# Proving Thermodynamics equations using partial derivatives

1. Apr 3, 2010

### cowcow8866

1. The problem statement, all variables and given/known data
Prove
(∂V/∂T)_s/(∂V/∂T)_p = 1/1-(gamma) (gamma = Cp/Cv)

2. Relevant equations
(∂V/∂T)_s = -C_v (kappa)/(beta)T (where beta = 1/V(∂V/∂T)_p, kappa = -1/V(∂V/∂P)_T

C_v= - T(∂P/∂T)_v(∂V/∂T)_s
3. The attempt at a solution
As part(a) ask me to find C_v, I do it similar for C_p

(∂S/∂T)_p=1/T(∂U/∂T)_p
C_p=T(∂S/∂T)_p=-T(∂P/∂T)_s/(∂P/∂S)_T=-T(∂P/∂T)_s(∂V/∂T)_p
(∂V/∂T)_s/(∂V/∂T)_p = -C_v/T(∂P/∂T)_v /C_p/-T(∂P/∂T)_s=C_v(∂P/∂T)_s/ C _p(∂P/∂T)_v

Then, i cannot figure out the remaining calculation out....

Last edited: Apr 3, 2010
2. Apr 3, 2010

### Mapes

How are you getting

$$\left(\frac{\partial S}{\partial T}\right)_P=\frac{1}{T}\left(\frac{\partial U}{\partial T}\right)_P\mathrm{?}$$

That's not correct; starting from $dU=T\,dS-P\,dV$, I get

$$\left(\frac{\partial U}{\partial T}\right)_P=T\left(\frac{\partial S}{\partial T}\right)_P-P\left(\frac{\partial V}{\partial T}\right)_P$$

$$\left(\frac{\partial S}{\partial T}\right)_P=\frac{1}{T}\left(\frac{\partial U}{\partial T}\right)_P+\frac{P}{T}\left(\frac{\partial V}{\partial T}\right)_P$$

Also, I see a sign error on at least one of your Maxwell relations. Not sure if this resolves the problem?

3. Apr 4, 2010

### cowcow8866

There is another derived equation in my book and I have been used it to prove it. However, I still have another question.
Since there is a previous question ask me to show
(∂C_p/∂P)_T = -T(∂^2V/∂T^2)_P
I use this method stated below but I get stuck.
http://i35.photobucket.com/albums/d199/cowcow8866/scan001.png

Actually, I derive this$$\left(\frac{\partial S}{\partial T}\right)_P=\frac{1}{T}\left(\frac{\partial U}{\partial T}\right)_P\mathrm{?}$$
from another equation TdS=dU-VdP and then prove(∂C_p/∂P)_T = -T(∂^2V/∂T^2)_P successfully.
However, it is very uncommon to be used or even it maybe wrong...

4. Apr 5, 2010

### Mapes

There is no such equation $T\,dS=dU-V\,dP$. Are you thinking of $T\,dS=dH-V\,dP$?

5. Apr 5, 2010

### cowcow8866

Maybe......
Anyway, I can solve this question now.

My next question is
Prove (∂C_p/∂P)_T = -T(∂^2V/∂T^2)_P
http://i35.photobucket.com/albums/d199/cowcow8866/scan001.png
I have done up to this step. How can I do the remaining part of this question??

6. Apr 5, 2010

### Cilabitaon

Sorry, I know I'm not a mod or anything, heck I've been here like 3 days =]

Isn't this kinda in the wrong section? Because tbh I'm currently doing A2 physics and I'd consider that to be kind of the end of introductory physics, what you're doing here seems a bit more advanced (yes, the maths is simple, but still beyond your average physics student imo).

Also, shouldn't you create a new thread for a new question.

Again, I'm not trying to be uppity, I'm just saying.

Last edited by a moderator: May 4, 2017
7. Apr 5, 2010

### Mapes

Your writeup contains the same mistake I pointed out in my post #2 for the previous problem. $(\partial U/\partial T)_P\neq C_P$.

8. Apr 5, 2010

### cowcow8866

I know the mistake now. Thank you.

9. Apr 5, 2010

### Mapes

You're welcome!