# Thermodynamics: integrating partial derivatives

1. Sep 12, 2007

### psid

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

Given that mechanical equation of state for a paramagnetic substance is
$$m=\left(\frac{DH}{T}\right)$$
where D is a constant, H is the magnetic field, m is molar magnetization and the molar heat capacity
$$c_{m}$$ is constant, find entropy and enthalpy

2. Relevant equations
$$\left(\frac{\partial S}{\partial T}\right)_{m}=\frac{C_{m}}{T}$$
$$\left(\frac{\partial S}{\partial m}\right)_{T}=-\left(\frac{\partial H}{\partial T}\right)_{m}$$

3. The attempt at a solution
Integrating above equations I found that
$$S=-\frac{m^{2}}{2D}+c_{m}\ln{\frac{T}{T_{0}}}.$$
Or alternatively
$$T=T_{0}\exp{(2DS+m^{2})/(2DC)}.$$
Now to the entalphy h. By definition
$$\left(\frac{\partial h}{\partial S}\right)_{H}=T=T_{0}\exp{(2DS+m^{2})/(2DC)}$$
and
$$\left(\frac{\partial h}{\partial H}\right)_{S}=-m=DH/T.$$
Now, I'm a bit confused here. When integrating these kind of partial derivatives, I guess that one should express the integrand using only those variables that are used to perform the differentiation, i.e. S and H in this particular case. If this is the case, one should probably eliminate m from above equation using the equation of state. But this gives
$$T=T_{0}\exp{(2DS+DH^{2}/(2CT^{2}))/(2DC)}$$
which makes no sense as T appears on both sides... What's going wrong?

Last edited: Sep 12, 2007