How to obtain maxwell relations

[B-80]

Hey, I have had a lot of trouble understanding how one obtains a Maxwell relation.

So let's say in general I know(from a specific problem)

T ds = dE - F dL

where F is a tension and L is a length, E is the energy T is the temperature and S is the entropy of a system.

In a specific problem I am asked to compute

ds/dL at constant Temp.

I have an expression for F and it turns out that ds/dL = dF/dT but I don't know how one arrives at this point.

Any help is appreciated, thanks.

 

[Bill_K]

Write it as dE = T ds + F dL. Define a function H = E - Ts, for which dH = - s dT + F dL. The Maxwell relation follows from the fact that the mixed second partial derivatives of H are equal:

∂²H/∂T∂L = ∂²H/∂L∂T which is - (∂s/∂L) = ∂F/∂T

(Did you have the signs right?)