# Clairaut's theorem?

1. Mar 15, 2009

### Miriverite

1. The problem statement, all variables and given/known data
Given dU = TdS - PdV, prove that:
1. (dT/dV)S = -(dP/dS)V
2. (dS/dV)T = (dP/dT)V

2. Relevant equations
Clairaut's states that d^2U/dXdY = d^2U/dYdX is true, or in my case:
[d/dX (dU/dY)X ]Y = [d/dY (dU/dX)Y ]X

3. The attempt at a solution
1. (dU/dS)V = T(S,V) and (dU/dV)S = -P(T,V)
so [d/dS (dU/dV)S]V = [d/dV (dU/dS)V]S
and that means [-P(S,V)/dS]V = [T(S,V)/dV]S
and thus, if we differentiate again (-dP/dS)V = (dT/dV)S

2. [d/dT (dU/dV)T]V = [d/dV (dU/dT)V]T
(dU/dV)T = -P(S,V) and (dU/dT)V = dS(T) (unsure of this)
so that means [-P(S,V)/dT]V = [dS(t)/dV)]T
This is where I get stuck, because there's a dS on the right hand side, but it's only P on the left hand, not dP. Am I missing an important step here?

Obviously we need to get the equation for dU into a form that depends not on S and V, but something that depends on T and V. I don't know how to do this, though..

Last edited: Mar 15, 2009