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

1. 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.

2. 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

3. 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.

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Mapes
Homework Helper
Gold Member
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?

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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 in to 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
Homework Helper
Gold Member
The first relation can't be true, the units don't match up.

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