# Maxwell relations Thermodynamics

1. Nov 29, 2014

### thonwer

1. The problem statement, all variables and given/known data
Show that: $(\frac{∂T} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n$

2. Relevant equations
$dU=TdS-PdV+μdn$

3. The attempt at a solution
$\frac {∂} {∂S} (\frac{∂U} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n$

$\frac {∂} {∂V} (\frac{∂U} {∂S})_V,_n=(\frac{∂T} {∂V})_S,_n$

I tried to isolate T and P, but I get stuck:
$TdS=dU+PdV$

$-PdV=dU-TdS$

How can I demonstrate that they are equal?

2. Nov 29, 2014

### CaptCoonoor

3. Nov 29, 2014

### Staff: Mentor

It looks like you already had it in your first two equations of Attempt at a Solution. You are aware that 2nd partial derivatives with respect to two variables are interchangeable (commutative), correct?

Chet

4. Nov 30, 2014

### thonwer

Yes I see that, but I ment to obtain $−(\frac {∂P} {∂S})_V,_n$ from $(\frac {∂T} {∂V})_S,_n$ or viceversa, demonstrating Schwartz relation in some way.

5. Nov 30, 2014

### Staff: Mentor

Isn't that what your first two equations under Attempt demonstrate?

Chet

6. Nov 30, 2014

### thonwer

If I say they are equal, I am assuming that Schwartz relation is valid in this case, or that U is a continuos function. I want to demonstrate why Schwartz relation is valid.

7. Nov 30, 2014

### Staff: Mentor

What's wrong with assuming that U is a continuous function for a single-component single-phase material?

Chet

8. Nov 30, 2014

### thonwer

Nothing, but in an exam I would have to reason why I assume that U is continuous, so if I could go from $-\left( {\frac{\partial P} {\partial S}}\right)_{V,n}$ to $\left( {\frac{\partial T} {\partial V}}\right)_{S,n}$ by using derivatives and their properties, the problem would be solved I think.

9. Nov 30, 2014

### Staff: Mentor

Why would you think it's not continuous?

Chet

10. Nov 30, 2014

### thonwer

I think it's continuous, I know from theory, but it's not a given in the problem, so if i use it, I have to say why. As I don't know how to justify that U is continuous, I think that, if I can demonstrate that Schwartz relation is valid, then, I can avoid justifying its continuity.

11. Nov 30, 2014

### Staff: Mentor

I don't know what to say. Are you a mathematician or a physicist? If the latter, why would you think that internal energy is not a continuous function of entropy and volume?

Chet

12. Nov 30, 2014

### thonwer

I'm studying Physics, and I know it is a continuous function of entropy and volume, but one thing is knowing and another is proving. I need to prove this knowledge.

13. Nov 30, 2014

### Staff: Mentor

It's observed experimentally. Does that count as proof? Otherwise you need to start looking into statistical thermo.

Chet

14. Nov 30, 2014

### thonwer

Mmm statiscal thermo is a subject I will study next year, so if there's no other proof I suppose experimental proof is what I need.Thank you.