# Thermodynamics Identity

1. Jun 9, 2007

### neelakash

1. The problem statement, all variables and given/known data

To prove that (∂u/∂T)_P=c_P – Pβv where _P =>P constant;β=>co-eff. of vol exp.

2. Relevant equations
3. The attempt at a solution

I proved it for ideal gases.
Write d'Q=dh-vdP
Now expand d'Q with 1st law and du(in 1st law) in terms of dP and dT.Since du is a total differential.
Expand dh as total differential in dP and dT.

Now I used the property of ideal gases:(∂u/∂v)_T=(∂h/∂P)_T=0
The rest is a bit manipulation.

Can anyone say how to prove this in general?
The book does not mention specifically that "use ideal behaviour" or so.
There must be some way to get it.

2. Jun 10, 2007

### siddharth

Use the first law.
Since $$dU = \delta Q - PdV$$, take the partial derivative wrt to T at constant pressure, and you get the answer.

For ideal gases, it's even easier, since U is only a function of T

3. Jun 11, 2007

### neelakash

Thank you...
Earlier I did not get this...