Editing Specific heat capacity (section)

==Physical basis ==
{{main|Molar heat capacity#Physical basis}}
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]].

===Monatomic gases===
[[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).<ref>{{Cite book |last=McQuarrie |first=Donald A. |title=Statistical Thermodynamics |publisher=[[University Science Books]] |year=1973 |location=New York, NY |pages=83–85}}</ref>  Thus, the [[molar heat capacity|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, <math>c_{V,\mathrm{m}} = 3R/2 \approx \mathrm{12.5 \, J \cdot K^{-1} \cdot mol^{-1}}</math> and <math>c_{P,\mathrm{m}} = 5R/2 \approx \mathrm{21 \, J \cdot K^{-1} \cdot mol^{-1}}</math>, where <math>R \approx \mathrm{8.31446 \, J \cdot K^{-1} \cdot mol^{-1}}</math> is the [[ideal gas constant|ideal gas unit]] (which is the product of [[Boltzmann constant|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]] <math>A</math>. That is, approximately,

<math display="block">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</math>

For the noble gases, from helium to xenon, these computed values are
{| class="wikitable"
!Gas
!He!!Ne!!Ar!!Kr!!Xe
|-
!<math>A</math>
| 4.00||20.17||39.95||83.80||131.29
|-
!<math>c_V</math> (J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>)
|3118||618.3||312.2||148.8||94.99
|-
!<math>c_p</math> (J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>)
|5197||1031||520.3||248.0||158.3
|}

===Polyatomic gases===
On the other hand, a polyatomic gas molecule (consisting of two or more atoms bound together) can store heat 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 [[Rotational energy|rotations]] of the molecule and vibration of the atoms relative to each other (including internal [[potential energy]]). 

There may also be contributions to the heat capacity from [[Excited state|excited electronic states]] for molecules where the energy gap between the ground state and the excited state is sufficiently small, such as {{chem|NO}}.<ref>{{Cite web |date=2020-11-26 |title=6.6: Electronic Partition Function |url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Statistical_Thermodynamics_(Jeschke)/06:_Partition_Functions_of_Gases/6.06:_Electronic_Partition_Function |access-date=2024-12-16 |website=Chemistry LibreTexts |language=en}}</ref> 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 {{chem|H|2}} due to ortho/para separation,<ref>{{Cite journal |last1=Bonhoeffer |first1=K.F. |last2=Harteck |first2=P. |date=1926 |title=Über Para- und Orthowasserstoff |url=https://www.degruyter.com/document/doi/10.1515/zpch-1929-0408/html |journal=Z. Phys. Chem. |volume=4B |pages=113–141|doi=10.1515/zpch-1929-0408 }}</ref> which arises from [[Spin quantum number|nuclear spin]] statistics, has been referred to as "one of the great triumphs of post-quantum mechanical statistical mechanics."<ref>{{Cite book |last=McQuarrie |first=Donald A. |title=Statistical Thermodynamics |publisher=University Science Books |year=1973 |location=New York, NY |pages=107}}</ref>

These extra [[degrees of freedom (physics and chemistry)|degrees of freedom]] or "modes" contribute to the specific heat capacity of the substance. Namely, when heat energy is injected into a gas with polyatomic molecules, only part of it will go into increasing 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 energy 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.<ref>Feynman, R., ''[[The Feynman Lectures on Physics]]'', Vol. 1, ch. 40, pp. 7–8</ref><ref>{{cite book |author= Reif, F. |year=1965 |title=Fundamentals of statistical and thermal physics |url= https://archive.org/details/fundamentalsofst00reif |url-access= registration |publisher=McGraw-Hill |pages =[https://archive.org/details/fundamentalsofst00reif/page/253 253–254]}}</ref><ref>{{cite book |last1=Kittel |first1=Charles |last2=Kroemer |first2=Herbert  |year=2000 |title=Thermal physics |publisher=W. H. Freeman |isbn=978-0-7167-1088-2 |page=78}}</ref>

[[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 heat 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 energy.

For example, the molar heat capacity of [[nitrogen]] {{chem|N|2}} at constant volume is <math>c_{V,\mathrm{m}} = \mathrm{20.6 \, J \cdot K^{-1} \cdot mol^{-1}}</math> (at 15&nbsp;°C, 1&nbsp;atm), which is <math>2.49 R</math>.<ref name="thor1993">Thornton, Steven T. and Rex, Andrew (1993) ''Modern Physics for Scientists and Engineers'', Saunders College Publishing</ref> That is the value expected from the [[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 <math>c_V</math> of {{chem|N|2}} (736&nbsp;J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>) is greater than that of an hypothetical monatomic gas with the same molecular mass 28 (445&nbsp;J⋅K<sup>−1</sup>⋅kg<sup>−1</sup>), by a factor of {{sfrac|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&nbsp;°C to about 300&nbsp;°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 <math>c_V</math> starts to increase rapidly at first, then slower as it tends to another constant value. It is 35.5&nbsp;J⋅K<sup>−1</sup>⋅mol<sup>−1</sup> at 1500&nbsp;°C, 36.9 at 2500&nbsp;°C, and 37.5 at 3500&nbsp;°C.<ref name=chas1998>Chase, M.W. Jr. (1998) ''[https://webbook.nist.gov/cgi/cbook.cgi?ID=C7727379&Type=JANAFG NIST-JANAF Themochemical Tables, Fourth Edition]'', In ''Journal of Physical and Chemical Reference Data'', Monograph 9, pages 1–1951.</ref>  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.