Past exam of the physics B course of the University of Cambridge / 2017 / Paper 2 / Section C / C11 / a / Solution

Start with the fundamental relation at fixed material amount,

d U = T d S - p d V

, and expand entropy in independent variables

T, V

:

d S = (\frac{\partial S}{\partial T})_{V} d T + (\frac{\partial S}{\partial V})_{T} d V = \frac{C _{V}}{T} d T + (\frac{\partial p}{\partial T})_{V} d V .

The last equality uses the supplied Maxwell relation. Consequently

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

For the excluded-volume ideal gas,

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

, so the bracket vanishes:

(\frac{\partial U}{\partial V})_{T} = 0, U = U (T), d U = C_{V} d T .

Once

U

depends only on temperature,

C_{V} = d U / d T

cannot depend on volume either. Its stipulated temperature independence therefore makes it a constant, giving

U = C_{V} T + U_{*}

with arbitrary energy reference

U_{*}

. Excluded volume changes the pressure without adding the volume-dependent attractive internal energy that a more general interacting-gas model might have.

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