Difference between constant-pressure and constant-volume heat capacities ( $C_{P} - C_{V}$ )

For a simple compressible system,

C_{P} - C_{V} = T (\frac{\partial V}{\partial T})_{P} (\frac{\partial P}{\partial T})_{V} .

This follows by expressing the entropy as a function of temperature and volume and using a Maxwell relation.

Heat-capacity difference and mechanical stability ( $C_{p} - C_{V} = T V α_{V}^{2} / κ_{T} \geq 0$ )

C_{p} - C_{V} = T (\partial p / \partial T)_{V} (\partial V / \partial T)_{p}

. At fixed volume, the differential equation of state gives

(\partial p / \partial T)_{V} = α_{V} / κ_{T}

, proving the displayed result. Positive absolute temperature and positive isothermal compressibility imply a nonnegative difference. It vanishes when thermal expansion does, so strict inequality requires a nonzero expansion coefficient. This stability argument includes nonideal internal-energy changes, unlike a purely ideal-gas expansion-work explanation.

For a fixed-composition simple compressible thermodynamic system with smooth equations of state, write the internal energy as

U (T, V)

H = U + p V

. Differentiating

H

at constant pressure and applying the multivariable chain rule gives

C_{p} = (\frac{\partial U}{\partial T})_{p} + p (\frac{\partial V}{\partial T})_{p}, (\frac{\partial U}{\partial T})_{p} = (\frac{\partial U}{\partial T})_{V} + (\frac{\partial U}{\partial V})_{T} (\frac{\partial V}{\partial T})_{p} .

Subtracting the heat capacity at constant volume proves the displayed identity. For an ideal gas,

(\partial U / \partial V)_{T} = 0

and

(\partial V / \partial T)_{p} = N k_{B} / p

, so

C_{p} - C_{V} = N k_{B}

. For a nonideal material the internal energy change with volume contributes as well as expansion work.

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