics, heat capacity – Physics World\nThe phonon theory of liquid thermodynamics | Scientific Reports","attrs_markdown":"[Jump to content](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#bodyContent)\n\nMain menu\n\nMain menu\n\nmove to sidebar\n\nhide\n\nNavigation\n\n- [Main page](https:\/\/en.wikipedia.org\/wiki\/Main_Page \"Visit the main page [z]\")\n- [Contents](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contents \"Guides to browsing Wikipedia\")\n- [Current events](https:\/\/en.wikipedia.org\/wiki\/Portal:Current_events \"Articles related to current events\")\n- [Random article](https:\/\/en.wikipedia.org\/wiki\/Special:Random \"Visit a randomly selected article [x]\")\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About \"Learn about Wikipedia and how it works\")\n- [Contact us](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us \"How to contact Wikipedia\")\n\nContribute\n\n- [Help](https:\/\/en.wikipedia.org\/wiki\/Help:Contents \"Guidance on how to use and edit Wikipedia\")\n- [Learn to edit](https:\/\/en.wikipedia.org\/wiki\/Help:Introduction \"Learn how to edit Wikipedia\")\n- [Community portal](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Community_portal \"The hub for editors\")\n- [Recent changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChanges \"A list of recent changes to Wikipedia [r]\")\n- [Upload file](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:File_upload_wizard \"Add images or other media for use on Wikipedia\")\n- [Special pages](https:\/\/en.wikipedia.org\/wiki\/Special:SpecialPages \"A list of all special pages [q]\")\n\n[![](https:\/\/en.wikipedia.org\/static\/images\/icons\/enwiki-25.svg) ![Wikipedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https:\/\/en.wikipedia.org\/static\/images\/mobile\/copyright\/wikipedia-tagline-en-25.svg)](https:\/\/en.wikipedia.org\/wiki\/Main_Page)\n\n[Search](https:\/\/en.wikipedia.org\/wiki\/Special:Search \"Search Wikipedia [f]\")\n\nAppearance\n\n- [Donate](https:\/\/donate.wikimedia.org\/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en)\n- [Create account](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:CreateAccount&returnto=Specific+heat+capacity \"You are encouraged to create an account and log in; however, it is not mandatory\")\n- [Log in](https:\/\/en.wikipedia.org\/w\/index.php?title=Special:UserLogin&returnto=Specific+heat+capacity \"You're encouraged to log in; however, it's not mandatory. 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[o]\")\n\n## Contents\nmove to sidebar\n\nhide\n\n- [(Top)](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity)\n- [1 History](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#History)\n  Toggle History subsection\n  - [1\\.1 Discovery of specific heat](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Discovery_of_specific_heat)\n- [2 Definition](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Definition)\n  Toggle Definition subsection\n  - [2\\.1 Variations](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Variations)\n  - [2\\.2 Applicability](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Applicability)\n- [3 Measurement](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Measurement)\n- [4 Units](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Units)\n  Toggle Units subsection\n  - [4\\.1 International system](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#International_system)\n  - [4\\.2 Imperial engineering units](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Imperial_engineering_units)\n  - [4\\.3 Calories](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Calories)\n- [5 Physical basis](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Physical_basis)\n  Toggle Physical basis subsection\n  - [5\\.1 Monatomic gases](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Monatomic_gases)\n  - [5\\.2 Polyatomic gases](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Polyatomic_gases)\n- [6 Derivations of heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Derivations_of_heat_capacity)\n  Toggle Derivations of heat capacity subsection\n  - [6\\.1 Relation between specific heat capacities](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Relation_between_specific_heat_capacities)\n  - [6\\.2 Specific heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Specific_heat_capacity)\n  - [6\\.3 Polytropic heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Polytropic_heat_capacity)\n  - [6\\.4 Dimensionless heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Dimensionless_heat_capacity)\n  - [6\\.5 Heat capacity at absolute zero](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Heat_capacity_at_absolute_zero)\n  - [6\\.6 Solid phase](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Solid_phase)\n  - [6\\.7 Theoretical estimation](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Theoretical_estimation)\n  - [6\\.8 Relation between heat capacities](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Relation_between_heat_capacities)\n    - [6\\.8.1 Ideal gas](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Ideal_gas)\n  - [6\\.9 Specific heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Specific_heat_capacity_2)\n  - [6\\.10 Polytropic heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Polytropic_heat_capacity_2)\n  - [6\\.11 Dimensionless heat capacity](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Dimensionless_heat_capacity_2)\n  - [6\\.12 Heat capacity at absolute zero](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Heat_capacity_at_absolute_zero_2)\n- [7 Thermodynamic derivation](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Thermodynamic_derivation)\n  Toggle Thermodynamic derivation subsection\n  - [7\\.1 State of matter in a homogeneous sample](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#State_of_matter_in_a_homogeneous_sample)\n  - [7\\.2 Conservation of energy](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Conservation_of_energy)\n  - [7\\.3 Connection to equation of state](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Connection_to_equation_of_state)\n  - [7\\.4 Relation between heat capacities](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Relation_between_heat_capacities_2)\n  - [7\\.5 Calculation from first principles](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Calculation_from_first_principles)\n    - [7\\.5.1 Ideal gas](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Ideal_gas_2)\n- [8 See also](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#See_also)\n- [9 Notes](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Notes)\n- [10 References](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#References)\n- [11 Further reading](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#Further_reading)\n- [12 External links](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#External_links)\n\nToggle the table of contents\n\n# Specific heat capacity\n61 languages\n\n- [Aragonés](https:\/\/an.wikipedia.org\/wiki\/Calor_especifica \"Calor especifica – Aragonese\")\n- [العربية](https:\/\/ar.wikipedia.org\/wiki\/%D8%AD%D8%B1%D8%A7%D8%B1%D8%A9_%D9%86%D9%88%D8%B9%D9%8A%D8%A9 \"حرارة نوعية – Arabic\")\n- [Asturianu](https:\/\/ast.wikipedia.org\/wiki\/Calor_espec%C3%ADfico \"Calor específico – Asturian\")\n- [Azərbaycanca](https:\/\/az.wikipedia.org\/wiki\/X%C3%BCsusi_istilik_tutumu \"Xüsusi istilik tutumu – Azerbaijani\")\n- [Boarisch](https:\/\/bar.wikipedia.org\/wiki\/Spezifische_W%C3%A4rmekapazit%C3%A4t \"Spezifische Wärmekapazität – Bavarian\")\n- [Беларуская](https:\/\/be.wikipedia.org\/wiki\/%D0%A3%D0%B4%D0%B7%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%86%D0%B5%D0%BF%D0%BB%D0%B0%D1%91%D0%BC%D1%96%D1%81%D1%82%D0%B0%D1%81%D1%86%D1%8C \"Удзельная цеплаёмістасць – Belarusian\")\n- [Български](https:\/\/bg.wikipedia.org\/wiki\/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%84%D0%B8%D1%87%D0%B5%D0%BD_%D1%82%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D0%B5%D0%BD_%D0%BA%D0%B0%D0%BF%D0%B0%D1%86%D0%B8%D1%82%D0%B5%D1%82 \"Специфичен топлинен капацитет – Bulgarian\")\n- [Bosanski](https:\/\/bs.wikipedia.org\/wiki\/Specifi%C4%8Dna_toplota \"Specifična toplota – Bosnian\")\n- [Català](https:\/\/ca.wikipedia.org\/wiki\/Capacitat_calor%C3%ADfica_espec%C3%ADfica_\\(termodin%C3%A0mica\\) \"Capacitat calorífica específica (termodinàmica) – Catalan\")\n- [کوردی](https:\/\/ckb.wikipedia.org\/wiki\/%DA%AF%DB%95%D8%B1%D9%85%DB%8C%DB%8C_%D8%AC%DB%86%D8%B1%DB%8C \"گەرمیی جۆری – Central Kurdish\")\n- [Čeština](https:\/\/cs.wikipedia.org\/wiki\/M%C4%9Brn%C3%A1_tepeln%C3%A1_kapacita \"Měrná tepelná kapacita – Czech\")\n- [Чӑвашла](https:\/\/cv.wikipedia.org\/wiki\/%D0%9F%D0%B0%D0%B9%D0%BB%D0%B0%D0%B2%D0%BB%D0%B0_%C4%83%D1%88%C4%83%D1%88%C4%83%D0%BD%C4%83%C3%A7%D1%82%D0%B0%D1%80%C4%83%D1%88 \"Пайлавла ăшăшăнăçтарăш – Chuvash\")\n- [Dansk](https:\/\/da.wikipedia.org\/wiki\/Varmefylde \"Varmefylde – Danish\")\n- [Deutsch](https:\/\/de.wikipedia.org\/wiki\/Spezifische_W%C3%A4rmekapazit%C3%A4t \"Spezifische Wärmekapazität – German\")\n- [Esperanto](https:\/\/eo.wikipedia.org\/wiki\/Masa_varmokapacito \"Masa varmokapacito – Esperanto\")\n- [Español](https:\/\/es.wikipedia.org\/wiki\/Calor_espec%C3%ADfico \"Calor específico – Spanish\")\n- [Eesti](https:\/\/et.wikipedia.org\/wiki\/Erisoojus \"Erisoojus – Estonian\")\n- [Euskara](https:\/\/eu.wikipedia.org\/wiki\/Bero_espezifiko \"Bero espezifiko – Basque\")\n- [فارسی](https:\/\/fa.wikipedia.org\/wiki\/%D8%B8%D8%B1%D9%81%DB%8C%D8%AA_%DA%AF%D8%B1%D9%85%D8%A7%DB%8C%DB%8C_%D9%88%DB%8C%DA%98%D9%87 \"ظرفیت گرمایی ویژه – Persian\")\n- [Suomi](https:\/\/fi.wikipedia.org\/wiki\/Ominaisl%C3%A4mp%C3%B6kapasiteetti \"Ominaislämpökapasiteetti – Finnish\")\n- [Français](https:\/\/fr.wikipedia.org\/wiki\/Capacit%C3%A9_thermique_massique \"Capacité thermique massique – French\")\n- [Galego](https:\/\/gl.wikipedia.org\/wiki\/Calor_espec%C3%ADfica \"Calor específica – Galician\")\n- [हिन्दी](https:\/\/hi.wikipedia.org\/wiki\/%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A4%BF%E0%A4%B7%E0%A5%8D%E0%A4%9F_%E0%A4%8A%E0%A4%B7%E0%A5%8D%E0%A4%AE%E0%A4%BE_%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%BF%E0%A4%A4%E0%A4%BE \"विशिष्ट ऊष्मा धारिता – Hindi\")\n- [Hrvatski](https:\/\/hr.wikipedia.org\/wiki\/Specifi%C4%8Dna_toplina \"Specifična toplina – Croatian\")\n- [Հայերեն](https:\/\/hy.wikipedia.org\/wiki\/%D5%8F%D5%A5%D5%BD%D5%A1%D5%AF%D5%A1%D6%80%D5%A1%D6%80_%D5%BB%D5%A5%D6%80%D5%B4%D5%B8%D6%82%D5%B6%D5%A1%D5%AF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6 \"Տեսակարար ջերմունակություն – Armenian\")\n- [Bahasa Indonesia](https:\/\/id.wikipedia.org\/wiki\/Kalor_jenis \"Kalor jenis – Indonesian\")\n- [Íslenska](https:\/\/is.wikipedia.org\/wiki\/E%C3%B0lisvarmi \"Eðlisvarmi – Icelandic\")\n- [Italiano](https:\/\/it.wikipedia.org\/wiki\/Calore_specifico \"Calore specifico – Italian\")\n- [日本語](https:\/\/ja.wikipedia.org\/wiki\/%E6%AF%94%E7%86%B1%E5%AE%B9%E9%87%8F \"比熱容量 – Japanese\")\n- [La .lojban.](https:\/\/jbo.wikipedia.org\/wiki\/glanejni_sroka%27e \"glanejni sroka'e – Lojban\")\n- [ქართული](https:\/\/ka.wikipedia.org\/wiki\/%E1%83%99%E1%83%A3%E1%83%97%E1%83%A0%E1%83%98_%E1%83%97%E1%83%91%E1%83%9D%E1%83%A2%E1%83%94%E1%83%95%E1%83%90%E1%83%93%E1%83%9D%E1%83%91%E1%83%90 \"კუთრი თბოტევადობა – Georgian\")\n- [한국어](https:\/\/ko.wikipedia.org\/wiki\/%EB%B9%84%EC%97%B4%EC%9A%A9%EB%9F%89 \"비열용량 – Korean\")\n- [Lietuvių](https:\/\/lt.wikipedia.org\/wiki\/Savitoji_%C5%A1ilumin%C4%97_talpa \"Savitoji šiluminė talpa – Lithuanian\")\n- [Македонски](https:\/\/mk.wikipedia.org\/wiki\/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%84%D0%B8%D1%87%D0%B5%D0%BD_%D1%82%D0%BE%D0%BF%D0%BB%D0%B8%D0%BD%D1%81%D0%BA%D0%B8_%D0%BA%D0%B0%D0%BF%D0%B0%D1%86%D0%B8%D1%82%D0%B5%D1%82 \"Специфичен топлински капацитет – Macedonian\")\n- [Bahasa Melayu](https:\/\/ms.wikipedia.org\/wiki\/Muatan_haba_tentu \"Muatan haba tentu – Malay\")\n- [Nederlands](https:\/\/nl.wikipedia.org\/wiki\/Soortelijke_warmte \"Soortelijke warmte – Dutch\")\n- [Norsk nynorsk](https:\/\/nn.wikipedia.org\/wiki\/Spesifikk_varmekapasitet \"Spesifikk varmekapasitet – Norwegian Nynorsk\")\n- [Norsk bokmål](https:\/\/no.wikipedia.org\/wiki\/Spesifikk_varmekapasitet \"Spesifikk varmekapasitet – Norwegian Bokmål\")\n- [Polski](https:\/\/pl.wikipedia.org\/wiki\/Ciep%C5%82o_w%C5%82a%C5%9Bciwe \"Ciepło właściwe – Polish\")\n- [Português](https:\/\/pt.wikipedia.org\/wiki\/Calor_espec%C3%ADfico \"Calor específico – Portuguese\")\n- [Română](https:\/\/ro.wikipedia.org\/wiki\/Capacitate_termic%C4%83_masic%C4%83 \"Capacitate termică masică – Romanian\")\n- [Русский](https:\/\/ru.wikipedia.org\/wiki\/%D0%A3%D0%B4%D0%B5%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D0%BF%D0%BB%D0%BE%D1%91%D0%BC%D0%BA%D0%BE%D1%81%D1%82%D1%8C \"Удельная теплоёмкость – Russian\")\n- [Sicilianu](https:\/\/scn.wikipedia.org\/wiki\/Caluri_spic%C3%ACficu \"Caluri spicìficu – Sicilian\")\n- [Srpskohrvatski \/ српскохрватски](https:\/\/sh.wikipedia.org\/wiki\/Specifi%C4%8Dna_toplota \"Specifična toplota – Serbo-Croatian\")\n- [Simple English](https:\/\/simple.wikipedia.org\/wiki\/Specific_heat \"Specific heat – Simple English\")\n- [Slovenčina](https:\/\/sk.wikipedia.org\/wiki\/Hmotnostn%C3%A1_tepeln%C3%A1_kapacita \"Hmotnostná tepelná kapacita – Slovak\")\n- [Slovenščina](https:\/\/sl.wikipedia.org\/wiki\/Specifi%C4%8Dna_toplota \"Specifična toplota – Slovenian\")\n- [Shqip](https:\/\/sq.wikipedia.org\/wiki\/Nxeht%C3%ABsia_specifike_e_trupit \"Nxehtësia specifike e trupit – Albanian\")\n- [Српски \/ srpski](https:\/\/sr.wikipedia.org\/wiki\/%D0%A1%D0%BF%D0%B5%D1%86%D0%B8%D1%84%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%BE%D0%BF%D0%BB%D0%BE%D1%82%D0%B0 \"Специфична топлота – Serbian\")\n- [Svenska](https:\/\/sv.wikipedia.org\/wiki\/Specifik_v%C3%A4rmekapacitet \"Specifik värmekapacitet – Swedish\")\n- [தமிழ்](https:\/\/ta.wikipedia.org\/wiki\/%E0%AE%A4%E0%AE%A9%E0%AF%8D%E0%AE%B5%E0%AF%86%E0%AE%AA%E0%AF%8D%E0%AE%AA_%E0%AE%8F%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B1%E0%AE%A9%E0%AF%8D \"தன்வெப்ப ஏற்புத்திறன் – Tamil\")\n- [ไทย](https:\/\/th.wikipedia.org\/wiki\/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%88%E0%B8%B8%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A3%E0%B9%89%E0%B8%AD%E0%B8%99%E0%B8%88%E0%B8%B3%E0%B9%80%E0%B8%9E%E0%B8%B2%E0%B8%B0 \"ความจุความร้อนจำเพาะ – Thai\")\n- [Türkçe](https:\/\/tr.wikipedia.org\/wiki\/%C3%96zg%C3%BCl_%C4%B1s%C4%B1 \"Özgül ısı – Turkish\")\n- [Українська](https:\/\/uk.wikipedia.org\/wiki\/%D0%9F%D0%B8%D1%82%D0%BE%D0%BC%D0%B0_%D1%82%D0%B5%D0%BF%D0%BB%D0%BE%D1%94%D0%BC%D0%BD%D1%96%D1%81%D1%82%D1%8C \"Питома теплоємність – Ukrainian\")\n- [اردو](https:\/\/ur.wikipedia.org\/wiki\/%D8%AD%D8%B1%D8%A7%D8%B1%D8%AA_%D8%A7%D8%B6%D8%A7%D9%81%DB%8C \"حرارت اضافی – Urdu\")\n- [Oʻzbekcha \/ ўзбекча](https:\/\/uz.wikipedia.org\/wiki\/Solishtirma_issiqlik_sig%CA%BBimi \"Solishtirma issiqlik sigʻimi – Uzbek\")\n- [Vèneto](https:\/\/vec.wikipedia.org\/wiki\/Ca%C5%82or_spesifego \"Całor spesifego – Venetian\")\n- [Tiếng Việt](https:\/\/vi.wikipedia.org\/wiki\/Nhi%E1%BB%87t_dung_ri%C3%AAng \"Nhiệt dung riêng – Vietnamese\")\n- [吴语](https:\/\/wuu.wikipedia.org\/wiki\/%E6%AF%94%E7%83%AD%E5%AE%B9 \"比热容 – Wu\")\n- [粵語](https:\/\/zh-yue.wikipedia.org\/wiki\/%E6%AF%94%E7%86%B1%E5%AE%B9%E9%87%8F \"比熱容量 – Cantonese\")\n- [中文](https:\/\/zh.wikipedia.org\/wiki\/%E6%AF%94%E7%86%B1%E5%AE%B9 \"比熱容 – Chinese\")\n\n[Edit links](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q487756#sitelinks-wikipedia \"Edit interlanguage links\")\n\n- [Article](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity \"View the content page [c]\")\n- [Talk](https:\/\/en.wikipedia.org\/wiki\/Talk:Specific_heat_capacity \"Discuss improvements to the content page [t]\")\n\nEnglish\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=history \"Past revisions of this page [h]\")\n\nTools\n\nTools\n\nmove to sidebar\n\nhide\n\nActions\n\n- [Read](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity)\n- [Edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit \"Edit this page [e]\")\n- [View history](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=history)\n\nGeneral\n\n- [What links here](https:\/\/en.wikipedia.org\/wiki\/Special:WhatLinksHere\/Specific_heat_capacity \"List of all English Wikipedia pages containing links to this page [j]\")\n- [Related changes](https:\/\/en.wikipedia.org\/wiki\/Special:RecentChangesLinked\/Specific_heat_capacity \"Recent changes in pages linked from 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version](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&printable=yes \"Printable version of this page [p]\")\n\nIn other projects\n\n- [Wikimedia Commons](https:\/\/commons.wikimedia.org\/wiki\/Category:Specific_heat_capacity)\n- [Wikidata item](https:\/\/www.wikidata.org\/wiki\/Special:EntityPage\/Q487756 \"Structured data on this page hosted by Wikidata [g]\")\n\nAppearance\n\nmove to sidebar\n\nhide\n\nFrom Wikipedia, the free encyclopedia\n\nHeat required to raise the temperature of a given unit of mass of a substance\n\nFor the specific heat capacities of particular substances, see [Table of specific heat capacities](https:\/\/en.wikipedia.org\/wiki\/Table_of_specific_heat_capacities \"Table of specific heat capacities\").\n\n| Specific heat capacity |   |\n|---|---|\n| Other names | Specific heat |\n| Common symbols | *c* |\n| [SI unit](https:\/\/en.wikipedia.org\/wiki\/SI_unit \"SI unit\") | J⋅kg−1⋅K−1 |\n| In [SI base units](https:\/\/en.wikipedia.org\/wiki\/SI_base_unit \"SI base unit\") | m2⋅K−1⋅s−2 |\n| [Intensive](https:\/\/en.wikipedia.org\/wiki\/Intensive_and_extensive_properties \"Intensive and extensive properties\")? | Yes |\n| [Dimension](https:\/\/en.wikipedia.org\/wiki\/Dimensional_analysis#Formulation \"Dimensional analysis\") | L2⋅T−2⋅Θ−1 |\n\n| [Thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\") |   |\n|---|---|\n| [![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/2\/22\/Carnot_heat_engine_2.svg\/250px-Carnot_heat_engine_2.svg.png)](https:\/\/en.wikipedia.org\/wiki\/Carnot_heat_engine \"Carnot heat engine\")The classical [Carnot heat engine](https:\/\/en.wikipedia.org\/wiki\/Carnot_heat_engine \"Carnot heat engine\") |   |\n| Branches [Classical](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\") [Statistical](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\") [Chemical](https:\/\/en.wikipedia.org\/wiki\/Chemical_thermodynamics \"Chemical thermodynamics\") [Quantum thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Quantum_thermodynamics \"Quantum thermodynamics\") [Equilibrium](https:\/\/en.wikipedia.org\/wiki\/Equilibrium_thermodynamics \"Equilibrium thermodynamics\") \/ [Non-equilibrium](https:\/\/en.wikipedia.org\/wiki\/Non-equilibrium_thermodynamics \"Non-equilibrium thermodynamics\") |   |\n| [Laws](https:\/\/en.wikipedia.org\/wiki\/Laws_of_thermodynamics \"Laws of thermodynamics\") [Zeroth](https:\/\/en.wikipedia.org\/wiki\/Zeroth_law_of_thermodynamics \"Zeroth law of thermodynamics\") [First](https:\/\/en.wikipedia.org\/wiki\/First_law_of_thermodynamics \"First law of thermodynamics\") [Second](https:\/\/en.wikipedia.org\/wiki\/Second_law_of_thermodynamics \"Second law of thermodynamics\") [Third](https:\/\/en.wikipedia.org\/wiki\/Third_law_of_thermodynamics \"Third law of thermodynamics\") |   |\n| [State](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_state \"Thermodynamic state\") |   |\n| [Equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\") [Ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") [Real gas](https:\/\/en.wikipedia.org\/wiki\/Real_gas \"Real gas\") [State of matter](https:\/\/en.wikipedia.org\/wiki\/State_of_matter \"State of matter\") [Phase (matter)](https:\/\/en.wikipedia.org\/wiki\/Phase_\\(matter\\) \"Phase (matter)\") [Equilibrium](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_equilibrium \"Thermodynamic equilibrium\") [Control volume](https:\/\/en.wikipedia.org\/wiki\/Control_volume \"Control volume\") [Instruments](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_instruments \"Thermodynamic instruments\") |   |\n| [Processes](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_process \"Thermodynamic process\") |   |\n| [Isobaric](https:\/\/en.wikipedia.org\/wiki\/Isobaric_process \"Isobaric process\") [Isochoric](https:\/\/en.wikipedia.org\/wiki\/Isochoric_process \"Isochoric process\") [Isothermal](https:\/\/en.wikipedia.org\/wiki\/Isothermal_process \"Isothermal process\") [Adiabatic](https:\/\/en.wikipedia.org\/wiki\/Adiabatic_process \"Adiabatic process\") [Isentropic](https:\/\/en.wikipedia.org\/wiki\/Isentropic_process \"Isentropic process\") [Isenthalpic](https:\/\/en.wikipedia.org\/wiki\/Isenthalpic_process \"Isenthalpic process\") [Quasistatic](https:\/\/en.wikipedia.org\/wiki\/Quasistatic_process \"Quasistatic process\") [Polytropic](https:\/\/en.wikipedia.org\/wiki\/Polytropic_process \"Polytropic process\") [Free expansion](https:\/\/en.wikipedia.org\/wiki\/Free_expansion \"Free expansion\") [Reversibility](https:\/\/en.wikipedia.org\/wiki\/Reversible_process_\\(thermodynamics\\) \"Reversible process (thermodynamics)\") [Irreversibility](https:\/\/en.wikipedia.org\/wiki\/Irreversible_process \"Irreversible process\") [Endoreversibility](https:\/\/en.wikipedia.org\/wiki\/Endoreversible_thermodynamics \"Endoreversible thermodynamics\") |   |\n| [Cycles](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_cycle \"Thermodynamic cycle\") |   |\n| [Heat engines](https:\/\/en.wikipedia.org\/wiki\/Heat_engine \"Heat engine\") [Heat pumps](https:\/\/en.wikipedia.org\/wiki\/Heat_pump_and_refrigeration_cycle \"Heat pump and refrigeration cycle\") [Thermal efficiency](https:\/\/en.wikipedia.org\/wiki\/Thermal_efficiency \"Thermal efficiency\") |   |\n|   |   |\n| [Property diagrams](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_diagrams \"Thermodynamic diagrams\") [Intensive and extensive properties](https:\/\/en.wikipedia.org\/wiki\/Intensive_and_extensive_properties \"Intensive and extensive properties\") |   |\n| [Process functions](https:\/\/en.wikipedia.org\/wiki\/Process_function \"Process function\") |   |\n| [Work](https:\/\/en.wikipedia.org\/wiki\/Work_\\(thermodynamics\\) \"Work (thermodynamics)\") [Heat](https:\/\/en.wikipedia.org\/wiki\/Heat \"Heat\") |   |\n| [Functions of state](https:\/\/en.wikipedia.org\/wiki\/State_function \"State function\") |   |\n| [Temperature](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_temperature \"Thermodynamic temperature\") \/ *[Entropy](https:\/\/en.wikipedia.org\/wiki\/Entropy \"Entropy\")* ([introduction](https:\/\/en.wikipedia.org\/wiki\/Introduction_to_entropy \"Introduction to entropy\")) [Pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") \/ *[Volume](https:\/\/en.wikipedia.org\/wiki\/Volume_\\(thermodynamics\\) \"Volume (thermodynamics)\")* [Chemical potential](https:\/\/en.wikipedia.org\/wiki\/Chemical_potential \"Chemical potential\") \/ *[Particle number](https:\/\/en.wikipedia.org\/wiki\/Particle_number \"Particle number\")* [Vapor quality](https:\/\/en.wikipedia.org\/wiki\/Vapor_quality \"Vapor quality\") [Reduced properties](https:\/\/en.wikipedia.org\/wiki\/Reduced_properties \"Reduced properties\") |   |\n|   |   |\n| [Specific heat capacity](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity \"Heat capacity\") | c \\= {\\\\displaystyle c=} ![{\\\\displaystyle c=}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/891d40a9b18752b04065caee655d008b3ec11428) |\n| Scientists [Bernoulli](https:\/\/en.wikipedia.org\/wiki\/Daniel_Bernoulli \"Daniel Bernoulli\") [Boltzmann](https:\/\/en.wikipedia.org\/wiki\/Ludwig_Boltzmann \"Ludwig Boltzmann\") [Bridgman](https:\/\/en.wikipedia.org\/wiki\/Percy_Williams_Bridgman \"Percy Williams Bridgman\") [Callen](https:\/\/en.wikipedia.org\/wiki\/Herbert_Callen \"Herbert Callen\") [Carathéodory](https:\/\/en.wikipedia.org\/wiki\/Constantin_Carath%C3%A9odory \"Constantin Carathéodory\") [Carnot](https:\/\/en.wikipedia.org\/wiki\/Nicolas_L%C3%A9onard_Sadi_Carnot \"Nicolas Léonard Sadi Carnot\") [Clapeyron](https:\/\/en.wikipedia.org\/wiki\/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron \"Benoît Paul Émile Clapeyron\") [Clausius](https:\/\/en.wikipedia.org\/wiki\/Rudolf_Clausius \"Rudolf Clausius\") [de Donder](https:\/\/en.wikipedia.org\/wiki\/Th%C3%A9ophile_de_Donder \"Théophile de Donder\") [Duhem](https:\/\/en.wikipedia.org\/wiki\/Pierre_Duhem \"Pierre Duhem\") [Gibbs](https:\/\/en.wikipedia.org\/wiki\/Josiah_Willard_Gibbs \"Josiah Willard Gibbs\") [von Helmholtz](https:\/\/en.wikipedia.org\/wiki\/Hermann_von_Helmholtz \"Hermann von Helmholtz\") [Joule](https:\/\/en.wikipedia.org\/wiki\/James_Prescott_Joule \"James Prescott Joule\") [Kelvin](https:\/\/en.wikipedia.org\/wiki\/Lord_Kelvin \"Lord Kelvin\") [Lewis](https:\/\/en.wikipedia.org\/wiki\/Gilbert_N._Lewis \"Gilbert N. Lewis\") [Massieu](https:\/\/en.wikipedia.org\/wiki\/Fran%C3%A7ois_Massieu \"François Massieu\") [Maxwell](https:\/\/en.wikipedia.org\/wiki\/James_Clerk_Maxwell \"James Clerk Maxwell\") [von Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_von_Mayer \"Julius von Mayer\") [Nernst](https:\/\/en.wikipedia.org\/wiki\/Walther_Nernst \"Walther Nernst\") [Onsager](https:\/\/en.wikipedia.org\/wiki\/Lars_Onsager \"Lars Onsager\") [Planck](https:\/\/en.wikipedia.org\/wiki\/Max_Planck \"Max Planck\") [Rankine](https:\/\/en.wikipedia.org\/wiki\/William_John_Macquorn_Rankine \"William John Macquorn Rankine\") [Smeaton](https:\/\/en.wikipedia.org\/wiki\/John_Smeaton \"John Smeaton\") [Stahl](https:\/\/en.wikipedia.org\/wiki\/Georg_Ernst_Stahl \"Georg Ernst Stahl\") [Tait](https:\/\/en.wikipedia.org\/wiki\/Peter_Tait_\\(physicist\\) \"Peter Tait (physicist)\") [Thompson](https:\/\/en.wikipedia.org\/wiki\/Benjamin_Thompson \"Benjamin Thompson\") [van der Waals](https:\/\/en.wikipedia.org\/wiki\/Johannes_Diderik_van_der_Waals \"Johannes Diderik van der Waals\") [Waterston](https:\/\/en.wikipedia.org\/wiki\/John_James_Waterston \"John James Waterston\") |   |\n| Other [Nucleation](https:\/\/en.wikipedia.org\/wiki\/Nucleation \"Nucleation\") [Self-assembly](https:\/\/en.wikipedia.org\/wiki\/Self-assembly \"Self-assembly\") [Self-organization](https:\/\/en.wikipedia.org\/wiki\/Self-organization \"Self-organization\") |   |\n| ![](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/9\/96\/Symbol_category_class.svg\/20px-Symbol_category_class.svg.png) [Category](https:\/\/en.wikipedia.org\/wiki\/Category:Thermodynamics \"Category:Thermodynamics\") |   |\n| [v](https:\/\/en.wikipedia.org\/wiki\/Template:Thermodynamics_sidebar \"Template:Thermodynamics sidebar\") [t](https:\/\/en.wikipedia.org\/wiki\/Template_talk:Thermodynamics_sidebar \"Template talk:Thermodynamics sidebar\") [e](https:\/\/en.wikipedia.org\/wiki\/Special:EditPage\/Template:Thermodynamics_sidebar \"Special:EditPage\/Template:Thermodynamics sidebar\") |   |\n\nIn [thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\"), the **specific heat capacity** (symbol *c*) of a substance is the amount of [heat](https:\/\/en.wikipedia.org\/wiki\/Heat \"Heat\") that must be added to one unit of mass of the substance in order to cause an increase of one unit in [temperature](https:\/\/en.wikipedia.org\/wiki\/Temperature \"Temperature\"). It is also referred to as **massic heat capacity** or as the **specific heat.** More formally it is the [heat capacity](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity \"Heat capacity\") of a sample of the substance divided by the [mass](https:\/\/en.wikipedia.org\/wiki\/Mass \"Mass\") of the sample.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-1) The [SI](https:\/\/en.wikipedia.org\/wiki\/International_System_of_Units \"International System of Units\") unit of specific heat capacity is [joule](https:\/\/en.wikipedia.org\/wiki\/Joule \"Joule\") per [kelvin](https:\/\/en.wikipedia.org\/wiki\/Kelvin \"Kelvin\") per [kilogram](https:\/\/en.wikipedia.org\/wiki\/Kilogram \"Kilogram\"), J⋅kg−1⋅K−1.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-2) For example, the heat required to raise the temperature of 1 kg of [water](https:\/\/en.wikipedia.org\/wiki\/Water \"Water\") by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-3)\n\nSpecific heat capacity often varies with temperature, and is different for each [state of matter](https:\/\/en.wikipedia.org\/wiki\/State_of_matter \"State of matter\"). Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C, but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1. The specific heat capacities of [iron](https:\/\/en.wikipedia.org\/wiki\/Iron \"Iron\"), [granite](https:\/\/en.wikipedia.org\/wiki\/Granite \"Granite\"), and [hydrogen](https:\/\/en.wikipedia.org\/wiki\/Hydrogen \"Hydrogen\") gas are about 449 J⋅kg−1⋅K−1, 790 J⋅kg−1⋅K−1, and 14300 J⋅kg−1⋅K−1, respectively.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-4) While the substance is undergoing a [phase transition](https:\/\/en.wikipedia.org\/wiki\/Phase_transition \"Phase transition\"), such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature.\n\nThe specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity *at constant pressure*) than when it is heated in a closed vessel that prevents expansion (specific heat capacity *at constant volume*). These two values are usually denoted by c p {\\\\displaystyle c\\_{p}} ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f) and c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082), respectively; their quotient γ \\= c p \/ c V {\\\\displaystyle \\\\gamma =c\\_{p}\/c\\_{V}} ![{\\\\displaystyle \\\\gamma =c\\_{p}\/c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dbe578fc940f94ff433f350a8b2d7cad4dbf01c0) is the [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\").\n\nThe term *specific heat* may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C;[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-colen2001-5) much in the fashion of [specific gravity](https:\/\/en.wikipedia.org\/wiki\/Specific_gravity \"Specific gravity\"). Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of [moles](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(unit\\) \"Mole (unit)\"), one gets the [molar heat capacity](https:\/\/en.wikipedia.org\/wiki\/Molar_heat_capacity \"Molar heat capacity\") instead, whose SI unit is joule per kelvin per mole, J⋅mol−1⋅K−1. If the amount is taken to be the [volume](https:\/\/en.wikipedia.org\/wiki\/Volume \"Volume\") of the sample (as is sometimes done in engineering), one gets the [volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\"), whose SI unit is joule per kelvin per [cubic meter](https:\/\/en.wikipedia.org\/wiki\/Cubic_meter \"Cubic meter\"), J⋅m−3⋅K−1.\n\n## History\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=1 \"Edit section: History\")\\]\n\n### Discovery of specific heat\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=2 \"Edit section: Discovery of specific heat\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/8c\/Black_Joseph_%28cropped%29.jpg\/250px-Black_Joseph_%28cropped%29.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Black_Joseph_\\(cropped\\).jpg)\n\nJoseph Black\n\nOne of the first scientists to use the concept was [Joseph Black](https:\/\/en.wikipedia.org\/wiki\/Joseph_Black \"Joseph Black\"), an 18th-century medical doctor and professor of medicine at [Glasgow University](https:\/\/en.wikipedia.org\/wiki\/Glasgow_University \"Glasgow University\"). He measured the specific heat capacities of many substances, using the term *capacity for heat*.[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-6) In 1756 or soon thereafter, Black began an extensive study of heat.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-:1-7) In 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by [Daniel Gabriel Fahrenheit](https:\/\/en.wikipedia.org\/wiki\/Daniel_Gabriel_Fahrenheit \"Daniel Gabriel Fahrenheit\") on behalf of Dutch physician [Herman Boerhaave](https:\/\/en.wikipedia.org\/wiki\/Herman_Boerhaave \"Herman Boerhaave\"). For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: \"Quicksilver \\[mercury\\] ... has less capacity for the matter of heat than water.\"[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-8)[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-:0-9)\n\n## Definition\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=3 \"Edit section: Definition\")\\]\n\nThe specific heat capacity of a substance, usually denoted by c {\\\\displaystyle c} ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) or s {\\\\displaystyle s} ![{\\\\displaystyle s}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/01d131dfd7673938b947072a13a9744fe997e632), is the heat capacity C {\\\\displaystyle C} ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) of a sample of the substance, divided by the mass M {\\\\displaystyle M} ![{\\\\displaystyle M}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f82cade9898ced02fdd08712e5f0c0151758a0dd) of the sample:[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-10) c \\= C M \\= 1 M ⋅ d Q d T , {\\\\displaystyle c={\\\\frac {C}{M}}={\\\\frac {1}{M}}\\\\cdot {\\\\frac {\\\\mathrm {d} Q}{\\\\mathrm {d} T}},} ![{\\\\displaystyle c={\\\\frac {C}{M}}={\\\\frac {1}{M}}\\\\cdot {\\\\frac {\\\\mathrm {d} Q}{\\\\mathrm {d} T}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a58291be3575e33a756ed894a18347c8eabbba29) where d Q {\\\\displaystyle \\\\mathrm {d} Q} ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739) [represents](https:\/\/en.wikipedia.org\/wiki\/Derivative \"Derivative\") the amount of heat needed to uniformly raise the temperature of the sample by a small increment d T {\\\\displaystyle \\\\mathrm {d} T} ![{\\\\displaystyle \\\\mathrm {d} T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/309833548168052f2696ca961149df099b5d1f80).\n\nLike the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) of the sample and the [pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") p {\\\\displaystyle p} ![{\\\\displaystyle p}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/81eac1e205430d1f40810df36a0edffdc367af36) applied to it. Therefore, it should be considered a function c ( p , T ) {\\\\displaystyle c(p,T)} ![{\\\\displaystyle c(p,T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e8bac9cbcaf700b5206116a980a24a84404ba2e5) of those two variables.\n\nThese parameters are usually specified when giving the specific heat capacity of a substance. For example, \"Water (liquid): c p {\\\\displaystyle c\\_{p}} ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f) = 4187 J⋅kg−1⋅K−1 (15 °C).\"[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-toolbox-11) When not specified, published values of the specific heat capacity c {\\\\displaystyle c} ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) generally are valid for some [standard conditions for temperature and pressure](https:\/\/en.wikipedia.org\/wiki\/Standard_conditions_for_temperature_and_pressure \"Standard conditions for temperature and pressure\").\n\nHowever, the dependency of c {\\\\displaystyle c} ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier ( p , T ) {\\\\displaystyle (p,T)} ![{\\\\displaystyle (p,T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/140e6e1e1f95c1aba3ffb6dc196a9be07a91d1a6) and approximates the specific heat capacity by a constant c {\\\\displaystyle c} ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) suitable for those ranges.\n\nSpecific heat capacity is an [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier \"specific\" in front of an extensive property often indicates an intensive property derived from it.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-12))\n\n### Variations\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=4 \"Edit section: Variations\")\\]\n\nThe injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and\/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure p {\\\\displaystyle p} ![{\\\\displaystyle p}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/81eac1e205430d1f40810df36a0edffdc367af36) and starting temperature T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0). Two particular choices are widely used:\n\n- If the pressure is kept constant (for instance, at the ambient atmospheric pressure), and the sample is allowed to expand, the expansion generates [work](https:\/\/en.wikipedia.org\/wiki\/Work_\\(thermodynamics\\) \"Work (thermodynamics)\"), as the force from the pressure displaces the enclosure or the surrounding fluid. That work must come from the heat energy provided. The specific heat capacity thus obtained is said to be measured **at constant pressure** (or **isobaric**) and is often denoted\n  c\n  \n  p\n  \n  {\\\\displaystyle c\\_{p}}\n  \n  ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f)\n  \n  .\n- On the other hand, if the expansion is prevented – for example, by a sufficiently rigid enclosure or by increasing the external pressure to counteract the internal one – no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the sample, including raising its temperature by an extra amount. The specific heat capacity obtained this way is said to be measured **at constant volume** (or **isochoric**) and denoted\n  c\n  \n  V\n  \n  {\\\\displaystyle c\\_{V}}\n  \n  ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082)\n  \n  .\n\nThe value of c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) is always less than the value of c p {\\\\displaystyle c\\_{p}} ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f) for all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\") of gases is typically between 1.3 and 1.67.[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-Lange-13)\n\n### Applicability\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=5 \"Edit section: Applicability\")\\]\n\nThe specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale.\n\nThe specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops.\n\nThe specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a [phase change](https:\/\/en.wikipedia.org\/wiki\/Phase_transition \"Phase transition\"), such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement.\n\n## Measurement\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=6 \"Edit section: Measurement\")\\]\n\nThe specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a [calorimeter](https:\/\/en.wikipedia.org\/wiki\/Calorimeter \"Calorimeter\"), and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as [differential scanning calorimetry](https:\/\/en.wikipedia.org\/wiki\/Differential_scanning_calorimetry \"Differential scanning calorimetry\").[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-14)[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-15)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fe\/Water_temperature_vs_heat_added.svg\/250px-Water_temperature_vs_heat_added.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Water_temperature_vs_heat_added.svg)\n\nGraph of temperature of phases of water heated from −100 °C to 200 °C – the dashed line example shows that melting and heating 1 kg of ice at −50 °C to water at 40 °C needs 600 kJ\n\nThe specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the [coefficient of thermal expansion](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_thermal_expansion \"Coefficient of thermal expansion\") and the [compressibility](https:\/\/en.wikipedia.org\/wiki\/Bulk_modulus \"Bulk modulus\") of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\n## Units\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=7 \"Edit section: Units\")\\]\n\n### International system\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=8 \"Edit section: International system\")\\]\n\nThe SI unit for specific heat capacity is joule per kelvin per kilogram ⁠J\/kg⋅K⁠, J⋅K−1⋅kg−1. Since an increment of temperature of one [degree Celsius](https:\/\/en.wikipedia.org\/wiki\/Celsius_scale \"Celsius scale\") is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J\/(kg⋅°C). Sometimes the [gram](https:\/\/en.wikipedia.org\/wiki\/Gram \"Gram\") is used instead of kilogram for the unit of mass: 1 J⋅g−1⋅K−1 = 1000 J⋅kg−1⋅K−1.\n\nThe specific heat capacity of a substance (per unit of mass) has [dimension](https:\/\/en.wikipedia.org\/wiki\/Dimensional_analysis \"Dimensional analysis\") L2⋅Θ−1⋅T−2, or (L\/T)2\/Θ. Therefore, the SI unit J⋅kg−1⋅K−1 is equivalent to [metre](https:\/\/en.wikipedia.org\/wiki\/Metre \"Metre\") squared per [second](https:\/\/en.wikipedia.org\/wiki\/Second \"Second\") squared per [kelvin](https:\/\/en.wikipedia.org\/wiki\/Kelvin \"Kelvin\") (m2⋅K−1⋅s−2).\n\n### Imperial engineering units\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=9 \"Edit section: Imperial engineering units\")\\]\n\nProfessionals in [construction](https:\/\/en.wikipedia.org\/wiki\/Construction \"Construction\"), [civil engineering](https:\/\/en.wikipedia.org\/wiki\/Civil_engineering \"Civil engineering\"), [chemical engineering](https:\/\/en.wikipedia.org\/wiki\/Chemical_engineering \"Chemical engineering\"), and other technical disciplines, especially in the [United States](https:\/\/en.wikipedia.org\/wiki\/United_States \"United States\"), may use [English Engineering units](https:\/\/en.wikipedia.org\/wiki\/English_Engineering_Units \"English Engineering Units\") including the [pound](https:\/\/en.wikipedia.org\/wiki\/Pound_\\(mass\\) \"Pound (mass)\") (lb = 0.45359237 kg) as the unit of mass, the [degree Fahrenheit](https:\/\/en.wikipedia.org\/wiki\/Fahrenheit \"Fahrenheit\") or [Rankine](https:\/\/en.wikipedia.org\/wiki\/Rankine_scale \"Rankine scale\") (°R = ⁠5\/9⁠ K, about 0.555556 K) as the unit of temperature increment, and the [British thermal unit](https:\/\/en.wikipedia.org\/wiki\/British_thermal_unit \"British thermal unit\") (BTU ≈ 1055.056 J),[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-Koch-16)[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-17) as the unit of heat.\n\nIn those contexts, the unit of specific heat capacity is BTU\/lb⋅°R, or 1 ⁠BTU\/lb⋅°R⁠ = 4186.68⁠J\/kg⋅K⁠.[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-18) The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU\/lb⋅°F.[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-19) Note the value's similarity to that of the calorie - 4187 J\/kg⋅°C ≈ 4184 J\/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C.\n\n### Calories\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=10 \"Edit section: Calories\")\\]\n\nIn chemistry, heat amounts were often measured in [calories](https:\/\/en.wikipedia.org\/wiki\/Calorie \"Calorie\"). Confusingly, there are two common units with that name, respectively denoted *cal* and *Cal*:\n\n- the *small calorie* (*gram-calorie, cal*) is 4.184 J exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal\/(°C⋅g).\n- The *grand calorie* (*kilocalorie, kilogram-calorie, food calorie, kcal, Cal*) is 1000 small calories, 4184 J exactly. It was defined so that the specific heat capacity of water would be 1 Cal\/(°C⋅kg).\n\nWhile these units are still used in some contexts (such as kilogram calorie in [nutrition](https:\/\/en.wikipedia.org\/wiki\/Nutrition \"Nutrition\")), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually:\n\n1\n\n⁠cal\/°C⋅g⁠ = 1\n\n⁠Cal\/°C⋅kg⁠ = 1\n\n⁠kcal\/°C⋅kg⁠ = 4184\n\n⁠J\/kg⋅K⁠[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-20) = 4.184\n\n⁠kJ\/kg⋅K⁠.\n\nNote that while cal is **1⁄1000** of a Cal or kcal, it is also per *gram* instead of **kilo***gram*: ergo, in either unit, the specific heat capacity of water is approximately 1.\n\n## Physical basis\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=11 \"Edit section: Physical basis\")\\]\n\nMain article: [Molar heat capacity § Physical basis](https:\/\/en.wikipedia.org\/wiki\/Molar_heat_capacity#Physical_basis \"Molar heat capacity\")\n\nThe temperature of a sample of a substance reflects the average [kinetic energy](https:\/\/en.wikipedia.org\/wiki\/Kinetic_energy \"Kinetic energy\") of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the [equipartition theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\").\n\n### Monatomic gases\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=12 \"Edit section: Monatomic gases\")\\]\n\n[Statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\") predicts that at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy, unless multiple electronic states are accessible at room temperature (such is the case for atomic fluorine).[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-21) Thus, the [heat capacity per mole](https:\/\/en.wikipedia.org\/wiki\/Molar_heat_capacity \"Molar heat capacity\") at room temperature is the same for all of the noble gases as well as for many other atomic vapors. More precisely, c V , m \\= 3 R \/ 2 ≈ 12\\.5 J ⋅ K − 1 ⋅ m o l − 1 {\\\\displaystyle c\\_{V,\\\\mathrm {m} }=3R\/2\\\\approx \\\\mathrm {12.5\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} } ![{\\\\displaystyle c\\_{V,\\\\mathrm {m} }=3R\/2\\\\approx \\\\mathrm {12.5\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dcee637c4392e7347a6b4b7d57395e90ef9b8f5b) and c P , m \\= 5 R \/ 2 ≈ 21 J ⋅ K − 1 ⋅ m o l − 1 {\\\\displaystyle c\\_{P,\\\\mathrm {m} }=5R\/2\\\\approx \\\\mathrm {21\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} } ![{\\\\displaystyle c\\_{P,\\\\mathrm {m} }=5R\/2\\\\approx \\\\mathrm {21\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f6fc485f5327a3d2857548cbe7c145147d826306), where R ≈ 8\\.31446 J ⋅ K − 1 ⋅ m o l − 1 {\\\\displaystyle R\\\\approx \\\\mathrm {8.31446\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} } ![{\\\\displaystyle R\\\\approx \\\\mathrm {8.31446\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b344e96636e94e8469d9e4a0eafa306b3bec07e) is the [ideal gas unit](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas_constant \"Ideal gas constant\") (which is the product of [Boltzmann conversion constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\") from [kelvin](https:\/\/en.wikipedia.org\/wiki\/Kelvin \"Kelvin\") microscopic energy unit to the macroscopic energy unit [joule](https:\/\/en.wikipedia.org\/wiki\/Joule \"Joule\"), and the [Avogadro number](https:\/\/en.wikipedia.org\/wiki\/Avogadro_number \"Avogadro number\")).\n\nTherefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional) [atomic weight](https:\/\/en.wikipedia.org\/wiki\/Atomic_weight \"Atomic weight\") A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3). That is, approximately,\n\nc V ≈ 12470 J ⋅ K − 1 ⋅ k g − 1 \/ A c p ≈ 20785 J ⋅ K − 1 ⋅ k g − 1 \/ A {\\\\displaystyle c\\_{V}\\\\approx \\\\mathrm {12470\\\\,J\\\\cdot K^{-1}\\\\cdot kg^{-1}} \/A\\\\quad \\\\quad \\\\quad c\\_{p}\\\\approx \\\\mathrm {20785\\\\,J\\\\cdot K^{-1}\\\\cdot kg^{-1}} \/A} ![{\\\\displaystyle c\\_{V}\\\\approx \\\\mathrm {12470\\\\,J\\\\cdot K^{-1}\\\\cdot kg^{-1}} \/A\\\\quad \\\\quad \\\\quad c\\_{p}\\\\approx \\\\mathrm {20785\\\\,J\\\\cdot K^{-1}\\\\cdot kg^{-1}} \/A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0733e9dc0db1000b284289198a6d8d9b4ed008d9)\n\nFor the noble gases, from helium to xenon, these computed values are\n\n| Gas | He | Ne | Ar | Kr | Xe |\n|---|---|---|---|---|---|\n| A {\\\\displaystyle A} ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) |   |   |   |   |   |\n\n### Polyatomic gases\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=13 \"Edit section: Polyatomic gases\")\\]\n\nA polyatomic gas molecule can store energy in additional degrees of freedom. Its kinetic energy contributes to the heat capacity in the same way as monatomic gases, but there are also contributions from the [rotations](https:\/\/en.wikipedia.org\/wiki\/Rotational_energy \"Rotational energy\") of the molecule and vibration of the atoms relative to each other (including internal [potential energy](https:\/\/en.wikipedia.org\/wiki\/Potential_energy \"Potential energy\")).\n\nThe heat capacity may also have contribution from [excited electronic states](https:\/\/en.wikipedia.org\/wiki\/Excited_state \"Excited state\") for molecules with a sufficiently small energy gap between the ground state and the excited state, such as in NO.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-22) For a few systems, quantum spin statistics can also be important contributions to the heat capacity, even at room temperature. The analysis of the heat capacity of H  \n2 due to ortho\/para separation,[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-23) which arises from [nuclear spin](https:\/\/en.wikipedia.org\/wiki\/Spin_quantum_number \"Spin quantum number\") statistics, has been referred to as \"one of the great triumphs of post-quantum mechanical statistical mechanics.\"[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-24)\n\nThese extra [degrees of freedom](https:\/\/en.wikipedia.org\/wiki\/Degrees_of_freedom_\\(physics_and_chemistry\\) \"Degrees of freedom (physics and chemistry)\") or \"modes\" contribute to the specific heat capacity of the substance. Namely, when energy is introduced into a gas with polyatomic molecules, only part of it will increase their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number of degrees of freedom of the molecules.[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-25)[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-26)[\\[27\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-27)\n\n[Quantum statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_statistical_mechanics \"Quantum statistical mechanics\") predicts that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (quanta), and that this affects the system's thermodynamic properties. Depending on the temperature, the average energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be \"frozen out\". In that case, the specific heat capacity of the substance increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat.\n\nFor example, the molar heat capacity of [nitrogen](https:\/\/en.wikipedia.org\/wiki\/Nitrogen \"Nitrogen\") N  \n2 at constant volume is c V , m \\= 20\\.6 J ⋅ K − 1 ⋅ m o l − 1 {\\\\displaystyle c\\_{V,\\\\mathrm {m} }=\\\\mathrm {20.6\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} } ![{\\\\displaystyle c\\_{V,\\\\mathrm {m} }=\\\\mathrm {20.6\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5c7c7437f35c01e65d4dc78ef1cbf4dfcf2fb023) (at 15 °C, 1 atm), which is 2\\.49 R {\\\\displaystyle 2.49R} ![{\\\\displaystyle 2.49R}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6691d251191f800f340af5ec9cd4e291d0a5344c).[\\[28\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-thor1993-28) That is the value expected from the [Equipartition Theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\") if each molecule had 5 kinetic degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat capacity c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) of N  \n2 (736 J⋅K−1⋅kg−1) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K−1⋅kg−1), by a factor of ⁠5\/3⁠. The vibrational and electronic degrees of freedom do not contribute significantly to the heat capacity in this case, due to the relatively large energy level gaps for both vibrational and electronic excitation in this molecule.\n\nThis value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still \"frozen out\". At about that temperature, those modes begin to \"un-freeze\" as vibrationally excited states become accessible. As a result c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-chas1998-29) The last value corresponds almost exactly to the value predicted by the Equipartition Theorem, since in the high-temperature limit the theorem predicts that the vibrational degree of freedom contributes twice as much to the heat capacity as any one of the translational or rotational degrees of freedom.\n\n## Derivations of heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=14 \"Edit section: Derivations of heat capacity\")\\]\n\n### Relation between specific heat capacities\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=15 \"Edit section: Relation between specific heat capacities\")\\]\n\nStarting from the [fundamental thermodynamic relation](https:\/\/en.wikipedia.org\/wiki\/Fundamental_thermodynamic_relation \"Fundamental thermodynamic relation\") one can show,\n\nc p − c v \\= α 2 T ρ β T {\\\\displaystyle c\\_{p}-c\\_{v}={\\\\frac {\\\\alpha ^{2}T}{\\\\rho \\\\beta \\_{T}}}} ![{\\\\displaystyle c\\_{p}-c\\_{v}={\\\\frac {\\\\alpha ^{2}T}{\\\\rho \\\\beta \\_{T}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/06bc97cafc646b40213b01cf9a30562276c7c744)\n\nwhere\n\n- α\n  \n  {\\\\displaystyle \\\\alpha }\n  \n  ![{\\\\displaystyle \\\\alpha }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)\n  is the [coefficient of thermal expansion](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_thermal_expansion \"Coefficient of thermal expansion\"),\n- β\n  \n  T\n  \n  {\\\\displaystyle \\\\beta \\_{T}}\n  \n  ![{\\\\displaystyle \\\\beta \\_{T}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3f2855fc59047bc00f5bdedbcad68550a7ac6d13)\n  is the [isothermal](https:\/\/en.wikipedia.org\/wiki\/Isothermal \"Isothermal\") [compressibility](https:\/\/en.wikipedia.org\/wiki\/Compressibility \"Compressibility\"), and\n- ρ\n  \n  {\\\\displaystyle \\\\rho }\n  \n  ![{\\\\displaystyle \\\\rho }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1f7d439671d1289b6a816e6af7a304be40608d64)\n  is [density](https:\/\/en.wikipedia.org\/wiki\/Density \"Density\").\n\nA derivation is discussed in the article [Relations between specific heats](https:\/\/en.wikipedia.org\/wiki\/Relations_between_specific_heats \"Relations between specific heats\").\n\nFor an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), if ρ {\\\\displaystyle \\\\rho } ![{\\\\displaystyle \\\\rho }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1f7d439671d1289b6a816e6af7a304be40608d64) is expressed as [molar](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") density in the above equation, this equation reduces simply to [Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_Robert_von_Mayer \"Julius Robert von Mayer\")'s relation,\n\nC p , m − C v , m \\= R {\\\\displaystyle C\\_{p,m}-C\\_{v,m}=R\\\\!} ![{\\\\displaystyle C\\_{p,m}-C\\_{v,m}=R\\\\!}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f23e5c5b22b24ffd7f1051a3e396784a902d867e)\n\nwhere C p , m {\\\\displaystyle C\\_{p,m}} ![{\\\\displaystyle C\\_{p,m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7ccf6b7671259c85a73e29bc870010db9a7d2288) and C v , m {\\\\displaystyle C\\_{v,m}} ![{\\\\displaystyle C\\_{v,m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9bc3350be2e3cdd42c3a445e5dae16ac2aa4a675) are [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively.\n\n### Specific heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=16 \"Edit section: Specific heat capacity\")\\]\n\nThe specific heat capacity of a material on a per mass basis is\n\nc \\= ∂ C ∂ m , {\\\\displaystyle c={\\\\partial C \\\\over \\\\partial m},} ![{\\\\displaystyle c={\\\\partial C \\\\over \\\\partial m},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/338148bac53d596abe5112d0891626997320114b)\n\nwhich in the absence of phase transitions is equivalent to\n\nc \\= E m \\= C m \\= C ρ V , {\\\\displaystyle c=E\\_{m}={C \\\\over m}={C \\\\over {\\\\rho V}},} ![{\\\\displaystyle c=E\\_{m}={C \\\\over m}={C \\\\over {\\\\rho V}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca1a8ab2bd40064b132ddd125dd7e96420c0337d)\n\nwhere\n\n- C\n  \n  {\\\\displaystyle C}\n  \n  ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029)\n  is the heat capacity of a body made of the material in question,\n- m\n  \n  {\\\\displaystyle m}\n  \n  ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)\n  is the mass of the body,\n- V\n  \n  {\\\\displaystyle V}\n  \n  ![{\\\\displaystyle V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/af0f6064540e84211d0ffe4dac72098adfa52845)\n  is the volume of the body, and\n- ρ\n  \n  \\=\n  \n  m\n  \n  V\n  \n  {\\\\displaystyle \\\\rho ={\\\\frac {m}{V}}}\n  \n  ![{\\\\displaystyle \\\\rho ={\\\\frac {m}{V}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f63465553e3f944d6ef79f90f992a02cf29c7f38)\n  is the density of the material.\n\nFor gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https:\/\/en.wikipedia.org\/wiki\/Isobaric_process \"Isobaric process\") (constant pressure, d p \\= 0 {\\\\displaystyle dp=0} ![{\\\\displaystyle dp=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/850e5511af66c84a989bf0e5d003852cb862778c)) or [isochoric](https:\/\/en.wikipedia.org\/wiki\/Isochoric_process \"Isochoric process\") (constant volume, d V \\= 0 {\\\\displaystyle dV=0} ![{\\\\displaystyle dV=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c1d505a8f349881feb179d02752b33e5ab95e035)) processes. The corresponding specific heat capacities are expressed as\n\nc p \\= ( ∂ C ∂ m ) p , c V \\= ( ∂ C ∂ m ) V . {\\\\displaystyle {\\\\begin{aligned}c\\_{p}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{p},\\\\\\\\c\\_{V}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{V}.\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}c\\_{p}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{p},\\\\\\\\c\\_{V}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{V}.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4576769aa96d12c2e062fe6e6a6ef4acfdc8b63b)\n\nA related parameter to c {\\\\displaystyle c} ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is C V − 1 {\\\\displaystyle CV^{-1}} ![{\\\\displaystyle CV^{-1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0155c68921e700c1325f32c73061273b2178796e), the [volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\"). In engineering practice, c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript m {\\\\displaystyle m} ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as c m {\\\\displaystyle c\\_{m}} ![{\\\\displaystyle c\\_{m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes\n\nc m \\= C m \\= c V ρ . {\\\\displaystyle c\\_{m}={\\\\frac {C}{m}}={\\\\frac {c\\_{V}}{\\\\rho }}.} ![{\\\\displaystyle c\\_{m}={\\\\frac {C}{m}}={\\\\frac {c\\_{V}}{\\\\rho }}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/551b63520ea921f8105a75ddf4ddb073eca1e9db)\n\nFor pure homogeneous [chemical compounds](https:\/\/en.wikipedia.org\/wiki\/Chemical_compound \"Chemical compound\") with established [molecular or molar mass](https:\/\/en.wikipedia.org\/wiki\/Molecular_mass \"Molecular mass\") or a [molar quantity](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") is established, heat capacity as an [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") can be expressed on a per [mole](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") basis instead of a per mass basis by the following equations analogous to the per mass equations:\n\nC p , m \\= ( ∂ C ∂ n ) p \\= molar heat capacity at constant pressure C V , m \\= ( ∂ C ∂ n ) V \\= molar heat capacity at constant volume {\\\\displaystyle {\\\\begin{alignedat}{3}C\\_{p,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{p}&={\\\\text{molar heat capacity at constant pressure}}\\\\\\\\C\\_{V,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{V}&={\\\\text{molar heat capacity at constant volume}}\\\\end{alignedat}}} ![{\\\\displaystyle {\\\\begin{alignedat}{3}C\\_{p,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{p}&={\\\\text{molar heat capacity at constant pressure}}\\\\\\\\C\\_{V,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{V}&={\\\\text{molar heat capacity at constant volume}}\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/39d63e3ec44e731e9eb128dcc30eaa9c87af059a)\n\nwhere *n* = number of moles in the body or [thermodynamic system](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_system \"Thermodynamic system\"). One may refer to such a *per mole* quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis.\n\n### Polytropic heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=17 \"Edit section: Polytropic heat capacity\")\\]\n\nThe [polytropic](https:\/\/en.wikipedia.org\/wiki\/Polytropic \"Polytropic\") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change\n\nC i , m \\= ( ∂ C ∂ n ) \\= molar heat capacity at polytropic process {\\\\displaystyle C\\_{i,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)={\\\\text{molar heat capacity at polytropic process}}} ![{\\\\displaystyle C\\_{i,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)={\\\\text{molar heat capacity at polytropic process}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/02f685610e35e01aa42e293380bdf94469ea8364)\n\nThe most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between **1** and the adiabatic exponent (*γ* or *κ*)\n\n### Dimensionless heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=18 \"Edit section: Dimensionless heat capacity\")\\]\n\nThe [dimensionless](https:\/\/en.wikipedia.org\/wiki\/Dimensionless_number \"Dimensionless number\") heat capacity of a material is\n\nC ∗ \\= C n R \\= C N k B {\\\\displaystyle C^{\\*}={\\\\frac {C}{nR}}={\\\\frac {C}{Nk\\_{\\\\text{B}}}}} ![{\\\\displaystyle C^{\\*}={\\\\frac {C}{nR}}={\\\\frac {C}{Nk\\_{\\\\text{B}}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d1beaede671f02ee56b81859fe98eed0feb26cf5)\n\nwhere\n\n- *C* is the heat capacity of a body made of the material in question (J\/K)\n- *n* is the [amount of substance](https:\/\/en.wikipedia.org\/wiki\/Amount_of_substance \"Amount of substance\") in the body ([mol](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(unit\\) \"Mole (unit)\"))\n- *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\") (J⋅K−1⋅mol−1)\n- *N* is the number of molecules in the body. (dimensionless)\n- *k*B is the [Boltzmann constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\") (J⋅K−1)\n\nAgain, [SI](https:\/\/en.wikipedia.org\/wiki\/SI \"SI\") units shown for example.\n\nIn the [Ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") article, dimensionless heat capacity C ∗ {\\\\displaystyle C^{\\*}} ![{\\\\displaystyle C^{\\*}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as c ^ {\\\\displaystyle {\\\\hat {c}}} ![{\\\\displaystyle {\\\\hat {c}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8417e85ae7f4eaee7df31347ce488f85c8884b93).\n\n### Heat capacity at absolute zero\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=19 \"Edit section: Heat capacity at absolute zero\")\\]\n\nFrom the definition of [entropy](https:\/\/en.wikipedia.org\/wiki\/Entropy#Thermodynamic_definition_of_entropy \"Entropy\")\n\nT d S \\= δ Q {\\\\displaystyle TdS=\\\\delta Q} ![{\\\\displaystyle TdS=\\\\delta Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e3ddb574969210251fd1e6d8847eb303ece6b6a7)\n\nthe absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature *Tf*\n\nS ( T f ) \\= ∫ T \\= 0 T f δ Q T \\= ∫ 0 T f δ Q d T d T T \\= ∫ 0 T f C ( T ) d T T . {\\\\displaystyle S(T\\_{f})=\\\\int \\_{T=0}^{T\\_{f}}{\\\\frac {\\\\delta Q}{T}}=\\\\int \\_{0}^{T\\_{f}}{\\\\frac {\\\\delta Q}{dT}}{\\\\frac {dT}{T}}=\\\\int \\_{0}^{T\\_{f}}C(T)\\\\,{\\\\frac {dT}{T}}.} ![{\\\\displaystyle S(T\\_{f})=\\\\int \\_{T=0}^{T\\_{f}}{\\\\frac {\\\\delta Q}{T}}=\\\\int \\_{0}^{T\\_{f}}{\\\\frac {\\\\delta Q}{dT}}{\\\\frac {dT}{T}}=\\\\int \\_{0}^{T\\_{f}}C(T)\\\\,{\\\\frac {dT}{T}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/25e81986ce6a91887325cc998b1245250f83a65a)\n\nThe heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the [third law of thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Third_law_of_thermodynamics \"Third law of thermodynamics\"). One of the strengths of the [Debye model](https:\/\/en.wikipedia.org\/wiki\/Debye_model \"Debye model\") is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached.\n\n### Solid phase\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=20 \"Edit section: Solid phase\")\\]\n\nThe theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3*R*, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas.\n\nThe Dulong–Petit limit results from the [equipartition theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\"), and as such is only valid in the classical limit of a [microstate continuum](https:\/\/en.wikipedia.org\/w\/index.php?title=Microstate_continuum&action=edit&redlink=1 \"Microstate continuum (page does not exist)\"), which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at [standard ambient temperature](https:\/\/en.wikipedia.org\/wiki\/Standard_ambient_temperature_and_pressure \"Standard ambient temperature and pressure\"), quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3*R* per mole of *atoms* in the solid, although in molecular solids, heat capacities calculated *per mole of molecules* in molecular solids may be more than 3*R*. For example, the heat capacity of water ice at the melting point is about 4.6*R* per mole of molecules, but only 1.5*R* per mole of atoms. The lower than 3*R* number \"per atom\" (as is the case with diamond and beryllium) results from the \"freezing out\" of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3*R* per mole of atoms of the Dulong–Petit theoretical maximum.\n\nFor a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of [phonons](https:\/\/en.wikipedia.org\/wiki\/Phonons \"Phonons\"). See [Debye model](https:\/\/en.wikipedia.org\/wiki\/Debye_model \"Debye model\").\n\n### Theoretical estimation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=21 \"Edit section: Theoretical estimation\")\\]\n\nThe path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \\> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below.\n\n- Water (liquid): CP = 4185.5 J⋅K−1⋅kg−1 (15 °C, 101.325 kPa)\n- Water (liquid): CVH = 74.539 J⋅K−1⋅mol−1 (25 °C)\n\nFor liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However, different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr).[\\[notes 1\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-gold-30)\n\n### Relation between heat capacities\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=22 \"Edit section: Relation between heat capacities\")\\]\n\nMain article: [Relations between heat capacities](https:\/\/en.wikipedia.org\/wiki\/Relations_between_heat_capacities \"Relations between heat capacities\")\n\nMeasuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see [coefficient of thermal expansion](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_thermal_expansion \"Coefficient of thermal expansion\") and [compressibility](https:\/\/en.wikipedia.org\/wiki\/Compressibility \"Compressibility\")). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws.\n\nThe [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\"), or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.\n\n#### Ideal gas\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=23 \"Edit section: Ideal gas\")\\]\n\nFor an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), evaluating the partial derivatives above according to the [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\"), where *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\"), for an ideal gas[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-31)\n\nP V \\= n R T , C P − C V \\= T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n , P \\= n R T V ⇒ ( ∂ P ∂ T ) V , n \\= n R V , V \\= n R T P ⇒ ( ∂ V ∂ T ) P , n \\= n R P . {\\\\displaystyle {\\\\begin{alignedat}{3}PV&=nRT,&\\\\\\\\C\\_{P}-C\\_{V}&=T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n},&\\\\\\\\P&={\\\\frac {nRT}{V}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}&={\\\\frac {nR}{V}},\\\\\\\\V&={\\\\frac {nRT}{P}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}&={\\\\frac {nR}{P}}.\\\\end{alignedat}}} ![{\\\\displaystyle {\\\\begin{alignedat}{3}PV&=nRT,&\\\\\\\\C\\_{P}-C\\_{V}&=T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n},&\\\\\\\\P&={\\\\frac {nRT}{V}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}&={\\\\frac {nR}{V}},\\\\\\\\V&={\\\\frac {nRT}{P}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}&={\\\\frac {nR}{P}}.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2fe220db0c71a99f43c20c80b90b430001c8aff8)\n\nSubstituting\n\nT ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n \\= T n R V n R P \\= n R T V n R P \\= P n R P \\= n R , {\\\\displaystyle T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}=T{\\\\frac {nR}{V}}{\\\\frac {nR}{P}}={\\\\frac {nRT}{V}}{\\\\frac {nR}{P}}=P{\\\\frac {nR}{P}}=nR,} ![{\\\\displaystyle T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}=T{\\\\frac {nR}{V}}{\\\\frac {nR}{P}}={\\\\frac {nRT}{V}}{\\\\frac {nR}{P}}=P{\\\\frac {nR}{P}}=nR,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e49d3f01b48a4771f06476e4922cd418e91e9354)\n\nthis equation reduces simply to [Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_Robert_von_Mayer \"Julius Robert von Mayer\")'s relation:\n\nC P , m − C V , m \\= R . {\\\\displaystyle C\\_{P,m}-C\\_{V,m}=R.} ![{\\\\displaystyle C\\_{P,m}-C\\_{V,m}=R.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1b0951113a93ae7d847ed0234a94039782ae0a5e)\n\nThe differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.\n\n### Specific heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=24 \"Edit section: Specific heat capacity\")\\]\n\nThe specific heat capacity of a material on a per mass basis is\n\nc \\= ∂ C ∂ m , {\\\\displaystyle c={\\\\frac {\\\\partial C}{\\\\partial m}},} ![{\\\\displaystyle c={\\\\frac {\\\\partial C}{\\\\partial m}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f6d4d3222fc01879cf5ebdcdd167c0f01338989)\n\nwhich in the absence of phase transitions is equivalent to\n\nc \\= E m \\= C m \\= C ρ V , {\\\\displaystyle c=E\\_{m}={\\\\frac {C}{m}}={\\\\frac {C}{\\\\rho V}},} ![{\\\\displaystyle c=E\\_{m}={\\\\frac {C}{m}}={\\\\frac {C}{\\\\rho V}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5a5685f216c3eaa301626d07a8dbeda4ec825d2a)\n\nwhere\n\n- C\n  \n  {\\\\displaystyle C}\n  \n  ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029)\n  is the heat capacity of a body made of the material in question,\n- m\n  \n  {\\\\displaystyle m}\n  \n  ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)\n  is the mass of the body,\n- V\n  \n  {\\\\displaystyle V}\n  \n  ![{\\\\displaystyle V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/af0f6064540e84211d0ffe4dac72098adfa52845)\n  is the volume of the body,\n- ρ\n  \n  \\=\n  \n  m\n  \n  V\n  \n  {\\\\displaystyle \\\\rho ={\\\\frac {m}{V}}}\n  \n  ![{\\\\displaystyle \\\\rho ={\\\\frac {m}{V}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f63465553e3f944d6ef79f90f992a02cf29c7f38)\n  is the density of the material.\n\nFor gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https:\/\/en.wikipedia.org\/wiki\/Isobaric_process \"Isobaric process\") (constant pressure, d P \\= 0 {\\\\displaystyle {\\\\text{d}}P=0} ![{\\\\displaystyle {\\\\text{d}}P=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2997415b7c242bcf6f4ebf7ef17957d9df03a9aa)) or [isochoric](https:\/\/en.wikipedia.org\/wiki\/Isochoric_process \"Isochoric process\") (constant volume, d V \\= 0 {\\\\displaystyle {\\\\text{d}}V=0} ![{\\\\displaystyle {\\\\text{d}}V=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e19ded03242b58e141dced9f6f46c3a8829e436)) processes. The corresponding specific heat capacities are expressed as\n\nc P \\= ( ∂ C ∂ m ) P , c V \\= ( ∂ C ∂ m ) V . {\\\\displaystyle {\\\\begin{aligned}c\\_{P}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{P},\\\\\\\\c\\_{V}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{V}.\\\\end{aligned}}} ![{\\\\displaystyle {\\\\begin{aligned}c\\_{P}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{P},\\\\\\\\c\\_{V}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{V}.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8d2cc9fc1a0307e6e726521189cf7e22158357b9)\n\nFrom the results of the previous section, dividing through by the mass gives the relation\n\nc P − c V \\= α 2 T ρ β T . {\\\\displaystyle c\\_{P}-c\\_{V}={\\\\frac {\\\\alpha ^{2}T}{\\\\rho \\\\beta \\_{T}}}.} ![{\\\\displaystyle c\\_{P}-c\\_{V}={\\\\frac {\\\\alpha ^{2}T}{\\\\rho \\\\beta \\_{T}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/058bcaccd4cbc0e8d16892f68bc68bf1e0062d7e)\n\nA related parameter to c {\\\\displaystyle c} ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is C \/ V {\\\\displaystyle C\/V} ![{\\\\displaystyle C\/V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4f43bc2f531ba84544e7e71ccab205ab4ad1fe5c), the [volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\"). In engineering practice, c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript m {\\\\displaystyle m} ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as c m {\\\\displaystyle c\\_{m}} ![{\\\\displaystyle c\\_{m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes\n\nc m \\= C m \\= c volumetric ρ . {\\\\displaystyle c\\_{m}={\\\\frac {C}{m}}={\\\\frac {c\\_{\\\\text{volumetric}}}{\\\\rho }}.} ![{\\\\displaystyle c\\_{m}={\\\\frac {C}{m}}={\\\\frac {c\\_{\\\\text{volumetric}}}{\\\\rho }}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e56b0a8c8ebcbe4ee889c7d963cb980a5d34f60b)\n\nFor pure [homogeneous](https:\/\/en.wikipedia.org\/wiki\/Homogeneous \"Homogeneous\") [chemical compounds](https:\/\/en.wikipedia.org\/wiki\/Chemical_compound \"Chemical compound\") with established [molecular or molar mass](https:\/\/en.wikipedia.org\/wiki\/Molecular_mass \"Molecular mass\"), or a [molar quantity](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\"), heat capacity as an [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") can be expressed on a per-[mole](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") basis instead of a per-mass basis by the following equations analogous to the per mass equations:\n\nC P , m \\= ( ∂ C ∂ n ) P \\= molar heat capacity at constant pressure, C V , m \\= ( ∂ C ∂ n ) V \\= molar heat capacity at constant volume, {\\\\displaystyle {\\\\begin{alignedat}{3}C\\_{P,m}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{P}&={\\\\text{molar heat capacity at constant pressure,}}\\\\\\\\C\\_{V,m}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{V}&={\\\\text{molar heat capacity at constant volume,}}\\\\end{alignedat}}} ![{\\\\displaystyle {\\\\begin{alignedat}{3}C\\_{P,m}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{P}&={\\\\text{molar heat capacity at constant pressure,}}\\\\\\\\C\\_{V,m}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{V}&={\\\\text{molar heat capacity at constant volume,}}\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a371ac38796d12f8beb3f30b4e16181a5e1981ba)\n\nwhere *n* is the number of moles in the body or [thermodynamic system](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_system \"Thermodynamic system\"). One may refer to such a per-mole quantity as **molar heat capacity** to distinguish it from specific heat capacity on a per-mass basis.\n\n### Polytropic heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=25 \"Edit section: Polytropic heat capacity\")\\]\n\nThe [polytropic](https:\/\/en.wikipedia.org\/wiki\/Polytropic \"Polytropic\") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change:\n\nC i , m \\= ( ∂ C ∂ n ) \\= molar heat capacity at polytropic process. {\\\\displaystyle C\\_{i,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)={\\\\text{molar heat capacity at polytropic process.}}} ![{\\\\displaystyle C\\_{i,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)={\\\\text{molar heat capacity at polytropic process.}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d94a2aa8fde269f13f73425bd4eb2e375717ad28)\n\nThe most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (*γ* or *κ*).\n\n### Dimensionless heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=26 \"Edit section: Dimensionless heat capacity\")\\]\n\nThe [dimensionless](https:\/\/en.wikipedia.org\/wiki\/Dimensionless_number \"Dimensionless number\") heat capacity of a material is\n\nC ∗ \\= C n R \\= C N k B , {\\\\displaystyle C^{\\*}={\\\\frac {C}{nR}}={\\\\frac {C}{Nk\\_{\\\\text{B}}}},} ![{\\\\displaystyle C^{\\*}={\\\\frac {C}{nR}}={\\\\frac {C}{Nk\\_{\\\\text{B}}}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/508315bfb919ceff9bf382f397926bccaa1168dc)\n\nwhere\n\n- C\n  \n  {\\\\displaystyle C}\n  \n  ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029)\n  is the heat capacity of a body made of the material in question (J\/K),\n- *n* is the [amount of substance](https:\/\/en.wikipedia.org\/wiki\/Amount_of_substance \"Amount of substance\") in the body ([mol](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(unit\\) \"Mole (unit)\")),\n- *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\") (J\/(K⋅mol)),\n- *N* is the number of molecules in the body (dimensionless),\n- *k*B is the [Boltzmann constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\") (J\/(K⋅molecule)).\n\nIn the [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") article, dimensionless heat capacity C ∗ {\\\\displaystyle C^{\\*}} ![{\\\\displaystyle C^{\\*}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as c ^ {\\\\displaystyle {\\\\hat {c}}} ![{\\\\displaystyle {\\\\hat {c}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8417e85ae7f4eaee7df31347ce488f85c8884b93) and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the [equipartition theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\").\n\nMore generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the [dimensionless entropy](https:\/\/en.wikipedia.org\/wiki\/Dimensionless_entropy \"Dimensionless entropy\") per particle S ∗ \\= S \/ N k B {\\\\displaystyle S^{\\*}=S\/Nk\\_{\\\\text{B}}} ![{\\\\displaystyle S^{\\*}=S\/Nk\\_{\\\\text{B}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/41f612e45f84ed9e293f85f75c6dae30943bca0b), measured in [nats](https:\/\/en.wikipedia.org\/wiki\/Nat_\\(unit\\) \"Nat (unit)\").\n\nC ∗ \\= d S ∗ d ( ln ⁡ T ) . {\\\\displaystyle C^{\\*}={\\\\frac {{\\\\text{d}}S^{\\*}}{{\\\\text{d}}(\\\\ln T)}}.} ![{\\\\displaystyle C^{\\*}={\\\\frac {{\\\\text{d}}S^{\\*}}{{\\\\text{d}}(\\\\ln T)}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/294a7378c60eaed01bcea2cb12c53dd0b5bf7f90)\n\nAlternatively, using base-2 logarithms, C ∗ {\\\\displaystyle C^{\\*}} ![{\\\\displaystyle C^{\\*}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in [bits](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\").[\\[31\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-32)\n\n### Heat capacity at absolute zero\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=27 \"Edit section: Heat capacity at absolute zero\")\\]\n\nFrom the definition of [entropy](https:\/\/en.wikipedia.org\/wiki\/Entropy#Thermodynamic_definition_of_entropy \"Entropy\")\n\nT d S \\= δ Q , {\\\\displaystyle T\\\\,{\\\\text{d}}S=\\\\delta Q,} ![{\\\\displaystyle T\\\\,{\\\\text{d}}S=\\\\delta Q,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5cca7738465956fa4c1d790333af389a15b667c7)\n\nthe absolute entropy can be calculated by integrating from zero to the final temperature *T*f:\n\nS ( T f ) \\= ∫ T \\= 0 T f δ Q T \\= ∫ 0 T f δ Q d T d T T \\= ∫ 0 T f C ( T ) d T T . {\\\\displaystyle S(T\\_{\\\\text{f}})=\\\\int \\_{T=0}^{T\\_{\\\\text{f}}}{\\\\frac {\\\\delta Q}{T}}=\\\\int \\_{0}^{T\\_{\\\\text{f}}}{\\\\frac {\\\\delta Q}{{\\\\text{d}}T}}{\\\\frac {{\\\\text{d}}T}{T}}=\\\\int \\_{0}^{T\\_{\\\\text{f}}}C(T)\\\\,{\\\\frac {{\\\\text{d}}T}{T}}.} ![{\\\\displaystyle S(T\\_{\\\\text{f}})=\\\\int \\_{T=0}^{T\\_{\\\\text{f}}}{\\\\frac {\\\\delta Q}{T}}=\\\\int \\_{0}^{T\\_{\\\\text{f}}}{\\\\frac {\\\\delta Q}{{\\\\text{d}}T}}{\\\\frac {{\\\\text{d}}T}{T}}=\\\\int \\_{0}^{T\\_{\\\\text{f}}}C(T)\\\\,{\\\\frac {{\\\\text{d}}T}{T}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/df93f18f951064e02f6128a3cd99ad4c72124aa1)\n\n## Thermodynamic derivation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=28 \"Edit section: Thermodynamic derivation\")\\]\n\nIn theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\") and an [internal energy function](https:\/\/en.wikipedia.org\/w\/index.php?title=Internal_energy_function&action=edit&redlink=1 \"Internal energy function (page does not exist)\").\n\n### State of matter in a homogeneous sample\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=29 \"Edit section: State of matter in a homogeneous sample\")\\]\n\nTo apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass M {\\\\displaystyle M} ![{\\\\displaystyle M}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f82cade9898ced02fdd08712e5f0c0151758a0dd). Assume that the evolution of the system is always slow enough for the internal pressure P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a) and temperature T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) be considered uniform throughout. The pressure P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a) would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air.\n\nThe state of the material can then be specified by three parameters: its temperature T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), the pressure P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), and its [specific volume](https:\/\/en.wikipedia.org\/wiki\/Specific_volume \"Specific volume\") ν \\= V \/ M {\\\\displaystyle \\\\nu =V\/M} ![{\\\\displaystyle \\\\nu =V\/M}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/df5e892352c30149ba6c93abe17145e15cdd73f5), where V {\\\\displaystyle V} ![{\\\\displaystyle V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the sample. (This quantity is the reciprocal 1 \/ ρ {\\\\displaystyle 1\/\\\\rho } ![{\\\\displaystyle 1\/\\\\rho }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/754f9b580e728c03bac621d1dd72cd606b6b5eef) of the material's [density](https:\/\/en.wikipedia.org\/wiki\/Density \"Density\") ρ \\= M \/ V {\\\\displaystyle \\\\rho =M\/V} ![{\\\\displaystyle \\\\rho =M\/V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2ed457a6aafb262f754ae8801d87eb036b8c5fe9).) Like T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) and P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), the specific volume ν {\\\\displaystyle \\\\nu } ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468) is an intensive property of the material and its state, that does not depend on the amount of substance in the sample.\n\nThose variables are not independent. The allowed states are defined by an [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\") relating those three variables: F ( T , P , ν ) \\= 0\\. {\\\\displaystyle F(T,P,\\\\nu )=0.} ![{\\\\displaystyle F(T,P,\\\\nu )=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/471651d7fefe7e18735444a61e0ffa3f6ff84876) The function F {\\\\displaystyle F} ![{\\\\displaystyle F}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/545fd099af8541605f7ee55f08225526be88ce57) depends on the material under consideration. The [specific internal energy](https:\/\/en.wikipedia.org\/wiki\/Specific_internal_energy \"Specific internal energy\") stored internally in the sample, per unit of mass, will then be another function U ( T , P , ν ) {\\\\displaystyle U(T,P,\\\\nu )} ![{\\\\displaystyle U(T,P,\\\\nu )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7003637ad40a88b0bf499d18811d1dab7f424ccf) of these state variables, that is also specific of the material. The total internal energy in the sample then will be M U ( T , P , ν ) {\\\\displaystyle M\\\\,U(T,P,\\\\nu )} ![{\\\\displaystyle M\\\\,U(T,P,\\\\nu )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/287870c17350c2311e2d86e64ec9dec7bfc90a75).\n\nFor some simple materials, like an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), one can derive from basic theory the equation of state F \\= 0 {\\\\displaystyle F=0} ![{\\\\displaystyle F=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/745afacbd4fd9affdc51ac09a0ecabae08da8676) and even the specific internal energy U {\\\\displaystyle U} ![{\\\\displaystyle U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/458a728f53b9a0274f059cd695e067c430956025) In general, these functions must be determined experimentally for each substance.\n\n### Conservation of energy\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=30 \"Edit section: Conservation of energy\")\\]\n\nThe absolute value of this quantity U {\\\\displaystyle U} ![{\\\\displaystyle U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/458a728f53b9a0274f059cd695e067c430956025) is undefined, and (for the purposes of thermodynamics) the state of \"zero internal energy\" can be chosen arbitrarily. However, by the [law of conservation of energy](https:\/\/en.wikipedia.org\/wiki\/Law_of_conservation_of_energy \"Law of conservation of energy\"), any infinitesimal increase M d U {\\\\displaystyle M\\\\,\\\\mathrm {d} U} ![{\\\\displaystyle M\\\\,\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fcddca6d7c51038a1a0134cde944e55e5bf81551) in the total internal energy M U {\\\\displaystyle MU} ![{\\\\displaystyle MU}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca0430adceb61aadc25575b6894c1633b41f8874) must be matched by the net flow of heat energy d Q {\\\\displaystyle \\\\mathrm {d} Q} ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739) into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is − P d V {\\\\displaystyle -P\\\\,\\\\mathrm {d} V} ![{\\\\displaystyle -P\\\\,\\\\mathrm {d} V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c7238b4c977539175fa28c0cff1658e50276637a), where d V {\\\\displaystyle \\\\mathrm {d} V} ![{\\\\displaystyle \\\\mathrm {d} V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b80507190aa9d38a279909db47b63657f2b62ba7) is the change in the sample's volume in that infinitesimal step.[\\[32\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-fein-33) Therefore\n\nd Q − P d V \\= M d U {\\\\displaystyle \\\\mathrm {d} Q-P\\\\,\\\\mathrm {d} V=M\\\\,\\\\mathrm {d} U} ![{\\\\displaystyle \\\\mathrm {d} Q-P\\\\,\\\\mathrm {d} V=M\\\\,\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3c3401e201975bf505b5fa40018ffd2bc7eab0a6)\n\nhence\n\nd Q M − P d ν \\= d U {\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M}}-P\\\\,\\\\mathrm {d} \\\\nu =\\\\mathrm {d} U} ![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M}}-P\\\\,\\\\mathrm {d} \\\\nu =\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/586d361903b89f3b074724dafc3680ca2cf85d26)\n\nIf the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount d Q {\\\\displaystyle \\\\mathrm {d} Q} ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739), then the term P d ν {\\\\displaystyle P\\\\,\\\\mathrm {d} \\\\nu } ![{\\\\displaystyle P\\\\,\\\\mathrm {d} \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e54803595e3392edc2125d5bbbfd6ee753922208) is zero (no mechanical work is done). Then, dividing by d T {\\\\displaystyle \\\\mathrm {d} T} ![{\\\\displaystyle \\\\mathrm {d} T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/309833548168052f2696ca961149df099b5d1f80),\n\nd Q M d T \\= d U d T {\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M\\\\,\\\\mathrm {d} T}}={\\\\frac {\\\\mathrm {d} U}{\\\\mathrm {d} T}}} ![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M\\\\,\\\\mathrm {d} T}}={\\\\frac {\\\\mathrm {d} U}{\\\\mathrm {d} T}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b32da0380fcf6d0c3bc87b7dbb0c7c0c02211f80)\n\nwhere d T {\\\\displaystyle \\\\mathrm {d} T} ![{\\\\displaystyle \\\\mathrm {d} T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/309833548168052f2696ca961149df099b5d1f80) is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume c V {\\\\displaystyle c\\_{V}} ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) of the material.\n\nFor the heat capacity at constant pressure, it is useful to define the [specific enthalpy](https:\/\/en.wikipedia.org\/wiki\/Specific_enthalpy \"Specific enthalpy\") of the system as the sum h ( T , P , ν ) \\= U ( T , P , ν ) \\+ P ν {\\\\displaystyle h(T,P,\\\\nu )=U(T,P,\\\\nu )+P\\\\nu } ![{\\\\displaystyle h(T,P,\\\\nu )=U(T,P,\\\\nu )+P\\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/38a0269a9025439e15bc408001d2ddd739e304a9). An infinitesimal change in the specific enthalpy will then be\n\nd h \\= d U \\+ V d P \\+ P d V {\\\\displaystyle \\\\mathrm {d} h=\\\\mathrm {d} U+V\\\\,\\\\mathrm {d} P+P\\\\,\\\\mathrm {d} V} ![{\\\\displaystyle \\\\mathrm {d} h=\\\\mathrm {d} U+V\\\\,\\\\mathrm {d} P+P\\\\,\\\\mathrm {d} V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/746c99ad25909e68e8e56ca8a0b6b9bcce84317c)\n\ntherefore\n\nd Q M \\+ V d P \\= d h {\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M}}+V\\\\,\\\\mathrm {d} P=\\\\mathrm {d} h} ![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M}}+V\\\\,\\\\mathrm {d} P=\\\\mathrm {d} h}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d1e31d5cd51830d6b5eb83bb7c77657c5be843c0)\n\nIf the pressure is kept constant, the second term on the left-hand side is zero, and\n\nd Q M d T \\= d h d T {\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M\\\\,\\\\mathrm {d} T}}={\\\\frac {\\\\mathrm {d} h}{\\\\mathrm {d} T}}} ![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M\\\\,\\\\mathrm {d} T}}={\\\\frac {\\\\mathrm {d} h}{\\\\mathrm {d} T}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6a4007c22be9c0d1e82c5a4a011c3bd298de9f44)\n\nThe left-hand side is the specific heat capacity at constant pressure c P {\\\\displaystyle c\\_{P}} ![{\\\\displaystyle c\\_{P}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/23f21e3d892369c6502159212f89643451ab67e4) of the material.\n\n### Connection to equation of state\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=31 \"Edit section: Connection to equation of state\")\\]\n\nIn general, the infinitesimal quantities d T , d P , d V , d U {\\\\displaystyle \\\\mathrm {d} T,\\\\mathrm {d} P,\\\\mathrm {d} V,\\\\mathrm {d} U} ![{\\\\displaystyle \\\\mathrm {d} T,\\\\mathrm {d} P,\\\\mathrm {d} V,\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/04ccfe4aed236cb970d27013b7e0fe2e3cbf7664) are constrained by the equation of state and the specific internal energy function. Namely,\n\n{ d T ∂ F ∂ T ( T , P , V ) \\+ d P ∂ F ∂ P ( T , P , V ) \\+ d V ∂ F ∂ V ( T , P , V ) \\= 0 d T ∂ U ∂ T ( T , P , V ) \\+ d P ∂ U ∂ P ( T , P , V ) \\+ d V ∂ U ∂ V ( T , P , V ) \\= d U {\\\\displaystyle {\\\\begin{cases}\\\\displaystyle \\\\mathrm {d} T{\\\\frac {\\\\partial F}{\\\\partial T}}(T,P,V)+\\\\mathrm {d} P{\\\\frac {\\\\partial F}{\\\\partial P}}(T,P,V)+\\\\mathrm {d} V{\\\\frac {\\\\partial F}{\\\\partial V}}(T,P,V)&=&0\\\\\\\\\\[2ex\\]\\\\displaystyle \\\\mathrm {d} T{\\\\frac {\\\\partial U}{\\\\partial T}}(T,P,V)+\\\\mathrm {d} P{\\\\frac {\\\\partial U}{\\\\partial P}}(T,P,V)+\\\\mathrm {d} V{\\\\frac {\\\\partial U}{\\\\partial V}}(T,P,V)&=&\\\\mathrm {d} U\\\\end{cases}}} ![{\\\\displaystyle {\\\\begin{cases}\\\\displaystyle \\\\mathrm {d} T{\\\\frac {\\\\partial F}{\\\\partial T}}(T,P,V)+\\\\mathrm {d} P{\\\\frac {\\\\partial F}{\\\\partial P}}(T,P,V)+\\\\mathrm {d} V{\\\\frac {\\\\partial F}{\\\\partial V}}(T,P,V)&=&0\\\\\\\\\\[2ex\\]\\\\displaystyle \\\\mathrm {d} T{\\\\frac {\\\\partial U}{\\\\partial T}}(T,P,V)+\\\\mathrm {d} P{\\\\frac {\\\\partial U}{\\\\partial P}}(T,P,V)+\\\\mathrm {d} V{\\\\frac {\\\\partial U}{\\\\partial V}}(T,P,V)&=&\\\\mathrm {d} U\\\\end{cases}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/957c9a178ff753bc04a30bed2819d7e5155314a0)\n\nHere ( ∂ F \/ ∂ T ) ( T , P , V ) {\\\\displaystyle (\\\\partial F\/\\\\partial T)(T,P,V)} ![{\\\\displaystyle (\\\\partial F\/\\\\partial T)(T,P,V)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24993db8c358df7a5c8539501cb70320459370e6) denotes the (partial) derivative of the state equation F {\\\\displaystyle F} ![{\\\\displaystyle F}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/545fd099af8541605f7ee55f08225526be88ce57) with respect to its T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) argument, keeping the other two arguments fixed, evaluated at the state ( T , P , V ) {\\\\displaystyle (T,P,V)} ![{\\\\displaystyle (T,P,V)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c41f5cdee6675ea3447082abaa34afbaf77e4a1f) in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space.\n\nThis analysis also holds no matter how the energy increment d Q {\\\\displaystyle \\\\mathrm {d} Q} ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739) is injected into the sample, namely by [heat conduction](https:\/\/en.wikipedia.org\/wiki\/Heat_conduction \"Heat conduction\"), irradiation, [electromagnetic induction](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_induction \"Electromagnetic induction\"), [radioactive decay](https:\/\/en.wikipedia.org\/wiki\/Radioactive_decay \"Radioactive decay\"), etc.\n\n### Relation between heat capacities\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=32 \"Edit section: Relation between heat capacities\")\\]\n\nFor any specific volume ν {\\\\displaystyle \\\\nu } ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468), denote p ν ( T ) {\\\\displaystyle p\\_{\\\\nu }(T)} ![{\\\\displaystyle p\\_{\\\\nu }(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fab6e24e625d12de49f61c60a3e95d19d5f305d) the function that describes how the pressure varies with the temperature T {\\\\displaystyle T} ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at ν {\\\\displaystyle \\\\nu } ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468). Analogously, for any pressure P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), let ν P ( T ) {\\\\displaystyle \\\\nu \\_{P}(T)} ![{\\\\displaystyle \\\\nu \\_{P}(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d920635881776d5b03f98ee5492f6e0549b3d2f7) be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a). Namely, those functions are such that\n\nF ( T , p ν ( T ) , ν ) \\= 0 {\\\\displaystyle F(T,p\\_{\\\\nu }(T),\\\\nu )=0} ![{\\\\displaystyle F(T,p\\_{\\\\nu }(T),\\\\nu )=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/84a1b582c06f3c19e7abe1e3728d16e072c1a51f)andF ( T , P , ν P ( T ) ) \\= 0 {\\\\displaystyle F(T,P,\\\\nu \\_{P}(T))=0} ![{\\\\displaystyle F(T,P,\\\\nu \\_{P}(T))=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b667b5d0711e09296d3fe4e9d3f6c395acd0f847)\n\nfor any values of T , P , ν {\\\\displaystyle T,P,\\\\nu } ![{\\\\displaystyle T,P,\\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f907ebc4eacdaceb323d1264f58855f033d41081). In other words, the graphs of p ν ( T ) {\\\\displaystyle p\\_{\\\\nu }(T)} ![{\\\\displaystyle p\\_{\\\\nu }(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fab6e24e625d12de49f61c60a3e95d19d5f305d) and ν P ( T ) {\\\\displaystyle \\\\nu \\_{P}(T)} ![{\\\\displaystyle \\\\nu \\_{P}(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d920635881776d5b03f98ee5492f6e0549b3d2f7) are slices of the surface defined by the state equation, cut by planes of constant ν {\\\\displaystyle \\\\nu } ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468) and constant P {\\\\displaystyle P} ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), respectively.\n\nThen, from the [fundamental thermodynamic relation](https:\/\/en.wikipedia.org\/wiki\/Fundamental_thermodynamic_relation \"Fundamental thermodynamic relation\") it follows that\n\nc P ( T , P , ν ) − c V ( T , P , ν ) \\= T \\[ d p ν d T ( T ) \\] \\[ d ν P d T ( T ) \\] {\\\\displaystyle c\\_{P}(T,P,\\\\nu )-c\\_{V}(T,P,\\\\nu )=T\\\\left\\[{\\\\frac {\\\\mathrm {d} p\\_{\\\\nu }}{\\\\mathrm {d} T}}(T)\\\\right\\]\\\\left\\[{\\\\frac {\\\\mathrm {d} \\\\nu \\_{P}}{\\\\mathrm {d} T}}(T)\\\\right\\]} ![{\\\\displaystyle c\\_{P}(T,P,\\\\nu )-c\\_{V}(T,P,\\\\nu )=T\\\\left\\[{\\\\frac {\\\\mathrm {d} p\\_{\\\\nu }}{\\\\mathrm {d} T}}(T)\\\\right\\]\\\\left\\[{\\\\frac {\\\\mathrm {d} \\\\nu \\_{P}}{\\\\mathrm {d} T}}(T)\\\\right\\]}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24554d2ae7d16cbb75b6ad485bf92856b55cf7bd)\n\nThis equation can be rewritten as\n\nc P ( T , P , ν ) − c V ( T , P , ν ) \\= ν T α 2 β T , {\\\\displaystyle c\\_{P}(T,P,\\\\nu )-c\\_{V}(T,P,\\\\nu )=\\\\nu T{\\\\frac {\\\\alpha ^{2}}{\\\\beta \\_{T}}},} ![{\\\\displaystyle c\\_{P}(T,P,\\\\nu )-c\\_{V}(T,P,\\\\nu )=\\\\nu T{\\\\frac {\\\\alpha ^{2}}{\\\\beta \\_{T}}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/27c864cf90d771a66e3a2fa50dec56b8d6c190ad)\n\nwhere\n\n- α\n  \n  {\\\\displaystyle \\\\alpha }\n  \n  ![{\\\\displaystyle \\\\alpha }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b79333175c8b3f0840bfb4ec41b8072c83ea88d3)\n  is the [coefficient of thermal expansion](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_thermal_expansion \"Coefficient of thermal expansion\"),\n- β\n  \n  T\n  \n  {\\\\displaystyle \\\\beta \\_{T}}\n  \n  ![{\\\\displaystyle \\\\beta \\_{T}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3f2855fc59047bc00f5bdedbcad68550a7ac6d13)\n  is the [isothermal](https:\/\/en.wikipedia.org\/wiki\/Isothermal \"Isothermal\") [compressibility](https:\/\/en.wikipedia.org\/wiki\/Compressibility \"Compressibility\"),\n\nboth depending on the state ( T , P , ν ) {\\\\displaystyle (T,P,\\\\nu )} ![{\\\\displaystyle (T,P,\\\\nu )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/847d2a615fb19d33916ce93a4fa935919abca4a7).\n\nThe [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\"), or adiabatic index, is the ratio c P \/ c V {\\\\displaystyle c\\_{P}\/c\\_{V}} ![{\\\\displaystyle c\\_{P}\/c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5bfcc80ce47374e8dfc55b32849c78810e782a87) of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.\n\n### Calculation from first principles\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=33 \"Edit section: Calculation from first principles\")\\]\n\nThe [path integral Monte Carlo](https:\/\/en.wikipedia.org\/wiki\/Path_integral_Monte_Carlo \"Path integral Monte Carlo\") method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \\> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3*R* = 24.94 joules per kelvin per mole of atoms ([Dulong–Petit law](https:\/\/en.wikipedia.org\/wiki\/Dulong%E2%80%93Petit_law \"Dulong–Petit law\"), *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\")). Low temperature approximations for both gases and solids at temperatures less than their characteristic [Einstein temperatures](https:\/\/en.wikipedia.org\/wiki\/Einstein_temperature \"Einstein temperature\") or [Debye temperatures](https:\/\/en.wikipedia.org\/wiki\/Debye_temperature \"Debye temperature\") can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material.[\\[33\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-Benjelloun-34)\n\n#### Ideal gas\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=34 \"Edit section: Ideal gas\")\\]\n\nFor an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), evaluating the partial derivatives above according to the [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\"), where *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\"), for an ideal gas[\\[34\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-35)\n\nP V \\= n R T , C P − C V \\= T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n , P \\= n R T V ⇒ ( ∂ P ∂ T ) V , n \\= n R V , V \\= n R T P ⇒ ( ∂ V ∂ T ) P , n \\= n R P . {\\\\displaystyle {\\\\begin{alignedat}{3}PV&=nRT,\\\\\\\\C\\_{P}-C\\_{V}&=T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n},\\\\\\\\P&={\\\\frac {nRT}{V}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}&={\\\\frac {nR}{V}},\\\\\\\\V&={\\\\frac {nRT}{P}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}&={\\\\frac {nR}{P}}.\\\\end{alignedat}}} ![{\\\\displaystyle {\\\\begin{alignedat}{3}PV&=nRT,\\\\\\\\C\\_{P}-C\\_{V}&=T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n},\\\\\\\\P&={\\\\frac {nRT}{V}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}&={\\\\frac {nR}{V}},\\\\\\\\V&={\\\\frac {nRT}{P}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}&={\\\\frac {nR}{P}}.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e2f8e4a06c2339161629da3ad59eddd0383a651f)\n\nSubstituting\n\nT ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n \\= T n R V n R P \\= n R T V n R P \\= P n R P \\= n R , {\\\\displaystyle T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}=T{\\\\frac {nR}{V}}{\\\\frac {nR}{P}}={\\\\frac {nRT}{V}}{\\\\frac {nR}{P}}=P{\\\\frac {nR}{P}}=nR,} ![{\\\\displaystyle T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}=T{\\\\frac {nR}{V}}{\\\\frac {nR}{P}}={\\\\frac {nRT}{V}}{\\\\frac {nR}{P}}=P{\\\\frac {nR}{P}}=nR,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e49d3f01b48a4771f06476e4922cd418e91e9354)\n\nthis equation reduces simply to [Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_Robert_von_Mayer \"Julius Robert von Mayer\")'s relation:\n\nC P , m − C V , m \\= R . {\\\\displaystyle C\\_{P,m}-C\\_{V,m}=R.} ![{\\\\displaystyle C\\_{P,m}-C\\_{V,m}=R.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1b0951113a93ae7d847ed0234a94039782ae0a5e)\n\nThe differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.\n\n## See also\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=35 \"Edit section: See also\")\\]\n\n[![icon](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/6f\/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg\/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg) [Physics portal](https:\/\/en.wikipedia.org\/wiki\/Portal:Physics \"Portal:Physics\")\n\n- [Enthalpy of fusion](https:\/\/en.wikipedia.org\/wiki\/Enthalpy_of_fusion \"Enthalpy of fusion\") (latent heat of melting)\n- [Enthalpy of vaporization](https:\/\/en.wikipedia.org\/wiki\/Enthalpy_of_vaporization \"Enthalpy of vaporization\") (latent heat of vaporization)\n- [Frenkel line](https:\/\/en.wikipedia.org\/wiki\/Frenkel_line \"Frenkel line\")\n- [Heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\")\n- [Heat equation](https:\/\/en.wikipedia.org\/wiki\/Heat_equation \"Heat equation\")\n- [Heat transfer coefficient](https:\/\/en.wikipedia.org\/wiki\/Heat_transfer_coefficient \"Heat transfer coefficient\")\n- [History of thermodynamics](https:\/\/en.wikipedia.org\/wiki\/History_of_thermodynamics \"History of thermodynamics\")\n- [Joback method](https:\/\/en.wikipedia.org\/wiki\/Joback_method \"Joback method\") (Estimation of heat capacities)\n- [Latent heat](https:\/\/en.wikipedia.org\/wiki\/Latent_heat \"Latent heat\")\n- [Material properties (thermodynamics)](https:\/\/en.wikipedia.org\/wiki\/Material_properties_\\(thermodynamics\\) \"Material properties (thermodynamics)\")\n- [Quantum statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_statistical_mechanics \"Quantum statistical mechanics\")\n- [R-value (insulation)](https:\/\/en.wikipedia.org\/wiki\/R-value_\\(insulation\\) \"R-value (insulation)\")\n- [Statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\")\n- [Table of specific heat capacities](https:\/\/en.wikipedia.org\/wiki\/Table_of_specific_heat_capacities \"Table of specific heat capacities\")\n- [Thermal mass](https:\/\/en.wikipedia.org\/wiki\/Thermal_mass \"Thermal mass\")\n- [Thermodynamic databases for pure substances](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_databases_for_pure_substances \"Thermodynamic databases for pure substances\")\n- [Thermodynamic equations](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_equations \"Thermodynamic equations\")\n- [Volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\")\n\n## Notes\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=36 \"Edit section: Notes\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-gold_30-0)** [IUPAC](https:\/\/en.wikipedia.org\/wiki\/International_Union_of_Pure_and_Applied_Chemistry \"International Union of Pure and Applied Chemistry\"), *[Compendium of Chemical Terminology](https:\/\/en.wikipedia.org\/wiki\/IUPAC_books#Gold_Book \"IUPAC books\")*, 5th ed. (the \"Gold Book\") (2025). Online version: (2006–) \"[Standard Pressure](https:\/\/goldbook.iupac.org\/terms\/view\/S05921.html)\". [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1351\/goldbook.S05921](https:\/\/doi.org\/10.1351%2Fgoldbook.S05921).\n\n## References\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=37 \"Edit section: References\")\\]\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-1)**\n   Halliday, David; Resnick, Robert; Walker, Jearl (2001). *Fundamentals of Physics* (6th ed.). New York, NY US: [John Wiley & Sons](https:\/\/en.wikipedia.org\/wiki\/John_Wiley_%26_Sons \"John Wiley & Sons\").\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-2)**\n   Open University (2008). *S104 Book 3 Energy and Light*, p. 59. [The Open University](https:\/\/en.wikipedia.org\/wiki\/The_Open_University \"The Open University\").\n   \n   [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [9781848731646](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781848731646 \"Special:BookSources\/9781848731646\")\n   \n   .\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-3)**\n   Open University (2008). *S104 Book 3 Energy and Light*, p. 179. [The Open University](https:\/\/en.wikipedia.org\/wiki\/The_Open_University \"The Open University\").\n   \n   [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [9781848731646](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781848731646 \"Special:BookSources\/9781848731646\")\n   \n   .\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-4)**\n   Engineering ToolBox (2003). [\"Specific Heat of some common Substances\"](https:\/\/www.engineeringtoolbox.com\/specific-heat-capacity-d_391.html).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-colen2001_5-0)** (2001): *Columbia Encyclopedia*, 6th ed.; as quoted by [Encyclopedia.com](https:\/\/www.encyclopedia.com\/science-and-technology\/physics\/physics\/specific-heat#1E1specheat). Columbia University Press. Accessed on 2019-04-11.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-6)**\n   Laidler, Keith J. (1993). [*The World of Physical Chemistry*](https:\/\/books.google.com\/books?id=01LRlPbH80cC). Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-19-855919-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-19-855919-4 \"Special:BookSources\/0-19-855919-4\")\n   \n   .\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-:1_7-0)**\n   [Ramsay, William](https:\/\/en.wikipedia.org\/wiki\/William_Ramsay \"William Ramsay\") (1918). *The life and letters of Joseph Black, M.D*. Constable. pp. 38–39\\.\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-8)**\n   Black, Joseph (1807). Robison, John (ed.). [*Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh*](https:\/\/books.google.com\/books?id=lqI9AQAAMAAJ&pg=PA76). Vol. 1. Mathew Carey. pp. 76–77\\.\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-:0_9-0)**\n   West, John B. (2014-06-15). [\"Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases\"](https:\/\/www.physiology.org\/doi\/10.1152\/ajplung.00020.2014). *American Journal of Physiology. Lung Cellular and Molecular Physiology*. **306** (12): L1057–L1063. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1152\/ajplung.00020.2014](https:\/\/doi.org\/10.1152%2Fajplung.00020.2014). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1040-0605](https:\/\/search.worldcat.org\/issn\/1040-0605). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24682452](https:\/\/pubmed.ncbi.nlm.nih.gov\/24682452).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-10)**\n    [International Bureau of Weights and Measures](https:\/\/en.wikipedia.org\/wiki\/International_Bureau_of_Weights_and_Measures \"International Bureau of Weights and Measures\") (2006), [*The International System of Units (SI)*](https:\/\/www.bipm.org\/documents\/20126\/41483022\/si_brochure_8.pdf) (PDF) (8th ed.), [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [92-822-2213-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/92-822-2213-6 \"Special:BookSources\/92-822-2213-6\")\n    \n    , [archived](https:\/\/web.archive.org\/web\/20210604163219\/https:\/\/www.bipm.org\/documents\/20126\/41483022\/si_brochure_8.pdf) (PDF) from the original on 2021-06-04, retrieved 2021-12-16\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-toolbox_11-0)**\n    [\"Water – Thermal Properties\"](http:\/\/www.engineeringtoolbox.com\/water-thermal-properties-d_162.html). Engineeringtoolbox.com. Retrieved 2021-03-29.\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-12)**\n    International Union of Pure and Applied Chemistry, Physical Chemistry Division. [\"Quantities, Units and Symbols in Physical Chemistry\"](http:\/\/old.iupac.org\/publications\/books\/gbook\/green_book_2ed.pdf) (PDF). Blackwell Sciences. p. 7. \"The adjective specific before the name of an extensive quantity is often used to mean divided by mass.\"\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-Lange_13-0)** Lange's Handbook of Chemistry, 10th ed., page 1524.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-14)**\n    Quick, C. R.; Schawe, J. E. K.; Uggowitzer, P. J.; Pogatscher, S. (2019-07-01). [\"Measurement of specific heat capacity via fast scanning calorimetry—Accuracy and loss corrections\"](https:\/\/doi.org\/10.1016%2Fj.tca.2019.03.021). *Thermochimica Acta*. Special Issue on occasion of the 65th birthday of Christoph Schick. **677**: 12–20\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019TcAc..677...12Q](https:\/\/ui.adsabs.harvard.edu\/abs\/2019TcAc..677...12Q). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.tca.2019.03.021](https:\/\/doi.org\/10.1016%2Fj.tca.2019.03.021). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0040-6031](https:\/\/search.worldcat.org\/issn\/0040-6031).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-15)**\n    Pogatscher, S.; Leutenegger, D.; Schawe, J. E. K.; Uggowitzer, P. J.; Löffler, J. F. (September 2016). [\"Solid–solid phase transitions via melting in metals\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC4844691). *Nature Communications*. **7** (1) 11113. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2016NatCo...711113P](https:\/\/ui.adsabs.harvard.edu\/abs\/2016NatCo...711113P). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/ncomms11113](https:\/\/doi.org\/10.1038%2Fncomms11113). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [2041-1723](https:\/\/search.worldcat.org\/issn\/2041-1723). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [4844691](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC4844691). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [27103085](https:\/\/pubmed.ncbi.nlm.nih.gov\/27103085).\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-Koch_16-0)**\n    Koch, Werner (2013). [*VDI Steam Tables*](https:\/\/books.google.com\/books?id=bJ_wBgAAQBAJ&pg=PA8) (4 ed.). Springer. p. 8. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-642-52941-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-642-52941-2 \"Special:BookSources\/978-3-642-52941-2\")\n    \n    .\n    \n    Published under the auspices of the *Verein Deutscher Ingenieure* (VDI).\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-17)**\n    Cardarelli, Francois (2012). [*Scientific Unit Conversion: A Practical Guide to Metrication*](https:\/\/books.google.com\/books?id=-ZveBwAAQBAJ&pg=PA19-IA35). M.J. Shields (translation) (2 ed.). Springer. p. 19. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4471-0805-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4471-0805-4 \"Special:BookSources\/978-1-4471-0805-4\")\n    \n    .\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-18)**\n    From direct values: 1\n    \n    ⁠BTU\/lb⋅°R⁠ × 1055.06\n    \n    ⁠J\/BTU⁠ × (\n    \n    ⁠1\/0\\.45359237⁠)\n    \n    ⁠lb\/kg⁠ x\n    \n    ⁠9\/5⁠\n    \n    ⁠°R\/K⁠ = 4186.82\n    \n    ⁠J\/kg⋅K⁠\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-19)** °F=°R\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-20)** °C=K\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-21)**\n    McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: [University Science Books](https:\/\/en.wikipedia.org\/w\/index.php?title=University_Science_Books&action=edit&redlink=1 \"University Science Books (page does not exist)\"). pp. 83–85\\.\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-22)**\n    [\"6.6: Electronic Partition Function\"](https:\/\/chem.libretexts.org\/Bookshelves\/Physical_and_Theoretical_Chemistry_Textbook_Maps\/Statistical_Thermodynamics_\\(Jeschke\\)\/06:_Partition_Functions_of_Gases\/6.06:_Electronic_Partition_Function). *Chemistry LibreTexts*. 2020-11-26. Retrieved 2024-12-16.\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-23)**\n    Bonhoeffer, K.F.; Harteck, P. (1926). [\"Über Para- und Orthowasserstoff\"](https:\/\/www.degruyter.com\/document\/doi\/10.1515\/zpch-1929-0408\/html). *Z. Phys. Chem*. **4B**: 113–141\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1515\/zpch-1929-0408](https:\/\/doi.org\/10.1515%2Fzpch-1929-0408).\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-24)**\n    McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: University Science Books. p. 107.\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-25)** Feynman, R., *[The Feynman Lectures on Physics](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\")*, Vol. 1, ch. 40, pp. 7–8\n26. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-26)**\n    Reif, F. (1965). [*Fundamentals of statistical and thermal physics*](https:\/\/archive.org\/details\/fundamentalsofst00reif). McGraw-Hill. pp. [253–254](https:\/\/archive.org\/details\/fundamentalsofst00reif\/page\/253).\n27. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-27)**\n    Kittel, Charles; Kroemer, Herbert (2000). *Thermal physics*. W. H. Freeman. p. 78. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-7167-1088-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-7167-1088-2 \"Special:BookSources\/978-0-7167-1088-2\")\n    \n    .\n28. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-thor1993_28-0)** Thornton, Steven T. and Rex, Andrew (1993) *Modern Physics for Scientists and Engineers*, Saunders College Publishing\n29. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-chas1998_29-0)** Chase, M.W. Jr. (1998) *[NIST-JANAF Themochemical Tables, Fourth Edition](https:\/\/webbook.nist.gov\/cgi\/cbook.cgi?ID=C7727379&Type=JANAFG)*, In *Journal of Physical and Chemical Reference Data*, Monograph 9, pages 1–1951.\n30. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-31)**\n    Yunus A. Cengel and Michael A. Boles, *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill, 2010,\n    \n    [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [007-352932-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/007-352932-X \"Special:BookSources\/007-352932-X\")\n    \n    .\n31. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-32)**\n    Fraundorf, P. (2003). \"Heat capacity in bits\". *American Journal of Physics*. **71** (11): 1142. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[cond-mat\/9711074](https:\/\/arxiv.org\/abs\/cond-mat\/9711074). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2003AmJPh..71.1142F](https:\/\/ui.adsabs.harvard.edu\/abs\/2003AmJPh..71.1142F). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.1593658](https:\/\/doi.org\/10.1119%2F1.1593658). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [18742525](https:\/\/api.semanticscholar.org\/CorpusID:18742525).\n32. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-fein_33-0)** Feynman, Richard, *[The Feynman Lectures on Physics](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\")*, Vol. 1, Ch. 45\n33. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-Benjelloun_34-0)** S. Benjelloun, \"Thermodynamic identities and thermodynamic consistency of Equation of States\", [Link to Archiv e-print](https:\/\/arxiv.org\/abs\/2105.04845) [Link to Hal e-print](https:\/\/hal.archives-ouvertes.fr\/hal-03216379\/)\n34. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-35)**\n    Cengel, Yunus A. and Boles, Michael A. (2010) *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill\n    \n    [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [007-352932-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/007-352932-X \"Special:BookSources\/007-352932-X\")\n    \n    .\n\n## Further reading\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=38 \"Edit section: Further reading\")\\]\n\n- Emmerich Wilhelm & Trevor M. Letcher, Eds., 2010, *Heat Capacities: Liquids, Solutions and Vapours*, Cambridge, U.K.:Royal Society of Chemistry,\n  [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-85404-176-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-85404-176-1 \"Special:BookSources\/0-85404-176-1\")\n  . A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, \"Heat Capacities: Introduction, Concepts, and Selected Applications\" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as [photoacoustic](https:\/\/en.wikipedia.org\/wiki\/Photoacoustic_effect \"Photoacoustic effect\") methods, e.g., Jan Thoen & Christ Glorieux, \"Photothermal Techniques for Heat Capacities,\" and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc.\n\n## External links\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=39 \"Edit section: External links\")\\]\n\n- (2012-05may-24) [Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World](https:\/\/physicsworld.com\/a\/phonon-theory-sheds-light-on-liquid-thermodynamics\/) [The phonon theory of liquid thermodynamics \\| Scientific Reports](https:\/\/www.nature.com\/articles\/srep00421)\n\n| [Authority control databases](https:\/\/en.wikipedia.org\/wiki\/Help:Authority_control \"Help:Authority control\") [![Edit this at Wikidata](https:\/\/upload.wikimedia.org\/wikipedia\/en\/thumb\/8\/8a\/OOjs_UI_icon_edit-ltr-progressive.svg\/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https:\/\/www.wikidata.org\/wiki\/Q487756#identifiers \"Edit this at Wikidata\") |   |\n|---|---|\n| International | [GND](https:\/\/d-nb.info\/gnd\/4182218-3) |\n| National | [United States](https:\/\/id.loc.gov\/authorities\/sh85126389) [France](https:\/\/catalogue.bnf.fr\/ark:\/12148\/cb12274423z) [BnF data](https:\/\/data.bnf.fr\/ark:\/12148\/cb12274423z) [Israel](https:\/\/www.nli.org.il\/en\/authorities\/987007565726505171) |\n| Other | [Yale LUX](https:\/\/lux.collections.yale.edu\/view\/concept\/4a484350-e872-49a3-87de-3b087bf8c5e5) |\n\n![](https:\/\/en.wikipedia.org\/wiki\/Special:CentralAutoLogin\/start?useformat=desktop&type=1x1&usesul3=1)\n\nRetrieved from \"<https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&oldid=1340264896>\"\n\n[Categories](https:\/\/en.wikipedia.org\/wiki\/Help:Category \"Help:Category\"):\n\n- [Physical quantities](https:\/\/en.wikipedia.org\/wiki\/Category:Physical_quantities \"Category:Physical quantities\")\n- [Thermodynamic properties](https:\/\/en.wikipedia.org\/wiki\/Category:Thermodynamic_properties \"Category:Thermodynamic properties\")\n\nHidden categories:\n\n- [Articles with short description](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_short_description \"Category:Articles with short description\")\n- [Short description is different from Wikidata](https:\/\/en.wikipedia.org\/wiki\/Category:Short_description_is_different_from_Wikidata \"Category:Short description is different from Wikidata\")\n- [All articles with unsourced statements](https:\/\/en.wikipedia.org\/wiki\/Category:All_articles_with_unsourced_statements \"Category:All articles with unsourced statements\")\n- [Articles with unsourced statements from April 2019](https:\/\/en.wikipedia.org\/wiki\/Category:Articles_with_unsourced_statements_from_April_2019 \"Category:Articles with unsourced statements from April 2019\")\n- [Pages using div col with small parameter](https:\/\/en.wikipedia.org\/wiki\/Category:Pages_using_div_col_with_small_parameter \"Category:Pages using div col with small parameter\")\n\n- This page was last edited on 24 February 2026, at 19:58 (UTC).\n- Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License \"Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License\"); additional terms may apply. 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Wikipedia® is a registered trademark of the [Wikimedia Foundation, Inc.](https:\/\/wikimediafoundation.org\/), a non-profit organization.\n\n- [Privacy policy](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Privacy_policy)\n- [About Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:About)\n- [Disclaimers](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:General_disclaimer)\n- [Contact Wikipedia](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Contact_us)\n- [Legal & safety contacts](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Legal:Wikimedia_Foundation_Legal_and_Safety_Contact_Information)\n- [Code of Conduct](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Universal_Code_of_Conduct)\n- [Developers](https:\/\/developer.wikimedia.org\/)\n- [Statistics](https:\/\/stats.wikimedia.org\/#\/en.wikipedia.org)\n- [Cookie statement](https:\/\/foundation.wikimedia.org\/wiki\/Special:MyLanguage\/Policy:Cookie_statement)\n- [Mobile view](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&mobileaction=toggle_view_mobile)\n\n- [![Wikimedia Foundation](https:\/\/en.wikipedia.org\/static\/images\/footer\/wikimedia.svg)](https:\/\/www.wikimedia.org\/)\n- [![Powered by MediaWiki](https:\/\/en.wikipedia.org\/w\/resources\/assets\/mediawiki_compact.svg)](https:\/\/www.mediawiki.org\/)\n\nSearch\n\nToggle the table of contents\n\nSpecific heat capacity\n\n61 languages\n\n[Add topic](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity)","attrs_readable_markdown":"| Specific heat capacity |   |\n|---|---|\n| Other names | Specific heat |\n| Common symbols | *c* |\n| [SI unit](https:\/\/en.wikipedia.org\/wiki\/SI_unit \"SI unit\") | J⋅kg−1⋅K−1 |\n| In [SI base units](https:\/\/en.wikipedia.org\/wiki\/SI_base_unit \"SI base unit\") | m2⋅K−1⋅s−2 |\n| [Intensive](https:\/\/en.wikipedia.org\/wiki\/Intensive_and_extensive_properties \"Intensive and extensive properties\")? | Yes |\n| [Dimension](https:\/\/en.wikipedia.org\/wiki\/Dimensional_analysis#Formulation \"Dimensional analysis\") | L2⋅T−2⋅Θ−1 |\n\nIn [thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Thermodynamics \"Thermodynamics\"), the **specific heat capacity** (symbol *c*) of a substance is the amount of [heat](https:\/\/en.wikipedia.org\/wiki\/Heat \"Heat\") that must be added to one unit of mass of the substance in order to cause an increase of one unit in [temperature](https:\/\/en.wikipedia.org\/wiki\/Temperature \"Temperature\"). It is also referred to as **massic heat capacity** or as the **specific heat.** More formally it is the [heat capacity](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity \"Heat capacity\") of a sample of the substance divided by the [mass](https:\/\/en.wikipedia.org\/wiki\/Mass \"Mass\") of the sample.[\\[1\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-1) The [SI](https:\/\/en.wikipedia.org\/wiki\/International_System_of_Units \"International System of Units\") unit of specific heat capacity is [joule](https:\/\/en.wikipedia.org\/wiki\/Joule \"Joule\") per [kelvin](https:\/\/en.wikipedia.org\/wiki\/Kelvin \"Kelvin\") per [kilogram](https:\/\/en.wikipedia.org\/wiki\/Kilogram \"Kilogram\"), J⋅kg−1⋅K−1.[\\[2\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-2) For example, the heat required to raise the temperature of 1 kg of [water](https:\/\/en.wikipedia.org\/wiki\/Water \"Water\") by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.[\\[3\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-3)\n\nSpecific heat capacity often varies with temperature, and is different for each [state of matter](https:\/\/en.wikipedia.org\/wiki\/State_of_matter \"State of matter\"). Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C, but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1. The specific heat capacities of [iron](https:\/\/en.wikipedia.org\/wiki\/Iron \"Iron\"), [granite](https:\/\/en.wikipedia.org\/wiki\/Granite \"Granite\"), and [hydrogen](https:\/\/en.wikipedia.org\/wiki\/Hydrogen \"Hydrogen\") gas are about 449 J⋅kg−1⋅K−1, 790 J⋅kg−1⋅K−1, and 14300 J⋅kg−1⋅K−1, respectively.[\\[4\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-4) While the substance is undergoing a [phase transition](https:\/\/en.wikipedia.org\/wiki\/Phase_transition \"Phase transition\"), such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature.\n\nThe specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity *at constant pressure*) than when it is heated in a closed vessel that prevents expansion (specific heat capacity *at constant volume*). These two values are usually denoted by ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f) and ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082), respectively; their quotient ![{\\\\displaystyle \\\\gamma =c\\_{p}\/c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dbe578fc940f94ff433f350a8b2d7cad4dbf01c0) is the [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\").\n\nThe term *specific heat* may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C;[\\[5\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-colen2001-5) much in the fashion of [specific gravity](https:\/\/en.wikipedia.org\/wiki\/Specific_gravity \"Specific gravity\"). Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of [moles](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(unit\\) \"Mole (unit)\"), one gets the [molar heat capacity](https:\/\/en.wikipedia.org\/wiki\/Molar_heat_capacity \"Molar heat capacity\") instead, whose SI unit is joule per kelvin per mole, J⋅mol−1⋅K−1. If the amount is taken to be the [volume](https:\/\/en.wikipedia.org\/wiki\/Volume \"Volume\") of the sample (as is sometimes done in engineering), one gets the [volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\"), whose SI unit is joule per kelvin per [cubic meter](https:\/\/en.wikipedia.org\/wiki\/Cubic_meter \"Cubic meter\"), J⋅m−3⋅K−1.\n\n### Discovery of specific heat\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=2 \"Edit section: Discovery of specific heat\")\\]\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/8c\/Black_Joseph_%28cropped%29.jpg\/250px-Black_Joseph_%28cropped%29.jpg)](https:\/\/en.wikipedia.org\/wiki\/File:Black_Joseph_\\(cropped\\).jpg)\n\nJoseph Black\n\nOne of the first scientists to use the concept was [Joseph Black](https:\/\/en.wikipedia.org\/wiki\/Joseph_Black \"Joseph Black\"), an 18th-century medical doctor and professor of medicine at [Glasgow University](https:\/\/en.wikipedia.org\/wiki\/Glasgow_University \"Glasgow University\"). He measured the specific heat capacities of many substances, using the term *capacity for heat*.[\\[6\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-6) In 1756 or soon thereafter, Black began an extensive study of heat.[\\[7\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-:1-7) In 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by [Daniel Gabriel Fahrenheit](https:\/\/en.wikipedia.org\/wiki\/Daniel_Gabriel_Fahrenheit \"Daniel Gabriel Fahrenheit\") on behalf of Dutch physician [Herman Boerhaave](https:\/\/en.wikipedia.org\/wiki\/Herman_Boerhaave \"Herman Boerhaave\"). For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: \"Quicksilver \\[mercury\\] ... has less capacity for the matter of heat than water.\"[\\[8\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-8)[\\[9\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-:0-9)\n\nThe specific heat capacity of a substance, usually denoted by ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) or ![{\\\\displaystyle s}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/01d131dfd7673938b947072a13a9744fe997e632), is the heat capacity ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) of a sample of the substance, divided by the mass ![{\\\\displaystyle M}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f82cade9898ced02fdd08712e5f0c0151758a0dd) of the sample:[\\[10\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-10) ![{\\\\displaystyle c={\\\\frac {C}{M}}={\\\\frac {1}{M}}\\\\cdot {\\\\frac {\\\\mathrm {d} Q}{\\\\mathrm {d} T}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a58291be3575e33a756ed894a18347c8eabbba29) where ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739) [represents](https:\/\/en.wikipedia.org\/wiki\/Derivative \"Derivative\") the amount of heat needed to uniformly raise the temperature of the sample by a small increment ![{\\\\displaystyle \\\\mathrm {d} T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/309833548168052f2696ca961149df099b5d1f80).\n\nLike the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) of the sample and the [pressure](https:\/\/en.wikipedia.org\/wiki\/Pressure \"Pressure\") ![{\\\\displaystyle p}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/81eac1e205430d1f40810df36a0edffdc367af36) applied to it. Therefore, it should be considered a function ![{\\\\displaystyle c(p,T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e8bac9cbcaf700b5206116a980a24a84404ba2e5) of those two variables.\n\nThese parameters are usually specified when giving the specific heat capacity of a substance. For example, \"Water (liquid): ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f) = 4187 J⋅kg−1⋅K−1 (15 °C).\"[\\[11\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-toolbox-11) When not specified, published values of the specific heat capacity ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) generally are valid for some [standard conditions for temperature and pressure](https:\/\/en.wikipedia.org\/wiki\/Standard_conditions_for_temperature_and_pressure \"Standard conditions for temperature and pressure\").\n\nHowever, the dependency of ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier ![{\\\\displaystyle (p,T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/140e6e1e1f95c1aba3ffb6dc196a9be07a91d1a6) and approximates the specific heat capacity by a constant ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) suitable for those ranges.\n\nSpecific heat capacity is an [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier \"specific\" in front of an extensive property often indicates an intensive property derived from it.[\\[12\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-12))\n\nThe injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and\/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure ![{\\\\displaystyle p}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/81eac1e205430d1f40810df36a0edffdc367af36) and starting temperature ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0). Two particular choices are widely used:\n\nThe value of ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) is always less than the value of ![{\\\\displaystyle c\\_{p}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/77463f4fbb953a6f1fe19d83708e553f6d21457f) for all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\") of gases is typically between 1.3 and 1.67.[\\[13\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-Lange-13)\n\nThe specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale.\n\nThe specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops.\n\nThe specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a [phase change](https:\/\/en.wikipedia.org\/wiki\/Phase_transition \"Phase transition\"), such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement.\n\nThe specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a [calorimeter](https:\/\/en.wikipedia.org\/wiki\/Calorimeter \"Calorimeter\"), and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as [differential scanning calorimetry](https:\/\/en.wikipedia.org\/wiki\/Differential_scanning_calorimetry \"Differential scanning calorimetry\").[\\[14\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-14)[\\[15\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-15)\n\n[![](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/f\/fe\/Water_temperature_vs_heat_added.svg\/250px-Water_temperature_vs_heat_added.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Water_temperature_vs_heat_added.svg)\n\nGraph of temperature of phases of water heated from −100 °C to 200 °C – the dashed line example shows that melting and heating 1 kg of ice at −50 °C to water at 40 °C needs 600 kJ\n\nThe specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the [coefficient of thermal expansion](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_thermal_expansion \"Coefficient of thermal expansion\") and the [compressibility](https:\/\/en.wikipedia.org\/wiki\/Bulk_modulus \"Bulk modulus\") of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.\\[*[citation needed](https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Citation_needed \"Wikipedia:Citation needed\")*\\]\n\n### International system\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=8 \"Edit section: International system\")\\]\n\nThe SI unit for specific heat capacity is joule per kelvin per kilogram ⁠J\/kg⋅K⁠, J⋅K−1⋅kg−1. Since an increment of temperature of one [degree Celsius](https:\/\/en.wikipedia.org\/wiki\/Celsius_scale \"Celsius scale\") is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J\/(kg⋅°C). Sometimes the [gram](https:\/\/en.wikipedia.org\/wiki\/Gram \"Gram\") is used instead of kilogram for the unit of mass: 1 J⋅g−1⋅K−1 = 1000 J⋅kg−1⋅K−1.\n\nThe specific heat capacity of a substance (per unit of mass) has [dimension](https:\/\/en.wikipedia.org\/wiki\/Dimensional_analysis \"Dimensional analysis\") L2⋅Θ−1⋅T−2, or (L\/T)2\/Θ. Therefore, the SI unit J⋅kg−1⋅K−1 is equivalent to [metre](https:\/\/en.wikipedia.org\/wiki\/Metre \"Metre\") squared per [second](https:\/\/en.wikipedia.org\/wiki\/Second \"Second\") squared per [kelvin](https:\/\/en.wikipedia.org\/wiki\/Kelvin \"Kelvin\") (m2⋅K−1⋅s−2).\n\n### Imperial engineering units\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=9 \"Edit section: Imperial engineering units\")\\]\n\nProfessionals in [construction](https:\/\/en.wikipedia.org\/wiki\/Construction \"Construction\"), [civil engineering](https:\/\/en.wikipedia.org\/wiki\/Civil_engineering \"Civil engineering\"), [chemical engineering](https:\/\/en.wikipedia.org\/wiki\/Chemical_engineering \"Chemical engineering\"), and other technical disciplines, especially in the [United States](https:\/\/en.wikipedia.org\/wiki\/United_States \"United States\"), may use [English Engineering units](https:\/\/en.wikipedia.org\/wiki\/English_Engineering_Units \"English Engineering Units\") including the [pound](https:\/\/en.wikipedia.org\/wiki\/Pound_\\(mass\\) \"Pound (mass)\") (lb = 0.45359237 kg) as the unit of mass, the [degree Fahrenheit](https:\/\/en.wikipedia.org\/wiki\/Fahrenheit \"Fahrenheit\") or [Rankine](https:\/\/en.wikipedia.org\/wiki\/Rankine_scale \"Rankine scale\") (°R = ⁠5\/9⁠ K, about 0.555556 K) as the unit of temperature increment, and the [British thermal unit](https:\/\/en.wikipedia.org\/wiki\/British_thermal_unit \"British thermal unit\") (BTU ≈ 1055.056 J),[\\[16\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-Koch-16)[\\[17\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-17) as the unit of heat.\n\nIn those contexts, the unit of specific heat capacity is BTU\/lb⋅°R, or 1 ⁠BTU\/lb⋅°R⁠ = 4186.68⁠J\/kg⋅K⁠.[\\[18\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-18) The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU\/lb⋅°F.[\\[19\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-19) Note the value's similarity to that of the calorie - 4187 J\/kg⋅°C ≈ 4184 J\/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C.\n\nIn chemistry, heat amounts were often measured in [calories](https:\/\/en.wikipedia.org\/wiki\/Calorie \"Calorie\"). Confusingly, there are two common units with that name, respectively denoted *cal* and *Cal*:\n\n- the *small calorie* (*gram-calorie, cal*) is 4.184 J exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal\/(°C⋅g).\n- The *grand calorie* (*kilocalorie, kilogram-calorie, food calorie, kcal, Cal*) is 1000 small calories, 4184 J exactly. It was defined so that the specific heat capacity of water would be 1 Cal\/(°C⋅kg).\n\nWhile these units are still used in some contexts (such as kilogram calorie in [nutrition](https:\/\/en.wikipedia.org\/wiki\/Nutrition \"Nutrition\")), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually:\n\n1\n\n⁠cal\/°C⋅g⁠ = 1\n\n⁠Cal\/°C⋅kg⁠ = 1\n\n⁠kcal\/°C⋅kg⁠ = 4184\n\n⁠J\/kg⋅K⁠[\\[20\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-20) = 4.184\n\n⁠kJ\/kg⋅K⁠.\n\nNote that while cal is **1⁄1000** of a Cal or kcal, it is also per *gram* instead of **kilo***gram*: ergo, in either unit, the specific heat capacity of water is approximately 1.\n\nThe temperature of a sample of a substance reflects the average [kinetic energy](https:\/\/en.wikipedia.org\/wiki\/Kinetic_energy \"Kinetic energy\") of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the [equipartition theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\").\n\n[Statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\") predicts that at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy, unless multiple electronic states are accessible at room temperature (such is the case for atomic fluorine).[\\[21\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-21) Thus, the [heat capacity per mole](https:\/\/en.wikipedia.org\/wiki\/Molar_heat_capacity \"Molar heat capacity\") at room temperature is the same for all of the noble gases as well as for many other atomic vapors. More precisely, ![{\\\\displaystyle c\\_{V,\\\\mathrm {m} }=3R\/2\\\\approx \\\\mathrm {12.5\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/dcee637c4392e7347a6b4b7d57395e90ef9b8f5b) and ![{\\\\displaystyle c\\_{P,\\\\mathrm {m} }=5R\/2\\\\approx \\\\mathrm {21\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f6fc485f5327a3d2857548cbe7c145147d826306), where ![{\\\\displaystyle R\\\\approx \\\\mathrm {8.31446\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2b344e96636e94e8469d9e4a0eafa306b3bec07e) is the [ideal gas unit](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas_constant \"Ideal gas constant\") (which is the product of [Boltzmann conversion constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\") from [kelvin](https:\/\/en.wikipedia.org\/wiki\/Kelvin \"Kelvin\") microscopic energy unit to the macroscopic energy unit [joule](https:\/\/en.wikipedia.org\/wiki\/Joule \"Joule\"), and the [Avogadro number](https:\/\/en.wikipedia.org\/wiki\/Avogadro_number \"Avogadro number\")).\n\nTherefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional) [atomic weight](https:\/\/en.wikipedia.org\/wiki\/Atomic_weight \"Atomic weight\") ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3). That is, approximately,\n\n![{\\\\displaystyle c\\_{V}\\\\approx \\\\mathrm {12470\\\\,J\\\\cdot K^{-1}\\\\cdot kg^{-1}} \/A\\\\quad \\\\quad \\\\quad c\\_{p}\\\\approx \\\\mathrm {20785\\\\,J\\\\cdot K^{-1}\\\\cdot kg^{-1}} \/A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0733e9dc0db1000b284289198a6d8d9b4ed008d9)\n\nFor the noble gases, from helium to xenon, these computed values are\n\n| Gas | He | Ne | Ar | Kr | Xe |\n|---|---|---|---|---|---|\n| ![{\\\\displaystyle A}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3) |   |   |   |   |   |\n\nA polyatomic gas molecule can store energy in additional degrees of freedom. Its kinetic energy contributes to the heat capacity in the same way as monatomic gases, but there are also contributions from the [rotations](https:\/\/en.wikipedia.org\/wiki\/Rotational_energy \"Rotational energy\") of the molecule and vibration of the atoms relative to each other (including internal [potential energy](https:\/\/en.wikipedia.org\/wiki\/Potential_energy \"Potential energy\")).\n\nThe heat capacity may also have contribution from [excited electronic states](https:\/\/en.wikipedia.org\/wiki\/Excited_state \"Excited state\") for molecules with a sufficiently small energy gap between the ground state and the excited state, such as in NO.[\\[22\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-22) For a few systems, quantum spin statistics can also be important contributions to the heat capacity, even at room temperature. The analysis of the heat capacity of H  \n2 due to ortho\/para separation,[\\[23\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-23) which arises from [nuclear spin](https:\/\/en.wikipedia.org\/wiki\/Spin_quantum_number \"Spin quantum number\") statistics, has been referred to as \"one of the great triumphs of post-quantum mechanical statistical mechanics.\"[\\[24\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-24)\n\nThese extra [degrees of freedom](https:\/\/en.wikipedia.org\/wiki\/Degrees_of_freedom_\\(physics_and_chemistry\\) \"Degrees of freedom (physics and chemistry)\") or \"modes\" contribute to the specific heat capacity of the substance. Namely, when energy is introduced into a gas with polyatomic molecules, only part of it will increase their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number of degrees of freedom of the molecules.[\\[25\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-25)[\\[26\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-26)[\\[27\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-27)\n\n[Quantum statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_statistical_mechanics \"Quantum statistical mechanics\") predicts that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (quanta), and that this affects the system's thermodynamic properties. Depending on the temperature, the average energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be \"frozen out\". In that case, the specific heat capacity of the substance increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat.\n\nFor example, the molar heat capacity of [nitrogen](https:\/\/en.wikipedia.org\/wiki\/Nitrogen \"Nitrogen\") N  \n2 at constant volume is ![{\\\\displaystyle c\\_{V,\\\\mathrm {m} }=\\\\mathrm {20.6\\\\,J\\\\cdot K^{-1}\\\\cdot mol^{-1}} }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5c7c7437f35c01e65d4dc78ef1cbf4dfcf2fb023) (at 15 °C, 1 atm), which is ![{\\\\displaystyle 2.49R}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6691d251191f800f340af5ec9cd4e291d0a5344c).[\\[28\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-thor1993-28) That is the value expected from the [Equipartition Theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\") if each molecule had 5 kinetic degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat capacity ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) of N  \n2 (736 J⋅K−1⋅kg−1) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K−1⋅kg−1), by a factor of ⁠5\/3⁠. The vibrational and electronic degrees of freedom do not contribute significantly to the heat capacity in this case, due to the relatively large energy level gaps for both vibrational and electronic excitation in this molecule.\n\nThis value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still \"frozen out\". At about that temperature, those modes begin to \"un-freeze\" as vibrationally excited states become accessible. As a result ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.[\\[29\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-chas1998-29) The last value corresponds almost exactly to the value predicted by the Equipartition Theorem, since in the high-temperature limit the theorem predicts that the vibrational degree of freedom contributes twice as much to the heat capacity as any one of the translational or rotational degrees of freedom.\n\n## Derivations of heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=14 \"Edit section: Derivations of heat capacity\")\\]\n\n### Relation between specific heat capacities\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=15 \"Edit section: Relation between specific heat capacities\")\\]\n\nStarting from the [fundamental thermodynamic relation](https:\/\/en.wikipedia.org\/wiki\/Fundamental_thermodynamic_relation \"Fundamental thermodynamic relation\") one can show,\n\n![{\\\\displaystyle c\\_{p}-c\\_{v}={\\\\frac {\\\\alpha ^{2}T}{\\\\rho \\\\beta \\_{T}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/06bc97cafc646b40213b01cf9a30562276c7c744)\n\nwhere\n\nA derivation is discussed in the article [Relations between specific heats](https:\/\/en.wikipedia.org\/wiki\/Relations_between_specific_heats \"Relations between specific heats\").\n\nFor an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), if ![{\\\\displaystyle \\\\rho }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1f7d439671d1289b6a816e6af7a304be40608d64) is expressed as [molar](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") density in the above equation, this equation reduces simply to [Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_Robert_von_Mayer \"Julius Robert von Mayer\")'s relation,\n\n![{\\\\displaystyle C\\_{p,m}-C\\_{v,m}=R\\\\!}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f23e5c5b22b24ffd7f1051a3e396784a902d867e)\n\nwhere ![{\\\\displaystyle C\\_{p,m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7ccf6b7671259c85a73e29bc870010db9a7d2288) and ![{\\\\displaystyle C\\_{v,m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9bc3350be2e3cdd42c3a445e5dae16ac2aa4a675) are [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively.\n\n### Specific heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=16 \"Edit section: Specific heat capacity\")\\]\n\nThe specific heat capacity of a material on a per mass basis is\n\n![{\\\\displaystyle c={\\\\partial C \\\\over \\\\partial m},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/338148bac53d596abe5112d0891626997320114b)\n\nwhich in the absence of phase transitions is equivalent to\n\n![{\\\\displaystyle c=E\\_{m}={C \\\\over m}={C \\\\over {\\\\rho V}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca1a8ab2bd40064b132ddd125dd7e96420c0337d)\n\nwhere\n\n- ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question,\n- ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is the mass of the body,\n- ![{\\\\displaystyle V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the body, and\n- ![{\\\\displaystyle \\\\rho ={\\\\frac {m}{V}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f63465553e3f944d6ef79f90f992a02cf29c7f38) is the density of the material.\n\nFor gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https:\/\/en.wikipedia.org\/wiki\/Isobaric_process \"Isobaric process\") (constant pressure, ![{\\\\displaystyle dp=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/850e5511af66c84a989bf0e5d003852cb862778c)) or [isochoric](https:\/\/en.wikipedia.org\/wiki\/Isochoric_process \"Isochoric process\") (constant volume, ![{\\\\displaystyle dV=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c1d505a8f349881feb179d02752b33e5ab95e035)) processes. The corresponding specific heat capacities are expressed as\n\n![{\\\\displaystyle {\\\\begin{aligned}c\\_{p}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{p},\\\\\\\\c\\_{V}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{V}.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4576769aa96d12c2e062fe6e6a6ef4acfdc8b63b)\n\nA related parameter to ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is ![{\\\\displaystyle CV^{-1}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0155c68921e700c1325f32c73061273b2178796e), the [volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\"). In engineering practice, ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as ![{\\\\displaystyle c\\_{m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes\n\n![{\\\\displaystyle c\\_{m}={\\\\frac {C}{m}}={\\\\frac {c\\_{V}}{\\\\rho }}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/551b63520ea921f8105a75ddf4ddb073eca1e9db)\n\nFor pure homogeneous [chemical compounds](https:\/\/en.wikipedia.org\/wiki\/Chemical_compound \"Chemical compound\") with established [molecular or molar mass](https:\/\/en.wikipedia.org\/wiki\/Molecular_mass \"Molecular mass\") or a [molar quantity](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") is established, heat capacity as an [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") can be expressed on a per [mole](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") basis instead of a per mass basis by the following equations analogous to the per mass equations:\n\n![{\\\\displaystyle {\\\\begin{alignedat}{3}C\\_{p,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{p}&={\\\\text{molar heat capacity at constant pressure}}\\\\\\\\C\\_{V,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{V}&={\\\\text{molar heat capacity at constant volume}}\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/39d63e3ec44e731e9eb128dcc30eaa9c87af059a)\n\nwhere *n* = number of moles in the body or [thermodynamic system](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_system \"Thermodynamic system\"). One may refer to such a *per mole* quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis.\n\n### Polytropic heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=17 \"Edit section: Polytropic heat capacity\")\\]\n\nThe [polytropic](https:\/\/en.wikipedia.org\/wiki\/Polytropic \"Polytropic\") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change\n\n![{\\\\displaystyle C\\_{i,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)={\\\\text{molar heat capacity at polytropic process}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/02f685610e35e01aa42e293380bdf94469ea8364)\n\nThe most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between **1** and the adiabatic exponent (*γ* or *κ*)\n\n### Dimensionless heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=18 \"Edit section: Dimensionless heat capacity\")\\]\n\nThe [dimensionless](https:\/\/en.wikipedia.org\/wiki\/Dimensionless_number \"Dimensionless number\") heat capacity of a material is\n\n![{\\\\displaystyle C^{\\*}={\\\\frac {C}{nR}}={\\\\frac {C}{Nk\\_{\\\\text{B}}}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d1beaede671f02ee56b81859fe98eed0feb26cf5)\n\nwhere\n\n- *C* is the heat capacity of a body made of the material in question (J\/K)\n- *n* is the [amount of substance](https:\/\/en.wikipedia.org\/wiki\/Amount_of_substance \"Amount of substance\") in the body ([mol](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(unit\\) \"Mole (unit)\"))\n- *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\") (J⋅K−1⋅mol−1)\n- *N* is the number of molecules in the body. (dimensionless)\n- *k*B is the [Boltzmann constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\") (J⋅K−1)\n\nAgain, [SI](https:\/\/en.wikipedia.org\/wiki\/SI \"SI\") units shown for example.\n\nIn the [Ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") article, dimensionless heat capacity ![{\\\\displaystyle C^{\\*}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as ![{\\\\displaystyle {\\\\hat {c}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8417e85ae7f4eaee7df31347ce488f85c8884b93).\n\n### Heat capacity at absolute zero\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=19 \"Edit section: Heat capacity at absolute zero\")\\]\n\nFrom the definition of [entropy](https:\/\/en.wikipedia.org\/wiki\/Entropy#Thermodynamic_definition_of_entropy \"Entropy\")\n\n![{\\\\displaystyle TdS=\\\\delta Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e3ddb574969210251fd1e6d8847eb303ece6b6a7)\n\nthe absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature *Tf*\n\n![{\\\\displaystyle S(T\\_{f})=\\\\int \\_{T=0}^{T\\_{f}}{\\\\frac {\\\\delta Q}{T}}=\\\\int \\_{0}^{T\\_{f}}{\\\\frac {\\\\delta Q}{dT}}{\\\\frac {dT}{T}}=\\\\int \\_{0}^{T\\_{f}}C(T)\\\\,{\\\\frac {dT}{T}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/25e81986ce6a91887325cc998b1245250f83a65a)\n\nThe heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the [third law of thermodynamics](https:\/\/en.wikipedia.org\/wiki\/Third_law_of_thermodynamics \"Third law of thermodynamics\"). One of the strengths of the [Debye model](https:\/\/en.wikipedia.org\/wiki\/Debye_model \"Debye model\") is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached.\n\nThe theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3*R*, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas.\n\nThe Dulong–Petit limit results from the [equipartition theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\"), and as such is only valid in the classical limit of a [microstate continuum](https:\/\/en.wikipedia.org\/w\/index.php?title=Microstate_continuum&action=edit&redlink=1 \"Microstate continuum (page does not exist)\"), which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at [standard ambient temperature](https:\/\/en.wikipedia.org\/wiki\/Standard_ambient_temperature_and_pressure \"Standard ambient temperature and pressure\"), quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3*R* per mole of *atoms* in the solid, although in molecular solids, heat capacities calculated *per mole of molecules* in molecular solids may be more than 3*R*. For example, the heat capacity of water ice at the melting point is about 4.6*R* per mole of molecules, but only 1.5*R* per mole of atoms. The lower than 3*R* number \"per atom\" (as is the case with diamond and beryllium) results from the \"freezing out\" of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3*R* per mole of atoms of the Dulong–Petit theoretical maximum.\n\nFor a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of [phonons](https:\/\/en.wikipedia.org\/wiki\/Phonons \"Phonons\"). See [Debye model](https:\/\/en.wikipedia.org\/wiki\/Debye_model \"Debye model\").\n\n### Theoretical estimation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=21 \"Edit section: Theoretical estimation\")\\]\n\nThe path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \\> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below.\n\n- Water (liquid): CP = 4185.5 J⋅K−1⋅kg−1 (15 °C, 101.325 kPa)\n- Water (liquid): CVH = 74.539 J⋅K−1⋅mol−1 (25 °C)\n\nFor liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However, different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr).[\\[notes 1\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-gold-30)\n\n### Relation between heat capacities\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=22 \"Edit section: Relation between heat capacities\")\\]\n\nMeasuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see [coefficient of thermal expansion](https:\/\/en.wikipedia.org\/wiki\/Coefficient_of_thermal_expansion \"Coefficient of thermal expansion\") and [compressibility](https:\/\/en.wikipedia.org\/wiki\/Compressibility \"Compressibility\")). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws.\n\nThe [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\"), or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.\n\nFor an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), evaluating the partial derivatives above according to the [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\"), where *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\"), for an ideal gas[\\[30\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-31)\n\n![{\\\\displaystyle {\\\\begin{alignedat}{3}PV&=nRT,&\\\\\\\\C\\_{P}-C\\_{V}&=T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n},&\\\\\\\\P&={\\\\frac {nRT}{V}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}&={\\\\frac {nR}{V}},\\\\\\\\V&={\\\\frac {nRT}{P}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}&={\\\\frac {nR}{P}}.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2fe220db0c71a99f43c20c80b90b430001c8aff8)\n\nSubstituting\n\n![{\\\\displaystyle T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}=T{\\\\frac {nR}{V}}{\\\\frac {nR}{P}}={\\\\frac {nRT}{V}}{\\\\frac {nR}{P}}=P{\\\\frac {nR}{P}}=nR,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e49d3f01b48a4771f06476e4922cd418e91e9354)\n\nthis equation reduces simply to [Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_Robert_von_Mayer \"Julius Robert von Mayer\")'s relation:\n\n![{\\\\displaystyle C\\_{P,m}-C\\_{V,m}=R.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1b0951113a93ae7d847ed0234a94039782ae0a5e)\n\nThe differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.\n\n### Specific heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=24 \"Edit section: Specific heat capacity\")\\]\n\nThe specific heat capacity of a material on a per mass basis is\n\n![{\\\\displaystyle c={\\\\frac {\\\\partial C}{\\\\partial m}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2f6d4d3222fc01879cf5ebdcdd167c0f01338989)\n\nwhich in the absence of phase transitions is equivalent to\n\n![{\\\\displaystyle c=E\\_{m}={\\\\frac {C}{m}}={\\\\frac {C}{\\\\rho V}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5a5685f216c3eaa301626d07a8dbeda4ec825d2a)\n\nwhere\n\n- ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question,\n- ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is the mass of the body,\n- ![{\\\\displaystyle V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the body,\n- ![{\\\\displaystyle \\\\rho ={\\\\frac {m}{V}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f63465553e3f944d6ef79f90f992a02cf29c7f38) is the density of the material.\n\nFor gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https:\/\/en.wikipedia.org\/wiki\/Isobaric_process \"Isobaric process\") (constant pressure, ![{\\\\displaystyle {\\\\text{d}}P=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2997415b7c242bcf6f4ebf7ef17957d9df03a9aa)) or [isochoric](https:\/\/en.wikipedia.org\/wiki\/Isochoric_process \"Isochoric process\") (constant volume, ![{\\\\displaystyle {\\\\text{d}}V=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3e19ded03242b58e141dced9f6f46c3a8829e436)) processes. The corresponding specific heat capacities are expressed as\n\n![{\\\\displaystyle {\\\\begin{aligned}c\\_{P}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{P},\\\\\\\\c\\_{V}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial m}}\\\\right)\\_{V}.\\\\end{aligned}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8d2cc9fc1a0307e6e726521189cf7e22158357b9)\n\nFrom the results of the previous section, dividing through by the mass gives the relation\n\n![{\\\\displaystyle c\\_{P}-c\\_{V}={\\\\frac {\\\\alpha ^{2}T}{\\\\rho \\\\beta \\_{T}}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/058bcaccd4cbc0e8d16892f68bc68bf1e0062d7e)\n\nA related parameter to ![{\\\\displaystyle c}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is ![{\\\\displaystyle C\/V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4f43bc2f531ba84544e7e71ccab205ab4ad1fe5c), the [volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\"). In engineering practice, ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript ![{\\\\displaystyle m}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as ![{\\\\displaystyle c\\_{m}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes\n\n![{\\\\displaystyle c\\_{m}={\\\\frac {C}{m}}={\\\\frac {c\\_{\\\\text{volumetric}}}{\\\\rho }}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e56b0a8c8ebcbe4ee889c7d963cb980a5d34f60b)\n\nFor pure [homogeneous](https:\/\/en.wikipedia.org\/wiki\/Homogeneous \"Homogeneous\") [chemical compounds](https:\/\/en.wikipedia.org\/wiki\/Chemical_compound \"Chemical compound\") with established [molecular or molar mass](https:\/\/en.wikipedia.org\/wiki\/Molecular_mass \"Molecular mass\"), or a [molar quantity](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\"), heat capacity as an [intensive property](https:\/\/en.wikipedia.org\/wiki\/Intensive_property \"Intensive property\") can be expressed on a per-[mole](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(chemistry\\) \"Mole (chemistry)\") basis instead of a per-mass basis by the following equations analogous to the per mass equations:\n\n![{\\\\displaystyle {\\\\begin{alignedat}{3}C\\_{P,m}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{P}&={\\\\text{molar heat capacity at constant pressure,}}\\\\\\\\C\\_{V,m}&=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)\\_{V}&={\\\\text{molar heat capacity at constant volume,}}\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a371ac38796d12f8beb3f30b4e16181a5e1981ba)\n\nwhere *n* is the number of moles in the body or [thermodynamic system](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_system \"Thermodynamic system\"). One may refer to such a per-mole quantity as **molar heat capacity** to distinguish it from specific heat capacity on a per-mass basis.\n\n### Polytropic heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=25 \"Edit section: Polytropic heat capacity\")\\]\n\nThe [polytropic](https:\/\/en.wikipedia.org\/wiki\/Polytropic \"Polytropic\") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change:\n\n![{\\\\displaystyle C\\_{i,m}=\\\\left({\\\\frac {\\\\partial C}{\\\\partial n}}\\\\right)={\\\\text{molar heat capacity at polytropic process.}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d94a2aa8fde269f13f73425bd4eb2e375717ad28)\n\nThe most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (*γ* or *κ*).\n\n### Dimensionless heat capacity\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=26 \"Edit section: Dimensionless heat capacity\")\\]\n\nThe [dimensionless](https:\/\/en.wikipedia.org\/wiki\/Dimensionless_number \"Dimensionless number\") heat capacity of a material is\n\n![{\\\\displaystyle C^{\\*}={\\\\frac {C}{nR}}={\\\\frac {C}{Nk\\_{\\\\text{B}}}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/508315bfb919ceff9bf382f397926bccaa1168dc)\n\nwhere\n\n- ![{\\\\displaystyle C}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question (J\/K),\n- *n* is the [amount of substance](https:\/\/en.wikipedia.org\/wiki\/Amount_of_substance \"Amount of substance\") in the body ([mol](https:\/\/en.wikipedia.org\/wiki\/Mole_\\(unit\\) \"Mole (unit)\")),\n- *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\") (J\/(K⋅mol)),\n- *N* is the number of molecules in the body (dimensionless),\n- *k*B is the [Boltzmann constant](https:\/\/en.wikipedia.org\/wiki\/Boltzmann_constant \"Boltzmann constant\") (J\/(K⋅molecule)).\n\nIn the [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\") article, dimensionless heat capacity ![{\\\\displaystyle C^{\\*}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as ![{\\\\displaystyle {\\\\hat {c}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8417e85ae7f4eaee7df31347ce488f85c8884b93) and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the [equipartition theorem](https:\/\/en.wikipedia.org\/wiki\/Equipartition_theorem \"Equipartition theorem\").\n\nMore generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the [dimensionless entropy](https:\/\/en.wikipedia.org\/wiki\/Dimensionless_entropy \"Dimensionless entropy\") per particle ![{\\\\displaystyle S^{\\*}=S\/Nk\\_{\\\\text{B}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/41f612e45f84ed9e293f85f75c6dae30943bca0b), measured in [nats](https:\/\/en.wikipedia.org\/wiki\/Nat_\\(unit\\) \"Nat (unit)\").\n\n![{\\\\displaystyle C^{\\*}={\\\\frac {{\\\\text{d}}S^{\\*}}{{\\\\text{d}}(\\\\ln T)}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/294a7378c60eaed01bcea2cb12c53dd0b5bf7f90)\n\nAlternatively, using base-2 logarithms, ![{\\\\displaystyle C^{\\*}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in [bits](https:\/\/en.wikipedia.org\/wiki\/Bit \"Bit\").[\\[31\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-32)\n\n### Heat capacity at absolute zero\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=27 \"Edit section: Heat capacity at absolute zero\")\\]\n\nFrom the definition of [entropy](https:\/\/en.wikipedia.org\/wiki\/Entropy#Thermodynamic_definition_of_entropy \"Entropy\")\n\n![{\\\\displaystyle T\\\\,{\\\\text{d}}S=\\\\delta Q,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5cca7738465956fa4c1d790333af389a15b667c7)\n\nthe absolute entropy can be calculated by integrating from zero to the final temperature *T*f:\n\n![{\\\\displaystyle S(T\\_{\\\\text{f}})=\\\\int \\_{T=0}^{T\\_{\\\\text{f}}}{\\\\frac {\\\\delta Q}{T}}=\\\\int \\_{0}^{T\\_{\\\\text{f}}}{\\\\frac {\\\\delta Q}{{\\\\text{d}}T}}{\\\\frac {{\\\\text{d}}T}{T}}=\\\\int \\_{0}^{T\\_{\\\\text{f}}}C(T)\\\\,{\\\\frac {{\\\\text{d}}T}{T}}.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/df93f18f951064e02f6128a3cd99ad4c72124aa1)\n\n## Thermodynamic derivation\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=28 \"Edit section: Thermodynamic derivation\")\\]\n\nIn theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\") and an [internal energy function](https:\/\/en.wikipedia.org\/w\/index.php?title=Internal_energy_function&action=edit&redlink=1 \"Internal energy function (page does not exist)\").\n\n### State of matter in a homogeneous sample\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=29 \"Edit section: State of matter in a homogeneous sample\")\\]\n\nTo apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass ![{\\\\displaystyle M}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f82cade9898ced02fdd08712e5f0c0151758a0dd). Assume that the evolution of the system is always slow enough for the internal pressure ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a) and temperature ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) be considered uniform throughout. The pressure ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a) would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air.\n\nThe state of the material can then be specified by three parameters: its temperature ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), the pressure ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), and its [specific volume](https:\/\/en.wikipedia.org\/wiki\/Specific_volume \"Specific volume\") ![{\\\\displaystyle \\\\nu =V\/M}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/df5e892352c30149ba6c93abe17145e15cdd73f5), where ![{\\\\displaystyle V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the sample. (This quantity is the reciprocal ![{\\\\displaystyle 1\/\\\\rho }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/754f9b580e728c03bac621d1dd72cd606b6b5eef) of the material's [density](https:\/\/en.wikipedia.org\/wiki\/Density \"Density\") ![{\\\\displaystyle \\\\rho =M\/V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/2ed457a6aafb262f754ae8801d87eb036b8c5fe9).) Like ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) and ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), the specific volume ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468) is an intensive property of the material and its state, that does not depend on the amount of substance in the sample.\n\nThose variables are not independent. The allowed states are defined by an [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\") relating those three variables: ![{\\\\displaystyle F(T,P,\\\\nu )=0.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/471651d7fefe7e18735444a61e0ffa3f6ff84876) The function ![{\\\\displaystyle F}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/545fd099af8541605f7ee55f08225526be88ce57) depends on the material under consideration. The [specific internal energy](https:\/\/en.wikipedia.org\/wiki\/Specific_internal_energy \"Specific internal energy\") stored internally in the sample, per unit of mass, will then be another function ![{\\\\displaystyle U(T,P,\\\\nu )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7003637ad40a88b0bf499d18811d1dab7f424ccf) of these state variables, that is also specific of the material. The total internal energy in the sample then will be ![{\\\\displaystyle M\\\\,U(T,P,\\\\nu )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/287870c17350c2311e2d86e64ec9dec7bfc90a75).\n\nFor some simple materials, like an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), one can derive from basic theory the equation of state ![{\\\\displaystyle F=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/745afacbd4fd9affdc51ac09a0ecabae08da8676) and even the specific internal energy ![{\\\\displaystyle U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/458a728f53b9a0274f059cd695e067c430956025) In general, these functions must be determined experimentally for each substance.\n\n### Conservation of energy\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=30 \"Edit section: Conservation of energy\")\\]\n\nThe absolute value of this quantity ![{\\\\displaystyle U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/458a728f53b9a0274f059cd695e067c430956025) is undefined, and (for the purposes of thermodynamics) the state of \"zero internal energy\" can be chosen arbitrarily. However, by the [law of conservation of energy](https:\/\/en.wikipedia.org\/wiki\/Law_of_conservation_of_energy \"Law of conservation of energy\"), any infinitesimal increase ![{\\\\displaystyle M\\\\,\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fcddca6d7c51038a1a0134cde944e55e5bf81551) in the total internal energy ![{\\\\displaystyle MU}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ca0430adceb61aadc25575b6894c1633b41f8874) must be matched by the net flow of heat energy ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739) into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is ![{\\\\displaystyle -P\\\\,\\\\mathrm {d} V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c7238b4c977539175fa28c0cff1658e50276637a), where ![{\\\\displaystyle \\\\mathrm {d} V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b80507190aa9d38a279909db47b63657f2b62ba7) is the change in the sample's volume in that infinitesimal step.[\\[32\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-fein-33) Therefore\n\n![{\\\\displaystyle \\\\mathrm {d} Q-P\\\\,\\\\mathrm {d} V=M\\\\,\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3c3401e201975bf505b5fa40018ffd2bc7eab0a6)\n\nhence\n\n![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M}}-P\\\\,\\\\mathrm {d} \\\\nu =\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/586d361903b89f3b074724dafc3680ca2cf85d26)\n\nIf the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739), then the term ![{\\\\displaystyle P\\\\,\\\\mathrm {d} \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e54803595e3392edc2125d5bbbfd6ee753922208) is zero (no mechanical work is done). Then, dividing by ![{\\\\displaystyle \\\\mathrm {d} T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/309833548168052f2696ca961149df099b5d1f80),\n\n![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M\\\\,\\\\mathrm {d} T}}={\\\\frac {\\\\mathrm {d} U}{\\\\mathrm {d} T}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b32da0380fcf6d0c3bc87b7dbb0c7c0c02211f80)\n\nwhere ![{\\\\displaystyle \\\\mathrm {d} T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/309833548168052f2696ca961149df099b5d1f80) is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume ![{\\\\displaystyle c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6c1ed5361b532828e1bda88a5282706a31b5d082) of the material.\n\nFor the heat capacity at constant pressure, it is useful to define the [specific enthalpy](https:\/\/en.wikipedia.org\/wiki\/Specific_enthalpy \"Specific enthalpy\") of the system as the sum ![{\\\\displaystyle h(T,P,\\\\nu )=U(T,P,\\\\nu )+P\\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/38a0269a9025439e15bc408001d2ddd739e304a9). An infinitesimal change in the specific enthalpy will then be\n\n![{\\\\displaystyle \\\\mathrm {d} h=\\\\mathrm {d} U+V\\\\,\\\\mathrm {d} P+P\\\\,\\\\mathrm {d} V}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/746c99ad25909e68e8e56ca8a0b6b9bcce84317c)\n\ntherefore\n\n![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M}}+V\\\\,\\\\mathrm {d} P=\\\\mathrm {d} h}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d1e31d5cd51830d6b5eb83bb7c77657c5be843c0)\n\nIf the pressure is kept constant, the second term on the left-hand side is zero, and\n\n![{\\\\displaystyle {\\\\frac {\\\\mathrm {d} Q}{M\\\\,\\\\mathrm {d} T}}={\\\\frac {\\\\mathrm {d} h}{\\\\mathrm {d} T}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6a4007c22be9c0d1e82c5a4a011c3bd298de9f44)\n\nThe left-hand side is the specific heat capacity at constant pressure ![{\\\\displaystyle c\\_{P}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/23f21e3d892369c6502159212f89643451ab67e4) of the material.\n\n### Connection to equation of state\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=31 \"Edit section: Connection to equation of state\")\\]\n\nIn general, the infinitesimal quantities ![{\\\\displaystyle \\\\mathrm {d} T,\\\\mathrm {d} P,\\\\mathrm {d} V,\\\\mathrm {d} U}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/04ccfe4aed236cb970d27013b7e0fe2e3cbf7664) are constrained by the equation of state and the specific internal energy function. Namely,\n\n![{\\\\displaystyle {\\\\begin{cases}\\\\displaystyle \\\\mathrm {d} T{\\\\frac {\\\\partial F}{\\\\partial T}}(T,P,V)+\\\\mathrm {d} P{\\\\frac {\\\\partial F}{\\\\partial P}}(T,P,V)+\\\\mathrm {d} V{\\\\frac {\\\\partial F}{\\\\partial V}}(T,P,V)&=&0\\\\\\\\\\[2ex\\]\\\\displaystyle \\\\mathrm {d} T{\\\\frac {\\\\partial U}{\\\\partial T}}(T,P,V)+\\\\mathrm {d} P{\\\\frac {\\\\partial U}{\\\\partial P}}(T,P,V)+\\\\mathrm {d} V{\\\\frac {\\\\partial U}{\\\\partial V}}(T,P,V)&=&\\\\mathrm {d} U\\\\end{cases}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/957c9a178ff753bc04a30bed2819d7e5155314a0)\n\nHere ![{\\\\displaystyle (\\\\partial F\/\\\\partial T)(T,P,V)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24993db8c358df7a5c8539501cb70320459370e6) denotes the (partial) derivative of the state equation ![{\\\\displaystyle F}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/545fd099af8541605f7ee55f08225526be88ce57) with respect to its ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) argument, keeping the other two arguments fixed, evaluated at the state ![{\\\\displaystyle (T,P,V)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c41f5cdee6675ea3447082abaa34afbaf77e4a1f) in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space.\n\nThis analysis also holds no matter how the energy increment ![{\\\\displaystyle \\\\mathrm {d} Q}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/aba942d26ee837fbee5ad2a635c34446afedd739) is injected into the sample, namely by [heat conduction](https:\/\/en.wikipedia.org\/wiki\/Heat_conduction \"Heat conduction\"), irradiation, [electromagnetic induction](https:\/\/en.wikipedia.org\/wiki\/Electromagnetic_induction \"Electromagnetic induction\"), [radioactive decay](https:\/\/en.wikipedia.org\/wiki\/Radioactive_decay \"Radioactive decay\"), etc.\n\n### Relation between heat capacities\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=32 \"Edit section: Relation between heat capacities\")\\]\n\nFor any specific volume ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468), denote ![{\\\\displaystyle p\\_{\\\\nu }(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fab6e24e625d12de49f61c60a3e95d19d5f305d) the function that describes how the pressure varies with the temperature ![{\\\\displaystyle T}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468). Analogously, for any pressure ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), let ![{\\\\displaystyle \\\\nu \\_{P}(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d920635881776d5b03f98ee5492f6e0549b3d2f7) be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a). Namely, those functions are such that\n\n![{\\\\displaystyle F(T,p\\_{\\\\nu }(T),\\\\nu )=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/84a1b582c06f3c19e7abe1e3728d16e072c1a51f)and![{\\\\displaystyle F(T,P,\\\\nu \\_{P}(T))=0}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b667b5d0711e09296d3fe4e9d3f6c395acd0f847)\n\nfor any values of ![{\\\\displaystyle T,P,\\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f907ebc4eacdaceb323d1264f58855f033d41081). In other words, the graphs of ![{\\\\displaystyle p\\_{\\\\nu }(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6fab6e24e625d12de49f61c60a3e95d19d5f305d) and ![{\\\\displaystyle \\\\nu \\_{P}(T)}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d920635881776d5b03f98ee5492f6e0549b3d2f7) are slices of the surface defined by the state equation, cut by planes of constant ![{\\\\displaystyle \\\\nu }](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468) and constant ![{\\\\displaystyle P}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4dc73bf40314945ff376bd363916a738548d40a), respectively.\n\nThen, from the [fundamental thermodynamic relation](https:\/\/en.wikipedia.org\/wiki\/Fundamental_thermodynamic_relation \"Fundamental thermodynamic relation\") it follows that\n\n![{\\\\displaystyle c\\_{P}(T,P,\\\\nu )-c\\_{V}(T,P,\\\\nu )=T\\\\left\\[{\\\\frac {\\\\mathrm {d} p\\_{\\\\nu }}{\\\\mathrm {d} T}}(T)\\\\right\\]\\\\left\\[{\\\\frac {\\\\mathrm {d} \\\\nu \\_{P}}{\\\\mathrm {d} T}}(T)\\\\right\\]}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24554d2ae7d16cbb75b6ad485bf92856b55cf7bd)\n\nThis equation can be rewritten as\n\n![{\\\\displaystyle c\\_{P}(T,P,\\\\nu )-c\\_{V}(T,P,\\\\nu )=\\\\nu T{\\\\frac {\\\\alpha ^{2}}{\\\\beta \\_{T}}},}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/27c864cf90d771a66e3a2fa50dec56b8d6c190ad)\n\nwhere\n\nboth depending on the state ![{\\\\displaystyle (T,P,\\\\nu )}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/847d2a615fb19d33916ce93a4fa935919abca4a7).\n\nThe [heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\"), or adiabatic index, is the ratio ![{\\\\displaystyle c\\_{P}\/c\\_{V}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5bfcc80ce47374e8dfc55b32849c78810e782a87) of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor.\n\n### Calculation from first principles\n\\[[edit](https:\/\/en.wikipedia.org\/w\/index.php?title=Specific_heat_capacity&action=edit&section=33 \"Edit section: Calculation from first principles\")\\]\n\nThe [path integral Monte Carlo](https:\/\/en.wikipedia.org\/wiki\/Path_integral_Monte_Carlo \"Path integral Monte Carlo\") method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \\> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3*R* = 24.94 joules per kelvin per mole of atoms ([Dulong–Petit law](https:\/\/en.wikipedia.org\/wiki\/Dulong%E2%80%93Petit_law \"Dulong–Petit law\"), *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\")). Low temperature approximations for both gases and solids at temperatures less than their characteristic [Einstein temperatures](https:\/\/en.wikipedia.org\/wiki\/Einstein_temperature \"Einstein temperature\") or [Debye temperatures](https:\/\/en.wikipedia.org\/wiki\/Debye_temperature \"Debye temperature\") can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material.[\\[33\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-Benjelloun-34)\n\nFor an [ideal gas](https:\/\/en.wikipedia.org\/wiki\/Ideal_gas \"Ideal gas\"), evaluating the partial derivatives above according to the [equation of state](https:\/\/en.wikipedia.org\/wiki\/Equation_of_state \"Equation of state\"), where *R* is the [gas constant](https:\/\/en.wikipedia.org\/wiki\/Gas_constant \"Gas constant\"), for an ideal gas[\\[34\\]](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_note-35)\n\n![{\\\\displaystyle {\\\\begin{alignedat}{3}PV&=nRT,\\\\\\\\C\\_{P}-C\\_{V}&=T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n},\\\\\\\\P&={\\\\frac {nRT}{V}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}&={\\\\frac {nR}{V}},\\\\\\\\V&={\\\\frac {nRT}{P}}\\\\Rightarrow \\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}&={\\\\frac {nR}{P}}.\\\\end{alignedat}}}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e2f8e4a06c2339161629da3ad59eddd0383a651f)\n\nSubstituting\n\n![{\\\\displaystyle T\\\\left({\\\\frac {\\\\partial P}{\\\\partial T}}\\\\right)\\_{V,n}\\\\left({\\\\frac {\\\\partial V}{\\\\partial T}}\\\\right)\\_{P,n}=T{\\\\frac {nR}{V}}{\\\\frac {nR}{P}}={\\\\frac {nRT}{V}}{\\\\frac {nR}{P}}=P{\\\\frac {nR}{P}}=nR,}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e49d3f01b48a4771f06476e4922cd418e91e9354)\n\nthis equation reduces simply to [Mayer](https:\/\/en.wikipedia.org\/wiki\/Julius_Robert_von_Mayer \"Julius Robert von Mayer\")'s relation:\n\n![{\\\\displaystyle C\\_{P,m}-C\\_{V,m}=R.}](https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1b0951113a93ae7d847ed0234a94039782ae0a5e)\n\nThe differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas.\n\n[![icon](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/6\/6f\/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg\/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png)](https:\/\/en.wikipedia.org\/wiki\/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg) [Physics portal](https:\/\/en.wikipedia.org\/wiki\/Portal:Physics \"Portal:Physics\")\n\n- [Enthalpy of fusion](https:\/\/en.wikipedia.org\/wiki\/Enthalpy_of_fusion \"Enthalpy of fusion\") (latent heat of melting)\n- [Enthalpy of vaporization](https:\/\/en.wikipedia.org\/wiki\/Enthalpy_of_vaporization \"Enthalpy of vaporization\") (latent heat of vaporization)\n- [Frenkel line](https:\/\/en.wikipedia.org\/wiki\/Frenkel_line \"Frenkel line\")\n- [Heat capacity ratio](https:\/\/en.wikipedia.org\/wiki\/Heat_capacity_ratio \"Heat capacity ratio\")\n- [Heat equation](https:\/\/en.wikipedia.org\/wiki\/Heat_equation \"Heat equation\")\n- [Heat transfer coefficient](https:\/\/en.wikipedia.org\/wiki\/Heat_transfer_coefficient \"Heat transfer coefficient\")\n- [History of thermodynamics](https:\/\/en.wikipedia.org\/wiki\/History_of_thermodynamics \"History of thermodynamics\")\n- [Joback method](https:\/\/en.wikipedia.org\/wiki\/Joback_method \"Joback method\") (Estimation of heat capacities)\n- [Latent heat](https:\/\/en.wikipedia.org\/wiki\/Latent_heat \"Latent heat\")\n- [Material properties (thermodynamics)](https:\/\/en.wikipedia.org\/wiki\/Material_properties_\\(thermodynamics\\) \"Material properties (thermodynamics)\")\n- [Quantum statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Quantum_statistical_mechanics \"Quantum statistical mechanics\")\n- [R-value (insulation)](https:\/\/en.wikipedia.org\/wiki\/R-value_\\(insulation\\) \"R-value (insulation)\")\n- [Statistical mechanics](https:\/\/en.wikipedia.org\/wiki\/Statistical_mechanics \"Statistical mechanics\")\n- [Table of specific heat capacities](https:\/\/en.wikipedia.org\/wiki\/Table_of_specific_heat_capacities \"Table of specific heat capacities\")\n- [Thermal mass](https:\/\/en.wikipedia.org\/wiki\/Thermal_mass \"Thermal mass\")\n- [Thermodynamic databases for pure substances](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_databases_for_pure_substances \"Thermodynamic databases for pure substances\")\n- [Thermodynamic equations](https:\/\/en.wikipedia.org\/wiki\/Thermodynamic_equations \"Thermodynamic equations\")\n- [Volumetric heat capacity](https:\/\/en.wikipedia.org\/wiki\/Volumetric_heat_capacity \"Volumetric heat capacity\")\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-gold_30-0)** [IUPAC](https:\/\/en.wikipedia.org\/wiki\/International_Union_of_Pure_and_Applied_Chemistry \"International Union of Pure and Applied Chemistry\"), *[Compendium of Chemical Terminology](https:\/\/en.wikipedia.org\/wiki\/IUPAC_books#Gold_Book \"IUPAC books\")*, 5th ed. (the \"Gold Book\") (2025). Online version: (2006–) \"[Standard Pressure](https:\/\/goldbook.iupac.org\/terms\/view\/S05921.html)\". [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1351\/goldbook.S05921](https:\/\/doi.org\/10.1351%2Fgoldbook.S05921).\n\n1. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-1)**\n   Halliday, David; Resnick, Robert; Walker, Jearl (2001). *Fundamentals of Physics* (6th ed.). New York, NY US: [John Wiley & Sons](https:\/\/en.wikipedia.org\/wiki\/John_Wiley_%26_Sons \"John Wiley & Sons\").\n2. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-2)**\n   Open University (2008). *S104 Book 3 Energy and Light*, p. 59. [The Open University](https:\/\/en.wikipedia.org\/wiki\/The_Open_University \"The Open University\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [9781848731646](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781848731646 \"Special:BookSources\/9781848731646\")\n   \n   .\n3. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-3)**\n   Open University (2008). *S104 Book 3 Energy and Light*, p. 179. [The Open University](https:\/\/en.wikipedia.org\/wiki\/The_Open_University \"The Open University\"). [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [9781848731646](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/9781848731646 \"Special:BookSources\/9781848731646\")\n   \n   .\n4. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-4)**\n   Engineering ToolBox (2003). [\"Specific Heat of some common Substances\"](https:\/\/www.engineeringtoolbox.com\/specific-heat-capacity-d_391.html).\n5. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-colen2001_5-0)** (2001): *Columbia Encyclopedia*, 6th ed.; as quoted by [Encyclopedia.com](https:\/\/www.encyclopedia.com\/science-and-technology\/physics\/physics\/specific-heat#1E1specheat). Columbia University Press. Accessed on 2019-04-11.\n6. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-6)**\n   Laidler, Keith J. (1993). [*The World of Physical Chemistry*](https:\/\/books.google.com\/books?id=01LRlPbH80cC). Oxford University Press. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n   \n   [0-19-855919-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-19-855919-4 \"Special:BookSources\/0-19-855919-4\")\n   \n   .\n7. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-:1_7-0)**\n   [Ramsay, William](https:\/\/en.wikipedia.org\/wiki\/William_Ramsay \"William Ramsay\") (1918). *The life and letters of Joseph Black, M.D*. Constable. pp. 38–39\\.\n8. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-8)**\n   Black, Joseph (1807). Robison, John (ed.). [*Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh*](https:\/\/books.google.com\/books?id=lqI9AQAAMAAJ&pg=PA76). Vol. 1. Mathew Carey. pp. 76–77\\.\n9. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-:0_9-0)**\n   West, John B. (2014-06-15). [\"Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases\"](https:\/\/www.physiology.org\/doi\/10.1152\/ajplung.00020.2014). *American Journal of Physiology. Lung Cellular and Molecular Physiology*. **306** (12): L1057–L1063. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1152\/ajplung.00020.2014](https:\/\/doi.org\/10.1152%2Fajplung.00020.2014). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [1040-0605](https:\/\/search.worldcat.org\/issn\/1040-0605). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [24682452](https:\/\/pubmed.ncbi.nlm.nih.gov\/24682452).\n10. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-10)**\n    [International Bureau of Weights and Measures](https:\/\/en.wikipedia.org\/wiki\/International_Bureau_of_Weights_and_Measures \"International Bureau of Weights and Measures\") (2006), [*The International System of Units (SI)*](https:\/\/www.bipm.org\/documents\/20126\/41483022\/si_brochure_8.pdf) (PDF) (8th ed.), [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [92-822-2213-6](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/92-822-2213-6 \"Special:BookSources\/92-822-2213-6\")\n    \n    , [archived](https:\/\/web.archive.org\/web\/20210604163219\/https:\/\/www.bipm.org\/documents\/20126\/41483022\/si_brochure_8.pdf) (PDF) from the original on 2021-06-04, retrieved 2021-12-16\n11. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-toolbox_11-0)**\n    [\"Water – Thermal Properties\"](http:\/\/www.engineeringtoolbox.com\/water-thermal-properties-d_162.html). Engineeringtoolbox.com. Retrieved 2021-03-29.\n12. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-12)**\n    International Union of Pure and Applied Chemistry, Physical Chemistry Division. [\"Quantities, Units and Symbols in Physical Chemistry\"](http:\/\/old.iupac.org\/publications\/books\/gbook\/green_book_2ed.pdf) (PDF). Blackwell Sciences. p. 7. \"The adjective specific before the name of an extensive quantity is often used to mean divided by mass.\"\n13. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-Lange_13-0)** Lange's Handbook of Chemistry, 10th ed., page 1524.\n14. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-14)**\n    Quick, C. R.; Schawe, J. E. K.; Uggowitzer, P. J.; Pogatscher, S. (2019-07-01). [\"Measurement of specific heat capacity via fast scanning calorimetry—Accuracy and loss corrections\"](https:\/\/doi.org\/10.1016%2Fj.tca.2019.03.021). *Thermochimica Acta*. Special Issue on occasion of the 65th birthday of Christoph Schick. **677**: 12–20\\. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2019TcAc..677...12Q](https:\/\/ui.adsabs.harvard.edu\/abs\/2019TcAc..677...12Q). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1016\/j.tca.2019.03.021](https:\/\/doi.org\/10.1016%2Fj.tca.2019.03.021). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [0040-6031](https:\/\/search.worldcat.org\/issn\/0040-6031).\n15. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-15)**\n    Pogatscher, S.; Leutenegger, D.; Schawe, J. E. K.; Uggowitzer, P. J.; Löffler, J. F. (September 2016). [\"Solid–solid phase transitions via melting in metals\"](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC4844691). *Nature Communications*. **7** (1) 11113. [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2016NatCo...711113P](https:\/\/ui.adsabs.harvard.edu\/abs\/2016NatCo...711113P). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1038\/ncomms11113](https:\/\/doi.org\/10.1038%2Fncomms11113). [ISSN](https:\/\/en.wikipedia.org\/wiki\/ISSN_\\(identifier\\) \"ISSN (identifier)\") [2041-1723](https:\/\/search.worldcat.org\/issn\/2041-1723). [PMC](https:\/\/en.wikipedia.org\/wiki\/PMC_\\(identifier\\) \"PMC (identifier)\") [4844691](https:\/\/www.ncbi.nlm.nih.gov\/pmc\/articles\/PMC4844691). [PMID](https:\/\/en.wikipedia.org\/wiki\/PMID_\\(identifier\\) \"PMID (identifier)\") [27103085](https:\/\/pubmed.ncbi.nlm.nih.gov\/27103085).\n16. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-Koch_16-0)**\n    Koch, Werner (2013). [*VDI Steam Tables*](https:\/\/books.google.com\/books?id=bJ_wBgAAQBAJ&pg=PA8) (4 ed.). Springer. p. 8. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-3-642-52941-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-3-642-52941-2 \"Special:BookSources\/978-3-642-52941-2\")\n    \n    .\n    \n    Published under the auspices of the *Verein Deutscher Ingenieure* (VDI).\n17. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-17)**\n    Cardarelli, Francois (2012). [*Scientific Unit Conversion: A Practical Guide to Metrication*](https:\/\/books.google.com\/books?id=-ZveBwAAQBAJ&pg=PA19-IA35). M.J. Shields (translation) (2 ed.). Springer. p. 19. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-1-4471-0805-4](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-1-4471-0805-4 \"Special:BookSources\/978-1-4471-0805-4\")\n    \n    .\n18. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-18)** From direct values: 1⁠BTU\/lb⋅°R⁠ × 1055.06⁠J\/BTU⁠ × (⁠1\/0\\.45359237⁠)⁠lb\/kg⁠ x ⁠9\/5⁠⁠°R\/K⁠ = 4186.82⁠J\/kg⋅K⁠\n19. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-19)** °F=°R\n20. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-20)** °C=K\n21. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-21)**\n    McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: [University Science Books](https:\/\/en.wikipedia.org\/w\/index.php?title=University_Science_Books&action=edit&redlink=1 \"University Science Books (page does not exist)\"). pp. 83–85\\.\n22. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-22)**\n    [\"6.6: Electronic Partition Function\"](https:\/\/chem.libretexts.org\/Bookshelves\/Physical_and_Theoretical_Chemistry_Textbook_Maps\/Statistical_Thermodynamics_\\(Jeschke\\)\/06:_Partition_Functions_of_Gases\/6.06:_Electronic_Partition_Function). *Chemistry LibreTexts*. 2020-11-26. Retrieved 2024-12-16.\n23. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-23)**\n    Bonhoeffer, K.F.; Harteck, P. (1926). [\"Über Para- und Orthowasserstoff\"](https:\/\/www.degruyter.com\/document\/doi\/10.1515\/zpch-1929-0408\/html). *Z. Phys. Chem*. **4B**: 113–141\\. [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1515\/zpch-1929-0408](https:\/\/doi.org\/10.1515%2Fzpch-1929-0408).\n24. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-24)**\n    McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: University Science Books. p. 107.\n25. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-25)** Feynman, R., *[The Feynman Lectures on Physics](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\")*, Vol. 1, ch. 40, pp. 7–8\n26. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-26)**\n    Reif, F. (1965). [*Fundamentals of statistical and thermal physics*](https:\/\/archive.org\/details\/fundamentalsofst00reif). McGraw-Hill. pp. [253–254](https:\/\/archive.org\/details\/fundamentalsofst00reif\/page\/253).\n27. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-27)**\n    Kittel, Charles; Kroemer, Herbert (2000). *Thermal physics*. W. H. Freeman. p. 78. [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [978-0-7167-1088-2](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/978-0-7167-1088-2 \"Special:BookSources\/978-0-7167-1088-2\")\n    \n    .\n28. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-thor1993_28-0)** Thornton, Steven T. and Rex, Andrew (1993) *Modern Physics for Scientists and Engineers*, Saunders College Publishing\n29. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-chas1998_29-0)** Chase, M.W. Jr. (1998) *[NIST-JANAF Themochemical Tables, Fourth Edition](https:\/\/webbook.nist.gov\/cgi\/cbook.cgi?ID=C7727379&Type=JANAFG)*, In *Journal of Physical and Chemical Reference Data*, Monograph 9, pages 1–1951.\n30. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-31)**\n    Yunus A. Cengel and Michael A. Boles, *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill, 2010, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [007-352932-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/007-352932-X \"Special:BookSources\/007-352932-X\")\n    \n    .\n31. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-32)**\n    Fraundorf, P. (2003). \"Heat capacity in bits\". *American Journal of Physics*. **71** (11): 1142. [arXiv](https:\/\/en.wikipedia.org\/wiki\/ArXiv_\\(identifier\\) \"ArXiv (identifier)\"):[cond-mat\/9711074](https:\/\/arxiv.org\/abs\/cond-mat\/9711074). [Bibcode](https:\/\/en.wikipedia.org\/wiki\/Bibcode_\\(identifier\\) \"Bibcode (identifier)\"):[2003AmJPh..71.1142F](https:\/\/ui.adsabs.harvard.edu\/abs\/2003AmJPh..71.1142F). [doi](https:\/\/en.wikipedia.org\/wiki\/Doi_\\(identifier\\) \"Doi (identifier)\"):[10\\.1119\/1.1593658](https:\/\/doi.org\/10.1119%2F1.1593658). [S2CID](https:\/\/en.wikipedia.org\/wiki\/S2CID_\\(identifier\\) \"S2CID (identifier)\") [18742525](https:\/\/api.semanticscholar.org\/CorpusID:18742525).\n32. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-fein_33-0)** Feynman, Richard, *[The Feynman Lectures on Physics](https:\/\/en.wikipedia.org\/wiki\/The_Feynman_Lectures_on_Physics \"The Feynman Lectures on Physics\")*, Vol. 1, Ch. 45\n33. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-Benjelloun_34-0)** S. Benjelloun, \"Thermodynamic identities and thermodynamic consistency of Equation of States\", [Link to Archiv e-print](https:\/\/arxiv.org\/abs\/2105.04845) [Link to Hal e-print](https:\/\/hal.archives-ouvertes.fr\/hal-03216379\/)\n34. **[^](https:\/\/en.wikipedia.org\/wiki\/Specific_heat_capacity#cite_ref-35)**\n    Cengel, Yunus A. and Boles, Michael A. (2010) *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n    \n    [007-352932-X](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/007-352932-X \"Special:BookSources\/007-352932-X\")\n    \n    .\n\n- Emmerich Wilhelm & Trevor M. Letcher, Eds., 2010, *Heat Capacities: Liquids, Solutions and Vapours*, Cambridge, U.K.:Royal Society of Chemistry, [ISBN](https:\/\/en.wikipedia.org\/wiki\/ISBN_\\(identifier\\) \"ISBN (identifier)\")\n  [0-85404-176-1](https:\/\/en.wikipedia.org\/wiki\/Special:BookSources\/0-85404-176-1 \"Special:BookSources\/0-85404-176-1\")\n  . A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, \"Heat Capacities: Introduction, Concepts, and Selected Applications\" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as [photoacoustic](https:\/\/en.wikipedia.org\/wiki\/Photoacoustic_effect \"Photoacoustic effect\") methods, e.g., Jan Thoen & Christ Glorieux, \"Photothermal Techniques for Heat Capacities,\" and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc.\n\n- (2012-05may-24) [Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World](https:\/\/physicsworld.com\/a\/phonon-theory-sheds-light-on-liquid-thermodynamics\/) [The phonon theory of liquid thermodynamics \\| Scientific Reports](https:\/\/www.nature.com\/articles\/srep00421)","meta_canonical":null,"ml_categories_json":"{\"\/Science\":989,\"\/Science\/Physics\":826}","ml_types_json":"{\"\/Article\":998,\"\/Article\/Wiki\":760}","ml_intent_types_json":"{\"Informational\":999}","meta_language":"en","attrs_author":null,"attrs_publish_time":0,"attrs_original_publish_time":1375989344,"attrs_is_republished":0,"attrs_nr_words":"9937","attrs_boilerpipe_nr_words":"6712","body_ext_links_number":87,"body_int_links_number":445,"meta_nofollow":0,"meta_noarchive":0,"props_was_rendered":0,"src_redirect":"","download_time_msec":56,"download_ttfb_msec":33,"download_size":70099}

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Specific heat capacity Other names Specific heat Common symbols c SI unit J⋅kg −1 ⋅K −1 In SI base units m 2 ⋅K −1 ⋅s −2 Intensive ? Yes Dimension L 2 ⋅T −2 ⋅Θ −1 In thermodynamics , the specific heat capacity (symbol c ) of a substance is the amount of heat that must be added to one unit of mass of the substance in order to cause an increase of one unit in temperature . It is also referred to as massic heat capacity or as the specific heat. More formally it is the heat capacity of a sample of the substance divided by the mass of the sample. [ 1 ] The SI unit of specific heat capacity is joule per kelvin per kilogram , J⋅kg −1 ⋅K −1 . [ 2 ] For example, the heat required to raise the temperature of 1 kg of water by 1 K is 4184 joules , so the specific heat capacity of water is 4184 J⋅kg −1 ⋅K −1 . [ 3 ] Specific heat capacity often varies with temperature, and is different for each state of matter . Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg −1 ⋅K −1 at 20 °C, but that of ice, just below 0 °C, is only 2093 J⋅kg −1 ⋅K −1 . The specific heat capacities of iron , granite , and hydrogen gas are about 449 J⋅kg −1 ⋅K −1 , 790 J⋅kg −1 ⋅K −1 , and 14300 J⋅kg −1 ⋅K −1 , respectively. [ 4 ] While the substance is undergoing a phase transition , such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature. The specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity at constant pressure ) than when it is heated in a closed vessel that prevents expansion (specific heat capacity at constant volume ). These two values are usually denoted by and , respectively; their quotient is the heat capacity ratio . The term specific heat may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C; [ 5 ] much in the fashion of specific gravity . Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of moles , one gets the molar heat capacity instead, whose SI unit is joule per kelvin per mole, J⋅mol −1 ⋅K −1 . If the amount is taken to be the volume of the sample (as is sometimes done in engineering), one gets the volumetric heat capacity , whose SI unit is joule per kelvin per cubic meter , J⋅m −3 ⋅K −1 . Discovery of specific heat [ edit ] Joseph Black One of the first scientists to use the concept was Joseph Black , an 18th-century medical doctor and professor of medicine at Glasgow University . He measured the specific heat capacities of many substances, using the term capacity for heat . [ 6 ] In 1756 or soon thereafter, Black began an extensive study of heat. [ 7 ] In 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by Daniel Gabriel Fahrenheit on behalf of Dutch physician Herman Boerhaave . For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: "Quicksilver [mercury] ... has less capacity for the matter of heat than water." [ 8 ] [ 9 ] The specific heat capacity of a substance, usually denoted by or , is the heat capacity of a sample of the substance, divided by the mass of the sample: [ 10 ] where represents the amount of heat needed to uniformly raise the temperature of the sample by a small increment . Like the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature of the sample and the pressure applied to it. Therefore, it should be considered a function of those two variables. These parameters are usually specified when giving the specific heat capacity of a substance. For example, "Water (liquid): = 4187 J⋅kg −1 ⋅K −1 (15 °C)." [ 11 ] When not specified, published values of the specific heat capacity generally are valid for some standard conditions for temperature and pressure . However, the dependency of on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier and approximates the specific heat capacity by a constant suitable for those ranges. Specific heat capacity is an intensive property of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier "specific" in front of an extensive property often indicates an intensive property derived from it. [ 12 ] ) The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure and starting temperature . Two particular choices are widely used: The value of is always less than the value of for all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the heat capacity ratio of gases is typically between 1.3 and 1.67. [ 13 ] The specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale. The specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops. The specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a phase change , such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement. The specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a calorimeter , and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as differential scanning calorimetry . [ 14 ] [ 15 ] Graph of temperature of phases of water heated from −100 °C to 200 °C – the dashed line example shows that melting and heating 1 kg of ice at −50 °C to water at 40 °C needs 600 kJ The specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the coefficient of thermal expansion and the compressibility of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics. [ citation needed ] International system [ edit ] The SI unit for specific heat capacity is joule per kelvin per kilogram ⁠ J / kg⋅K ⁠ , J⋅K −1 ⋅kg −1 . Since an increment of temperature of one degree Celsius is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J/(kg⋅°C). Sometimes the gram is used instead of kilogram for the unit of mass: 1 J⋅g −1 ⋅K −1 = 1000 J⋅kg −1 ⋅K −1 . The specific heat capacity of a substance (per unit of mass) has dimension L 2 ⋅Θ −1 ⋅T −2 , or (L/T) 2 /Θ. Therefore, the SI unit J⋅kg −1 ⋅K −1 is equivalent to metre squared per second squared per kelvin (m 2 ⋅K −1 ⋅s −2 ). Imperial engineering units [ edit ] Professionals in construction , civil engineering , chemical engineering , and other technical disciplines, especially in the United States , may use English Engineering units including the pound (lb = 0.45359237 kg) as the unit of mass, the degree Fahrenheit or Rankine (°R = ⁠ 5 / 9 ⁠ K, about 0.555556 K) as the unit of temperature increment, and the British thermal unit (BTU ≈ 1055.056 J), [ 16 ] [ 17 ] as the unit of heat. In those contexts, the unit of specific heat capacity is BTU/lb⋅°R, or 1 ⁠ BTU / lb⋅°R ⁠ = 4186.68 ⁠ J / kg⋅K ⁠ . [ 18 ] The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU/lb⋅°F. [ 19 ] Note the value's similarity to that of the calorie - 4187 J/kg⋅°C ≈ 4184 J/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C. In chemistry, heat amounts were often measured in calories . Confusingly, there are two common units with that name, respectively denoted cal and Cal : the small calorie ( gram-calorie, cal ) is 4.184 J exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal/(°C⋅g). The grand calorie ( kilocalorie, kilogram-calorie, food calorie, kcal, Cal ) is 1000 small calories, 4184 J exactly. It was defined so that the specific heat capacity of water would be 1 Cal/(°C⋅kg). While these units are still used in some contexts (such as kilogram calorie in nutrition ), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually: 1 ⁠ cal / °C⋅g ⁠ = 1 ⁠ Cal / °C⋅kg ⁠ = 1 ⁠ kcal / °C⋅kg ⁠ = 4184 ⁠ J / kg⋅K ⁠ [ 20 ] = 4.184 ⁠ kJ / kg⋅K ⁠ . Note that while cal is 1 ⁄ 1000 of a Cal or kcal, it is also per gram instead of kilo gram : ergo, in either unit, the specific heat capacity of water is approximately 1. The temperature of a sample of a substance reflects the average kinetic energy of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the equipartition theorem . Statistical mechanics predicts that at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy, unless multiple electronic states are accessible at room temperature (such is the case for atomic fluorine). [ 21 ] Thus, the heat capacity per mole at room temperature is the same for all of the noble gases as well as for many other atomic vapors. More precisely, and , where is the ideal gas unit (which is the product of Boltzmann conversion constant from kelvin microscopic energy unit to the macroscopic energy unit joule , and the Avogadro number ). Therefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional) atomic weight . That is, approximately, For the noble gases, from helium to xenon, these computed values are Gas He Ne Ar Kr Xe 4.00 20.17 39.95 83.80 131.29 (J⋅K −1 ⋅kg −1 ) 3118 618.3 312.2 148.8 94.99 (J⋅K −1 ⋅kg −1 ) 5197 1031 520.3 248.0 158.3 A polyatomic gas molecule can store energy in additional degrees of freedom. Its kinetic energy contributes to the heat capacity in the same way as monatomic gases, but there are also contributions from the rotations of the molecule and vibration of the atoms relative to each other (including internal potential energy ). The heat capacity may also have contribution from excited electronic states for molecules with a sufficiently small energy gap between the ground state and the excited state, such as in NO . [ 22 ] For a few systems, quantum spin statistics can also be important contributions to the heat capacity, even at room temperature. The analysis of the heat capacity of H 2 due to ortho/para separation, [ 23 ] which arises from nuclear spin statistics, has been referred to as "one of the great triumphs of post-quantum mechanical statistical mechanics." [ 24 ] These extra degrees of freedom or "modes" contribute to the specific heat capacity of the substance. Namely, when energy is introduced into a gas with polyatomic molecules, only part of it will increase their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number of degrees of freedom of the molecules. [ 25 ] [ 26 ] [ 27 ] Quantum statistical mechanics predicts that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (quanta), and that this affects the system's thermodynamic properties. Depending on the temperature, the average energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be "frozen out". In that case, the specific heat capacity of the substance increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat. For example, the molar heat capacity of nitrogen N 2 at constant volume is (at 15 °C, 1 atm), which is . [ 28 ] That is the value expected from the Equipartition Theorem if each molecule had 5 kinetic degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat capacity of N 2 (736 J⋅K −1 ⋅kg −1 ) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K −1 ⋅kg −1 ), by a factor of ⁠ 5 / 3 ⁠ . The vibrational and electronic degrees of freedom do not contribute significantly to the heat capacity in this case, due to the relatively large energy level gaps for both vibrational and electronic excitation in this molecule. This value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still "frozen out". At about that temperature, those modes begin to "un-freeze" as vibrationally excited states become accessible. As a result starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K −1 ⋅mol −1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C. [ 29 ] The last value corresponds almost exactly to the value predicted by the Equipartition Theorem, since in the high-temperature limit the theorem predicts that the vibrational degree of freedom contributes twice as much to the heat capacity as any one of the translational or rotational degrees of freedom. Derivations of heat capacity [ edit ] Relation between specific heat capacities [ edit ] Starting from the fundamental thermodynamic relation one can show, where A derivation is discussed in the article Relations between specific heats . For an ideal gas , if is expressed as molar density in the above equation, this equation reduces simply to Mayer 's relation, where and are intensive property heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively. Specific heat capacity [ edit ] The specific heat capacity of a material on a per mass basis is which in the absence of phase transitions is equivalent to where is the heat capacity of a body made of the material in question, is the mass of the body, is the volume of the body, and is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include isobaric (constant pressure, ) or isochoric (constant volume, ) processes. The corresponding specific heat capacities are expressed as A related parameter to is , the volumetric heat capacity . In engineering practice, for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript , as . Of course, from the above relationships, for solids one writes For pure homogeneous chemical compounds with established molecular or molar mass or a molar quantity is established, heat capacity as an intensive property can be expressed on a per mole basis instead of a per mass basis by the following equations analogous to the per mass equations: where n = number of moles in the body or thermodynamic system . One may refer to such a per mole quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis. Polytropic heat capacity [ edit ] The polytropic heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent ( γ or κ ) Dimensionless heat capacity [ edit ] The dimensionless heat capacity of a material is where C is the heat capacity of a body made of the material in question (J/K) n is the amount of substance in the body ( mol ) R is the gas constant (J⋅K −1 ⋅mol −1 ) N is the number of molecules in the body. (dimensionless) k B is the Boltzmann constant (J⋅K −1 ) Again, SI units shown for example. In the Ideal gas article, dimensionless heat capacity is expressed as . Heat capacity at absolute zero [ edit ] From the definition of entropy the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature T f The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the third law of thermodynamics . One of the strengths of the Debye model is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached. The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3 R , so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas. The Dulong–Petit limit results from the equipartition theorem , and as such is only valid in the classical limit of a microstate continuum , which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at standard ambient temperature , quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3 R per mole of atoms in the solid, although in molecular solids, heat capacities calculated per mole of molecules in molecular solids may be more than 3 R . For example, the heat capacity of water ice at the melting point is about 4.6 R per mole of molecules, but only 1.5 R per mole of atoms. The lower than 3 R number "per atom" (as is the case with diamond and beryllium) results from the "freezing out" of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3 R per mole of atoms of the Dulong–Petit theoretical maximum. For a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of phonons . See Debye model . Theoretical estimation [ edit ] The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. Water (liquid): CP = 4185.5 J⋅K −1 ⋅kg −1 (15 °C, 101.325 kPa) Water (liquid): CVH = 74.539 J⋅K −1 ⋅mol −1 (25 °C) For liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However, different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr). [ notes 1 ] Relation between heat capacities [ edit ] Measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see coefficient of thermal expansion and compressibility ). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. The heat capacity ratio , or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. For an ideal gas , evaluating the partial derivatives above according to the equation of state , where R is the gas constant , for an ideal gas [ 30 ] Substituting this equation reduces simply to Mayer 's relation: The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. Specific heat capacity [ edit ] The specific heat capacity of a material on a per mass basis is which in the absence of phase transitions is equivalent to where is the heat capacity of a body made of the material in question, is the mass of the body, is the volume of the body, is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include isobaric (constant pressure, ) or isochoric (constant volume, ) processes. The corresponding specific heat capacities are expressed as From the results of the previous section, dividing through by the mass gives the relation A related parameter to is , the volumetric heat capacity . In engineering practice, for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript , as . Of course, from the above relationships, for solids one writes For pure homogeneous chemical compounds with established molecular or molar mass , or a molar quantity , heat capacity as an intensive property can be expressed on a per- mole basis instead of a per-mass basis by the following equations analogous to the per mass equations: where n is the number of moles in the body or thermodynamic system . One may refer to such a per-mole quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis. Polytropic heat capacity [ edit ] The polytropic heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change: The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent ( γ or κ ). Dimensionless heat capacity [ edit ] The dimensionless heat capacity of a material is where is the heat capacity of a body made of the material in question (J/K), n is the amount of substance in the body ( mol ), R is the gas constant (J/(K⋅mol)), N is the number of molecules in the body (dimensionless), k B is the Boltzmann constant (J/(K⋅molecule)). In the ideal gas article, dimensionless heat capacity is expressed as and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the equipartition theorem . More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the dimensionless entropy per particle , measured in nats . Alternatively, using base-2 logarithms, relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in bits . [ 31 ] Heat capacity at absolute zero [ edit ] From the definition of entropy the absolute entropy can be calculated by integrating from zero to the final temperature T f : Thermodynamic derivation [ edit ] In theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an equation of state and an internal energy function . State of matter in a homogeneous sample [ edit ] To apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass . Assume that the evolution of the system is always slow enough for the internal pressure and temperature be considered uniform throughout. The pressure would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air. The state of the material can then be specified by three parameters: its temperature , the pressure , and its specific volume , where is the volume of the sample. (This quantity is the reciprocal of the material's density .) Like and , the specific volume is an intensive property of the material and its state, that does not depend on the amount of substance in the sample. Those variables are not independent. The allowed states are defined by an equation of state relating those three variables: The function depends on the material under consideration. The specific internal energy stored internally in the sample, per unit of mass, will then be another function of these state variables, that is also specific of the material. The total internal energy in the sample then will be . For some simple materials, like an ideal gas , one can derive from basic theory the equation of state and even the specific internal energy In general, these functions must be determined experimentally for each substance. Conservation of energy [ edit ] The absolute value of this quantity is undefined, and (for the purposes of thermodynamics) the state of "zero internal energy" can be chosen arbitrarily. However, by the law of conservation of energy , any infinitesimal increase in the total internal energy must be matched by the net flow of heat energy into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is , where is the change in the sample's volume in that infinitesimal step. [ 32 ] Therefore hence If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount , then the term is zero (no mechanical work is done). Then, dividing by , where is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume of the material. For the heat capacity at constant pressure, it is useful to define the specific enthalpy of the system as the sum . An infinitesimal change in the specific enthalpy will then be therefore If the pressure is kept constant, the second term on the left-hand side is zero, and The left-hand side is the specific heat capacity at constant pressure of the material. Connection to equation of state [ edit ] In general, the infinitesimal quantities are constrained by the equation of state and the specific internal energy function. Namely, Here denotes the (partial) derivative of the state equation with respect to its argument, keeping the other two arguments fixed, evaluated at the state in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space. This analysis also holds no matter how the energy increment is injected into the sample, namely by heat conduction , irradiation, electromagnetic induction , radioactive decay , etc. Relation between heat capacities [ edit ] For any specific volume , denote the function that describes how the pressure varies with the temperature , as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at . Analogously, for any pressure , let be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at . Namely, those functions are such that and for any values of . In other words, the graphs of and are slices of the surface defined by the state equation, cut by planes of constant and constant , respectively. Then, from the fundamental thermodynamic relation it follows that This equation can be rewritten as where both depending on the state . The heat capacity ratio , or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. Calculation from first principles [ edit ] The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number > iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3 R = 24.94 joules per kelvin per mole of atoms ( Dulong–Petit law , R is the gas constant ). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material. [ 33 ] For an ideal gas , evaluating the partial derivatives above according to the equation of state , where R is the gas constant , for an ideal gas [ 34 ] Substituting this equation reduces simply to Mayer 's relation: The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. Physics portal Enthalpy of fusion (latent heat of melting) Enthalpy of vaporization (latent heat of vaporization) Frenkel line Heat capacity ratio Heat equation Heat transfer coefficient History of thermodynamics Joback method (Estimation of heat capacities) Latent heat Material properties (thermodynamics) Quantum statistical mechanics R-value (insulation) Statistical mechanics Table of specific heat capacities Thermal mass Thermodynamic databases for pure substances Thermodynamic equations Volumetric heat capacity ^ IUPAC , Compendium of Chemical Terminology , 5th ed. (the "Gold Book") (2025). Online version: (2006–) " Standard Pressure ". doi : 10.1351/goldbook.S05921 . ^ Halliday, David; Resnick, Robert; Walker, Jearl (2001). Fundamentals of Physics (6th ed.). New York, NY US: John Wiley & Sons . ^ Open University (2008). S104 Book 3 Energy and Light , p. 59. The Open University . ISBN 9781848731646 . ^ Open University (2008). S104 Book 3 Energy and Light , p. 179. The Open University . ISBN 9781848731646 . ^ Engineering ToolBox (2003). "Specific Heat of some common Substances" . ^ (2001): Columbia Encyclopedia , 6th ed.; as quoted by Encyclopedia.com . Columbia University Press. Accessed on 2019-04-11. ^ Laidler, Keith J. (1993). The World of Physical Chemistry . Oxford University Press. ISBN 0-19-855919-4 . ^ Ramsay, William (1918). The life and letters of Joseph Black, M.D . Constable. pp. 38– 39. ^ Black, Joseph (1807). Robison, John (ed.). Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh . Vol. 1. Mathew Carey. pp. 76– 77. ^ West, John B. (2014-06-15). "Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases" . American Journal of Physiology. Lung Cellular and Molecular Physiology . 306 (12): L1057– L1063. doi : 10.1152/ajplung.00020.2014 . ISSN 1040-0605 . PMID 24682452 . ^ International Bureau of Weights and Measures (2006), The International System of Units (SI) (PDF) (8th ed.), ISBN 92-822-2213-6 , archived (PDF) from the original on 2021-06-04 , retrieved 2021-12-16 ^ "Water – Thermal Properties" . Engineeringtoolbox.com . Retrieved 2021-03-29 . ^ International Union of Pure and Applied Chemistry, Physical Chemistry Division. "Quantities, Units and Symbols in Physical Chemistry" (PDF) . Blackwell Sciences. p. 7. The adjective specific before the name of an extensive quantity is often used to mean divided by mass. ^ Lange's Handbook of Chemistry, 10th ed., page 1524. ^ Quick, C. R.; Schawe, J. E. K.; Uggowitzer, P. J.; Pogatscher, S. (2019-07-01). "Measurement of specific heat capacity via fast scanning calorimetry—Accuracy and loss corrections" . Thermochimica Acta . Special Issue on occasion of the 65th birthday of Christoph Schick. 677 : 12– 20. Bibcode : 2019TcAc..677...12Q . doi : 10.1016/j.tca.2019.03.021 . ISSN 0040-6031 . ^ Pogatscher, S.; Leutenegger, D.; Schawe, J. E. K.; Uggowitzer, P. J.; Löffler, J. F. (September 2016). "Solid–solid phase transitions via melting in metals" . Nature Communications . 7 (1) 11113. Bibcode : 2016NatCo...711113P . doi : 10.1038/ncomms11113 . ISSN 2041-1723 . PMC 4844691 . PMID 27103085 . ^ Koch, Werner (2013). VDI Steam Tables (4 ed.). Springer. p. 8. ISBN 978-3-642-52941-2 . Published under the auspices of the Verein Deutscher Ingenieure (VDI). ^ Cardarelli, Francois (2012). Scientific Unit Conversion: A Practical Guide to Metrication . M.J. Shields (translation) (2 ed.). Springer. p. 19. ISBN 978-1-4471-0805-4 . ^ From direct values: 1 ⁠ BTU / lb⋅°R ⁠ × 1055.06 ⁠ J / BTU ⁠ × ( ⁠ 1 / 0.45359237 ⁠ ) ⁠ lb / kg ⁠ x ⁠ 9 / 5 ⁠ ⁠ °R / K ⁠ = 4186.82 ⁠ J / kg⋅K ⁠ ^ °F=°R ^ °C=K ^ McQuarrie, Donald A. (1973). Statistical Thermodynamics . New York, NY: University Science Books . pp. 83– 85. ^ "6.6: Electronic Partition Function" . Chemistry LibreTexts . 2020-11-26 . Retrieved 2024-12-16 . ^ Bonhoeffer, K.F.; Harteck, P. (1926). "Über Para- und Orthowasserstoff" . Z. Phys. Chem . 4B : 113– 141. doi : 10.1515/zpch-1929-0408 . ^ McQuarrie, Donald A. (1973). Statistical Thermodynamics . New York, NY: University Science Books. p. 107. ^ Feynman, R., The Feynman Lectures on Physics , Vol. 1, ch. 40, pp. 7–8 ^ Reif, F. (1965). Fundamentals of statistical and thermal physics . McGraw-Hill. pp. 253–254 . ^ Kittel, Charles; Kroemer, Herbert (2000). Thermal physics . W. H. Freeman. p. 78. ISBN 978-0-7167-1088-2 . ^ Thornton, Steven T. and Rex, Andrew (1993) Modern Physics for Scientists and Engineers , Saunders College Publishing ^ Chase, M.W. Jr. (1998) NIST-JANAF Themochemical Tables, Fourth Edition , In Journal of Physical and Chemical Reference Data , Monograph 9, pages 1–1951. ^ Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach , 7th Edition, McGraw-Hill, 2010, ISBN 007-352932-X . ^ Fraundorf, P. (2003). "Heat capacity in bits". American Journal of Physics . 71 (11): 1142. arXiv : cond-mat/9711074 . Bibcode : 2003AmJPh..71.1142F . doi : 10.1119/1.1593658 . S2CID 18742525 . ^ Feynman, Richard, The Feynman Lectures on Physics , Vol. 1, Ch. 45 ^ S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", Link to Archiv e-print Link to Hal e-print ^ Cengel, Yunus A. and Boles, Michael A. (2010) Thermodynamics: An Engineering Approach , 7th Edition, McGraw-Hill ISBN 007-352932-X . Emmerich Wilhelm & Trevor M. Letcher, Eds., 2010, Heat Capacities: Liquids, Solutions and Vapours , Cambridge, U.K.:Royal Society of Chemistry, ISBN 0-85404-176-1 . A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, "Heat Capacities: Introduction, Concepts, and Selected Applications" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as photoacoustic methods, e.g., Jan Thoen & Christ Glorieux, "Photothermal Techniques for Heat Capacities," and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc. (2012-05may-24) Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World The phonon theory of liquid thermodynamics | Scientific Reports

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[Jump to content](https://en.wikipedia.org/wiki/Specific_heat_capacity#bodyContent) Main menu Main menu move to sidebar hide Navigation - [Main page](https://en.wikipedia.org/wiki/Main_Page "Visit the main page [z]") - [Contents](https://en.wikipedia.org/wiki/Wikipedia:Contents "Guides to browsing Wikipedia") - [Current events](https://en.wikipedia.org/wiki/Portal:Current_events "Articles related to current events") - [Random article](https://en.wikipedia.org/wiki/Special:Random "Visit a randomly selected article [x]") - [About Wikipedia](https://en.wikipedia.org/wiki/Wikipedia:About "Learn about Wikipedia and how it works") - [Contact us](https://en.wikipedia.org/wiki/Wikipedia:Contact_us "How to contact Wikipedia") Contribute - [Help](https://en.wikipedia.org/wiki/Help:Contents "Guidance on how to use and edit Wikipedia") - [Learn to edit](https://en.wikipedia.org/wiki/Help:Introduction "Learn how to edit Wikipedia") - [Community portal](https://en.wikipedia.org/wiki/Wikipedia:Community_portal "The hub for editors") - [Recent changes](https://en.wikipedia.org/wiki/Special:RecentChanges "A list of recent changes to Wikipedia [r]") - [Upload file](https://en.wikipedia.org/wiki/Wikipedia:File_upload_wizard "Add images or other media for use on Wikipedia") - [Special pages](https://en.wikipedia.org/wiki/Special:SpecialPages "A list of all special pages [q]") [![](https://en.wikipedia.org/static/images/icons/enwiki-25.svg) ![Wikipedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en-25.svg) ![The Free Encyclopedia](https://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en-25.svg)](https://en.wikipedia.org/wiki/Main_Page) [Search](https://en.wikipedia.org/wiki/Special:Search "Search Wikipedia [f]") Appearance - [Donate](https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en) - [Create account](https://en.wikipedia.org/w/index.php?title=Special:CreateAccount&returnto=Specific+heat+capacity "You are encouraged to create an account and log in; however, it is not mandatory") - [Log in](https://en.wikipedia.org/w/index.php?title=Special:UserLogin&returnto=Specific+heat+capacity "You're encouraged to log in; however, it's not mandatory. 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[o]") ## Contents move to sidebar hide - [(Top)](https://en.wikipedia.org/wiki/Specific_heat_capacity) - [1 History](https://en.wikipedia.org/wiki/Specific_heat_capacity#History) Toggle History subsection - [1\.1 Discovery of specific heat](https://en.wikipedia.org/wiki/Specific_heat_capacity#Discovery_of_specific_heat) - [2 Definition](https://en.wikipedia.org/wiki/Specific_heat_capacity#Definition) Toggle Definition subsection - [2\.1 Variations](https://en.wikipedia.org/wiki/Specific_heat_capacity#Variations) - [2\.2 Applicability](https://en.wikipedia.org/wiki/Specific_heat_capacity#Applicability) - [3 Measurement](https://en.wikipedia.org/wiki/Specific_heat_capacity#Measurement) - [4 Units](https://en.wikipedia.org/wiki/Specific_heat_capacity#Units) Toggle Units subsection - [4\.1 International system](https://en.wikipedia.org/wiki/Specific_heat_capacity#International_system) - [4\.2 Imperial engineering units](https://en.wikipedia.org/wiki/Specific_heat_capacity#Imperial_engineering_units) - [4\.3 Calories](https://en.wikipedia.org/wiki/Specific_heat_capacity#Calories) - [5 Physical basis](https://en.wikipedia.org/wiki/Specific_heat_capacity#Physical_basis) Toggle Physical basis subsection - [5\.1 Monatomic gases](https://en.wikipedia.org/wiki/Specific_heat_capacity#Monatomic_gases) - [5\.2 Polyatomic gases](https://en.wikipedia.org/wiki/Specific_heat_capacity#Polyatomic_gases) - [6 Derivations of heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Derivations_of_heat_capacity) Toggle Derivations of heat capacity subsection - [6\.1 Relation between specific heat capacities](https://en.wikipedia.org/wiki/Specific_heat_capacity#Relation_between_specific_heat_capacities) - [6\.2 Specific heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Specific_heat_capacity) - [6\.3 Polytropic heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Polytropic_heat_capacity) - [6\.4 Dimensionless heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Dimensionless_heat_capacity) - [6\.5 Heat capacity at absolute zero](https://en.wikipedia.org/wiki/Specific_heat_capacity#Heat_capacity_at_absolute_zero) - [6\.6 Solid phase](https://en.wikipedia.org/wiki/Specific_heat_capacity#Solid_phase) - [6\.7 Theoretical estimation](https://en.wikipedia.org/wiki/Specific_heat_capacity#Theoretical_estimation) - [6\.8 Relation between heat capacities](https://en.wikipedia.org/wiki/Specific_heat_capacity#Relation_between_heat_capacities) - [6\.8.1 Ideal gas](https://en.wikipedia.org/wiki/Specific_heat_capacity#Ideal_gas) - [6\.9 Specific heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Specific_heat_capacity_2) - [6\.10 Polytropic heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Polytropic_heat_capacity_2) - [6\.11 Dimensionless heat capacity](https://en.wikipedia.org/wiki/Specific_heat_capacity#Dimensionless_heat_capacity_2) - [6\.12 Heat capacity at absolute zero](https://en.wikipedia.org/wiki/Specific_heat_capacity#Heat_capacity_at_absolute_zero_2) - [7 Thermodynamic derivation](https://en.wikipedia.org/wiki/Specific_heat_capacity#Thermodynamic_derivation) Toggle Thermodynamic derivation subsection - [7\.1 State of matter in a homogeneous sample](https://en.wikipedia.org/wiki/Specific_heat_capacity#State_of_matter_in_a_homogeneous_sample) - [7\.2 Conservation of energy](https://en.wikipedia.org/wiki/Specific_heat_capacity#Conservation_of_energy) - [7\.3 Connection to equation of state](https://en.wikipedia.org/wiki/Specific_heat_capacity#Connection_to_equation_of_state) - [7\.4 Relation between heat capacities](https://en.wikipedia.org/wiki/Specific_heat_capacity#Relation_between_heat_capacities_2) - [7\.5 Calculation from first principles](https://en.wikipedia.org/wiki/Specific_heat_capacity#Calculation_from_first_principles) - [7\.5.1 Ideal gas](https://en.wikipedia.org/wiki/Specific_heat_capacity#Ideal_gas_2) - [8 See also](https://en.wikipedia.org/wiki/Specific_heat_capacity#See_also) - [9 Notes](https://en.wikipedia.org/wiki/Specific_heat_capacity#Notes) - [10 References](https://en.wikipedia.org/wiki/Specific_heat_capacity#References) - [11 Further reading](https://en.wikipedia.org/wiki/Specific_heat_capacity#Further_reading) - [12 External links](https://en.wikipedia.org/wiki/Specific_heat_capacity#External_links) Toggle the table of contents # Specific heat capacity 61 languages - [Aragonés](https://an.wikipedia.org/wiki/Calor_especifica "Calor especifica – Aragonese") - [العربية](https://ar.wikipedia.org/wiki/%D8%AD%D8%B1%D8%A7%D8%B1%D8%A9_%D9%86%D9%88%D8%B9%D9%8A%D8%A9 "حرارة نوعية – Arabic") - [Asturianu](https://ast.wikipedia.org/wiki/Calor_espec%C3%ADfico "Calor específico – Asturian") - 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unit](https://en.wikipedia.org/wiki/SI_unit "SI unit") | J⋅kg−1⋅K−1 | | In [SI base units](https://en.wikipedia.org/wiki/SI_base_unit "SI base unit") | m2⋅K−1⋅s−2 | | [Intensive](https://en.wikipedia.org/wiki/Intensive_and_extensive_properties "Intensive and extensive properties")? | Yes | | [Dimension](https://en.wikipedia.org/wiki/Dimensional_analysis#Formulation "Dimensional analysis") | L2⋅T−2⋅Θ−1 | | [Thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics") | | |---|---| | [![](https://upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/250px-Carnot_heat_engine_2.svg.png)](https://en.wikipedia.org/wiki/Carnot_heat_engine "Carnot heat engine")The classical [Carnot heat engine](https://en.wikipedia.org/wiki/Carnot_heat_engine "Carnot heat engine") | | | Branches [Classical](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics") [Statistical](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") [Chemical](https://en.wikipedia.org/wiki/Chemical_thermodynamics "Chemical thermodynamics") [Quantum thermodynamics](https://en.wikipedia.org/wiki/Quantum_thermodynamics "Quantum thermodynamics") [Equilibrium](https://en.wikipedia.org/wiki/Equilibrium_thermodynamics "Equilibrium thermodynamics") / [Non-equilibrium](https://en.wikipedia.org/wiki/Non-equilibrium_thermodynamics "Non-equilibrium thermodynamics") | | | [Laws](https://en.wikipedia.org/wiki/Laws_of_thermodynamics "Laws of thermodynamics") [Zeroth](https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics "Zeroth law of thermodynamics") [First](https://en.wikipedia.org/wiki/First_law_of_thermodynamics "First law of thermodynamics") [Second](https://en.wikipedia.org/wiki/Second_law_of_thermodynamics "Second law of thermodynamics") [Third](https://en.wikipedia.org/wiki/Third_law_of_thermodynamics "Third law of thermodynamics") | | | [State](https://en.wikipedia.org/wiki/Thermodynamic_state "Thermodynamic state") | | | [Equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state") [Ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") [Real gas](https://en.wikipedia.org/wiki/Real_gas "Real gas") [State of matter](https://en.wikipedia.org/wiki/State_of_matter "State of matter") [Phase (matter)](https://en.wikipedia.org/wiki/Phase_\(matter\) "Phase (matter)") [Equilibrium](https://en.wikipedia.org/wiki/Thermodynamic_equilibrium "Thermodynamic equilibrium") [Control volume](https://en.wikipedia.org/wiki/Control_volume "Control volume") [Instruments](https://en.wikipedia.org/wiki/Thermodynamic_instruments "Thermodynamic instruments") | | | [Processes](https://en.wikipedia.org/wiki/Thermodynamic_process "Thermodynamic process") | | | [Isobaric](https://en.wikipedia.org/wiki/Isobaric_process "Isobaric process") [Isochoric](https://en.wikipedia.org/wiki/Isochoric_process "Isochoric process") [Isothermal](https://en.wikipedia.org/wiki/Isothermal_process "Isothermal process") [Adiabatic](https://en.wikipedia.org/wiki/Adiabatic_process "Adiabatic process") [Isentropic](https://en.wikipedia.org/wiki/Isentropic_process "Isentropic process") [Isenthalpic](https://en.wikipedia.org/wiki/Isenthalpic_process "Isenthalpic process") [Quasistatic](https://en.wikipedia.org/wiki/Quasistatic_process "Quasistatic process") [Polytropic](https://en.wikipedia.org/wiki/Polytropic_process "Polytropic process") [Free expansion](https://en.wikipedia.org/wiki/Free_expansion "Free expansion") [Reversibility](https://en.wikipedia.org/wiki/Reversible_process_\(thermodynamics\) "Reversible process (thermodynamics)") [Irreversibility](https://en.wikipedia.org/wiki/Irreversible_process "Irreversible process") [Endoreversibility](https://en.wikipedia.org/wiki/Endoreversible_thermodynamics "Endoreversible thermodynamics") | | | [Cycles](https://en.wikipedia.org/wiki/Thermodynamic_cycle "Thermodynamic cycle") | | | [Heat engines](https://en.wikipedia.org/wiki/Heat_engine "Heat engine") [Heat pumps](https://en.wikipedia.org/wiki/Heat_pump_and_refrigeration_cycle "Heat pump and refrigeration cycle") [Thermal efficiency](https://en.wikipedia.org/wiki/Thermal_efficiency "Thermal efficiency") | | | | | | [Property diagrams](https://en.wikipedia.org/wiki/Thermodynamic_diagrams "Thermodynamic diagrams") [Intensive and extensive properties](https://en.wikipedia.org/wiki/Intensive_and_extensive_properties "Intensive and extensive properties") | | | [Process functions](https://en.wikipedia.org/wiki/Process_function "Process function") | | | [Work](https://en.wikipedia.org/wiki/Work_\(thermodynamics\) "Work (thermodynamics)") [Heat](https://en.wikipedia.org/wiki/Heat "Heat") | | | [Functions of state](https://en.wikipedia.org/wiki/State_function "State function") | | | [Temperature](https://en.wikipedia.org/wiki/Thermodynamic_temperature "Thermodynamic temperature") / *[Entropy](https://en.wikipedia.org/wiki/Entropy "Entropy")* ([introduction](https://en.wikipedia.org/wiki/Introduction_to_entropy "Introduction to entropy")) [Pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") / *[Volume](https://en.wikipedia.org/wiki/Volume_\(thermodynamics\) "Volume (thermodynamics)")* [Chemical potential](https://en.wikipedia.org/wiki/Chemical_potential "Chemical potential") / *[Particle number](https://en.wikipedia.org/wiki/Particle_number "Particle number")* [Vapor quality](https://en.wikipedia.org/wiki/Vapor_quality "Vapor quality") [Reduced properties](https://en.wikipedia.org/wiki/Reduced_properties "Reduced properties") | | | | | | [Specific heat capacity](https://en.wikipedia.org/wiki/Heat_capacity "Heat capacity") | c \= {\\displaystyle c=} ![{\\displaystyle c=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428) | | Scientists [Bernoulli](https://en.wikipedia.org/wiki/Daniel_Bernoulli "Daniel Bernoulli") [Boltzmann](https://en.wikipedia.org/wiki/Ludwig_Boltzmann "Ludwig Boltzmann") [Bridgman](https://en.wikipedia.org/wiki/Percy_Williams_Bridgman "Percy Williams Bridgman") [Callen](https://en.wikipedia.org/wiki/Herbert_Callen "Herbert Callen") [Carathéodory](https://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odory "Constantin Carathéodory") [Carnot](https://en.wikipedia.org/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot "Nicolas Léonard Sadi Carnot") [Clapeyron](https://en.wikipedia.org/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron "Benoît Paul Émile Clapeyron") [Clausius](https://en.wikipedia.org/wiki/Rudolf_Clausius "Rudolf Clausius") [de Donder](https://en.wikipedia.org/wiki/Th%C3%A9ophile_de_Donder "Théophile de Donder") [Duhem](https://en.wikipedia.org/wiki/Pierre_Duhem "Pierre Duhem") [Gibbs](https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs "Josiah Willard Gibbs") [von Helmholtz](https://en.wikipedia.org/wiki/Hermann_von_Helmholtz "Hermann von Helmholtz") [Joule](https://en.wikipedia.org/wiki/James_Prescott_Joule "James Prescott Joule") [Kelvin](https://en.wikipedia.org/wiki/Lord_Kelvin "Lord Kelvin") [Lewis](https://en.wikipedia.org/wiki/Gilbert_N._Lewis "Gilbert N. Lewis") [Massieu](https://en.wikipedia.org/wiki/Fran%C3%A7ois_Massieu "François Massieu") [Maxwell](https://en.wikipedia.org/wiki/James_Clerk_Maxwell "James Clerk Maxwell") [von Mayer](https://en.wikipedia.org/wiki/Julius_von_Mayer "Julius von Mayer") [Nernst](https://en.wikipedia.org/wiki/Walther_Nernst "Walther Nernst") [Onsager](https://en.wikipedia.org/wiki/Lars_Onsager "Lars Onsager") [Planck](https://en.wikipedia.org/wiki/Max_Planck "Max Planck") [Rankine](https://en.wikipedia.org/wiki/William_John_Macquorn_Rankine "William John Macquorn Rankine") [Smeaton](https://en.wikipedia.org/wiki/John_Smeaton "John Smeaton") [Stahl](https://en.wikipedia.org/wiki/Georg_Ernst_Stahl "Georg Ernst Stahl") [Tait](https://en.wikipedia.org/wiki/Peter_Tait_\(physicist\) "Peter Tait (physicist)") [Thompson](https://en.wikipedia.org/wiki/Benjamin_Thompson "Benjamin Thompson") [van der Waals](https://en.wikipedia.org/wiki/Johannes_Diderik_van_der_Waals "Johannes Diderik van der Waals") [Waterston](https://en.wikipedia.org/wiki/John_James_Waterston "John James Waterston") | | | Other [Nucleation](https://en.wikipedia.org/wiki/Nucleation "Nucleation") [Self-assembly](https://en.wikipedia.org/wiki/Self-assembly "Self-assembly") [Self-organization](https://en.wikipedia.org/wiki/Self-organization "Self-organization") | | | ![](https://upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/20px-Symbol_category_class.svg.png) [Category](https://en.wikipedia.org/wiki/Category:Thermodynamics "Category:Thermodynamics") | | | [v](https://en.wikipedia.org/wiki/Template:Thermodynamics_sidebar "Template:Thermodynamics sidebar") [t](https://en.wikipedia.org/wiki/Template_talk:Thermodynamics_sidebar "Template talk:Thermodynamics sidebar") [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Thermodynamics_sidebar "Special:EditPage/Template:Thermodynamics sidebar") | | In [thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics"), the **specific heat capacity** (symbol *c*) of a substance is the amount of [heat](https://en.wikipedia.org/wiki/Heat "Heat") that must be added to one unit of mass of the substance in order to cause an increase of one unit in [temperature](https://en.wikipedia.org/wiki/Temperature "Temperature"). It is also referred to as **massic heat capacity** or as the **specific heat.** More formally it is the [heat capacity](https://en.wikipedia.org/wiki/Heat_capacity "Heat capacity") of a sample of the substance divided by the [mass](https://en.wikipedia.org/wiki/Mass "Mass") of the sample.[\[1\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-1) The [SI](https://en.wikipedia.org/wiki/International_System_of_Units "International System of Units") unit of specific heat capacity is [joule](https://en.wikipedia.org/wiki/Joule "Joule") per [kelvin](https://en.wikipedia.org/wiki/Kelvin "Kelvin") per [kilogram](https://en.wikipedia.org/wiki/Kilogram "Kilogram"), J⋅kg−1⋅K−1.[\[2\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-2) For example, the heat required to raise the temperature of 1 kg of [water](https://en.wikipedia.org/wiki/Water "Water") by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.[\[3\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-3) Specific heat capacity often varies with temperature, and is different for each [state of matter](https://en.wikipedia.org/wiki/State_of_matter "State of matter"). Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C, but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1. The specific heat capacities of [iron](https://en.wikipedia.org/wiki/Iron "Iron"), [granite](https://en.wikipedia.org/wiki/Granite "Granite"), and [hydrogen](https://en.wikipedia.org/wiki/Hydrogen "Hydrogen") gas are about 449 J⋅kg−1⋅K−1, 790 J⋅kg−1⋅K−1, and 14300 J⋅kg−1⋅K−1, respectively.[\[4\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-4) While the substance is undergoing a [phase transition](https://en.wikipedia.org/wiki/Phase_transition "Phase transition"), such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature. The specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity *at constant pressure*) than when it is heated in a closed vessel that prevents expansion (specific heat capacity *at constant volume*). These two values are usually denoted by c p {\\displaystyle c\_{p}} ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) and c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082), respectively; their quotient γ \= c p / c V {\\displaystyle \\gamma =c\_{p}/c\_{V}} ![{\\displaystyle \\gamma =c\_{p}/c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe578fc940f94ff433f350a8b2d7cad4dbf01c0) is the [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio"). The term *specific heat* may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C;[\[5\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-colen2001-5) much in the fashion of [specific gravity](https://en.wikipedia.org/wiki/Specific_gravity "Specific gravity"). Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of [moles](https://en.wikipedia.org/wiki/Mole_\(unit\) "Mole (unit)"), one gets the [molar heat capacity](https://en.wikipedia.org/wiki/Molar_heat_capacity "Molar heat capacity") instead, whose SI unit is joule per kelvin per mole, J⋅mol−1⋅K−1. If the amount is taken to be the [volume](https://en.wikipedia.org/wiki/Volume "Volume") of the sample (as is sometimes done in engineering), one gets the [volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity"), whose SI unit is joule per kelvin per [cubic meter](https://en.wikipedia.org/wiki/Cubic_meter "Cubic meter"), J⋅m−3⋅K−1. ## History \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=1 "Edit section: History")\] ### Discovery of specific heat \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=2 "Edit section: Discovery of specific heat")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Black_Joseph_%28cropped%29.jpg/250px-Black_Joseph_%28cropped%29.jpg)](https://en.wikipedia.org/wiki/File:Black_Joseph_\(cropped\).jpg) Joseph Black One of the first scientists to use the concept was [Joseph Black](https://en.wikipedia.org/wiki/Joseph_Black "Joseph Black"), an 18th-century medical doctor and professor of medicine at [Glasgow University](https://en.wikipedia.org/wiki/Glasgow_University "Glasgow University"). He measured the specific heat capacities of many substances, using the term *capacity for heat*.[\[6\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-6) In 1756 or soon thereafter, Black began an extensive study of heat.[\[7\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-:1-7) In 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by [Daniel Gabriel Fahrenheit](https://en.wikipedia.org/wiki/Daniel_Gabriel_Fahrenheit "Daniel Gabriel Fahrenheit") on behalf of Dutch physician [Herman Boerhaave](https://en.wikipedia.org/wiki/Herman_Boerhaave "Herman Boerhaave"). For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: "Quicksilver \[mercury\] ... has less capacity for the matter of heat than water."[\[8\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-8)[\[9\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-:0-9) ## Definition \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=3 "Edit section: Definition")\] The specific heat capacity of a substance, usually denoted by c {\\displaystyle c} ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) or s {\\displaystyle s} ![{\\displaystyle s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632), is the heat capacity C {\\displaystyle C} ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) of a sample of the substance, divided by the mass M {\\displaystyle M} ![{\\displaystyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd) of the sample:[\[10\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-10) c \= C M \= 1 M ⋅ d Q d T , {\\displaystyle c={\\frac {C}{M}}={\\frac {1}{M}}\\cdot {\\frac {\\mathrm {d} Q}{\\mathrm {d} T}},} ![{\\displaystyle c={\\frac {C}{M}}={\\frac {1}{M}}\\cdot {\\frac {\\mathrm {d} Q}{\\mathrm {d} T}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58291be3575e33a756ed894a18347c8eabbba29) where d Q {\\displaystyle \\mathrm {d} Q} ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739) [represents](https://en.wikipedia.org/wiki/Derivative "Derivative") the amount of heat needed to uniformly raise the temperature of the sample by a small increment d T {\\displaystyle \\mathrm {d} T} ![{\\displaystyle \\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80). Like the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) of the sample and the [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") p {\\displaystyle p} ![{\\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) applied to it. Therefore, it should be considered a function c ( p , T ) {\\displaystyle c(p,T)} ![{\\displaystyle c(p,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bac9cbcaf700b5206116a980a24a84404ba2e5) of those two variables. These parameters are usually specified when giving the specific heat capacity of a substance. For example, "Water (liquid): c p {\\displaystyle c\_{p}} ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) = 4187 J⋅kg−1⋅K−1 (15 °C)."[\[11\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-toolbox-11) When not specified, published values of the specific heat capacity c {\\displaystyle c} ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) generally are valid for some [standard conditions for temperature and pressure](https://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressure "Standard conditions for temperature and pressure"). However, the dependency of c {\\displaystyle c} ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier ( p , T ) {\\displaystyle (p,T)} ![{\\displaystyle (p,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140e6e1e1f95c1aba3ffb6dc196a9be07a91d1a6) and approximates the specific heat capacity by a constant c {\\displaystyle c} ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) suitable for those ranges. Specific heat capacity is an [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier "specific" in front of an extensive property often indicates an intensive property derived from it.[\[12\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-12)) ### Variations \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=4 "Edit section: Variations")\] The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure p {\\displaystyle p} ![{\\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) and starting temperature T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0). Two particular choices are widely used: - If the pressure is kept constant (for instance, at the ambient atmospheric pressure), and the sample is allowed to expand, the expansion generates [work](https://en.wikipedia.org/wiki/Work_\(thermodynamics\) "Work (thermodynamics)"), as the force from the pressure displaces the enclosure or the surrounding fluid. That work must come from the heat energy provided. The specific heat capacity thus obtained is said to be measured **at constant pressure** (or **isobaric**) and is often denoted c p {\\displaystyle c\_{p}} ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) . - On the other hand, if the expansion is prevented – for example, by a sufficiently rigid enclosure or by increasing the external pressure to counteract the internal one – no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the sample, including raising its temperature by an extra amount. The specific heat capacity obtained this way is said to be measured **at constant volume** (or **isochoric**) and denoted c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) . The value of c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) is always less than the value of c p {\\displaystyle c\_{p}} ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) for all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio") of gases is typically between 1.3 and 1.67.[\[13\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-Lange-13) ### Applicability \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=5 "Edit section: Applicability")\] The specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale. The specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops. The specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a [phase change](https://en.wikipedia.org/wiki/Phase_transition "Phase transition"), such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement. ## Measurement \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=6 "Edit section: Measurement")\] The specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a [calorimeter](https://en.wikipedia.org/wiki/Calorimeter "Calorimeter"), and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as [differential scanning calorimetry](https://en.wikipedia.org/wiki/Differential_scanning_calorimetry "Differential scanning calorimetry").[\[14\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-15) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Water_temperature_vs_heat_added.svg/250px-Water_temperature_vs_heat_added.svg.png)](https://en.wikipedia.org/wiki/File:Water_temperature_vs_heat_added.svg) Graph of temperature of phases of water heated from −100 °C to 200 °C – the dashed line example shows that melting and heating 1 kg of ice at −50 °C to water at 40 °C needs 600 kJ The specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the [coefficient of thermal expansion](https://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion "Coefficient of thermal expansion") and the [compressibility](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] ## Units \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=7 "Edit section: Units")\] ### International system \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=8 "Edit section: International system")\] The SI unit for specific heat capacity is joule per kelvin per kilogram ⁠J/kg⋅K⁠, J⋅K−1⋅kg−1. Since an increment of temperature of one [degree Celsius](https://en.wikipedia.org/wiki/Celsius_scale "Celsius scale") is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J/(kg⋅°C). Sometimes the [gram](https://en.wikipedia.org/wiki/Gram "Gram") is used instead of kilogram for the unit of mass: 1 J⋅g−1⋅K−1 = 1000 J⋅kg−1⋅K−1. The specific heat capacity of a substance (per unit of mass) has [dimension](https://en.wikipedia.org/wiki/Dimensional_analysis "Dimensional analysis") L2⋅Θ−1⋅T−2, or (L/T)2/Θ. Therefore, the SI unit J⋅kg−1⋅K−1 is equivalent to [metre](https://en.wikipedia.org/wiki/Metre "Metre") squared per [second](https://en.wikipedia.org/wiki/Second "Second") squared per [kelvin](https://en.wikipedia.org/wiki/Kelvin "Kelvin") (m2⋅K−1⋅s−2). ### Imperial engineering units \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=9 "Edit section: Imperial engineering units")\] Professionals in [construction](https://en.wikipedia.org/wiki/Construction "Construction"), [civil engineering](https://en.wikipedia.org/wiki/Civil_engineering "Civil engineering"), [chemical engineering](https://en.wikipedia.org/wiki/Chemical_engineering "Chemical engineering"), and other technical disciplines, especially in the [United States](https://en.wikipedia.org/wiki/United_States "United States"), may use [English Engineering units](https://en.wikipedia.org/wiki/English_Engineering_Units "English Engineering Units") including the [pound](https://en.wikipedia.org/wiki/Pound_\(mass\) "Pound (mass)") (lb = 0.45359237 kg) as the unit of mass, the [degree Fahrenheit](https://en.wikipedia.org/wiki/Fahrenheit "Fahrenheit") or [Rankine](https://en.wikipedia.org/wiki/Rankine_scale "Rankine scale") (°R = ⁠5/9⁠ K, about 0.555556 K) as the unit of temperature increment, and the [British thermal unit](https://en.wikipedia.org/wiki/British_thermal_unit "British thermal unit") (BTU ≈ 1055.056 J),[\[16\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-Koch-16)[\[17\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-17) as the unit of heat. In those contexts, the unit of specific heat capacity is BTU/lb⋅°R, or 1 ⁠BTU/lb⋅°R⁠ = 4186.68⁠J/kg⋅K⁠.[\[18\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-18) The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU/lb⋅°F.[\[19\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-19) Note the value's similarity to that of the calorie - 4187 J/kg⋅°C ≈ 4184 J/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C. ### Calories \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=10 "Edit section: Calories")\] In chemistry, heat amounts were often measured in [calories](https://en.wikipedia.org/wiki/Calorie "Calorie"). Confusingly, there are two common units with that name, respectively denoted *cal* and *Cal*: - the *small calorie* (*gram-calorie, cal*) is 4.184 J exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal/(°C⋅g). - The *grand calorie* (*kilocalorie, kilogram-calorie, food calorie, kcal, Cal*) is 1000 small calories, 4184 J exactly. It was defined so that the specific heat capacity of water would be 1 Cal/(°C⋅kg). While these units are still used in some contexts (such as kilogram calorie in [nutrition](https://en.wikipedia.org/wiki/Nutrition "Nutrition")), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually: 1 ⁠cal/°C⋅g⁠ = 1 ⁠Cal/°C⋅kg⁠ = 1 ⁠kcal/°C⋅kg⁠ = 4184 ⁠J/kg⋅K⁠[\[20\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-20) = 4.184 ⁠kJ/kg⋅K⁠. Note that while cal is **1⁄1000** of a Cal or kcal, it is also per *gram* instead of **kilo***gram*: ergo, in either unit, the specific heat capacity of water is approximately 1. ## Physical basis \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=11 "Edit section: Physical basis")\] Main article: [Molar heat capacity § Physical basis](https://en.wikipedia.org/wiki/Molar_heat_capacity#Physical_basis "Molar heat capacity") The temperature of a sample of a substance reflects the average [kinetic energy](https://en.wikipedia.org/wiki/Kinetic_energy "Kinetic energy") of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem"). ### Monatomic gases \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=12 "Edit section: Monatomic gases")\] [Statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") predicts that at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy, unless multiple electronic states are accessible at room temperature (such is the case for atomic fluorine).[\[21\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-21) Thus, the [heat capacity per mole](https://en.wikipedia.org/wiki/Molar_heat_capacity "Molar heat capacity") at room temperature is the same for all of the noble gases as well as for many other atomic vapors. More precisely, c V , m \= 3 R / 2 ≈ 12\.5 J ⋅ K − 1 ⋅ m o l − 1 {\\displaystyle c\_{V,\\mathrm {m} }=3R/2\\approx \\mathrm {12.5\\,J\\cdot K^{-1}\\cdot mol^{-1}} } ![{\\displaystyle c\_{V,\\mathrm {m} }=3R/2\\approx \\mathrm {12.5\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcee637c4392e7347a6b4b7d57395e90ef9b8f5b) and c P , m \= 5 R / 2 ≈ 21 J ⋅ K − 1 ⋅ m o l − 1 {\\displaystyle c\_{P,\\mathrm {m} }=5R/2\\approx \\mathrm {21\\,J\\cdot K^{-1}\\cdot mol^{-1}} } ![{\\displaystyle c\_{P,\\mathrm {m} }=5R/2\\approx \\mathrm {21\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6fc485f5327a3d2857548cbe7c145147d826306), where R ≈ 8\.31446 J ⋅ K − 1 ⋅ m o l − 1 {\\displaystyle R\\approx \\mathrm {8.31446\\,J\\cdot K^{-1}\\cdot mol^{-1}} } ![{\\displaystyle R\\approx \\mathrm {8.31446\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b344e96636e94e8469d9e4a0eafa306b3bec07e) is the [ideal gas unit](https://en.wikipedia.org/wiki/Ideal_gas_constant "Ideal gas constant") (which is the product of [Boltzmann conversion constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") from [kelvin](https://en.wikipedia.org/wiki/Kelvin "Kelvin") microscopic energy unit to the macroscopic energy unit [joule](https://en.wikipedia.org/wiki/Joule "Joule"), and the [Avogadro number](https://en.wikipedia.org/wiki/Avogadro_number "Avogadro number")). Therefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional) [atomic weight](https://en.wikipedia.org/wiki/Atomic_weight "Atomic weight") A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3). That is, approximately, c V ≈ 12470 J ⋅ K − 1 ⋅ k g − 1 / A c p ≈ 20785 J ⋅ K − 1 ⋅ k g − 1 / A {\\displaystyle c\_{V}\\approx \\mathrm {12470\\,J\\cdot K^{-1}\\cdot kg^{-1}} /A\\quad \\quad \\quad c\_{p}\\approx \\mathrm {20785\\,J\\cdot K^{-1}\\cdot kg^{-1}} /A} ![{\\displaystyle c\_{V}\\approx \\mathrm {12470\\,J\\cdot K^{-1}\\cdot kg^{-1}} /A\\quad \\quad \\quad c\_{p}\\approx \\mathrm {20785\\,J\\cdot K^{-1}\\cdot kg^{-1}} /A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0733e9dc0db1000b284289198a6d8d9b4ed008d9) For the noble gases, from helium to xenon, these computed values are | Gas | He | Ne | Ar | Kr | Xe | |---|---|---|---|---|---| | A {\\displaystyle A} ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) | | | | | | ### Polyatomic gases \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=13 "Edit section: Polyatomic gases")\] A polyatomic gas molecule can store energy in additional degrees of freedom. Its kinetic energy contributes to the heat capacity in the same way as monatomic gases, but there are also contributions from the [rotations](https://en.wikipedia.org/wiki/Rotational_energy "Rotational energy") of the molecule and vibration of the atoms relative to each other (including internal [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy")). The heat capacity may also have contribution from [excited electronic states](https://en.wikipedia.org/wiki/Excited_state "Excited state") for molecules with a sufficiently small energy gap between the ground state and the excited state, such as in NO.[\[22\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-22) For a few systems, quantum spin statistics can also be important contributions to the heat capacity, even at room temperature. The analysis of the heat capacity of H 2 due to ortho/para separation,[\[23\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-23) which arises from [nuclear spin](https://en.wikipedia.org/wiki/Spin_quantum_number "Spin quantum number") statistics, has been referred to as "one of the great triumphs of post-quantum mechanical statistical mechanics."[\[24\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-24) These extra [degrees of freedom](https://en.wikipedia.org/wiki/Degrees_of_freedom_\(physics_and_chemistry\) "Degrees of freedom (physics and chemistry)") or "modes" contribute to the specific heat capacity of the substance. Namely, when energy is introduced into a gas with polyatomic molecules, only part of it will increase their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number of degrees of freedom of the molecules.[\[25\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-25)[\[26\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-26)[\[27\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-27) [Quantum statistical mechanics](https://en.wikipedia.org/wiki/Quantum_statistical_mechanics "Quantum statistical mechanics") predicts that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (quanta), and that this affects the system's thermodynamic properties. Depending on the temperature, the average energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be "frozen out". In that case, the specific heat capacity of the substance increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat. For example, the molar heat capacity of [nitrogen](https://en.wikipedia.org/wiki/Nitrogen "Nitrogen") N 2 at constant volume is c V , m \= 20\.6 J ⋅ K − 1 ⋅ m o l − 1 {\\displaystyle c\_{V,\\mathrm {m} }=\\mathrm {20.6\\,J\\cdot K^{-1}\\cdot mol^{-1}} } ![{\\displaystyle c\_{V,\\mathrm {m} }=\\mathrm {20.6\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7c7437f35c01e65d4dc78ef1cbf4dfcf2fb023) (at 15 °C, 1 atm), which is 2\.49 R {\\displaystyle 2.49R} ![{\\displaystyle 2.49R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6691d251191f800f340af5ec9cd4e291d0a5344c).[\[28\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-thor1993-28) That is the value expected from the [Equipartition Theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem") if each molecule had 5 kinetic degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat capacity c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) of N 2 (736 J⋅K−1⋅kg−1) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K−1⋅kg−1), by a factor of ⁠5/3⁠. The vibrational and electronic degrees of freedom do not contribute significantly to the heat capacity in this case, due to the relatively large energy level gaps for both vibrational and electronic excitation in this molecule. This value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still "frozen out". At about that temperature, those modes begin to "un-freeze" as vibrationally excited states become accessible. As a result c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.[\[29\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-chas1998-29) The last value corresponds almost exactly to the value predicted by the Equipartition Theorem, since in the high-temperature limit the theorem predicts that the vibrational degree of freedom contributes twice as much to the heat capacity as any one of the translational or rotational degrees of freedom. ## Derivations of heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=14 "Edit section: Derivations of heat capacity")\] ### Relation between specific heat capacities \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=15 "Edit section: Relation between specific heat capacities")\] Starting from the [fundamental thermodynamic relation](https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation "Fundamental thermodynamic relation") one can show, c p − c v \= α 2 T ρ β T {\\displaystyle c\_{p}-c\_{v}={\\frac {\\alpha ^{2}T}{\\rho \\beta \_{T}}}} ![{\\displaystyle c\_{p}-c\_{v}={\\frac {\\alpha ^{2}T}{\\rho \\beta \_{T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06bc97cafc646b40213b01cf9a30562276c7c744) where - α {\\displaystyle \\alpha } ![{\\displaystyle \\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3) is the [coefficient of thermal expansion](https://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion "Coefficient of thermal expansion"), - β T {\\displaystyle \\beta \_{T}} ![{\\displaystyle \\beta \_{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2855fc59047bc00f5bdedbcad68550a7ac6d13) is the [isothermal](https://en.wikipedia.org/wiki/Isothermal "Isothermal") [compressibility](https://en.wikipedia.org/wiki/Compressibility "Compressibility"), and - ρ {\\displaystyle \\rho } ![{\\displaystyle \\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64) is [density](https://en.wikipedia.org/wiki/Density "Density"). A derivation is discussed in the article [Relations between specific heats](https://en.wikipedia.org/wiki/Relations_between_specific_heats "Relations between specific heats"). For an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), if ρ {\\displaystyle \\rho } ![{\\displaystyle \\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64) is expressed as [molar](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") density in the above equation, this equation reduces simply to [Mayer](https://en.wikipedia.org/wiki/Julius_Robert_von_Mayer "Julius Robert von Mayer")'s relation, C p , m − C v , m \= R {\\displaystyle C\_{p,m}-C\_{v,m}=R\\!} ![{\\displaystyle C\_{p,m}-C\_{v,m}=R\\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f23e5c5b22b24ffd7f1051a3e396784a902d867e) where C p , m {\\displaystyle C\_{p,m}} ![{\\displaystyle C\_{p,m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccf6b7671259c85a73e29bc870010db9a7d2288) and C v , m {\\displaystyle C\_{v,m}} ![{\\displaystyle C\_{v,m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc3350be2e3cdd42c3a445e5dae16ac2aa4a675) are [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively. ### Specific heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=16 "Edit section: Specific heat capacity")\] The specific heat capacity of a material on a per mass basis is c \= ∂ C ∂ m , {\\displaystyle c={\\partial C \\over \\partial m},} ![{\\displaystyle c={\\partial C \\over \\partial m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/338148bac53d596abe5112d0891626997320114b) which in the absence of phase transitions is equivalent to c \= E m \= C m \= C ρ V , {\\displaystyle c=E\_{m}={C \\over m}={C \\over {\\rho V}},} ![{\\displaystyle c=E\_{m}={C \\over m}={C \\over {\\rho V}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1a8ab2bd40064b132ddd125dd7e96420c0337d) where - C {\\displaystyle C} ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question, - m {\\displaystyle m} ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is the mass of the body, - V {\\displaystyle V} ![{\\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the body, and - ρ \= m V {\\displaystyle \\rho ={\\frac {m}{V}}} ![{\\displaystyle \\rho ={\\frac {m}{V}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63465553e3f944d6ef79f90f992a02cf29c7f38) is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https://en.wikipedia.org/wiki/Isobaric_process "Isobaric process") (constant pressure, d p \= 0 {\\displaystyle dp=0} ![{\\displaystyle dp=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850e5511af66c84a989bf0e5d003852cb862778c)) or [isochoric](https://en.wikipedia.org/wiki/Isochoric_process "Isochoric process") (constant volume, d V \= 0 {\\displaystyle dV=0} ![{\\displaystyle dV=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d505a8f349881feb179d02752b33e5ab95e035)) processes. The corresponding specific heat capacities are expressed as c p \= ( ∂ C ∂ m ) p , c V \= ( ∂ C ∂ m ) V . {\\displaystyle {\\begin{aligned}c\_{p}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{p},\\\\c\_{V}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{V}.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}c\_{p}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{p},\\\\c\_{V}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{V}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4576769aa96d12c2e062fe6e6a6ef4acfdc8b63b) A related parameter to c {\\displaystyle c} ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is C V − 1 {\\displaystyle CV^{-1}} ![{\\displaystyle CV^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0155c68921e700c1325f32c73061273b2178796e), the [volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity"). In engineering practice, c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript m {\\displaystyle m} ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as c m {\\displaystyle c\_{m}} ![{\\displaystyle c\_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes c m \= C m \= c V ρ . {\\displaystyle c\_{m}={\\frac {C}{m}}={\\frac {c\_{V}}{\\rho }}.} ![{\\displaystyle c\_{m}={\\frac {C}{m}}={\\frac {c\_{V}}{\\rho }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/551b63520ea921f8105a75ddf4ddb073eca1e9db) For pure homogeneous [chemical compounds](https://en.wikipedia.org/wiki/Chemical_compound "Chemical compound") with established [molecular or molar mass](https://en.wikipedia.org/wiki/Molecular_mass "Molecular mass") or a [molar quantity](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") is established, heat capacity as an [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") can be expressed on a per [mole](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") basis instead of a per mass basis by the following equations analogous to the per mass equations: C p , m \= ( ∂ C ∂ n ) p \= molar heat capacity at constant pressure C V , m \= ( ∂ C ∂ n ) V \= molar heat capacity at constant volume {\\displaystyle {\\begin{alignedat}{3}C\_{p,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{p}&={\\text{molar heat capacity at constant pressure}}\\\\C\_{V,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{V}&={\\text{molar heat capacity at constant volume}}\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{3}C\_{p,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{p}&={\\text{molar heat capacity at constant pressure}}\\\\C\_{V,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{V}&={\\text{molar heat capacity at constant volume}}\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d63e3ec44e731e9eb128dcc30eaa9c87af059a) where *n* = number of moles in the body or [thermodynamic system](https://en.wikipedia.org/wiki/Thermodynamic_system "Thermodynamic system"). One may refer to such a *per mole* quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis. ### Polytropic heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=17 "Edit section: Polytropic heat capacity")\] The [polytropic](https://en.wikipedia.org/wiki/Polytropic "Polytropic") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change C i , m \= ( ∂ C ∂ n ) \= molar heat capacity at polytropic process {\\displaystyle C\_{i,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)={\\text{molar heat capacity at polytropic process}}} ![{\\displaystyle C\_{i,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)={\\text{molar heat capacity at polytropic process}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02f685610e35e01aa42e293380bdf94469ea8364) The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between **1** and the adiabatic exponent (*γ* or *κ*) ### Dimensionless heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=18 "Edit section: Dimensionless heat capacity")\] The [dimensionless](https://en.wikipedia.org/wiki/Dimensionless_number "Dimensionless number") heat capacity of a material is C ∗ \= C n R \= C N k B {\\displaystyle C^{\*}={\\frac {C}{nR}}={\\frac {C}{Nk\_{\\text{B}}}}} ![{\\displaystyle C^{\*}={\\frac {C}{nR}}={\\frac {C}{Nk\_{\\text{B}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1beaede671f02ee56b81859fe98eed0feb26cf5) where - *C* is the heat capacity of a body made of the material in question (J/K) - *n* is the [amount of substance](https://en.wikipedia.org/wiki/Amount_of_substance "Amount of substance") in the body ([mol](https://en.wikipedia.org/wiki/Mole_\(unit\) "Mole (unit)")) - *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant") (J⋅K−1⋅mol−1) - *N* is the number of molecules in the body. (dimensionless) - *k*B is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") (J⋅K−1) Again, [SI](https://en.wikipedia.org/wiki/SI "SI") units shown for example. In the [Ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") article, dimensionless heat capacity C ∗ {\\displaystyle C^{\*}} ![{\\displaystyle C^{\*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as c ^ {\\displaystyle {\\hat {c}}} ![{\\displaystyle {\\hat {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8417e85ae7f4eaee7df31347ce488f85c8884b93). ### Heat capacity at absolute zero \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=19 "Edit section: Heat capacity at absolute zero")\] From the definition of [entropy](https://en.wikipedia.org/wiki/Entropy#Thermodynamic_definition_of_entropy "Entropy") T d S \= δ Q {\\displaystyle TdS=\\delta Q} ![{\\displaystyle TdS=\\delta Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ddb574969210251fd1e6d8847eb303ece6b6a7) the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature *Tf* S ( T f ) \= ∫ T \= 0 T f δ Q T \= ∫ 0 T f δ Q d T d T T \= ∫ 0 T f C ( T ) d T T . {\\displaystyle S(T\_{f})=\\int \_{T=0}^{T\_{f}}{\\frac {\\delta Q}{T}}=\\int \_{0}^{T\_{f}}{\\frac {\\delta Q}{dT}}{\\frac {dT}{T}}=\\int \_{0}^{T\_{f}}C(T)\\,{\\frac {dT}{T}}.} ![{\\displaystyle S(T\_{f})=\\int \_{T=0}^{T\_{f}}{\\frac {\\delta Q}{T}}=\\int \_{0}^{T\_{f}}{\\frac {\\delta Q}{dT}}{\\frac {dT}{T}}=\\int \_{0}^{T\_{f}}C(T)\\,{\\frac {dT}{T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e81986ce6a91887325cc998b1245250f83a65a) The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the [third law of thermodynamics](https://en.wikipedia.org/wiki/Third_law_of_thermodynamics "Third law of thermodynamics"). One of the strengths of the [Debye model](https://en.wikipedia.org/wiki/Debye_model "Debye model") is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached. ### Solid phase \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=20 "Edit section: Solid phase")\] The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3*R*, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas. The Dulong–Petit limit results from the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem"), and as such is only valid in the classical limit of a [microstate continuum](https://en.wikipedia.org/w/index.php?title=Microstate_continuum&action=edit&redlink=1 "Microstate continuum (page does not exist)"), which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at [standard ambient temperature](https://en.wikipedia.org/wiki/Standard_ambient_temperature_and_pressure "Standard ambient temperature and pressure"), quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3*R* per mole of *atoms* in the solid, although in molecular solids, heat capacities calculated *per mole of molecules* in molecular solids may be more than 3*R*. For example, the heat capacity of water ice at the melting point is about 4.6*R* per mole of molecules, but only 1.5*R* per mole of atoms. The lower than 3*R* number "per atom" (as is the case with diamond and beryllium) results from the "freezing out" of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3*R* per mole of atoms of the Dulong–Petit theoretical maximum. For a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of [phonons](https://en.wikipedia.org/wiki/Phonons "Phonons"). See [Debye model](https://en.wikipedia.org/wiki/Debye_model "Debye model"). ### Theoretical estimation \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=21 "Edit section: Theoretical estimation")\] The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. - Water (liquid): CP = 4185.5 J⋅K−1⋅kg−1 (15 °C, 101.325 kPa) - Water (liquid): CVH = 74.539 J⋅K−1⋅mol−1 (25 °C) For liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However, different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr).[\[notes 1\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-gold-30) ### Relation between heat capacities \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=22 "Edit section: Relation between heat capacities")\] Main article: [Relations between heat capacities](https://en.wikipedia.org/wiki/Relations_between_heat_capacities "Relations between heat capacities") Measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see [coefficient of thermal expansion](https://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion "Coefficient of thermal expansion") and [compressibility](https://en.wikipedia.org/wiki/Compressibility "Compressibility")). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. The [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio"), or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. #### Ideal gas \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=23 "Edit section: Ideal gas")\] For an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), evaluating the partial derivatives above according to the [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state"), where *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant"), for an ideal gas[\[30\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-31) P V \= n R T , C P − C V \= T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n , P \= n R T V ⇒ ( ∂ P ∂ T ) V , n \= n R V , V \= n R T P ⇒ ( ∂ V ∂ T ) P , n \= n R P . {\\displaystyle {\\begin{alignedat}{3}PV&=nRT,&\\\\C\_{P}-C\_{V}&=T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n},&\\\\P&={\\frac {nRT}{V}}\\Rightarrow \\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}&={\\frac {nR}{V}},\\\\V&={\\frac {nRT}{P}}\\Rightarrow \\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}&={\\frac {nR}{P}}.\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{3}PV&=nRT,&\\\\C\_{P}-C\_{V}&=T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n},&\\\\P&={\\frac {nRT}{V}}\\Rightarrow \\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}&={\\frac {nR}{V}},\\\\V&={\\frac {nRT}{P}}\\Rightarrow \\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}&={\\frac {nR}{P}}.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe220db0c71a99f43c20c80b90b430001c8aff8) Substituting T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n \= T n R V n R P \= n R T V n R P \= P n R P \= n R , {\\displaystyle T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}=T{\\frac {nR}{V}}{\\frac {nR}{P}}={\\frac {nRT}{V}}{\\frac {nR}{P}}=P{\\frac {nR}{P}}=nR,} ![{\\displaystyle T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}=T{\\frac {nR}{V}}{\\frac {nR}{P}}={\\frac {nRT}{V}}{\\frac {nR}{P}}=P{\\frac {nR}{P}}=nR,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49d3f01b48a4771f06476e4922cd418e91e9354) this equation reduces simply to [Mayer](https://en.wikipedia.org/wiki/Julius_Robert_von_Mayer "Julius Robert von Mayer")'s relation: C P , m − C V , m \= R . {\\displaystyle C\_{P,m}-C\_{V,m}=R.} ![{\\displaystyle C\_{P,m}-C\_{V,m}=R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0951113a93ae7d847ed0234a94039782ae0a5e) The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. ### Specific heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=24 "Edit section: Specific heat capacity")\] The specific heat capacity of a material on a per mass basis is c \= ∂ C ∂ m , {\\displaystyle c={\\frac {\\partial C}{\\partial m}},} ![{\\displaystyle c={\\frac {\\partial C}{\\partial m}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6d4d3222fc01879cf5ebdcdd167c0f01338989) which in the absence of phase transitions is equivalent to c \= E m \= C m \= C ρ V , {\\displaystyle c=E\_{m}={\\frac {C}{m}}={\\frac {C}{\\rho V}},} ![{\\displaystyle c=E\_{m}={\\frac {C}{m}}={\\frac {C}{\\rho V}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5685f216c3eaa301626d07a8dbeda4ec825d2a) where - C {\\displaystyle C} ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question, - m {\\displaystyle m} ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is the mass of the body, - V {\\displaystyle V} ![{\\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the body, - ρ \= m V {\\displaystyle \\rho ={\\frac {m}{V}}} ![{\\displaystyle \\rho ={\\frac {m}{V}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63465553e3f944d6ef79f90f992a02cf29c7f38) is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https://en.wikipedia.org/wiki/Isobaric_process "Isobaric process") (constant pressure, d P \= 0 {\\displaystyle {\\text{d}}P=0} ![{\\displaystyle {\\text{d}}P=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2997415b7c242bcf6f4ebf7ef17957d9df03a9aa)) or [isochoric](https://en.wikipedia.org/wiki/Isochoric_process "Isochoric process") (constant volume, d V \= 0 {\\displaystyle {\\text{d}}V=0} ![{\\displaystyle {\\text{d}}V=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e19ded03242b58e141dced9f6f46c3a8829e436)) processes. The corresponding specific heat capacities are expressed as c P \= ( ∂ C ∂ m ) P , c V \= ( ∂ C ∂ m ) V . {\\displaystyle {\\begin{aligned}c\_{P}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{P},\\\\c\_{V}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{V}.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}c\_{P}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{P},\\\\c\_{V}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{V}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2cc9fc1a0307e6e726521189cf7e22158357b9) From the results of the previous section, dividing through by the mass gives the relation c P − c V \= α 2 T ρ β T . {\\displaystyle c\_{P}-c\_{V}={\\frac {\\alpha ^{2}T}{\\rho \\beta \_{T}}}.} ![{\\displaystyle c\_{P}-c\_{V}={\\frac {\\alpha ^{2}T}{\\rho \\beta \_{T}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058bcaccd4cbc0e8d16892f68bc68bf1e0062d7e) A related parameter to c {\\displaystyle c} ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is C / V {\\displaystyle C/V} ![{\\displaystyle C/V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f43bc2f531ba84544e7e71ccab205ab4ad1fe5c), the [volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity"). In engineering practice, c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript m {\\displaystyle m} ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as c m {\\displaystyle c\_{m}} ![{\\displaystyle c\_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes c m \= C m \= c volumetric ρ . {\\displaystyle c\_{m}={\\frac {C}{m}}={\\frac {c\_{\\text{volumetric}}}{\\rho }}.} ![{\\displaystyle c\_{m}={\\frac {C}{m}}={\\frac {c\_{\\text{volumetric}}}{\\rho }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56b0a8c8ebcbe4ee889c7d963cb980a5d34f60b) For pure [homogeneous](https://en.wikipedia.org/wiki/Homogeneous "Homogeneous") [chemical compounds](https://en.wikipedia.org/wiki/Chemical_compound "Chemical compound") with established [molecular or molar mass](https://en.wikipedia.org/wiki/Molecular_mass "Molecular mass"), or a [molar quantity](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)"), heat capacity as an [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") can be expressed on a per-[mole](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") basis instead of a per-mass basis by the following equations analogous to the per mass equations: C P , m \= ( ∂ C ∂ n ) P \= molar heat capacity at constant pressure, C V , m \= ( ∂ C ∂ n ) V \= molar heat capacity at constant volume, {\\displaystyle {\\begin{alignedat}{3}C\_{P,m}&=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{P}&={\\text{molar heat capacity at constant pressure,}}\\\\C\_{V,m}&=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{V}&={\\text{molar heat capacity at constant volume,}}\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{3}C\_{P,m}&=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{P}&={\\text{molar heat capacity at constant pressure,}}\\\\C\_{V,m}&=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{V}&={\\text{molar heat capacity at constant volume,}}\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a371ac38796d12f8beb3f30b4e16181a5e1981ba) where *n* is the number of moles in the body or [thermodynamic system](https://en.wikipedia.org/wiki/Thermodynamic_system "Thermodynamic system"). One may refer to such a per-mole quantity as **molar heat capacity** to distinguish it from specific heat capacity on a per-mass basis. ### Polytropic heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=25 "Edit section: Polytropic heat capacity")\] The [polytropic](https://en.wikipedia.org/wiki/Polytropic "Polytropic") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change: C i , m \= ( ∂ C ∂ n ) \= molar heat capacity at polytropic process. {\\displaystyle C\_{i,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)={\\text{molar heat capacity at polytropic process.}}} ![{\\displaystyle C\_{i,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)={\\text{molar heat capacity at polytropic process.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d94a2aa8fde269f13f73425bd4eb2e375717ad28) The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (*γ* or *κ*). ### Dimensionless heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=26 "Edit section: Dimensionless heat capacity")\] The [dimensionless](https://en.wikipedia.org/wiki/Dimensionless_number "Dimensionless number") heat capacity of a material is C ∗ \= C n R \= C N k B , {\\displaystyle C^{\*}={\\frac {C}{nR}}={\\frac {C}{Nk\_{\\text{B}}}},} ![{\\displaystyle C^{\*}={\\frac {C}{nR}}={\\frac {C}{Nk\_{\\text{B}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/508315bfb919ceff9bf382f397926bccaa1168dc) where - C {\\displaystyle C} ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question (J/K), - *n* is the [amount of substance](https://en.wikipedia.org/wiki/Amount_of_substance "Amount of substance") in the body ([mol](https://en.wikipedia.org/wiki/Mole_\(unit\) "Mole (unit)")), - *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant") (J/(K⋅mol)), - *N* is the number of molecules in the body (dimensionless), - *k*B is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") (J/(K⋅molecule)). In the [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") article, dimensionless heat capacity C ∗ {\\displaystyle C^{\*}} ![{\\displaystyle C^{\*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as c ^ {\\displaystyle {\\hat {c}}} ![{\\displaystyle {\\hat {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8417e85ae7f4eaee7df31347ce488f85c8884b93) and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem"). More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the [dimensionless entropy](https://en.wikipedia.org/wiki/Dimensionless_entropy "Dimensionless entropy") per particle S ∗ \= S / N k B {\\displaystyle S^{\*}=S/Nk\_{\\text{B}}} ![{\\displaystyle S^{\*}=S/Nk\_{\\text{B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f612e45f84ed9e293f85f75c6dae30943bca0b), measured in [nats](https://en.wikipedia.org/wiki/Nat_\(unit\) "Nat (unit)"). C ∗ \= d S ∗ d ( ln ⁡ T ) . {\\displaystyle C^{\*}={\\frac {{\\text{d}}S^{\*}}{{\\text{d}}(\\ln T)}}.} ![{\\displaystyle C^{\*}={\\frac {{\\text{d}}S^{\*}}{{\\text{d}}(\\ln T)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/294a7378c60eaed01bcea2cb12c53dd0b5bf7f90) Alternatively, using base-2 logarithms, C ∗ {\\displaystyle C^{\*}} ![{\\displaystyle C^{\*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in [bits](https://en.wikipedia.org/wiki/Bit "Bit").[\[31\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-32) ### Heat capacity at absolute zero \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=27 "Edit section: Heat capacity at absolute zero")\] From the definition of [entropy](https://en.wikipedia.org/wiki/Entropy#Thermodynamic_definition_of_entropy "Entropy") T d S \= δ Q , {\\displaystyle T\\,{\\text{d}}S=\\delta Q,} ![{\\displaystyle T\\,{\\text{d}}S=\\delta Q,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cca7738465956fa4c1d790333af389a15b667c7) the absolute entropy can be calculated by integrating from zero to the final temperature *T*f: S ( T f ) \= ∫ T \= 0 T f δ Q T \= ∫ 0 T f δ Q d T d T T \= ∫ 0 T f C ( T ) d T T . {\\displaystyle S(T\_{\\text{f}})=\\int \_{T=0}^{T\_{\\text{f}}}{\\frac {\\delta Q}{T}}=\\int \_{0}^{T\_{\\text{f}}}{\\frac {\\delta Q}{{\\text{d}}T}}{\\frac {{\\text{d}}T}{T}}=\\int \_{0}^{T\_{\\text{f}}}C(T)\\,{\\frac {{\\text{d}}T}{T}}.} ![{\\displaystyle S(T\_{\\text{f}})=\\int \_{T=0}^{T\_{\\text{f}}}{\\frac {\\delta Q}{T}}=\\int \_{0}^{T\_{\\text{f}}}{\\frac {\\delta Q}{{\\text{d}}T}}{\\frac {{\\text{d}}T}{T}}=\\int \_{0}^{T\_{\\text{f}}}C(T)\\,{\\frac {{\\text{d}}T}{T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df93f18f951064e02f6128a3cd99ad4c72124aa1) ## Thermodynamic derivation \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=28 "Edit section: Thermodynamic derivation")\] In theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state") and an [internal energy function](https://en.wikipedia.org/w/index.php?title=Internal_energy_function&action=edit&redlink=1 "Internal energy function (page does not exist)"). ### State of matter in a homogeneous sample \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=29 "Edit section: State of matter in a homogeneous sample")\] To apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass M {\\displaystyle M} ![{\\displaystyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd). Assume that the evolution of the system is always slow enough for the internal pressure P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a) and temperature T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) be considered uniform throughout. The pressure P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a) would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air. The state of the material can then be specified by three parameters: its temperature T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), the pressure P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), and its [specific volume](https://en.wikipedia.org/wiki/Specific_volume "Specific volume") ν \= V / M {\\displaystyle \\nu =V/M} ![{\\displaystyle \\nu =V/M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e892352c30149ba6c93abe17145e15cdd73f5), where V {\\displaystyle V} ![{\\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the sample. (This quantity is the reciprocal 1 / ρ {\\displaystyle 1/\\rho } ![{\\displaystyle 1/\\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/754f9b580e728c03bac621d1dd72cd606b6b5eef) of the material's [density](https://en.wikipedia.org/wiki/Density "Density") ρ \= M / V {\\displaystyle \\rho =M/V} ![{\\displaystyle \\rho =M/V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed457a6aafb262f754ae8801d87eb036b8c5fe9).) Like T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) and P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), the specific volume ν {\\displaystyle \\nu } ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) is an intensive property of the material and its state, that does not depend on the amount of substance in the sample. Those variables are not independent. The allowed states are defined by an [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state") relating those three variables: F ( T , P , ν ) \= 0\. {\\displaystyle F(T,P,\\nu )=0.} ![{\\displaystyle F(T,P,\\nu )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/471651d7fefe7e18735444a61e0ffa3f6ff84876) The function F {\\displaystyle F} ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) depends on the material under consideration. The [specific internal energy](https://en.wikipedia.org/wiki/Specific_internal_energy "Specific internal energy") stored internally in the sample, per unit of mass, will then be another function U ( T , P , ν ) {\\displaystyle U(T,P,\\nu )} ![{\\displaystyle U(T,P,\\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7003637ad40a88b0bf499d18811d1dab7f424ccf) of these state variables, that is also specific of the material. The total internal energy in the sample then will be M U ( T , P , ν ) {\\displaystyle M\\,U(T,P,\\nu )} ![{\\displaystyle M\\,U(T,P,\\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/287870c17350c2311e2d86e64ec9dec7bfc90a75). For some simple materials, like an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), one can derive from basic theory the equation of state F \= 0 {\\displaystyle F=0} ![{\\displaystyle F=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745afacbd4fd9affdc51ac09a0ecabae08da8676) and even the specific internal energy U {\\displaystyle U} ![{\\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025) In general, these functions must be determined experimentally for each substance. ### Conservation of energy \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=30 "Edit section: Conservation of energy")\] The absolute value of this quantity U {\\displaystyle U} ![{\\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025) is undefined, and (for the purposes of thermodynamics) the state of "zero internal energy" can be chosen arbitrarily. However, by the [law of conservation of energy](https://en.wikipedia.org/wiki/Law_of_conservation_of_energy "Law of conservation of energy"), any infinitesimal increase M d U {\\displaystyle M\\,\\mathrm {d} U} ![{\\displaystyle M\\,\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcddca6d7c51038a1a0134cde944e55e5bf81551) in the total internal energy M U {\\displaystyle MU} ![{\\displaystyle MU}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0430adceb61aadc25575b6894c1633b41f8874) must be matched by the net flow of heat energy d Q {\\displaystyle \\mathrm {d} Q} ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739) into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is − P d V {\\displaystyle -P\\,\\mathrm {d} V} ![{\\displaystyle -P\\,\\mathrm {d} V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7238b4c977539175fa28c0cff1658e50276637a), where d V {\\displaystyle \\mathrm {d} V} ![{\\displaystyle \\mathrm {d} V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80507190aa9d38a279909db47b63657f2b62ba7) is the change in the sample's volume in that infinitesimal step.[\[32\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-fein-33) Therefore d Q − P d V \= M d U {\\displaystyle \\mathrm {d} Q-P\\,\\mathrm {d} V=M\\,\\mathrm {d} U} ![{\\displaystyle \\mathrm {d} Q-P\\,\\mathrm {d} V=M\\,\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3401e201975bf505b5fa40018ffd2bc7eab0a6) hence d Q M − P d ν \= d U {\\displaystyle {\\frac {\\mathrm {d} Q}{M}}-P\\,\\mathrm {d} \\nu =\\mathrm {d} U} ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M}}-P\\,\\mathrm {d} \\nu =\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/586d361903b89f3b074724dafc3680ca2cf85d26) If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount d Q {\\displaystyle \\mathrm {d} Q} ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739), then the term P d ν {\\displaystyle P\\,\\mathrm {d} \\nu } ![{\\displaystyle P\\,\\mathrm {d} \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e54803595e3392edc2125d5bbbfd6ee753922208) is zero (no mechanical work is done). Then, dividing by d T {\\displaystyle \\mathrm {d} T} ![{\\displaystyle \\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80), d Q M d T \= d U d T {\\displaystyle {\\frac {\\mathrm {d} Q}{M\\,\\mathrm {d} T}}={\\frac {\\mathrm {d} U}{\\mathrm {d} T}}} ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M\\,\\mathrm {d} T}}={\\frac {\\mathrm {d} U}{\\mathrm {d} T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b32da0380fcf6d0c3bc87b7dbb0c7c0c02211f80) where d T {\\displaystyle \\mathrm {d} T} ![{\\displaystyle \\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80) is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume c V {\\displaystyle c\_{V}} ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) of the material. For the heat capacity at constant pressure, it is useful to define the [specific enthalpy](https://en.wikipedia.org/wiki/Specific_enthalpy "Specific enthalpy") of the system as the sum h ( T , P , ν ) \= U ( T , P , ν ) \+ P ν {\\displaystyle h(T,P,\\nu )=U(T,P,\\nu )+P\\nu } ![{\\displaystyle h(T,P,\\nu )=U(T,P,\\nu )+P\\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a0269a9025439e15bc408001d2ddd739e304a9). An infinitesimal change in the specific enthalpy will then be d h \= d U \+ V d P \+ P d V {\\displaystyle \\mathrm {d} h=\\mathrm {d} U+V\\,\\mathrm {d} P+P\\,\\mathrm {d} V} ![{\\displaystyle \\mathrm {d} h=\\mathrm {d} U+V\\,\\mathrm {d} P+P\\,\\mathrm {d} V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/746c99ad25909e68e8e56ca8a0b6b9bcce84317c) therefore d Q M \+ V d P \= d h {\\displaystyle {\\frac {\\mathrm {d} Q}{M}}+V\\,\\mathrm {d} P=\\mathrm {d} h} ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M}}+V\\,\\mathrm {d} P=\\mathrm {d} h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e31d5cd51830d6b5eb83bb7c77657c5be843c0) If the pressure is kept constant, the second term on the left-hand side is zero, and d Q M d T \= d h d T {\\displaystyle {\\frac {\\mathrm {d} Q}{M\\,\\mathrm {d} T}}={\\frac {\\mathrm {d} h}{\\mathrm {d} T}}} ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M\\,\\mathrm {d} T}}={\\frac {\\mathrm {d} h}{\\mathrm {d} T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4007c22be9c0d1e82c5a4a011c3bd298de9f44) The left-hand side is the specific heat capacity at constant pressure c P {\\displaystyle c\_{P}} ![{\\displaystyle c\_{P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f21e3d892369c6502159212f89643451ab67e4) of the material. ### Connection to equation of state \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=31 "Edit section: Connection to equation of state")\] In general, the infinitesimal quantities d T , d P , d V , d U {\\displaystyle \\mathrm {d} T,\\mathrm {d} P,\\mathrm {d} V,\\mathrm {d} U} ![{\\displaystyle \\mathrm {d} T,\\mathrm {d} P,\\mathrm {d} V,\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ccfe4aed236cb970d27013b7e0fe2e3cbf7664) are constrained by the equation of state and the specific internal energy function. Namely, { d T ∂ F ∂ T ( T , P , V ) \+ d P ∂ F ∂ P ( T , P , V ) \+ d V ∂ F ∂ V ( T , P , V ) \= 0 d T ∂ U ∂ T ( T , P , V ) \+ d P ∂ U ∂ P ( T , P , V ) \+ d V ∂ U ∂ V ( T , P , V ) \= d U {\\displaystyle {\\begin{cases}\\displaystyle \\mathrm {d} T{\\frac {\\partial F}{\\partial T}}(T,P,V)+\\mathrm {d} P{\\frac {\\partial F}{\\partial P}}(T,P,V)+\\mathrm {d} V{\\frac {\\partial F}{\\partial V}}(T,P,V)&=&0\\\\\[2ex\]\\displaystyle \\mathrm {d} T{\\frac {\\partial U}{\\partial T}}(T,P,V)+\\mathrm {d} P{\\frac {\\partial U}{\\partial P}}(T,P,V)+\\mathrm {d} V{\\frac {\\partial U}{\\partial V}}(T,P,V)&=&\\mathrm {d} U\\end{cases}}} ![{\\displaystyle {\\begin{cases}\\displaystyle \\mathrm {d} T{\\frac {\\partial F}{\\partial T}}(T,P,V)+\\mathrm {d} P{\\frac {\\partial F}{\\partial P}}(T,P,V)+\\mathrm {d} V{\\frac {\\partial F}{\\partial V}}(T,P,V)&=&0\\\\\[2ex\]\\displaystyle \\mathrm {d} T{\\frac {\\partial U}{\\partial T}}(T,P,V)+\\mathrm {d} P{\\frac {\\partial U}{\\partial P}}(T,P,V)+\\mathrm {d} V{\\frac {\\partial U}{\\partial V}}(T,P,V)&=&\\mathrm {d} U\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/957c9a178ff753bc04a30bed2819d7e5155314a0) Here ( ∂ F / ∂ T ) ( T , P , V ) {\\displaystyle (\\partial F/\\partial T)(T,P,V)} ![{\\displaystyle (\\partial F/\\partial T)(T,P,V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24993db8c358df7a5c8539501cb70320459370e6) denotes the (partial) derivative of the state equation F {\\displaystyle F} ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) with respect to its T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) argument, keeping the other two arguments fixed, evaluated at the state ( T , P , V ) {\\displaystyle (T,P,V)} ![{\\displaystyle (T,P,V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41f5cdee6675ea3447082abaa34afbaf77e4a1f) in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space. This analysis also holds no matter how the energy increment d Q {\\displaystyle \\mathrm {d} Q} ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739) is injected into the sample, namely by [heat conduction](https://en.wikipedia.org/wiki/Heat_conduction "Heat conduction"), irradiation, [electromagnetic induction](https://en.wikipedia.org/wiki/Electromagnetic_induction "Electromagnetic induction"), [radioactive decay](https://en.wikipedia.org/wiki/Radioactive_decay "Radioactive decay"), etc. ### Relation between heat capacities \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=32 "Edit section: Relation between heat capacities")\] For any specific volume ν {\\displaystyle \\nu } ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468), denote p ν ( T ) {\\displaystyle p\_{\\nu }(T)} ![{\\displaystyle p\_{\\nu }(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fab6e24e625d12de49f61c60a3e95d19d5f305d) the function that describes how the pressure varies with the temperature T {\\displaystyle T} ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at ν {\\displaystyle \\nu } ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468). Analogously, for any pressure P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), let ν P ( T ) {\\displaystyle \\nu \_{P}(T)} ![{\\displaystyle \\nu \_{P}(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d920635881776d5b03f98ee5492f6e0549b3d2f7) be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a). Namely, those functions are such that F ( T , p ν ( T ) , ν ) \= 0 {\\displaystyle F(T,p\_{\\nu }(T),\\nu )=0} ![{\\displaystyle F(T,p\_{\\nu }(T),\\nu )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84a1b582c06f3c19e7abe1e3728d16e072c1a51f)andF ( T , P , ν P ( T ) ) \= 0 {\\displaystyle F(T,P,\\nu \_{P}(T))=0} ![{\\displaystyle F(T,P,\\nu \_{P}(T))=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b667b5d0711e09296d3fe4e9d3f6c395acd0f847) for any values of T , P , ν {\\displaystyle T,P,\\nu } ![{\\displaystyle T,P,\\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f907ebc4eacdaceb323d1264f58855f033d41081). In other words, the graphs of p ν ( T ) {\\displaystyle p\_{\\nu }(T)} ![{\\displaystyle p\_{\\nu }(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fab6e24e625d12de49f61c60a3e95d19d5f305d) and ν P ( T ) {\\displaystyle \\nu \_{P}(T)} ![{\\displaystyle \\nu \_{P}(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d920635881776d5b03f98ee5492f6e0549b3d2f7) are slices of the surface defined by the state equation, cut by planes of constant ν {\\displaystyle \\nu } ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) and constant P {\\displaystyle P} ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), respectively. Then, from the [fundamental thermodynamic relation](https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation "Fundamental thermodynamic relation") it follows that c P ( T , P , ν ) − c V ( T , P , ν ) \= T \[ d p ν d T ( T ) \] \[ d ν P d T ( T ) \] {\\displaystyle c\_{P}(T,P,\\nu )-c\_{V}(T,P,\\nu )=T\\left\[{\\frac {\\mathrm {d} p\_{\\nu }}{\\mathrm {d} T}}(T)\\right\]\\left\[{\\frac {\\mathrm {d} \\nu \_{P}}{\\mathrm {d} T}}(T)\\right\]} ![{\\displaystyle c\_{P}(T,P,\\nu )-c\_{V}(T,P,\\nu )=T\\left\[{\\frac {\\mathrm {d} p\_{\\nu }}{\\mathrm {d} T}}(T)\\right\]\\left\[{\\frac {\\mathrm {d} \\nu \_{P}}{\\mathrm {d} T}}(T)\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24554d2ae7d16cbb75b6ad485bf92856b55cf7bd) This equation can be rewritten as c P ( T , P , ν ) − c V ( T , P , ν ) \= ν T α 2 β T , {\\displaystyle c\_{P}(T,P,\\nu )-c\_{V}(T,P,\\nu )=\\nu T{\\frac {\\alpha ^{2}}{\\beta \_{T}}},} ![{\\displaystyle c\_{P}(T,P,\\nu )-c\_{V}(T,P,\\nu )=\\nu T{\\frac {\\alpha ^{2}}{\\beta \_{T}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27c864cf90d771a66e3a2fa50dec56b8d6c190ad) where - α {\\displaystyle \\alpha } ![{\\displaystyle \\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3) is the [coefficient of thermal expansion](https://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion "Coefficient of thermal expansion"), - β T {\\displaystyle \\beta \_{T}} ![{\\displaystyle \\beta \_{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2855fc59047bc00f5bdedbcad68550a7ac6d13) is the [isothermal](https://en.wikipedia.org/wiki/Isothermal "Isothermal") [compressibility](https://en.wikipedia.org/wiki/Compressibility "Compressibility"), both depending on the state ( T , P , ν ) {\\displaystyle (T,P,\\nu )} ![{\\displaystyle (T,P,\\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/847d2a615fb19d33916ce93a4fa935919abca4a7). The [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio"), or adiabatic index, is the ratio c P / c V {\\displaystyle c\_{P}/c\_{V}} ![{\\displaystyle c\_{P}/c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfcc80ce47374e8dfc55b32849c78810e782a87) of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. ### Calculation from first principles \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=33 "Edit section: Calculation from first principles")\] The [path integral Monte Carlo](https://en.wikipedia.org/wiki/Path_integral_Monte_Carlo "Path integral Monte Carlo") method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3*R* = 24.94 joules per kelvin per mole of atoms ([Dulong–Petit law](https://en.wikipedia.org/wiki/Dulong%E2%80%93Petit_law "Dulong–Petit law"), *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant")). Low temperature approximations for both gases and solids at temperatures less than their characteristic [Einstein temperatures](https://en.wikipedia.org/wiki/Einstein_temperature "Einstein temperature") or [Debye temperatures](https://en.wikipedia.org/wiki/Debye_temperature "Debye temperature") can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material.[\[33\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-Benjelloun-34) #### Ideal gas \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=34 "Edit section: Ideal gas")\] For an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), evaluating the partial derivatives above according to the [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state"), where *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant"), for an ideal gas[\[34\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-35) P V \= n R T , C P − C V \= T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n , P \= n R T V ⇒ ( ∂ P ∂ T ) V , n \= n R V , V \= n R T P ⇒ ( ∂ V ∂ T ) P , n \= n R P . {\\displaystyle {\\begin{alignedat}{3}PV&=nRT,\\\\C\_{P}-C\_{V}&=T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n},\\\\P&={\\frac {nRT}{V}}\\Rightarrow \\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}&={\\frac {nR}{V}},\\\\V&={\\frac {nRT}{P}}\\Rightarrow \\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}&={\\frac {nR}{P}}.\\end{alignedat}}} ![{\\displaystyle {\\begin{alignedat}{3}PV&=nRT,\\\\C\_{P}-C\_{V}&=T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n},\\\\P&={\\frac {nRT}{V}}\\Rightarrow \\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}&={\\frac {nR}{V}},\\\\V&={\\frac {nRT}{P}}\\Rightarrow \\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}&={\\frac {nR}{P}}.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f8e4a06c2339161629da3ad59eddd0383a651f) Substituting T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n \= T n R V n R P \= n R T V n R P \= P n R P \= n R , {\\displaystyle T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}=T{\\frac {nR}{V}}{\\frac {nR}{P}}={\\frac {nRT}{V}}{\\frac {nR}{P}}=P{\\frac {nR}{P}}=nR,} ![{\\displaystyle T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}=T{\\frac {nR}{V}}{\\frac {nR}{P}}={\\frac {nRT}{V}}{\\frac {nR}{P}}=P{\\frac {nR}{P}}=nR,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49d3f01b48a4771f06476e4922cd418e91e9354) this equation reduces simply to [Mayer](https://en.wikipedia.org/wiki/Julius_Robert_von_Mayer "Julius Robert von Mayer")'s relation: C P , m − C V , m \= R . {\\displaystyle C\_{P,m}-C\_{V,m}=R.} ![{\\displaystyle C\_{P,m}-C\_{V,m}=R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0951113a93ae7d847ed0234a94039782ae0a5e) The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. ## See also \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=35 "Edit section: See also")\] [![icon](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png)](https://en.wikipedia.org/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg) [Physics portal](https://en.wikipedia.org/wiki/Portal:Physics "Portal:Physics") - [Enthalpy of fusion](https://en.wikipedia.org/wiki/Enthalpy_of_fusion "Enthalpy of fusion") (latent heat of melting) - [Enthalpy of vaporization](https://en.wikipedia.org/wiki/Enthalpy_of_vaporization "Enthalpy of vaporization") (latent heat of vaporization) - [Frenkel line](https://en.wikipedia.org/wiki/Frenkel_line "Frenkel line") - [Heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio") - [Heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") - [Heat transfer coefficient](https://en.wikipedia.org/wiki/Heat_transfer_coefficient "Heat transfer coefficient") - [History of thermodynamics](https://en.wikipedia.org/wiki/History_of_thermodynamics "History of thermodynamics") - [Joback method](https://en.wikipedia.org/wiki/Joback_method "Joback method") (Estimation of heat capacities) - [Latent heat](https://en.wikipedia.org/wiki/Latent_heat "Latent heat") - [Material properties (thermodynamics)](https://en.wikipedia.org/wiki/Material_properties_\(thermodynamics\) "Material properties (thermodynamics)") - [Quantum statistical mechanics](https://en.wikipedia.org/wiki/Quantum_statistical_mechanics "Quantum statistical mechanics") - [R-value (insulation)](https://en.wikipedia.org/wiki/R-value_\(insulation\) "R-value (insulation)") - [Statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") - [Table of specific heat capacities](https://en.wikipedia.org/wiki/Table_of_specific_heat_capacities "Table of specific heat capacities") - [Thermal mass](https://en.wikipedia.org/wiki/Thermal_mass "Thermal mass") - [Thermodynamic databases for pure substances](https://en.wikipedia.org/wiki/Thermodynamic_databases_for_pure_substances "Thermodynamic databases for pure substances") - [Thermodynamic equations](https://en.wikipedia.org/wiki/Thermodynamic_equations "Thermodynamic equations") - [Volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity") ## Notes \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=36 "Edit section: Notes")\] 1. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-gold_30-0)** [IUPAC](https://en.wikipedia.org/wiki/International_Union_of_Pure_and_Applied_Chemistry "International Union of Pure and Applied Chemistry"), *[Compendium of Chemical Terminology](https://en.wikipedia.org/wiki/IUPAC_books#Gold_Book "IUPAC books")*, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "[Standard Pressure](https://goldbook.iupac.org/terms/view/S05921.html)". [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1351/goldbook.S05921](https://doi.org/10.1351%2Fgoldbook.S05921). ## References \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=37 "Edit section: References")\] 1. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-1)** Halliday, David; Resnick, Robert; Walker, Jearl (2001). *Fundamentals of Physics* (6th ed.). New York, NY US: [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"). 2. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-2)** Open University (2008). *S104 Book 3 Energy and Light*, p. 59. [The Open University](https://en.wikipedia.org/wiki/The_Open_University "The Open University"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781848731646](https://en.wikipedia.org/wiki/Special:BookSources/9781848731646 "Special:BookSources/9781848731646") . 3. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-3)** Open University (2008). *S104 Book 3 Energy and Light*, p. 179. [The Open University](https://en.wikipedia.org/wiki/The_Open_University "The Open University"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781848731646](https://en.wikipedia.org/wiki/Special:BookSources/9781848731646 "Special:BookSources/9781848731646") . 4. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-4)** Engineering ToolBox (2003). ["Specific Heat of some common Substances"](https://www.engineeringtoolbox.com/specific-heat-capacity-d_391.html). 5. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-colen2001_5-0)** (2001): *Columbia Encyclopedia*, 6th ed.; as quoted by [Encyclopedia.com](https://www.encyclopedia.com/science-and-technology/physics/physics/specific-heat#1E1specheat). Columbia University Press. Accessed on 2019-04-11. 6. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-6)** Laidler, Keith J. (1993). [*The World of Physical Chemistry*](https://books.google.com/books?id=01LRlPbH80cC). Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-19-855919-4](https://en.wikipedia.org/wiki/Special:BookSources/0-19-855919-4 "Special:BookSources/0-19-855919-4") . 7. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-:1_7-0)** [Ramsay, William](https://en.wikipedia.org/wiki/William_Ramsay "William Ramsay") (1918). *The life and letters of Joseph Black, M.D*. Constable. pp. 38–39\. 8. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-8)** Black, Joseph (1807). Robison, John (ed.). [*Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh*](https://books.google.com/books?id=lqI9AQAAMAAJ&pg=PA76). Vol. 1. Mathew Carey. pp. 76–77\. 9. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-:0_9-0)** West, John B. (2014-06-15). ["Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases"](https://www.physiology.org/doi/10.1152/ajplung.00020.2014). *American Journal of Physiology. Lung Cellular and Molecular Physiology*. **306** (12): L1057–L1063. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1152/ajplung.00020.2014](https://doi.org/10.1152%2Fajplung.00020.2014). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1040-0605](https://search.worldcat.org/issn/1040-0605). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24682452](https://pubmed.ncbi.nlm.nih.gov/24682452). 10. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-10)** [International Bureau of Weights and Measures](https://en.wikipedia.org/wiki/International_Bureau_of_Weights_and_Measures "International Bureau of Weights and Measures") (2006), [*The International System of Units (SI)*](https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf) (PDF) (8th ed.), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [92-822-2213-6](https://en.wikipedia.org/wiki/Special:BookSources/92-822-2213-6 "Special:BookSources/92-822-2213-6") , [archived](https://web.archive.org/web/20210604163219/https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf) (PDF) from the original on 2021-06-04, retrieved 2021-12-16 11. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-toolbox_11-0)** ["Water – Thermal Properties"](http://www.engineeringtoolbox.com/water-thermal-properties-d_162.html). Engineeringtoolbox.com. Retrieved 2021-03-29. 12. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-12)** International Union of Pure and Applied Chemistry, Physical Chemistry Division. ["Quantities, Units and Symbols in Physical Chemistry"](http://old.iupac.org/publications/books/gbook/green_book_2ed.pdf) (PDF). Blackwell Sciences. p. 7. "The adjective specific before the name of an extensive quantity is often used to mean divided by mass." 13. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-Lange_13-0)** Lange's Handbook of Chemistry, 10th ed., page 1524. 14. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-14)** Quick, C. R.; Schawe, J. E. K.; Uggowitzer, P. J.; Pogatscher, S. (2019-07-01). ["Measurement of specific heat capacity via fast scanning calorimetry—Accuracy and loss corrections"](https://doi.org/10.1016%2Fj.tca.2019.03.021). *Thermochimica Acta*. Special Issue on occasion of the 65th birthday of Christoph Schick. **677**: 12–20\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019TcAc..677...12Q](https://ui.adsabs.harvard.edu/abs/2019TcAc..677...12Q). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.tca.2019.03.021](https://doi.org/10.1016%2Fj.tca.2019.03.021). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0040-6031](https://search.worldcat.org/issn/0040-6031). 15. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-15)** Pogatscher, S.; Leutenegger, D.; Schawe, J. E. K.; Uggowitzer, P. J.; Löffler, J. F. (September 2016). ["Solid–solid phase transitions via melting in metals"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4844691). *Nature Communications*. **7** (1) 11113. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2016NatCo...711113P](https://ui.adsabs.harvard.edu/abs/2016NatCo...711113P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/ncomms11113](https://doi.org/10.1038%2Fncomms11113). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [2041-1723](https://search.worldcat.org/issn/2041-1723). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [4844691](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4844691). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [27103085](https://pubmed.ncbi.nlm.nih.gov/27103085). 16. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-Koch_16-0)** Koch, Werner (2013). [*VDI Steam Tables*](https://books.google.com/books?id=bJ_wBgAAQBAJ&pg=PA8) (4 ed.). Springer. p. 8. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-642-52941-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-52941-2 "Special:BookSources/978-3-642-52941-2") . Published under the auspices of the *Verein Deutscher Ingenieure* (VDI). 17. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-17)** Cardarelli, Francois (2012). [*Scientific Unit Conversion: A Practical Guide to Metrication*](https://books.google.com/books?id=-ZveBwAAQBAJ&pg=PA19-IA35). M.J. Shields (translation) (2 ed.). Springer. p. 19. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4471-0805-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4471-0805-4 "Special:BookSources/978-1-4471-0805-4") . 18. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-18)** From direct values: 1 ⁠BTU/lb⋅°R⁠ × 1055.06 ⁠J/BTU⁠ × ( ⁠1/0\.45359237⁠) ⁠lb/kg⁠ x ⁠9/5⁠ ⁠°R/K⁠ = 4186.82 ⁠J/kg⋅K⁠ 19. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-19)** °F=°R 20. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-20)** °C=K 21. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-21)** McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: [University Science Books](https://en.wikipedia.org/w/index.php?title=University_Science_Books&action=edit&redlink=1 "University Science Books (page does not exist)"). pp. 83–85\. 22. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-22)** ["6.6: Electronic Partition Function"](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Statistical_Thermodynamics_\(Jeschke\)/06:_Partition_Functions_of_Gases/6.06:_Electronic_Partition_Function). *Chemistry LibreTexts*. 2020-11-26. Retrieved 2024-12-16. 23. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-23)** Bonhoeffer, K.F.; Harteck, P. (1926). ["Über Para- und Orthowasserstoff"](https://www.degruyter.com/document/doi/10.1515/zpch-1929-0408/html). *Z. Phys. Chem*. **4B**: 113–141\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1515/zpch-1929-0408](https://doi.org/10.1515%2Fzpch-1929-0408). 24. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-24)** McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: University Science Books. p. 107. 25. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-25)** Feynman, R., *[The Feynman Lectures on Physics](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics")*, Vol. 1, ch. 40, pp. 7–8 26. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-26)** Reif, F. (1965). [*Fundamentals of statistical and thermal physics*](https://archive.org/details/fundamentalsofst00reif). McGraw-Hill. pp. [253–254](https://archive.org/details/fundamentalsofst00reif/page/253). 27. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-27)** Kittel, Charles; Kroemer, Herbert (2000). *Thermal physics*. W. H. Freeman. p. 78. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-7167-1088-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7167-1088-2 "Special:BookSources/978-0-7167-1088-2") . 28. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-thor1993_28-0)** Thornton, Steven T. and Rex, Andrew (1993) *Modern Physics for Scientists and Engineers*, Saunders College Publishing 29. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-chas1998_29-0)** Chase, M.W. Jr. (1998) *[NIST-JANAF Themochemical Tables, Fourth Edition](https://webbook.nist.gov/cgi/cbook.cgi?ID=C7727379&Type=JANAFG)*, In *Journal of Physical and Chemical Reference Data*, Monograph 9, pages 1–1951. 30. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-31)** Yunus A. Cengel and Michael A. Boles, *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill, 2010, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [007-352932-X](https://en.wikipedia.org/wiki/Special:BookSources/007-352932-X "Special:BookSources/007-352932-X") . 31. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-32)** Fraundorf, P. (2003). "Heat capacity in bits". *American Journal of Physics*. **71** (11): 1142. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[cond-mat/9711074](https://arxiv.org/abs/cond-mat/9711074). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2003AmJPh..71.1142F](https://ui.adsabs.harvard.edu/abs/2003AmJPh..71.1142F). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.1593658](https://doi.org/10.1119%2F1.1593658). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [18742525](https://api.semanticscholar.org/CorpusID:18742525). 32. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-fein_33-0)** Feynman, Richard, *[The Feynman Lectures on Physics](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics")*, Vol. 1, Ch. 45 33. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-Benjelloun_34-0)** S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", [Link to Archiv e-print](https://arxiv.org/abs/2105.04845) [Link to Hal e-print](https://hal.archives-ouvertes.fr/hal-03216379/) 34. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-35)** Cengel, Yunus A. and Boles, Michael A. (2010) *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [007-352932-X](https://en.wikipedia.org/wiki/Special:BookSources/007-352932-X "Special:BookSources/007-352932-X") . ## Further reading \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=38 "Edit section: Further reading")\] - Emmerich Wilhelm & Trevor M. Letcher, Eds., 2010, *Heat Capacities: Liquids, Solutions and Vapours*, Cambridge, U.K.:Royal Society of Chemistry, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-85404-176-1](https://en.wikipedia.org/wiki/Special:BookSources/0-85404-176-1 "Special:BookSources/0-85404-176-1") . A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, "Heat Capacities: Introduction, Concepts, and Selected Applications" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as [photoacoustic](https://en.wikipedia.org/wiki/Photoacoustic_effect "Photoacoustic effect") methods, e.g., Jan Thoen & Christ Glorieux, "Photothermal Techniques for Heat Capacities," and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc. ## External links \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=39 "Edit section: External links")\] - (2012-05may-24) [Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World](https://physicsworld.com/a/phonon-theory-sheds-light-on-liquid-thermodynamics/) [The phonon theory of liquid thermodynamics \| Scientific Reports](https://www.nature.com/articles/srep00421) | [Authority control databases](https://en.wikipedia.org/wiki/Help:Authority_control "Help:Authority control") [![Edit this at Wikidata](https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png)](https://www.wikidata.org/wiki/Q487756#identifiers "Edit this at Wikidata") | | |---|---| | International | [GND](https://d-nb.info/gnd/4182218-3) | | National | [United States](https://id.loc.gov/authorities/sh85126389) [France](https://catalogue.bnf.fr/ark:/12148/cb12274423z) [BnF data](https://data.bnf.fr/ark:/12148/cb12274423z) [Israel](https://www.nli.org.il/en/authorities/987007565726505171) | | Other | [Yale LUX](https://lux.collections.yale.edu/view/concept/4a484350-e872-49a3-87de-3b087bf8c5e5) | ![](https://en.wikipedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1) Retrieved from "<https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&oldid=1340264896>" [Categories](https://en.wikipedia.org/wiki/Help:Category "Help:Category"): - [Physical quantities](https://en.wikipedia.org/wiki/Category:Physical_quantities "Category:Physical quantities") - [Thermodynamic properties](https://en.wikipedia.org/wiki/Category:Thermodynamic_properties "Category:Thermodynamic properties") Hidden categories: - [Articles with short description](https://en.wikipedia.org/wiki/Category:Articles_with_short_description "Category:Articles with short description") - [Short description is different from Wikidata](https://en.wikipedia.org/wiki/Category:Short_description_is_different_from_Wikidata "Category:Short description is different from Wikidata") - [All articles with unsourced statements](https://en.wikipedia.org/wiki/Category:All_articles_with_unsourced_statements "Category:All articles with unsourced statements") - [Articles with unsourced statements from April 2019](https://en.wikipedia.org/wiki/Category:Articles_with_unsourced_statements_from_April_2019 "Category:Articles with unsourced statements from April 2019") - [Pages using div col with small parameter](https://en.wikipedia.org/wiki/Category:Pages_using_div_col_with_small_parameter "Category:Pages using div col with small parameter") - This page was last edited on 24 February 2026, at 19:58 (UTC). - Text is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License "Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License"); additional terms may apply. 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| Specific heat capacity | | |---|---| | Other names | Specific heat | | Common symbols | *c* | | [SI unit](https://en.wikipedia.org/wiki/SI_unit "SI unit") | J⋅kg−1⋅K−1 | | In [SI base units](https://en.wikipedia.org/wiki/SI_base_unit "SI base unit") | m2⋅K−1⋅s−2 | | [Intensive](https://en.wikipedia.org/wiki/Intensive_and_extensive_properties "Intensive and extensive properties")? | Yes | | [Dimension](https://en.wikipedia.org/wiki/Dimensional_analysis#Formulation "Dimensional analysis") | L2⋅T−2⋅Θ−1 | In [thermodynamics](https://en.wikipedia.org/wiki/Thermodynamics "Thermodynamics"), the **specific heat capacity** (symbol *c*) of a substance is the amount of [heat](https://en.wikipedia.org/wiki/Heat "Heat") that must be added to one unit of mass of the substance in order to cause an increase of one unit in [temperature](https://en.wikipedia.org/wiki/Temperature "Temperature"). It is also referred to as **massic heat capacity** or as the **specific heat.** More formally it is the [heat capacity](https://en.wikipedia.org/wiki/Heat_capacity "Heat capacity") of a sample of the substance divided by the [mass](https://en.wikipedia.org/wiki/Mass "Mass") of the sample.[\[1\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-1) The [SI](https://en.wikipedia.org/wiki/International_System_of_Units "International System of Units") unit of specific heat capacity is [joule](https://en.wikipedia.org/wiki/Joule "Joule") per [kelvin](https://en.wikipedia.org/wiki/Kelvin "Kelvin") per [kilogram](https://en.wikipedia.org/wiki/Kilogram "Kilogram"), J⋅kg−1⋅K−1.[\[2\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-2) For example, the heat required to raise the temperature of 1 kg of [water](https://en.wikipedia.org/wiki/Water "Water") by 1 K is 4184 joules, so the specific heat capacity of water is 4184 J⋅kg−1⋅K−1.[\[3\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-3) Specific heat capacity often varies with temperature, and is different for each [state of matter](https://en.wikipedia.org/wiki/State_of_matter "State of matter"). Liquid water has one of the highest specific heat capacities among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C, but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1. The specific heat capacities of [iron](https://en.wikipedia.org/wiki/Iron "Iron"), [granite](https://en.wikipedia.org/wiki/Granite "Granite"), and [hydrogen](https://en.wikipedia.org/wiki/Hydrogen "Hydrogen") gas are about 449 J⋅kg−1⋅K−1, 790 J⋅kg−1⋅K−1, and 14300 J⋅kg−1⋅K−1, respectively.[\[4\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-4) While the substance is undergoing a [phase transition](https://en.wikipedia.org/wiki/Phase_transition "Phase transition"), such as melting or boiling, its specific heat capacity is technically undefined, because the heat goes into changing its state rather than raising its temperature. The specific heat capacity of a substance, especially a gas, may be significantly higher when it is allowed to expand as it is heated (specific heat capacity *at constant pressure*) than when it is heated in a closed vessel that prevents expansion (specific heat capacity *at constant volume*). These two values are usually denoted by ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) and ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082), respectively; their quotient ![{\\displaystyle \\gamma =c\_{p}/c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbe578fc940f94ff433f350a8b2d7cad4dbf01c0) is the [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio"). The term *specific heat* may also refer to the ratio between the specific heat capacities of a substance at a given temperature and of a reference substance at a reference temperature, such as water at 15 °C;[\[5\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-colen2001-5) much in the fashion of [specific gravity](https://en.wikipedia.org/wiki/Specific_gravity "Specific gravity"). Specific heat capacity is also related to other intensive measures of heat capacity with other denominators. If the amount of substance is measured as a number of [moles](https://en.wikipedia.org/wiki/Mole_\(unit\) "Mole (unit)"), one gets the [molar heat capacity](https://en.wikipedia.org/wiki/Molar_heat_capacity "Molar heat capacity") instead, whose SI unit is joule per kelvin per mole, J⋅mol−1⋅K−1. If the amount is taken to be the [volume](https://en.wikipedia.org/wiki/Volume "Volume") of the sample (as is sometimes done in engineering), one gets the [volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity"), whose SI unit is joule per kelvin per [cubic meter](https://en.wikipedia.org/wiki/Cubic_meter "Cubic meter"), J⋅m−3⋅K−1. ### Discovery of specific heat \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=2 "Edit section: Discovery of specific heat")\] [![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Black_Joseph_%28cropped%29.jpg/250px-Black_Joseph_%28cropped%29.jpg)](https://en.wikipedia.org/wiki/File:Black_Joseph_\(cropped\).jpg) Joseph Black One of the first scientists to use the concept was [Joseph Black](https://en.wikipedia.org/wiki/Joseph_Black "Joseph Black"), an 18th-century medical doctor and professor of medicine at [Glasgow University](https://en.wikipedia.org/wiki/Glasgow_University "Glasgow University"). He measured the specific heat capacities of many substances, using the term *capacity for heat*.[\[6\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-6) In 1756 or soon thereafter, Black began an extensive study of heat.[\[7\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-:1-7) In 1760 he realized that when two different substances of equal mass but different temperatures are mixed, the changes in number of degrees in the two substances differ, though the heat gained by the cooler substance and lost by the hotter is the same. Black related an experiment conducted by [Daniel Gabriel Fahrenheit](https://en.wikipedia.org/wiki/Daniel_Gabriel_Fahrenheit "Daniel Gabriel Fahrenheit") on behalf of Dutch physician [Herman Boerhaave](https://en.wikipedia.org/wiki/Herman_Boerhaave "Herman Boerhaave"). For clarity, he then described a hypothetical, but realistic variant of the experiment: If equal masses of 100 °F water and 150 °F mercury are mixed, the water temperature increases by 20 ° and the mercury temperature decreases by 30 ° (both arriving at 120 °F), even though the heat gained by the water and lost by the mercury is the same. This clarified the distinction between heat and temperature. It also introduced the concept of specific heat capacity, being different for different substances. Black wrote: "Quicksilver \[mercury\] ... has less capacity for the matter of heat than water."[\[8\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-8)[\[9\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-:0-9) The specific heat capacity of a substance, usually denoted by ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) or ![{\\displaystyle s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632), is the heat capacity ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) of a sample of the substance, divided by the mass ![{\\displaystyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd) of the sample:[\[10\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-10) ![{\\displaystyle c={\\frac {C}{M}}={\\frac {1}{M}}\\cdot {\\frac {\\mathrm {d} Q}{\\mathrm {d} T}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a58291be3575e33a756ed894a18347c8eabbba29) where ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739) [represents](https://en.wikipedia.org/wiki/Derivative "Derivative") the amount of heat needed to uniformly raise the temperature of the sample by a small increment ![{\\displaystyle \\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80). Like the heat capacity of an object, the specific heat capacity of a substance may vary, sometimes substantially, depending on the starting temperature ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) of the sample and the [pressure](https://en.wikipedia.org/wiki/Pressure "Pressure") ![{\\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) applied to it. Therefore, it should be considered a function ![{\\displaystyle c(p,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bac9cbcaf700b5206116a980a24a84404ba2e5) of those two variables. These parameters are usually specified when giving the specific heat capacity of a substance. For example, "Water (liquid): ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) = 4187 J⋅kg−1⋅K−1 (15 °C)."[\[11\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-toolbox-11) When not specified, published values of the specific heat capacity ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) generally are valid for some [standard conditions for temperature and pressure](https://en.wikipedia.org/wiki/Standard_conditions_for_temperature_and_pressure "Standard conditions for temperature and pressure"). However, the dependency of ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) on starting temperature and pressure can often be ignored in practical contexts, e.g. when working in narrow ranges of those variables. In those contexts one usually omits the qualifier ![{\\displaystyle (p,T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/140e6e1e1f95c1aba3ffb6dc196a9be07a91d1a6) and approximates the specific heat capacity by a constant ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) suitable for those ranges. Specific heat capacity is an [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") of a substance, an intrinsic characteristic that does not depend on the size or shape of the amount in consideration. (The qualifier "specific" in front of an extensive property often indicates an intensive property derived from it.[\[12\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-12)) The injection of heat energy into a substance, besides raising its temperature, usually causes an increase in its volume and/or its pressure, depending on how the sample is confined. The choice made about the latter affects the measured specific heat capacity, even for the same starting pressure ![{\\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36) and starting temperature ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0). Two particular choices are widely used: The value of ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) is always less than the value of ![{\\displaystyle c\_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77463f4fbb953a6f1fe19d83708e553f6d21457f) for all fluids. This difference is particularly notable in gases where values under constant pressure are typically 30% to 66.7% greater than those at constant volume. Hence the [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio") of gases is typically between 1.3 and 1.67.[\[13\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-Lange-13) The specific heat capacity can be defined and measured for gases, liquids, and solids of fairly general composition and molecular structure. These include gas mixtures, solutions and alloys, or heterogenous materials such as milk, sand, granite, and concrete, if considered at a sufficiently large scale. The specific heat capacity can be defined also for materials that change state or composition as the temperature and pressure change, as long as the changes are reversible and gradual. Thus, for example, the concepts are definable for a gas or liquid that dissociates as the temperature increases, as long as the products of the dissociation promptly and completely recombine when it drops. The specific heat capacity is not meaningful if the substance undergoes irreversible chemical changes, or if there is a [phase change](https://en.wikipedia.org/wiki/Phase_transition "Phase transition"), such as melting or boiling, at a sharp temperature within the range of temperatures spanned by the measurement. The specific heat capacity of a substance is typically determined according to the definition; namely, by measuring the heat capacity of a sample of the substance, usually with a [calorimeter](https://en.wikipedia.org/wiki/Calorimeter "Calorimeter"), and dividing by the sample's mass. Several techniques can be applied for estimating the heat capacity of a substance, such as [differential scanning calorimetry](https://en.wikipedia.org/wiki/Differential_scanning_calorimetry "Differential scanning calorimetry").[\[14\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-14)[\[15\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-15) [![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Water_temperature_vs_heat_added.svg/250px-Water_temperature_vs_heat_added.svg.png)](https://en.wikipedia.org/wiki/File:Water_temperature_vs_heat_added.svg) Graph of temperature of phases of water heated from −100 °C to 200 °C – the dashed line example shows that melting and heating 1 kg of ice at −50 °C to water at 40 °C needs 600 kJ The specific heat capacities of gases can be measured at constant volume, by enclosing the sample in a rigid container. On the other hand, measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids, since one often would need impractical pressures in order to prevent the expansion that would be caused by even small increases in temperature. Instead, the common practice is to measure the specific heat capacity at constant pressure (allowing the material to expand or contract as it wishes), determine separately the [coefficient of thermal expansion](https://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion "Coefficient of thermal expansion") and the [compressibility](https://en.wikipedia.org/wiki/Bulk_modulus "Bulk modulus") of the material, and compute the specific heat capacity at constant volume from these data according to the laws of thermodynamics.\[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")*\] ### International system \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=8 "Edit section: International system")\] The SI unit for specific heat capacity is joule per kelvin per kilogram ⁠J/kg⋅K⁠, J⋅K−1⋅kg−1. Since an increment of temperature of one [degree Celsius](https://en.wikipedia.org/wiki/Celsius_scale "Celsius scale") is the same as an increment of one kelvin, that is the same as joule per degree Celsius per kilogram: J/(kg⋅°C). Sometimes the [gram](https://en.wikipedia.org/wiki/Gram "Gram") is used instead of kilogram for the unit of mass: 1 J⋅g−1⋅K−1 = 1000 J⋅kg−1⋅K−1. The specific heat capacity of a substance (per unit of mass) has [dimension](https://en.wikipedia.org/wiki/Dimensional_analysis "Dimensional analysis") L2⋅Θ−1⋅T−2, or (L/T)2/Θ. Therefore, the SI unit J⋅kg−1⋅K−1 is equivalent to [metre](https://en.wikipedia.org/wiki/Metre "Metre") squared per [second](https://en.wikipedia.org/wiki/Second "Second") squared per [kelvin](https://en.wikipedia.org/wiki/Kelvin "Kelvin") (m2⋅K−1⋅s−2). ### Imperial engineering units \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=9 "Edit section: Imperial engineering units")\] Professionals in [construction](https://en.wikipedia.org/wiki/Construction "Construction"), [civil engineering](https://en.wikipedia.org/wiki/Civil_engineering "Civil engineering"), [chemical engineering](https://en.wikipedia.org/wiki/Chemical_engineering "Chemical engineering"), and other technical disciplines, especially in the [United States](https://en.wikipedia.org/wiki/United_States "United States"), may use [English Engineering units](https://en.wikipedia.org/wiki/English_Engineering_Units "English Engineering Units") including the [pound](https://en.wikipedia.org/wiki/Pound_\(mass\) "Pound (mass)") (lb = 0.45359237 kg) as the unit of mass, the [degree Fahrenheit](https://en.wikipedia.org/wiki/Fahrenheit "Fahrenheit") or [Rankine](https://en.wikipedia.org/wiki/Rankine_scale "Rankine scale") (°R = ⁠5/9⁠ K, about 0.555556 K) as the unit of temperature increment, and the [British thermal unit](https://en.wikipedia.org/wiki/British_thermal_unit "British thermal unit") (BTU ≈ 1055.056 J),[\[16\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-Koch-16)[\[17\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-17) as the unit of heat. In those contexts, the unit of specific heat capacity is BTU/lb⋅°R, or 1 ⁠BTU/lb⋅°R⁠ = 4186.68⁠J/kg⋅K⁠.[\[18\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-18) The BTU was originally defined so that the average specific heat capacity of water would be 1 BTU/lb⋅°F.[\[19\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-19) Note the value's similarity to that of the calorie - 4187 J/kg⋅°C ≈ 4184 J/kg⋅°C (~.07%) - as they are essentially measuring the same energy, using water as a basis reference, scaled to their systems' respective lbs and °F, or kg and °C. In chemistry, heat amounts were often measured in [calories](https://en.wikipedia.org/wiki/Calorie "Calorie"). Confusingly, there are two common units with that name, respectively denoted *cal* and *Cal*: - the *small calorie* (*gram-calorie, cal*) is 4.184 J exactly. It was originally defined so that the specific heat capacity of liquid water would be 1 cal/(°C⋅g). - The *grand calorie* (*kilocalorie, kilogram-calorie, food calorie, kcal, Cal*) is 1000 small calories, 4184 J exactly. It was defined so that the specific heat capacity of water would be 1 Cal/(°C⋅kg). While these units are still used in some contexts (such as kilogram calorie in [nutrition](https://en.wikipedia.org/wiki/Nutrition "Nutrition")), their use is now deprecated in technical and scientific fields. When heat is measured in these units, the unit of specific heat capacity is usually: 1 ⁠cal/°C⋅g⁠ = 1 ⁠Cal/°C⋅kg⁠ = 1 ⁠kcal/°C⋅kg⁠ = 4184 ⁠J/kg⋅K⁠[\[20\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-20) = 4.184 ⁠kJ/kg⋅K⁠. Note that while cal is **1⁄1000** of a Cal or kcal, it is also per *gram* instead of **kilo***gram*: ergo, in either unit, the specific heat capacity of water is approximately 1. The temperature of a sample of a substance reflects the average [kinetic energy](https://en.wikipedia.org/wiki/Kinetic_energy "Kinetic energy") of its constituent particles (atoms or molecules) relative to its center of mass. However, not all energy provided to a sample of a substance will go into raising its temperature, exemplified via the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem"). [Statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") predicts that at room temperature and ordinary pressures, an isolated atom in a gas cannot store any significant amount of energy except in the form of kinetic energy, unless multiple electronic states are accessible at room temperature (such is the case for atomic fluorine).[\[21\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-21) Thus, the [heat capacity per mole](https://en.wikipedia.org/wiki/Molar_heat_capacity "Molar heat capacity") at room temperature is the same for all of the noble gases as well as for many other atomic vapors. More precisely, ![{\\displaystyle c\_{V,\\mathrm {m} }=3R/2\\approx \\mathrm {12.5\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcee637c4392e7347a6b4b7d57395e90ef9b8f5b) and ![{\\displaystyle c\_{P,\\mathrm {m} }=5R/2\\approx \\mathrm {21\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6fc485f5327a3d2857548cbe7c145147d826306), where ![{\\displaystyle R\\approx \\mathrm {8.31446\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b344e96636e94e8469d9e4a0eafa306b3bec07e) is the [ideal gas unit](https://en.wikipedia.org/wiki/Ideal_gas_constant "Ideal gas constant") (which is the product of [Boltzmann conversion constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") from [kelvin](https://en.wikipedia.org/wiki/Kelvin "Kelvin") microscopic energy unit to the macroscopic energy unit [joule](https://en.wikipedia.org/wiki/Joule "Joule"), and the [Avogadro number](https://en.wikipedia.org/wiki/Avogadro_number "Avogadro number")). Therefore, the specific heat capacity (per gram, not per mole) of a monatomic gas will be inversely proportional to its (adimensional) [atomic weight](https://en.wikipedia.org/wiki/Atomic_weight "Atomic weight") ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3). That is, approximately, ![{\\displaystyle c\_{V}\\approx \\mathrm {12470\\,J\\cdot K^{-1}\\cdot kg^{-1}} /A\\quad \\quad \\quad c\_{p}\\approx \\mathrm {20785\\,J\\cdot K^{-1}\\cdot kg^{-1}} /A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0733e9dc0db1000b284289198a6d8d9b4ed008d9) For the noble gases, from helium to xenon, these computed values are | Gas | He | Ne | Ar | Kr | Xe | |---|---|---|---|---|---| | ![{\\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3) | | | | | | A polyatomic gas molecule can store energy in additional degrees of freedom. Its kinetic energy contributes to the heat capacity in the same way as monatomic gases, but there are also contributions from the [rotations](https://en.wikipedia.org/wiki/Rotational_energy "Rotational energy") of the molecule and vibration of the atoms relative to each other (including internal [potential energy](https://en.wikipedia.org/wiki/Potential_energy "Potential energy")). The heat capacity may also have contribution from [excited electronic states](https://en.wikipedia.org/wiki/Excited_state "Excited state") for molecules with a sufficiently small energy gap between the ground state and the excited state, such as in NO.[\[22\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-22) For a few systems, quantum spin statistics can also be important contributions to the heat capacity, even at room temperature. The analysis of the heat capacity of H 2 due to ortho/para separation,[\[23\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-23) which arises from [nuclear spin](https://en.wikipedia.org/wiki/Spin_quantum_number "Spin quantum number") statistics, has been referred to as "one of the great triumphs of post-quantum mechanical statistical mechanics."[\[24\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-24) These extra [degrees of freedom](https://en.wikipedia.org/wiki/Degrees_of_freedom_\(physics_and_chemistry\) "Degrees of freedom (physics and chemistry)") or "modes" contribute to the specific heat capacity of the substance. Namely, when energy is introduced into a gas with polyatomic molecules, only part of it will increase their kinetic energy, and hence the temperature; the rest will go to into the other degrees of freedom. To achieve the same increase in temperature, more heat is needed for a gram of that substance than for a gram of a monatomic gas. Thus, the specific heat capacity per mole of a polyatomic gas depends both on the molecular mass and the number of degrees of freedom of the molecules.[\[25\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-25)[\[26\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-26)[\[27\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-27) [Quantum statistical mechanics](https://en.wikipedia.org/wiki/Quantum_statistical_mechanics "Quantum statistical mechanics") predicts that each rotational or vibrational mode can only take or lose energy in certain discrete amounts (quanta), and that this affects the system's thermodynamic properties. Depending on the temperature, the average energy per molecule may be too small compared to the quanta needed to activate some of those degrees of freedom. Those modes are said to be "frozen out". In that case, the specific heat capacity of the substance increases with temperature, sometimes in a step-like fashion as mode becomes unfrozen and starts absorbing part of the input heat. For example, the molar heat capacity of [nitrogen](https://en.wikipedia.org/wiki/Nitrogen "Nitrogen") N 2 at constant volume is ![{\\displaystyle c\_{V,\\mathrm {m} }=\\mathrm {20.6\\,J\\cdot K^{-1}\\cdot mol^{-1}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7c7437f35c01e65d4dc78ef1cbf4dfcf2fb023) (at 15 °C, 1 atm), which is ![{\\displaystyle 2.49R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6691d251191f800f340af5ec9cd4e291d0a5344c).[\[28\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-thor1993-28) That is the value expected from the [Equipartition Theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem") if each molecule had 5 kinetic degrees of freedom. These turn out to be three degrees of the molecule's velocity vector, plus two degrees from its rotation about an axis through the center of mass and perpendicular to the line of the two atoms. Because of those two extra degrees of freedom, the specific heat capacity ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) of N 2 (736 J⋅K−1⋅kg−1) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445 J⋅K−1⋅kg−1), by a factor of ⁠5/3⁠. The vibrational and electronic degrees of freedom do not contribute significantly to the heat capacity in this case, due to the relatively large energy level gaps for both vibrational and electronic excitation in this molecule. This value for the specific heat capacity of nitrogen is practically constant from below −150 °C to about 300 °C. In that temperature range, the two additional degrees of freedom that correspond to vibrations of the atoms, stretching and compressing the bond, are still "frozen out". At about that temperature, those modes begin to "un-freeze" as vibrationally excited states become accessible. As a result ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5 J⋅K−1⋅mol−1 at 1500 °C, 36.9 at 2500 °C, and 37.5 at 3500 °C.[\[29\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-chas1998-29) The last value corresponds almost exactly to the value predicted by the Equipartition Theorem, since in the high-temperature limit the theorem predicts that the vibrational degree of freedom contributes twice as much to the heat capacity as any one of the translational or rotational degrees of freedom. ## Derivations of heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=14 "Edit section: Derivations of heat capacity")\] ### Relation between specific heat capacities \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=15 "Edit section: Relation between specific heat capacities")\] Starting from the [fundamental thermodynamic relation](https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation "Fundamental thermodynamic relation") one can show, ![{\\displaystyle c\_{p}-c\_{v}={\\frac {\\alpha ^{2}T}{\\rho \\beta \_{T}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06bc97cafc646b40213b01cf9a30562276c7c744) where A derivation is discussed in the article [Relations between specific heats](https://en.wikipedia.org/wiki/Relations_between_specific_heats "Relations between specific heats"). For an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), if ![{\\displaystyle \\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64) is expressed as [molar](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") density in the above equation, this equation reduces simply to [Mayer](https://en.wikipedia.org/wiki/Julius_Robert_von_Mayer "Julius Robert von Mayer")'s relation, ![{\\displaystyle C\_{p,m}-C\_{v,m}=R\\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f23e5c5b22b24ffd7f1051a3e396784a902d867e) where ![{\\displaystyle C\_{p,m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccf6b7671259c85a73e29bc870010db9a7d2288) and ![{\\displaystyle C\_{v,m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bc3350be2e3cdd42c3a445e5dae16ac2aa4a675) are [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") heat capacities expressed on a per mole basis at constant pressure and constant volume, respectively. ### Specific heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=16 "Edit section: Specific heat capacity")\] The specific heat capacity of a material on a per mass basis is ![{\\displaystyle c={\\partial C \\over \\partial m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/338148bac53d596abe5112d0891626997320114b) which in the absence of phase transitions is equivalent to ![{\\displaystyle c=E\_{m}={C \\over m}={C \\over {\\rho V}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1a8ab2bd40064b132ddd125dd7e96420c0337d) where - ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question, - ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is the mass of the body, - ![{\\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the body, and - ![{\\displaystyle \\rho ={\\frac {m}{V}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63465553e3f944d6ef79f90f992a02cf29c7f38) is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https://en.wikipedia.org/wiki/Isobaric_process "Isobaric process") (constant pressure, ![{\\displaystyle dp=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850e5511af66c84a989bf0e5d003852cb862778c)) or [isochoric](https://en.wikipedia.org/wiki/Isochoric_process "Isochoric process") (constant volume, ![{\\displaystyle dV=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d505a8f349881feb179d02752b33e5ab95e035)) processes. The corresponding specific heat capacities are expressed as ![{\\displaystyle {\\begin{aligned}c\_{p}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{p},\\\\c\_{V}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{V}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4576769aa96d12c2e062fe6e6a6ef4acfdc8b63b) A related parameter to ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is ![{\\displaystyle CV^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0155c68921e700c1325f32c73061273b2178796e), the [volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity"). In engineering practice, ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity is often explicitly written with the subscript ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as ![{\\displaystyle c\_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes ![{\\displaystyle c\_{m}={\\frac {C}{m}}={\\frac {c\_{V}}{\\rho }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/551b63520ea921f8105a75ddf4ddb073eca1e9db) For pure homogeneous [chemical compounds](https://en.wikipedia.org/wiki/Chemical_compound "Chemical compound") with established [molecular or molar mass](https://en.wikipedia.org/wiki/Molecular_mass "Molecular mass") or a [molar quantity](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") is established, heat capacity as an [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") can be expressed on a per [mole](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") basis instead of a per mass basis by the following equations analogous to the per mass equations: ![{\\displaystyle {\\begin{alignedat}{3}C\_{p,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{p}&={\\text{molar heat capacity at constant pressure}}\\\\C\_{V,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{V}&={\\text{molar heat capacity at constant volume}}\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d63e3ec44e731e9eb128dcc30eaa9c87af059a) where *n* = number of moles in the body or [thermodynamic system](https://en.wikipedia.org/wiki/Thermodynamic_system "Thermodynamic system"). One may refer to such a *per mole* quantity as molar heat capacity to distinguish it from specific heat capacity on a per-mass basis. ### Polytropic heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=17 "Edit section: Polytropic heat capacity")\] The [polytropic](https://en.wikipedia.org/wiki/Polytropic "Polytropic") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change ![{\\displaystyle C\_{i,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)={\\text{molar heat capacity at polytropic process}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02f685610e35e01aa42e293380bdf94469ea8364) The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between **1** and the adiabatic exponent (*γ* or *κ*) ### Dimensionless heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=18 "Edit section: Dimensionless heat capacity")\] The [dimensionless](https://en.wikipedia.org/wiki/Dimensionless_number "Dimensionless number") heat capacity of a material is ![{\\displaystyle C^{\*}={\\frac {C}{nR}}={\\frac {C}{Nk\_{\\text{B}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1beaede671f02ee56b81859fe98eed0feb26cf5) where - *C* is the heat capacity of a body made of the material in question (J/K) - *n* is the [amount of substance](https://en.wikipedia.org/wiki/Amount_of_substance "Amount of substance") in the body ([mol](https://en.wikipedia.org/wiki/Mole_\(unit\) "Mole (unit)")) - *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant") (J⋅K−1⋅mol−1) - *N* is the number of molecules in the body. (dimensionless) - *k*B is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") (J⋅K−1) Again, [SI](https://en.wikipedia.org/wiki/SI "SI") units shown for example. In the [Ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") article, dimensionless heat capacity ![{\\displaystyle C^{\*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as ![{\\displaystyle {\\hat {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8417e85ae7f4eaee7df31347ce488f85c8884b93). ### Heat capacity at absolute zero \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=19 "Edit section: Heat capacity at absolute zero")\] From the definition of [entropy](https://en.wikipedia.org/wiki/Entropy#Thermodynamic_definition_of_entropy "Entropy") ![{\\displaystyle TdS=\\delta Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ddb574969210251fd1e6d8847eb303ece6b6a7) the absolute entropy can be calculated by integrating from zero kelvins temperature to the final temperature *Tf* ![{\\displaystyle S(T\_{f})=\\int \_{T=0}^{T\_{f}}{\\frac {\\delta Q}{T}}=\\int \_{0}^{T\_{f}}{\\frac {\\delta Q}{dT}}{\\frac {dT}{T}}=\\int \_{0}^{T\_{f}}C(T)\\,{\\frac {dT}{T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e81986ce6a91887325cc998b1245250f83a65a) The heat capacity must be zero at zero temperature in order for the above integral not to yield an infinite absolute entropy, thus violating the [third law of thermodynamics](https://en.wikipedia.org/wiki/Third_law_of_thermodynamics "Third law of thermodynamics"). One of the strengths of the [Debye model](https://en.wikipedia.org/wiki/Debye_model "Debye model") is that (unlike the preceding Einstein model) it predicts the proper mathematical form of the approach of heat capacity toward zero, as absolute zero temperature is approached. The theoretical maximum heat capacity for larger and larger multi-atomic gases at higher temperatures, also approaches the Dulong–Petit limit of 3*R*, so long as this is calculated per mole of atoms, not molecules. The reason is that gases with very large molecules, in theory have almost the same high-temperature heat capacity as solids, lacking only the (small) heat capacity contribution that comes from potential energy that cannot be stored between separate molecules in a gas. The Dulong–Petit limit results from the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem"), and as such is only valid in the classical limit of a [microstate continuum](https://en.wikipedia.org/w/index.php?title=Microstate_continuum&action=edit&redlink=1 "Microstate continuum (page does not exist)"), which is a high temperature limit. For light and non-metallic elements, as well as most of the common molecular solids based on carbon compounds at [standard ambient temperature](https://en.wikipedia.org/wiki/Standard_ambient_temperature_and_pressure "Standard ambient temperature and pressure"), quantum effects may also play an important role, as they do in multi-atomic gases. These effects usually combine to give heat capacities lower than 3*R* per mole of *atoms* in the solid, although in molecular solids, heat capacities calculated *per mole of molecules* in molecular solids may be more than 3*R*. For example, the heat capacity of water ice at the melting point is about 4.6*R* per mole of molecules, but only 1.5*R* per mole of atoms. The lower than 3*R* number "per atom" (as is the case with diamond and beryllium) results from the "freezing out" of possible vibration modes for light atoms at suitably low temperatures, just as in many low-mass-atom gases at room temperatures. Because of high crystal binding energies, these effects are seen in solids more often than liquids: for example the heat capacity of liquid water is twice that of ice at near the same temperature, and is again close to the 3*R* per mole of atoms of the Dulong–Petit theoretical maximum. For a more modern and precise analysis of the heat capacities of solids, especially at low temperatures, it is useful to use the idea of [phonons](https://en.wikipedia.org/wiki/Phonons "Phonons"). See [Debye model](https://en.wikipedia.org/wiki/Debye_model "Debye model"). ### Theoretical estimation \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=21 "Edit section: Theoretical estimation")\] The path integral Monte Carlo method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3R = 24.94 joules per kelvin per mole of atoms (Dulong–Petit law, R is the gas constant). Low temperature approximations for both gases and solids at temperatures less than their characteristic Einstein temperatures or Debye temperatures can be made by the methods of Einstein and Debye discussed below. - Water (liquid): CP = 4185.5 J⋅K−1⋅kg−1 (15 °C, 101.325 kPa) - Water (liquid): CVH = 74.539 J⋅K−1⋅mol−1 (25 °C) For liquids and gases, it is important to know the pressure to which given heat capacity data refer. Most published data are given for standard pressure. However, different standard conditions for temperature and pressure have been defined by different organizations. The International Union of Pure and Applied Chemistry (IUPAC) changed its recommendation from one atmosphere to the round value 100 kPa (≈750.062 Torr).[\[notes 1\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-gold-30) ### Relation between heat capacities \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=22 "Edit section: Relation between heat capacities")\] Measuring the specific heat capacity at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see [coefficient of thermal expansion](https://en.wikipedia.org/wiki/Coefficient_of_thermal_expansion "Coefficient of thermal expansion") and [compressibility](https://en.wikipedia.org/wiki/Compressibility "Compressibility")). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. The [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio"), or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. For an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), evaluating the partial derivatives above according to the [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state"), where *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant"), for an ideal gas[\[30\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-31) ![{\\displaystyle {\\begin{alignedat}{3}PV&=nRT,&\\\\C\_{P}-C\_{V}&=T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n},&\\\\P&={\\frac {nRT}{V}}\\Rightarrow \\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}&={\\frac {nR}{V}},\\\\V&={\\frac {nRT}{P}}\\Rightarrow \\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}&={\\frac {nR}{P}}.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe220db0c71a99f43c20c80b90b430001c8aff8) Substituting ![{\\displaystyle T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}=T{\\frac {nR}{V}}{\\frac {nR}{P}}={\\frac {nRT}{V}}{\\frac {nR}{P}}=P{\\frac {nR}{P}}=nR,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49d3f01b48a4771f06476e4922cd418e91e9354) this equation reduces simply to [Mayer](https://en.wikipedia.org/wiki/Julius_Robert_von_Mayer "Julius Robert von Mayer")'s relation: ![{\\displaystyle C\_{P,m}-C\_{V,m}=R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0951113a93ae7d847ed0234a94039782ae0a5e) The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. ### Specific heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=24 "Edit section: Specific heat capacity")\] The specific heat capacity of a material on a per mass basis is ![{\\displaystyle c={\\frac {\\partial C}{\\partial m}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6d4d3222fc01879cf5ebdcdd167c0f01338989) which in the absence of phase transitions is equivalent to ![{\\displaystyle c=E\_{m}={\\frac {C}{m}}={\\frac {C}{\\rho V}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5685f216c3eaa301626d07a8dbeda4ec825d2a) where - ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question, - ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc) is the mass of the body, - ![{\\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the body, - ![{\\displaystyle \\rho ={\\frac {m}{V}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63465553e3f944d6ef79f90f992a02cf29c7f38) is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include [isobaric](https://en.wikipedia.org/wiki/Isobaric_process "Isobaric process") (constant pressure, ![{\\displaystyle {\\text{d}}P=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2997415b7c242bcf6f4ebf7ef17957d9df03a9aa)) or [isochoric](https://en.wikipedia.org/wiki/Isochoric_process "Isochoric process") (constant volume, ![{\\displaystyle {\\text{d}}V=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e19ded03242b58e141dced9f6f46c3a8829e436)) processes. The corresponding specific heat capacities are expressed as ![{\\displaystyle {\\begin{aligned}c\_{P}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{P},\\\\c\_{V}&=\\left({\\frac {\\partial C}{\\partial m}}\\right)\_{V}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d2cc9fc1a0307e6e726521189cf7e22158357b9) From the results of the previous section, dividing through by the mass gives the relation ![{\\displaystyle c\_{P}-c\_{V}={\\frac {\\alpha ^{2}T}{\\rho \\beta \_{T}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/058bcaccd4cbc0e8d16892f68bc68bf1e0062d7e) A related parameter to ![{\\displaystyle c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455) is ![{\\displaystyle C/V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f43bc2f531ba84544e7e71ccab205ab4ad1fe5c), the [volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity"). In engineering practice, ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the specific heat capacity is often explicitly written with the subscript ![{\\displaystyle m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc), as ![{\\displaystyle c\_{m}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92f980a7ccf6827b6925c6d6421984d9c5859). Of course, from the above relationships, for solids one writes ![{\\displaystyle c\_{m}={\\frac {C}{m}}={\\frac {c\_{\\text{volumetric}}}{\\rho }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56b0a8c8ebcbe4ee889c7d963cb980a5d34f60b) For pure [homogeneous](https://en.wikipedia.org/wiki/Homogeneous "Homogeneous") [chemical compounds](https://en.wikipedia.org/wiki/Chemical_compound "Chemical compound") with established [molecular or molar mass](https://en.wikipedia.org/wiki/Molecular_mass "Molecular mass"), or a [molar quantity](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)"), heat capacity as an [intensive property](https://en.wikipedia.org/wiki/Intensive_property "Intensive property") can be expressed on a per-[mole](https://en.wikipedia.org/wiki/Mole_\(chemistry\) "Mole (chemistry)") basis instead of a per-mass basis by the following equations analogous to the per mass equations: ![{\\displaystyle {\\begin{alignedat}{3}C\_{P,m}&=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{P}&={\\text{molar heat capacity at constant pressure,}}\\\\C\_{V,m}&=\\left({\\frac {\\partial C}{\\partial n}}\\right)\_{V}&={\\text{molar heat capacity at constant volume,}}\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a371ac38796d12f8beb3f30b4e16181a5e1981ba) where *n* is the number of moles in the body or [thermodynamic system](https://en.wikipedia.org/wiki/Thermodynamic_system "Thermodynamic system"). One may refer to such a per-mole quantity as **molar heat capacity** to distinguish it from specific heat capacity on a per-mass basis. ### Polytropic heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=25 "Edit section: Polytropic heat capacity")\] The [polytropic](https://en.wikipedia.org/wiki/Polytropic "Polytropic") heat capacity is calculated at processes if all the thermodynamic properties (pressure, volume, temperature) change: ![{\\displaystyle C\_{i,m}=\\left({\\frac {\\partial C}{\\partial n}}\\right)={\\text{molar heat capacity at polytropic process.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d94a2aa8fde269f13f73425bd4eb2e375717ad28) The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (*γ* or *κ*). ### Dimensionless heat capacity \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=26 "Edit section: Dimensionless heat capacity")\] The [dimensionless](https://en.wikipedia.org/wiki/Dimensionless_number "Dimensionless number") heat capacity of a material is ![{\\displaystyle C^{\*}={\\frac {C}{nR}}={\\frac {C}{Nk\_{\\text{B}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/508315bfb919ceff9bf382f397926bccaa1168dc) where - ![{\\displaystyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029) is the heat capacity of a body made of the material in question (J/K), - *n* is the [amount of substance](https://en.wikipedia.org/wiki/Amount_of_substance "Amount of substance") in the body ([mol](https://en.wikipedia.org/wiki/Mole_\(unit\) "Mole (unit)")), - *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant") (J/(K⋅mol)), - *N* is the number of molecules in the body (dimensionless), - *k*B is the [Boltzmann constant](https://en.wikipedia.org/wiki/Boltzmann_constant "Boltzmann constant") (J/(K⋅molecule)). In the [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas") article, dimensionless heat capacity ![{\\displaystyle C^{\*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) is expressed as ![{\\displaystyle {\\hat {c}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8417e85ae7f4eaee7df31347ce488f85c8884b93) and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the [equipartition theorem](https://en.wikipedia.org/wiki/Equipartition_theorem "Equipartition theorem"). More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the [dimensionless entropy](https://en.wikipedia.org/wiki/Dimensionless_entropy "Dimensionless entropy") per particle ![{\\displaystyle S^{\*}=S/Nk\_{\\text{B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41f612e45f84ed9e293f85f75c6dae30943bca0b), measured in [nats](https://en.wikipedia.org/wiki/Nat_\(unit\) "Nat (unit)"). ![{\\displaystyle C^{\*}={\\frac {{\\text{d}}S^{\*}}{{\\text{d}}(\\ln T)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/294a7378c60eaed01bcea2cb12c53dd0b5bf7f90) Alternatively, using base-2 logarithms, ![{\\displaystyle C^{\*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d) relates the base-2 logarithmic increase in temperature to the increase in the dimensionless entropy measured in [bits](https://en.wikipedia.org/wiki/Bit "Bit").[\[31\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-32) ### Heat capacity at absolute zero \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=27 "Edit section: Heat capacity at absolute zero")\] From the definition of [entropy](https://en.wikipedia.org/wiki/Entropy#Thermodynamic_definition_of_entropy "Entropy") ![{\\displaystyle T\\,{\\text{d}}S=\\delta Q,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cca7738465956fa4c1d790333af389a15b667c7) the absolute entropy can be calculated by integrating from zero to the final temperature *T*f: ![{\\displaystyle S(T\_{\\text{f}})=\\int \_{T=0}^{T\_{\\text{f}}}{\\frac {\\delta Q}{T}}=\\int \_{0}^{T\_{\\text{f}}}{\\frac {\\delta Q}{{\\text{d}}T}}{\\frac {{\\text{d}}T}{T}}=\\int \_{0}^{T\_{\\text{f}}}C(T)\\,{\\frac {{\\text{d}}T}{T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df93f18f951064e02f6128a3cd99ad4c72124aa1) ## Thermodynamic derivation \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=28 "Edit section: Thermodynamic derivation")\] In theory, the specific heat capacity of a substance can also be derived from its abstract thermodynamic modeling by an [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state") and an [internal energy function](https://en.wikipedia.org/w/index.php?title=Internal_energy_function&action=edit&redlink=1 "Internal energy function (page does not exist)"). ### State of matter in a homogeneous sample \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=29 "Edit section: State of matter in a homogeneous sample")\] To apply the theory, one considers the sample of the substance (solid, liquid, or gas) for which the specific heat capacity can be defined; in particular, that it has homogeneous composition and fixed mass ![{\\displaystyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd). Assume that the evolution of the system is always slow enough for the internal pressure ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a) and temperature ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) be considered uniform throughout. The pressure ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a) would be equal to the pressure applied to it by the enclosure or some surrounding fluid, such as air. The state of the material can then be specified by three parameters: its temperature ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), the pressure ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), and its [specific volume](https://en.wikipedia.org/wiki/Specific_volume "Specific volume") ![{\\displaystyle \\nu =V/M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5e892352c30149ba6c93abe17145e15cdd73f5), where ![{\\displaystyle V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845) is the volume of the sample. (This quantity is the reciprocal ![{\\displaystyle 1/\\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/754f9b580e728c03bac621d1dd72cd606b6b5eef) of the material's [density](https://en.wikipedia.org/wiki/Density "Density") ![{\\displaystyle \\rho =M/V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed457a6aafb262f754ae8801d87eb036b8c5fe9).) Like ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) and ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), the specific volume ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) is an intensive property of the material and its state, that does not depend on the amount of substance in the sample. Those variables are not independent. The allowed states are defined by an [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state") relating those three variables: ![{\\displaystyle F(T,P,\\nu )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/471651d7fefe7e18735444a61e0ffa3f6ff84876) The function ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) depends on the material under consideration. The [specific internal energy](https://en.wikipedia.org/wiki/Specific_internal_energy "Specific internal energy") stored internally in the sample, per unit of mass, will then be another function ![{\\displaystyle U(T,P,\\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7003637ad40a88b0bf499d18811d1dab7f424ccf) of these state variables, that is also specific of the material. The total internal energy in the sample then will be ![{\\displaystyle M\\,U(T,P,\\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/287870c17350c2311e2d86e64ec9dec7bfc90a75). For some simple materials, like an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), one can derive from basic theory the equation of state ![{\\displaystyle F=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/745afacbd4fd9affdc51ac09a0ecabae08da8676) and even the specific internal energy ![{\\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025) In general, these functions must be determined experimentally for each substance. ### Conservation of energy \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=30 "Edit section: Conservation of energy")\] The absolute value of this quantity ![{\\displaystyle U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025) is undefined, and (for the purposes of thermodynamics) the state of "zero internal energy" can be chosen arbitrarily. However, by the [law of conservation of energy](https://en.wikipedia.org/wiki/Law_of_conservation_of_energy "Law of conservation of energy"), any infinitesimal increase ![{\\displaystyle M\\,\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcddca6d7c51038a1a0134cde944e55e5bf81551) in the total internal energy ![{\\displaystyle MU}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca0430adceb61aadc25575b6894c1633b41f8874) must be matched by the net flow of heat energy ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739) into the sample, plus any net mechanical energy provided to it by enclosure or surrounding medium on it. The latter is ![{\\displaystyle -P\\,\\mathrm {d} V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7238b4c977539175fa28c0cff1658e50276637a), where ![{\\displaystyle \\mathrm {d} V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80507190aa9d38a279909db47b63657f2b62ba7) is the change in the sample's volume in that infinitesimal step.[\[32\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-fein-33) Therefore ![{\\displaystyle \\mathrm {d} Q-P\\,\\mathrm {d} V=M\\,\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3401e201975bf505b5fa40018ffd2bc7eab0a6) hence ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M}}-P\\,\\mathrm {d} \\nu =\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/586d361903b89f3b074724dafc3680ca2cf85d26) If the volume of the sample (hence the specific volume of the material) is kept constant during the injection of the heat amount ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739), then the term ![{\\displaystyle P\\,\\mathrm {d} \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e54803595e3392edc2125d5bbbfd6ee753922208) is zero (no mechanical work is done). Then, dividing by ![{\\displaystyle \\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80), ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M\\,\\mathrm {d} T}}={\\frac {\\mathrm {d} U}{\\mathrm {d} T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b32da0380fcf6d0c3bc87b7dbb0c7c0c02211f80) where ![{\\displaystyle \\mathrm {d} T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/309833548168052f2696ca961149df099b5d1f80) is the change in temperature that resulted from the heat input. The left-hand side is the specific heat capacity at constant volume ![{\\displaystyle c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c1ed5361b532828e1bda88a5282706a31b5d082) of the material. For the heat capacity at constant pressure, it is useful to define the [specific enthalpy](https://en.wikipedia.org/wiki/Specific_enthalpy "Specific enthalpy") of the system as the sum ![{\\displaystyle h(T,P,\\nu )=U(T,P,\\nu )+P\\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38a0269a9025439e15bc408001d2ddd739e304a9). An infinitesimal change in the specific enthalpy will then be ![{\\displaystyle \\mathrm {d} h=\\mathrm {d} U+V\\,\\mathrm {d} P+P\\,\\mathrm {d} V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/746c99ad25909e68e8e56ca8a0b6b9bcce84317c) therefore ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M}}+V\\,\\mathrm {d} P=\\mathrm {d} h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1e31d5cd51830d6b5eb83bb7c77657c5be843c0) If the pressure is kept constant, the second term on the left-hand side is zero, and ![{\\displaystyle {\\frac {\\mathrm {d} Q}{M\\,\\mathrm {d} T}}={\\frac {\\mathrm {d} h}{\\mathrm {d} T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4007c22be9c0d1e82c5a4a011c3bd298de9f44) The left-hand side is the specific heat capacity at constant pressure ![{\\displaystyle c\_{P}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f21e3d892369c6502159212f89643451ab67e4) of the material. ### Connection to equation of state \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=31 "Edit section: Connection to equation of state")\] In general, the infinitesimal quantities ![{\\displaystyle \\mathrm {d} T,\\mathrm {d} P,\\mathrm {d} V,\\mathrm {d} U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04ccfe4aed236cb970d27013b7e0fe2e3cbf7664) are constrained by the equation of state and the specific internal energy function. Namely, ![{\\displaystyle {\\begin{cases}\\displaystyle \\mathrm {d} T{\\frac {\\partial F}{\\partial T}}(T,P,V)+\\mathrm {d} P{\\frac {\\partial F}{\\partial P}}(T,P,V)+\\mathrm {d} V{\\frac {\\partial F}{\\partial V}}(T,P,V)&=&0\\\\\[2ex\]\\displaystyle \\mathrm {d} T{\\frac {\\partial U}{\\partial T}}(T,P,V)+\\mathrm {d} P{\\frac {\\partial U}{\\partial P}}(T,P,V)+\\mathrm {d} V{\\frac {\\partial U}{\\partial V}}(T,P,V)&=&\\mathrm {d} U\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/957c9a178ff753bc04a30bed2819d7e5155314a0) Here ![{\\displaystyle (\\partial F/\\partial T)(T,P,V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24993db8c358df7a5c8539501cb70320459370e6) denotes the (partial) derivative of the state equation ![{\\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57) with respect to its ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0) argument, keeping the other two arguments fixed, evaluated at the state ![{\\displaystyle (T,P,V)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41f5cdee6675ea3447082abaa34afbaf77e4a1f) in question. The other partial derivatives are defined in the same way. These two equations on the four infinitesimal increments normally constrain them to a two-dimensional linear subspace space of possible infinitesimal state changes, that depends on the material and on the state. The constant-volume and constant-pressure changes are only two particular directions in this space. This analysis also holds no matter how the energy increment ![{\\displaystyle \\mathrm {d} Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba942d26ee837fbee5ad2a635c34446afedd739) is injected into the sample, namely by [heat conduction](https://en.wikipedia.org/wiki/Heat_conduction "Heat conduction"), irradiation, [electromagnetic induction](https://en.wikipedia.org/wiki/Electromagnetic_induction "Electromagnetic induction"), [radioactive decay](https://en.wikipedia.org/wiki/Radioactive_decay "Radioactive decay"), etc. ### Relation between heat capacities \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=32 "Edit section: Relation between heat capacities")\] For any specific volume ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468), denote ![{\\displaystyle p\_{\\nu }(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fab6e24e625d12de49f61c60a3e95d19d5f305d) the function that describes how the pressure varies with the temperature ![{\\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0), as allowed by the equation of state, when the specific volume of the material is forcefully kept constant at ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468). Analogously, for any pressure ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), let ![{\\displaystyle \\nu \_{P}(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d920635881776d5b03f98ee5492f6e0549b3d2f7) be the function that describes how the specific volume varies with the temperature, when the pressure is kept constant at ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a). Namely, those functions are such that ![{\\displaystyle F(T,p\_{\\nu }(T),\\nu )=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84a1b582c06f3c19e7abe1e3728d16e072c1a51f)and![{\\displaystyle F(T,P,\\nu \_{P}(T))=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b667b5d0711e09296d3fe4e9d3f6c395acd0f847) for any values of ![{\\displaystyle T,P,\\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f907ebc4eacdaceb323d1264f58855f033d41081). In other words, the graphs of ![{\\displaystyle p\_{\\nu }(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fab6e24e625d12de49f61c60a3e95d19d5f305d) and ![{\\displaystyle \\nu \_{P}(T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d920635881776d5b03f98ee5492f6e0549b3d2f7) are slices of the surface defined by the state equation, cut by planes of constant ![{\\displaystyle \\nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468) and constant ![{\\displaystyle P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a), respectively. Then, from the [fundamental thermodynamic relation](https://en.wikipedia.org/wiki/Fundamental_thermodynamic_relation "Fundamental thermodynamic relation") it follows that ![{\\displaystyle c\_{P}(T,P,\\nu )-c\_{V}(T,P,\\nu )=T\\left\[{\\frac {\\mathrm {d} p\_{\\nu }}{\\mathrm {d} T}}(T)\\right\]\\left\[{\\frac {\\mathrm {d} \\nu \_{P}}{\\mathrm {d} T}}(T)\\right\]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24554d2ae7d16cbb75b6ad485bf92856b55cf7bd) This equation can be rewritten as ![{\\displaystyle c\_{P}(T,P,\\nu )-c\_{V}(T,P,\\nu )=\\nu T{\\frac {\\alpha ^{2}}{\\beta \_{T}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27c864cf90d771a66e3a2fa50dec56b8d6c190ad) where both depending on the state ![{\\displaystyle (T,P,\\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/847d2a615fb19d33916ce93a4fa935919abca4a7). The [heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio"), or adiabatic index, is the ratio ![{\\displaystyle c\_{P}/c\_{V}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfcc80ce47374e8dfc55b32849c78810e782a87) of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. ### Calculation from first principles \[[edit](https://en.wikipedia.org/w/index.php?title=Specific_heat_capacity&action=edit&section=33 "Edit section: Calculation from first principles")\] The [path integral Monte Carlo](https://en.wikipedia.org/wiki/Path_integral_Monte_Carlo "Path integral Monte Carlo") method is a numerical approach for determining the values of heat capacity, based on quantum dynamical principles. However, good approximations can be made for gases in many states using simpler methods outlined below. For many solids composed of relatively heavy atoms (atomic number \> iron), at non-cryogenic temperatures, the heat capacity at room temperature approaches 3*R* = 24.94 joules per kelvin per mole of atoms ([Dulong–Petit law](https://en.wikipedia.org/wiki/Dulong%E2%80%93Petit_law "Dulong–Petit law"), *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant")). Low temperature approximations for both gases and solids at temperatures less than their characteristic [Einstein temperatures](https://en.wikipedia.org/wiki/Einstein_temperature "Einstein temperature") or [Debye temperatures](https://en.wikipedia.org/wiki/Debye_temperature "Debye temperature") can be made by the methods of Einstein and Debye discussed below. However, attention should be made for the consistency of such ab-initio considerations when used along with an equation of state for the considered material.[\[33\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-Benjelloun-34) For an [ideal gas](https://en.wikipedia.org/wiki/Ideal_gas "Ideal gas"), evaluating the partial derivatives above according to the [equation of state](https://en.wikipedia.org/wiki/Equation_of_state "Equation of state"), where *R* is the [gas constant](https://en.wikipedia.org/wiki/Gas_constant "Gas constant"), for an ideal gas[\[34\]](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_note-35) ![{\\displaystyle {\\begin{alignedat}{3}PV&=nRT,\\\\C\_{P}-C\_{V}&=T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n},\\\\P&={\\frac {nRT}{V}}\\Rightarrow \\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}&={\\frac {nR}{V}},\\\\V&={\\frac {nRT}{P}}\\Rightarrow \\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}&={\\frac {nR}{P}}.\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f8e4a06c2339161629da3ad59eddd0383a651f) Substituting ![{\\displaystyle T\\left({\\frac {\\partial P}{\\partial T}}\\right)\_{V,n}\\left({\\frac {\\partial V}{\\partial T}}\\right)\_{P,n}=T{\\frac {nR}{V}}{\\frac {nR}{P}}={\\frac {nRT}{V}}{\\frac {nR}{P}}=P{\\frac {nR}{P}}=nR,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e49d3f01b48a4771f06476e4922cd418e91e9354) this equation reduces simply to [Mayer](https://en.wikipedia.org/wiki/Julius_Robert_von_Mayer "Julius Robert von Mayer")'s relation: ![{\\displaystyle C\_{P,m}-C\_{V,m}=R.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b0951113a93ae7d847ed0234a94039782ae0a5e) The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. [![icon](https://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/40px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png)](https://en.wikipedia.org/wiki/File:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg) [Physics portal](https://en.wikipedia.org/wiki/Portal:Physics "Portal:Physics") - [Enthalpy of fusion](https://en.wikipedia.org/wiki/Enthalpy_of_fusion "Enthalpy of fusion") (latent heat of melting) - [Enthalpy of vaporization](https://en.wikipedia.org/wiki/Enthalpy_of_vaporization "Enthalpy of vaporization") (latent heat of vaporization) - [Frenkel line](https://en.wikipedia.org/wiki/Frenkel_line "Frenkel line") - [Heat capacity ratio](https://en.wikipedia.org/wiki/Heat_capacity_ratio "Heat capacity ratio") - [Heat equation](https://en.wikipedia.org/wiki/Heat_equation "Heat equation") - [Heat transfer coefficient](https://en.wikipedia.org/wiki/Heat_transfer_coefficient "Heat transfer coefficient") - [History of thermodynamics](https://en.wikipedia.org/wiki/History_of_thermodynamics "History of thermodynamics") - [Joback method](https://en.wikipedia.org/wiki/Joback_method "Joback method") (Estimation of heat capacities) - [Latent heat](https://en.wikipedia.org/wiki/Latent_heat "Latent heat") - [Material properties (thermodynamics)](https://en.wikipedia.org/wiki/Material_properties_\(thermodynamics\) "Material properties (thermodynamics)") - [Quantum statistical mechanics](https://en.wikipedia.org/wiki/Quantum_statistical_mechanics "Quantum statistical mechanics") - [R-value (insulation)](https://en.wikipedia.org/wiki/R-value_\(insulation\) "R-value (insulation)") - [Statistical mechanics](https://en.wikipedia.org/wiki/Statistical_mechanics "Statistical mechanics") - [Table of specific heat capacities](https://en.wikipedia.org/wiki/Table_of_specific_heat_capacities "Table of specific heat capacities") - [Thermal mass](https://en.wikipedia.org/wiki/Thermal_mass "Thermal mass") - [Thermodynamic databases for pure substances](https://en.wikipedia.org/wiki/Thermodynamic_databases_for_pure_substances "Thermodynamic databases for pure substances") - [Thermodynamic equations](https://en.wikipedia.org/wiki/Thermodynamic_equations "Thermodynamic equations") - [Volumetric heat capacity](https://en.wikipedia.org/wiki/Volumetric_heat_capacity "Volumetric heat capacity") 1. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-gold_30-0)** [IUPAC](https://en.wikipedia.org/wiki/International_Union_of_Pure_and_Applied_Chemistry "International Union of Pure and Applied Chemistry"), *[Compendium of Chemical Terminology](https://en.wikipedia.org/wiki/IUPAC_books#Gold_Book "IUPAC books")*, 5th ed. (the "Gold Book") (2025). Online version: (2006–) "[Standard Pressure](https://goldbook.iupac.org/terms/view/S05921.html)". [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1351/goldbook.S05921](https://doi.org/10.1351%2Fgoldbook.S05921). 1. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-1)** Halliday, David; Resnick, Robert; Walker, Jearl (2001). *Fundamentals of Physics* (6th ed.). New York, NY US: [John Wiley & Sons](https://en.wikipedia.org/wiki/John_Wiley_%26_Sons "John Wiley & Sons"). 2. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-2)** Open University (2008). *S104 Book 3 Energy and Light*, p. 59. [The Open University](https://en.wikipedia.org/wiki/The_Open_University "The Open University"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781848731646](https://en.wikipedia.org/wiki/Special:BookSources/9781848731646 "Special:BookSources/9781848731646") . 3. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-3)** Open University (2008). *S104 Book 3 Energy and Light*, p. 179. [The Open University](https://en.wikipedia.org/wiki/The_Open_University "The Open University"). [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [9781848731646](https://en.wikipedia.org/wiki/Special:BookSources/9781848731646 "Special:BookSources/9781848731646") . 4. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-4)** Engineering ToolBox (2003). ["Specific Heat of some common Substances"](https://www.engineeringtoolbox.com/specific-heat-capacity-d_391.html). 5. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-colen2001_5-0)** (2001): *Columbia Encyclopedia*, 6th ed.; as quoted by [Encyclopedia.com](https://www.encyclopedia.com/science-and-technology/physics/physics/specific-heat#1E1specheat). Columbia University Press. Accessed on 2019-04-11. 6. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-6)** Laidler, Keith J. (1993). [*The World of Physical Chemistry*](https://books.google.com/books?id=01LRlPbH80cC). Oxford University Press. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-19-855919-4](https://en.wikipedia.org/wiki/Special:BookSources/0-19-855919-4 "Special:BookSources/0-19-855919-4") . 7. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-:1_7-0)** [Ramsay, William](https://en.wikipedia.org/wiki/William_Ramsay "William Ramsay") (1918). *The life and letters of Joseph Black, M.D*. Constable. pp. 38–39\. 8. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-8)** Black, Joseph (1807). Robison, John (ed.). [*Lectures on the Elements of Chemistry: Delivered in the University of Edinburgh*](https://books.google.com/books?id=lqI9AQAAMAAJ&pg=PA76). Vol. 1. Mathew Carey. pp. 76–77\. 9. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-:0_9-0)** West, John B. (2014-06-15). ["Joseph Black, carbon dioxide, latent heat, and the beginnings of the discovery of the respiratory gases"](https://www.physiology.org/doi/10.1152/ajplung.00020.2014). *American Journal of Physiology. Lung Cellular and Molecular Physiology*. **306** (12): L1057–L1063. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1152/ajplung.00020.2014](https://doi.org/10.1152%2Fajplung.00020.2014). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [1040-0605](https://search.worldcat.org/issn/1040-0605). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [24682452](https://pubmed.ncbi.nlm.nih.gov/24682452). 10. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-10)** [International Bureau of Weights and Measures](https://en.wikipedia.org/wiki/International_Bureau_of_Weights_and_Measures "International Bureau of Weights and Measures") (2006), [*The International System of Units (SI)*](https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf) (PDF) (8th ed.), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [92-822-2213-6](https://en.wikipedia.org/wiki/Special:BookSources/92-822-2213-6 "Special:BookSources/92-822-2213-6") , [archived](https://web.archive.org/web/20210604163219/https://www.bipm.org/documents/20126/41483022/si_brochure_8.pdf) (PDF) from the original on 2021-06-04, retrieved 2021-12-16 11. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-toolbox_11-0)** ["Water – Thermal Properties"](http://www.engineeringtoolbox.com/water-thermal-properties-d_162.html). Engineeringtoolbox.com. Retrieved 2021-03-29. 12. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-12)** International Union of Pure and Applied Chemistry, Physical Chemistry Division. ["Quantities, Units and Symbols in Physical Chemistry"](http://old.iupac.org/publications/books/gbook/green_book_2ed.pdf) (PDF). Blackwell Sciences. p. 7. "The adjective specific before the name of an extensive quantity is often used to mean divided by mass." 13. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-Lange_13-0)** Lange's Handbook of Chemistry, 10th ed., page 1524. 14. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-14)** Quick, C. R.; Schawe, J. E. K.; Uggowitzer, P. J.; Pogatscher, S. (2019-07-01). ["Measurement of specific heat capacity via fast scanning calorimetry—Accuracy and loss corrections"](https://doi.org/10.1016%2Fj.tca.2019.03.021). *Thermochimica Acta*. Special Issue on occasion of the 65th birthday of Christoph Schick. **677**: 12–20\. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2019TcAc..677...12Q](https://ui.adsabs.harvard.edu/abs/2019TcAc..677...12Q). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1016/j.tca.2019.03.021](https://doi.org/10.1016%2Fj.tca.2019.03.021). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [0040-6031](https://search.worldcat.org/issn/0040-6031). 15. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-15)** Pogatscher, S.; Leutenegger, D.; Schawe, J. E. K.; Uggowitzer, P. J.; Löffler, J. F. (September 2016). ["Solid–solid phase transitions via melting in metals"](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4844691). *Nature Communications*. **7** (1) 11113. [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2016NatCo...711113P](https://ui.adsabs.harvard.edu/abs/2016NatCo...711113P). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1038/ncomms11113](https://doi.org/10.1038%2Fncomms11113). [ISSN](https://en.wikipedia.org/wiki/ISSN_\(identifier\) "ISSN (identifier)") [2041-1723](https://search.worldcat.org/issn/2041-1723). [PMC](https://en.wikipedia.org/wiki/PMC_\(identifier\) "PMC (identifier)") [4844691](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4844691). [PMID](https://en.wikipedia.org/wiki/PMID_\(identifier\) "PMID (identifier)") [27103085](https://pubmed.ncbi.nlm.nih.gov/27103085). 16. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-Koch_16-0)** Koch, Werner (2013). [*VDI Steam Tables*](https://books.google.com/books?id=bJ_wBgAAQBAJ&pg=PA8) (4 ed.). Springer. p. 8. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-3-642-52941-2](https://en.wikipedia.org/wiki/Special:BookSources/978-3-642-52941-2 "Special:BookSources/978-3-642-52941-2") . Published under the auspices of the *Verein Deutscher Ingenieure* (VDI). 17. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-17)** Cardarelli, Francois (2012). [*Scientific Unit Conversion: A Practical Guide to Metrication*](https://books.google.com/books?id=-ZveBwAAQBAJ&pg=PA19-IA35). M.J. Shields (translation) (2 ed.). Springer. p. 19. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-1-4471-0805-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4471-0805-4 "Special:BookSources/978-1-4471-0805-4") . 18. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-18)** From direct values: 1⁠BTU/lb⋅°R⁠ × 1055.06⁠J/BTU⁠ × (⁠1/0\.45359237⁠)⁠lb/kg⁠ x ⁠9/5⁠⁠°R/K⁠ = 4186.82⁠J/kg⋅K⁠ 19. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-19)** °F=°R 20. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-20)** °C=K 21. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-21)** McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: [University Science Books](https://en.wikipedia.org/w/index.php?title=University_Science_Books&action=edit&redlink=1 "University Science Books (page does not exist)"). pp. 83–85\. 22. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-22)** ["6.6: Electronic Partition Function"](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Statistical_Thermodynamics_\(Jeschke\)/06:_Partition_Functions_of_Gases/6.06:_Electronic_Partition_Function). *Chemistry LibreTexts*. 2020-11-26. Retrieved 2024-12-16. 23. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-23)** Bonhoeffer, K.F.; Harteck, P. (1926). ["Über Para- und Orthowasserstoff"](https://www.degruyter.com/document/doi/10.1515/zpch-1929-0408/html). *Z. Phys. Chem*. **4B**: 113–141\. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1515/zpch-1929-0408](https://doi.org/10.1515%2Fzpch-1929-0408). 24. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-24)** McQuarrie, Donald A. (1973). *Statistical Thermodynamics*. New York, NY: University Science Books. p. 107. 25. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-25)** Feynman, R., *[The Feynman Lectures on Physics](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics")*, Vol. 1, ch. 40, pp. 7–8 26. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-26)** Reif, F. (1965). [*Fundamentals of statistical and thermal physics*](https://archive.org/details/fundamentalsofst00reif). McGraw-Hill. pp. [253–254](https://archive.org/details/fundamentalsofst00reif/page/253). 27. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-27)** Kittel, Charles; Kroemer, Herbert (2000). *Thermal physics*. W. H. Freeman. p. 78. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [978-0-7167-1088-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7167-1088-2 "Special:BookSources/978-0-7167-1088-2") . 28. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-thor1993_28-0)** Thornton, Steven T. and Rex, Andrew (1993) *Modern Physics for Scientists and Engineers*, Saunders College Publishing 29. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-chas1998_29-0)** Chase, M.W. Jr. (1998) *[NIST-JANAF Themochemical Tables, Fourth Edition](https://webbook.nist.gov/cgi/cbook.cgi?ID=C7727379&Type=JANAFG)*, In *Journal of Physical and Chemical Reference Data*, Monograph 9, pages 1–1951. 30. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-31)** Yunus A. Cengel and Michael A. Boles, *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill, 2010, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [007-352932-X](https://en.wikipedia.org/wiki/Special:BookSources/007-352932-X "Special:BookSources/007-352932-X") . 31. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-32)** Fraundorf, P. (2003). "Heat capacity in bits". *American Journal of Physics*. **71** (11): 1142. [arXiv](https://en.wikipedia.org/wiki/ArXiv_\(identifier\) "ArXiv (identifier)"):[cond-mat/9711074](https://arxiv.org/abs/cond-mat/9711074). [Bibcode](https://en.wikipedia.org/wiki/Bibcode_\(identifier\) "Bibcode (identifier)"):[2003AmJPh..71.1142F](https://ui.adsabs.harvard.edu/abs/2003AmJPh..71.1142F). [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.1119/1.1593658](https://doi.org/10.1119%2F1.1593658). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [18742525](https://api.semanticscholar.org/CorpusID:18742525). 32. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-fein_33-0)** Feynman, Richard, *[The Feynman Lectures on Physics](https://en.wikipedia.org/wiki/The_Feynman_Lectures_on_Physics "The Feynman Lectures on Physics")*, Vol. 1, Ch. 45 33. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-Benjelloun_34-0)** S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", [Link to Archiv e-print](https://arxiv.org/abs/2105.04845) [Link to Hal e-print](https://hal.archives-ouvertes.fr/hal-03216379/) 34. **[^](https://en.wikipedia.org/wiki/Specific_heat_capacity#cite_ref-35)** Cengel, Yunus A. and Boles, Michael A. (2010) *Thermodynamics: An Engineering Approach*, 7th Edition, McGraw-Hill [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [007-352932-X](https://en.wikipedia.org/wiki/Special:BookSources/007-352932-X "Special:BookSources/007-352932-X") . - Emmerich Wilhelm & Trevor M. Letcher, Eds., 2010, *Heat Capacities: Liquids, Solutions and Vapours*, Cambridge, U.K.:Royal Society of Chemistry, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)") [0-85404-176-1](https://en.wikipedia.org/wiki/Special:BookSources/0-85404-176-1 "Special:BookSources/0-85404-176-1") . A very recent outline of selected traditional aspects of the title subject, including a recent specialist introduction to its theory, Emmerich Wilhelm, "Heat Capacities: Introduction, Concepts, and Selected Applications" (Chapter 1, pp. 1–27), chapters on traditional and more contemporary experimental methods such as [photoacoustic](https://en.wikipedia.org/wiki/Photoacoustic_effect "Photoacoustic effect") methods, e.g., Jan Thoen & Christ Glorieux, "Photothermal Techniques for Heat Capacities," and chapters on newer research interests, including on the heat capacities of proteins and other polymeric systems (Chs. 16, 15), of liquid crystals (Ch. 17), etc. - (2012-05may-24) [Phonon theory sheds light on liquid thermodynamics, heat capacity – Physics World](https://physicsworld.com/a/phonon-theory-sheds-light-on-liquid-thermodynamics/) [The phonon theory of liquid thermodynamics \| Scientific Reports](https://www.nature.com/articles/srep00421)

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