# $(\frac{\partial U}{\partial P})_T$ derivation

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1. Dec 3, 2015

### wololo

1. The problem statement, all variables and given/known data

2. Relevant equations
Maxwell relations

3. The attempt at a solution
Here is how I proceeded. Am I allowed to go from line 1 to 2? It almost seems too simple.
$$dU=TdS-PdV \\ (\frac{\partial U}{\partial P})_T=T(\frac{\partial S}{\partial P})_T-P(\frac{\partial V}{\partial P})_T \\ \text{We know that}-(\frac{\partial S}{\partial P})_T=(\frac{\partial V}{\partial T})_P \\ (\frac{\partial U}{\partial P})_T=-T(\frac{\partial V}{\partial T})_P-P(\frac{\partial V}{\partial P})_T \\ (\frac{\partial U}{\partial P})_T=-[T(\frac{\partial V}{\partial T})_P+P(\frac{\partial V}{\partial P})_T]$$

2. Dec 3, 2015

### Staff: Mentor

This seems fine, as long as you are allowed to use the "we know that" equation, which comes from the Maxwell relations.