# Thermodynamics exercise

1. Sep 23, 2014

### prehisto

1. The problem statement, all variables and given/known data
Hi!
My problem :
There is given substance for which (dV/dT)_p =ap^2,where V-volume,T-temperature,p-pressure,a-const.
How much the entropy S will increase if T=const and pressure changes from p1 to p2?

2. Relevant equations

3. The attempt at a solution
So
1) Mathematically the question of this excersie could be represented as
dS=Int [(dS/dp)_T dp] from p1 to p2
2) The next step should be to use Gibs potential : dG=Vdp-SdT
From Gibbs potential i should derive something like this (dV/dT)_p = -(dS/dp)_T (*)

But i can go as far as dG=(dG/dp)dp -(dG/dT)dT -> (dG/dp)_T=V ;(dG/dT)_p=-S
I think that the problem is mathematical: i can't figure out how from dG i can get (*)

P.S. I'm sorry about the equations, i cant find any option for equation builder.

2. Sep 23, 2014

### Staff: Mentor

You're very close to having the answer. Here's what you have so far:
$$\left(\frac{\partial G}{\partial P}\right)_T=V$$
$$\left(\frac{\partial G}{\partial T}\right)_P=-S$$
Now, what happens if you take the partial of the first equation with respect to T and the partial of the second equation with respect to P?

Chet

3. Sep 24, 2014

### prehisto

it s dG/dpdT=dV/dP and dG/dTdp=-dS/dp
I assume there is property of dG when dG/dpdT=dG/dTdp and so -dS/dp=dV/dP! Which allows me to substitute dS/dp=-dV/dp in integral and i can now easily integrate.
Thanks for the help!

P.S. Still having problems with formulas ,do you use "latex"code in insert "code" section ? I does not work for me :(

4. Sep 24, 2014

### Staff: Mentor

Yes. That's a property of partial differentiation. The second partial with respect to two independent variables is independent of the order of the partial differentiation.
I've just been doing the latex in the original message. When you preview the actual message, it comes out looking the way it is supposed to look. Also, it comes out looking right if, after you post the reply, you click on the refresh "circle" next to the back arrow (next to the url).

chet