Editing Specific heat capacity (section)

=== Calculation from first principles ===
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 temperature]]s or [[Debye temperature]]s 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.<ref name="Benjelloun">S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", [https://arxiv.org/abs/2105.04845 Link to Archiv e-print] [https://hal.archives-ouvertes.fr/hal-03216379/ Link to Hal e-print]</ref>

==== Ideal gas ====
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<ref>Cengel, Yunus A. and Boles, Michael A. (2010) ''Thermodynamics: An Engineering Approach'', 7th Edition, McGraw-Hill {{ISBN|007-352932-X}}.</ref>

<math>\begin{alignat}{3}
P V &= n R T,\\
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{alignat}</math>

Substituting

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

this equation reduces simply to [[Julius Robert von Mayer|Mayer]]'s relation:

<math display="block">C_{P,m} - C_{V,m} = R.</math>

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.