Maxwell relations Thermodynamics

[thonwer]

Homework Statement

Show that: [itex] (\frac{∂T} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n [/itex]

Homework Equations

[itex] dU=TdS-PdV+μdn [/itex]

The Attempt at a Solution

[itex] \frac {∂} {∂S} (\frac{∂U} {∂V})_S,_n=-(\frac {∂P} {∂S})_V,_n [/itex]

[itex] \frac {∂} {∂V} (\frac{∂U} {∂S})_V,_n=(\frac{∂T} {∂V})_S,_n [/itex]

I tried to isolate T and P, but I get stuck:
[itex] TdS=dU+PdV [/itex]

[itex] -PdV=dU-TdS [/itex]

How can I demonstrate that they are equal?

 

[CaptCoonoor]

 

[Chestermiller]

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

 

[thonwer]

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

 

[Chestermiller]

Yes I see that, but I ment to obtain [itex] −(\frac {∂P} {∂S})_V,_n [/itex] from [itex] (\frac {∂T} {∂V})_S,_n [/itex] or viceversa.

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

Chet

 

[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.

 

[Chestermiller]

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.

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

Chet

 

[thonwer]

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

 

[Chestermiller]

Why would you think it's not continuous?

Chet

 

[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.

 

[Chestermiller]

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.

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

 

[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.

 

[Chestermiller]

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.

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

Chet

 

[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.