Thermodynamics - Calorific Capacity

[Jalo]

Homework Statement

Consider a system with a constant number of particles.
Write the total differential dS in terms of the derivarives \frac{∂S}{∂T} and \frac{∂S}{∂V}. Introduce C_V (calorific capacity at constant volume).
Next write the total differential of the volume dV in terms of the parcial derivatives \frac{∂V}{∂T} and \frac{∂V}{∂P}. Assume that the pressure is constant. Show that the result comes in the form of:

C_P-C_V= Expression

Homework Equations

C_P=T\frac{∂S}{∂T} , P and N Constant
C_V=T\frac{∂S}{∂T} , V and N Constant

The Attempt at a Solution

First I wrote the differential of the entropy as asked:

dS = \frac{∂S}{∂T}dT + \frac{∂S}{∂V}dV

I know that \frac{∂S}{∂V} = C_V/T. Substituting I get:

dS = C_V/T dT + \frac{∂S}{∂V}dV

Next I found the differential of the volume:

dV = \frac{∂V}{∂T}dT + \frac{∂V}{∂P}dP

Since the pressure is constant it reduces to the form

dV = \frac{∂V}{∂T}dT

Substituting in our dS expression we get:

dS = C_V/T dT + \frac{∂S}{∂V}\frac{∂V}{∂T}dT =
= C_V/T dT + \frac{∂S}{∂T}dT =
= C_V/T dT + C_P/T dT ⇔
⇔ T\frac{dS}{dT} = C_V + C_P

I'm making some mistake. If anyone could point me in the right direction I'd appreciate.

Thanks!

 

[haruspex]

I think your problem is a confusion over the meanings of different partial derivatives. Your use of $\frac{∂S}{∂V}$ refers to changing volume, keeping temperature constant.
That's inconsistent with later equating $\frac{∂S}{∂V}\frac{∂V}{∂T} = \frac{∂S}{∂T}$, which, to be valid, assumes pressure constant in all terms.