Hi I am confuse with thermodynamical potentials $F$ and $G$. I am trying to find a expression for the heat capacity at constant magnetic field $H$. So I start at:
$$C_h=\frac{1}{V}\left(\frac{dQ}{dT}\right)_h
=\frac{T}{V}\left(\frac{dS}{dT}\right)_h.$$
Now in some books (https://theory.physics.manchester.ac.uk/~judith/stat_therm/node72.html) they use the Helmholtz potential $F=-k_\mathrm{B}T\ln(Z_N)$ with $Z_N$ the partion function in the canonical. So $F=-\frac{dS}{dT}$ and
$C_h=-\frac{T}{V}\left(\frac{d^2F}{dT^2}\right)_h$
But in most of the bibliography, when they do the same proccess to find $C_p$ they always use the Gibbs potential $G$. Since $p$ and $h$ are the extensive varible I think that is more correct to use $G$ also in the magentism counterpart of $C_p$.
Another way to see the same is that $dF=\mu_0HdM-SdT$ so I think that the expresion $F=-\frac{dS}{dT}$ is not correct when $H$ is constant but when $M$ is.
In conclusion, why do books use Helmholtz energy to find $C_h$ when they must use the Gibbs potential.
Best Answer
The thermodynamics of magnetic systems can be somewhat subtle. Even discussing something as seemingly basic as the energy of such systems requires extreme care. This is because energy is stored in the "distortion" of the system itself as well as the externally applied field, and the energies are usually comparable in systems of interest (consider for example a piston compressing an ideal gas, which may be held in place by a conservative force like a spring; without some kind of implicit or explicit convention in place, would it be immediately clear whether to include the energy of the piston in the total energy of the 'system'?) Another factor to consider is that many magnetic systems are 'typically' found in condensed phases (solids especially), and under 'typical' conditions (i.e. atmospheric ambient pressure) it is reasonable to treat them as incompressible (i.e. $p\delta V \ll U_{magnetic}$, and so $U_M\approx H_M$ and $F_M\approx G_M$.) There are times when one might be interested in the magnetic properties of gases, but without extremely powerful fields, the magnetic contribution to the total energy would be tiny compared to the translational and angular energies, and/or internal bond energies (the subtle interplay between elastic/Coulomb energies and the electronic spin degrees of freedom is what allows condensed phases to exhibit such striking magnetic phenomena.)
So in short, it is probably more accurate to use the Gibbs free energy when calculating a specific heat to use for various queries, given that pressure can be easier to measure than the at times microscopic compression of a magnetic sample, but in situations where you expect $F_M$ to be large (and not a very subtle correction to a somewhat turbid free energy landscape dominated by other phenomena), it is probably forgivable to use the Helmholtz free energy instead.