# Thermodynamics, magnetic system

1. Jul 20, 2013

### fluidistic

1. The problem statement, all variables and given/known data
Hi guys, once again I'm having troubles with a magnetic system problem. Here it is:
Consider a magnetic system described by the state equation $M=f(B,T)$ where M is the magnetization of the system and B is the applied magnetic field; f is differentiable.
1)Express the differential of internal energy in funtion of T and M and determine which condition the specific heat at constant magnetization must satisfy in terms of the state equation.
2)Assume now that the state equation has the form M=f(B/T) where f is continuously differentiable and only depends on the ratio B/T. Verify that under this assumption U and $C_M$ only depend on T.
3)Assume now that the state function has the particular form $M=DB/T$ (Curie's law) and $C_M$ is constant. Write down the fundamental equation in the entropy representation.

2. Relevant equations
Lots I guess.

3. The attempt at a solution
I'm stuck at part 1), unfortunately.
So first, I think it is fair if I consider that U depends only on M, T and n where n is fixed. I don't know if I can make this assumption or start with the fact that U is a function of S, V, M and n; where V and n are fixed.
Assuming the former, then $dU=\left ( \frac{\partial U}{\partial M} \right ) _{T,n} dM+ \left ( \frac{\partial U}{\partial T} \right ) _{M,n} dT$$=C_MdT+\left [ B-T \left ( \frac{\partial B}{\partial T} \right ) _{M,n} \right ] dM$.
Using a cyclic relation on the $\left ( \frac{\partial B}{\partial T} \right ) _{M,n}$ term I reach that $dU=C_MdT+ \left [ B+T \frac{\left ( \frac{\partial M}{\partial T} \right ) _B }{\left ( \frac{\partial M}{\partial B} \right ) _T} \right ] dM$. The $C_M$ bothers me, but I don't think I could rewrite it with M's and T's as they are asking. The partial derivatives also bother me, I don't know if I must rewrite that expression.
I don't really know the form of the expression they are asking me. Any tip is appreciated. Thank you.