= Mayer's relation
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= Mayer relation
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Mayer's relation states that the difference between <heat capacity at constant pressure> and <heat capacity at constant volume> for a fixed-composition <ideal gas> is
$$
C_p(T)-C_V(T)=nR=Nk_B.
$$
Here both capacities are total <heat capacities> in $\mathrm{J\,K^{-1}}$, $n$ is <amount of substance>, $N$ is the number of particles, $R$ is the <molar gas constant> and $k_B$ is the <Boltzmann constant>. In other normalizations the same statement is
$$
C_{p,\mathrm m}(T)-C_{V,\mathrm m}(T)=R,\qquad
c_p(T)-c_V(T)=\frac{nR}{m}.
$$
The first is a difference of <molar heat capacities>, in $\mathrm{J\,mol^{-1}\,K^{-1}}$; the second uses <specific heat capacities>, in $\mathrm{J\,kg^{-1}\,K^{-1}}$. Writing $C_p-C_V=R$ with total capacities would omit the sample's <amount of substance>. The relation holds pointwise even when the capacities depend on <temperature>. https://openstax.org/books/university-physics-volume-2/pages/3-5-heat-capacities-of-an-ideal-gas[Samuel J. Ling, William Moebs and Jeff Sanny, University Physics Volume 2, §3.5: ideal-gas capacity difference]{ref}
The proof outline is short: <enthalpy> is $H=U+pV=U+nRT$, and <internal energy of an ideal gas> is independent of <volume (thermodynamics)> at fixed <temperature>. Differentiating at fixed composition and <amount of substance> therefore gives $C_p=dH/dT=dU/dT+nR=C_V+nR$. It uses the <ideal gas law>, rather than assuming any particular number of molecular <degrees of freedom>. At fixed <pressure>, the additional <heat> supplies expansion <work>, $p\,dV=nR\,dT$. The two processes can have the same <temperature> endpoints and equal $\Delta U$ while requiring different <heat> inputs.
For one <mole> of the rigid-rotation diatomic <ideal gas> model, $C_V=5R/2$ and $C_p=7R/2$. Heating by $10\,\mathrm K$ with $R\simeq8.31446\,\mathrm{J\,mol^{-1}\,K^{-1}}$ gives $Q_V\simeq207.86\,\mathrm J$ in a rigid sealed vessel and $Q_p\simeq291.01\,\mathrm J$ under a frictionless piston at fixed <pressure>. Both gain $207.86\,\mathrm J$ of <internal energy>, while the piston delivers approximately $83.14\,\mathrm J$ of <work>. This is a calculated model budget. If the capacities vary with <temperature>, the corresponding finite statement is still
$$
Q_p-Q_V=nR(T_2-T_1),\qquad
Q_V=n\int_{T_1}^{T_2}C_{V,\mathrm m}(T)\,dT,
$$
under the fixed-<pressure> and fixed-<volume (thermodynamics)> heating assumptions in <heat capacity>. Their final <volume (thermodynamics)> values generally differ. Combining <Mayer's relation> with <heat capacity ratio> also gives $C_V=nR/(\gamma-1)$ and $C_p=\gamma nR/(\gamma-1)$ locally; constant values cannot be extrapolated across intervals in which $\gamma(T)$ changes.
For a smooth stable single <thermodynamic phase> of a general <simple compressible system>, the broader <Maxwell relation> identity is
$$
C_p-C_V=T\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_p
=\frac{TV\alpha^2}{\kappa_T},
$$
where $\alpha$ is the <coefficient of thermal expansion> and $\kappa_T$ the <isothermal compressibility>, with composition fixed throughout. Substituting the <ideal gas law> gives $nR$; a <real gas> need not do so. This distinction matters near a <phase transition> or when interactions appreciably change the <equation of state>. It explains why <Mayer's relation> is useful: thermal measurements under different constraints become linked to a mechanical response, while the difference retains a particularly simple value for an <ideal gas>.<Statistical Physics by David Tong/4 Classical Thermodynamics>[§4.4.2: Maxwell relations and the general heat-capacity difference]{ref}
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