[Thermo] - Maxwells Relations? Prove the Validity of the Equations

[tesla22]

Homework Statement

We are working on some problems for class and we are given statements which I accept as valid but don't know how to prove they are valid. I believe I have to utilize the maxwell relations but the terms seem unfamiliar to me.

Homework Equations

(1)
Partial
(d^2f / ds^2)_T = T / Kv

(2)
Partial
(d^2h / ds^2)_P = T / C_p

(3)
Partial
(d^2u / ds^2)_v = T / C_v

The Attempt at a Solution

For the first equation, I know that the isothermal compressibiity K = -1/v partial(dv/dP)_T

For the second equation I also know that C_p/T = partial(ds/dT)_P = 1/T * partial(dh/dT)_P and i need to take this knowledge to combine the equations but I don't see how I would be getting the square on the ds out of this. Obviously I'm missing something.

C_v/T should be the same proof with u in place of h holding v constant.

Thanks for any help.

 

[Mapes]

Take a look at https://www.physicsforums.com/showpost.php?p=1681973&postcount=4". To handle the second derivative, note that \left(\frac{\partial H}{\partial S}\right)_P is also known as T. So differentiate T with respect to S again at constant pressure. Does this help?

 

[tesla22]

Thank You, I think the answers were staring me in the face but in a different format and i didn't put 2 and 2 together until your response. I now have a solution where T=du/ds with v constant and T=dh/ds with p constant (correct?) and plugging these into the equations cv=T(ds/dt)_v and cp=T(dt/ds)_p to get the desired proof. I think its right.

I'm still struggling with the third one however so any assistance would be appreciated. How do you make the proper symbols appear?

 

[Mapes]

The first relation can't be true, the units don't match up.

 

[tesla22]

The first relation can't be true, the units don't match up.

Yes that's the one I'm struggling with sorry, the first one. The 2nd and 3rd I believe I have proven as said in the previous post.

I completely screwed up the first one, the actual problem is as follows:

(d^2f / dv^2)_T = 1 / Kv

This is the one I am struggling with at the moment. Sorry I was trying too hard to convey the format of the problem that i got the variables wrong somehow.

 

[Mapes]

OK, that makes a lot more sense. Try writing out the differential form of the Helmholtz free energy dF and taking the derivative of that twice with respect to V at constant T. The first derivative equals a well-known parameter.