Problem 5.12 (Schroeder’s Intro to Thermal Physics)

Problem 5.12

Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. Therefore, for instance,

$\dfrac{\partial }{\partial V} \Bigl( \dfrac{\partial U}{\partial S} \Bigr) = \dfrac{\partial }{\partial S} \Bigl( \dfrac{\partial U}{\partial V} \Bigr) ,$

where each $\partial / \partial V$ is taken with $S$ fixed, each $\partial / \partial S$ is taken with $V$ fixed, and $N$ is always held fixed. From the thermodynamic identity (for $U$ ) you can evaluate the partial derivatives in parentheses to obtain

$\Bigl( \dfrac{\partial T}{\partial V} \Bigr)_S = - \Bigl( \dfrac{\partial P}{\partial S} \Bigr)_V ,$

a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Then derive an analogous Maxwell relation from each of the other three thermodynamic identities discussed in the text (for H, F, and G). Hold N fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to N, but after you’ve done four of them the novelty begins to wear off. For applications of these Maxwell relations, see the next four problems.

Solution:

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Problem 5.12 (Schroeder’s Intro to Thermal Physics)

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Problem 5.12 (Schroeder’s Intro to Thermal Physics)

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