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**1. Homework Statement**

This is question 2.18 from Bowley and Sanchez, "Introductory Statistical Mechanics" .

Show with the help of Maxwell's Relations that

$$T dS = C_v dT + T (\frac{\partial P}{\partial T})_V dV$$

and

$$TdS = C_p dT - T( \frac{\partial V}{\partial T})_P dP.$$

Then, prove that

$$(\frac{\partial U}{\partial V})_T = T (\frac{\partial P}{\partial T})_V - P$$

**2. Homework Equations**

The four common Maxwell's Relations. They can be found here: https://en.wikipedia.org/wiki/Maxwell_relations.

**3. The Attempt at a Solution**

The issue for me is the first part. I have managed to prove that the second is true.

My attempt thus far:

Using the fact that ##dU = TdS - PdV##, we can rewrite the equation as $$TdS = dU + PdV.$$

We can then multiply and divide dU by dT to get $$TdS = (\frac{\partial U}{\partial T})_V dT + PdV.$$ However, ##\frac{\partial U}{\partial T}_V = C_v##, and so we have $$TdS = C_v dT + PdV.$$

This then leaves me with the issue of how to convert the term PdV ##\to T (\frac{\partial P}{\partial T})_V dV##. I was considering using the relation ##\frac{\partial P}{\partial T} = \frac{\partial S}{\partial V}##, but I'm not sure how to get that to work. I'll keep working at it, but any help is appreciated!