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Problem 5.14. The partial-derivative relations derived in Problems 1.46, 3.33, and 5.12, plus a bit more partial-derivative trickery, can be used to derive a completely general relation between Cp and Cv. a) With the heat capacity expressions from Problem 3.33 in mind, first consider S to be a function of T and V. Expand dS in terms of the partial derivatives S/Tv and S/VT. Note that one of these derivatives is related to Cv. b) To bring in Cp, consider V to be a function of T and P and expand dV in terms of partial derivatives in a similar way. Plug this expression for dV into the result of part a), then set dP = 0 and note that you have derived a nontrivial expression for (S/Tp). This derivative is related to Cp, so you now have a formula for the difference Cp - Cv. c) Write the remaining partial derivatives in terms of measurable quantities using a Maxwell relation and the result of Problem 1.46. Your final result should be KT. d) Check that this formula gives the correct value of Cp - Cv for an ideal gas. e) Use this formula to argue that Cp cannot be less than Cv.

          Problem 5.14. The partial-derivative relations derived in Problems 1.46, 3.33, and 5.12, plus a bit more partial-derivative trickery, can be used to derive a completely general relation between Cp and Cv. 
a) With the heat capacity expressions from Problem 3.33 in mind, first consider S to be a function of T and V. Expand dS in terms of the partial derivatives S/Tv and S/VT. Note that one of these derivatives is related to Cv. 
b) To bring in Cp, consider V to be a function of T and P and expand dV in terms of partial derivatives in a similar way. Plug this expression for dV into the result of part a), then set dP = 0 and note that you have derived a nontrivial expression for (S/Tp). This derivative is related to Cp, so you now have a formula for the difference Cp - Cv.
c) Write the remaining partial derivatives in terms of measurable quantities using a Maxwell relation and the result of Problem 1.46. Your final result should be KT.
d) Check that this formula gives the correct value of Cp - Cv for an ideal gas.
e) Use this formula to argue that Cp cannot be less than Cv.

problem 514the partial derivative relations derived in problems l46333 and 512plus a bit more partial derivative trickerycan be used to derive a com pletely general relation between cp and c 96405

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