Blundell Thermal Physics · Problem 16.2

Thermodynamic potentials

Physics — Statistical Mechanics · thermodynamic-expansions-partial-derivatives

Problem

(a) Derive the following general relations (i)(∂T∂V)U=−1CV[T(∂p∂T)V−p],(ii)(∂T∂V)S=−1CVT(∂p∂T)V,(iii)(∂T∂p)H=1Cp[T(∂V∂T)p−V].\begin{aligned} \text{(i)}\quad &\left(\frac{\partial T}{\partial V}\right)_U = -\frac{1}{C_V}\left[T\left(\frac{\partial p}{\partial T}\right)_V - p\right],\\ \text{(ii)}\quad &\left(\frac{\partial T}{\partial V}\right)_S = -\frac{1}{C_V}T\left(\frac{\partial p}{\partial T}\right)_V,\\ \text{(iii)}\quad &\left(\frac{\partial T}{\partial p}\right)_H = \frac{1}{C_p}\left[T\left(\frac{\partial V}{\partial T}\right)_p - V\right]. \end{aligned} In each case the quantity on the left-hand side is the appropriate thing to consider for a particular type of expansion. State what type of expansion each refers to. (b) Using these relations, verify that for an ideal gas (∂T/∂V)U=0(\partial T/\partial V)_U = 0 and (∂T/∂p)H=0(\partial T/\partial p)_H = 0, and that (∂T/∂V)S(\partial T/\partial V)_S leads to the familiar relation pVγ=constantpV^\gamma = \text{constant} along an isentrope (a curve of constant entropy).

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