Relation between specific heats for a magnetic system

Question

I have been trying to prove the relations between specific heats at constant magnetization and magnetic field H, i.e. $C_H$ and $C_M$. I know that the relationship between $C_V$ and $C_V$ is given as $$\kappa_T (C_P-C_V) = TV\alpha_P^2$$ and I want to prove similar relations for magnetic systems. Problem is, I have been stuck at it for the past two days and am getting increasingly exasperated.

How do I go about doing this? No matter what I try, I end up getting stuck somewhere. My starting point is to write entropy as $S\equiv S(T,V,M)$ but even that leaves me nowhere.

Is there a source I can refer to that lists out the correct formulae for the four thermodynamic potentials in a magnetic system. I know that $dU=TdS-PdV+HdM$, and I want to find such equations for all thermodynamic potentials.

Answer 1

Neglecting pressure and volume, your energy is: $$ dU=TdS+HdM $$ You can therefore apply the same method by formally substituting $P\to-H$ and $V\to M$. The analogue for enthalpy is: $$ \mathcal H = U-HM \\ d\mathcal H = TdS-MdH $$ and define the respective heat capacities: $$ C_M = \left(\frac{\partial U}{\partial T}\right)_M \quad C_H = \left(\frac{\partial \mathcal H}{\partial T}\right)_H $$ By analogy, you can immediately write Mayer’s relation: $$ C_H-C_M=T\frac{\alpha^2}{\chi}\\ \alpha = \left(\frac{\partial M}{\partial T}\right)_H \quad \chi= \left(\frac{\partial M}{\partial H}\right)_T $$ with $\chi$ magnetic susceptibility replacing compressibility and $\alpha$ magnetic "expansion" (there must be a more common name in the literature) replacing thermal expansion.

The proof is the same, the goal is to express $C_M$ in terms of $T,H$. You just need to use Maxwell's relation, i.e. the existence of state functions. For this, it is best to rewrite the expressions of heat capacities: $$ C_M = T\left(\frac{\partial S}{\partial T}\right)_M \quad C_H = T\left(\frac{\partial S}{\partial T}\right)_H $$ Using the chain rule: $$ \left(\frac{\partial S}{\partial T}\right)_M = \left(\frac{\partial S}{\partial T}\right)_H+\left(\frac{\partial S}{\partial H}\right)_T\left(\frac{\partial H}{\partial T}\right)_M $$ From Maxwell's relation on free enthalpy: $$ G = \mathcal H-TS \quad dG = -SdT-MdH \\ \left(\frac{\partial S}{\partial H}\right)_T = \left(\frac{\partial M}{\partial T}\right)_H $$ Then, using: $$ dM = \left(\frac{\partial M}{\partial T}\right)_HdT+\left(\frac{\partial M}{\partial H}\right)_TdH \\ \left(\frac{\partial H}{\partial T}\right)_M = -\frac{\left(\frac{\partial M}{\partial T}\right)_H}{\left(\frac{\partial M}{\partial H}\right)_T} $$ so you get: $$ T\left(\frac{\partial S}{\partial T}\right)_M = T\left(\frac{\partial S}{\partial T}\right)_H-T\frac{\left(\frac{\partial M}{\partial T}\right)_H}{\left(\frac{\partial M}{\partial H}\right)_T}\left(\frac{\partial M}{\partial T}\right)_H $$ Substituting the definition of the various coefficients, you recover Mayer's relation.