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Application of Maxwell's Relations

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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!
 
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We can then multiply and divide dU by dT to get $$TdS = (\frac{\partial U}{\partial T})_V dT + PdV.$$
Well, this is certainly not true! Recall that given a multivariable function ##f(x,y)##, we have
[tex]df(x,y) = \left(\frac{\partial f}{\partial x}\right)_{y} dx + \left(\frac{\partial f}{\partial y}\right)_{x} dy [/tex]
 
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Well, this is certainly not true! Recall that given a multivariable function ##f(x,y)##, we have
[tex]df(x,y) = \left(\frac{\partial f}{\partial x}\right)_{y} dx + \left(\frac{\partial f}{\partial y}\right)_{x} dy [/tex]
I solved it!
Consider the fact that $$ dU = (\frac{\partial U}{\partial S}) dS + (\frac{\partial U}P\partial V}) dV.$$ Then since ##dU = TdS - PdV##, we can see that ##\frac{\partial U}{\partial S} = T##, or ##\frac{\partial S}{\partial U} = \frac{1}{T}##. Now, consider a function ##S=S(U,V)##.
$$dS = (\frac{\partial S}{\partial U}) dU_V + (\frac{\partial S}{\partial V}) dV$$
Using one of the Maxwell relations, we know that ##\frac{\partial S}{\partial V} = \frac{\partial P}{\partial T}##. We also know that ##dU_V = C_V dT##. Therefore, we have
$$dS = \frac{1}{T} C_V dT + (\frac{\partial P}{\partial T}) dV$$
and multiplying through by T gives
$$TdS = C_V dT + T(\frac{\partial P}{\partial T}) dV$$

A similar process using S(U,P) will give the other. Thank you!
 

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