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 Incoming links: Volume derivative form of the heat-capacity difference

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 2 / 9 / b / Solution Created 2026-10-18 Updated 2026-10-20

Differentiate

H = U + p V

at fixed pressure and subtract the heat capacity at constant volume:

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

Now use

U = U (T, V)

and the multivariable chain rule along a constant-pressure path:

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

Substitution cancels the fixed-volume derivative and gives the volume derivative form of the heat-capacity difference,

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

For the ideal gas, the first term in brackets vanishes and

V = N k T / p

, so

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

. Consequently

C_{p} - C_{V} = N k

for the entire gas, with fixed particle number

N

. The extra heat capacity accounts for expansion work at constant pressure.

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