The enthalpy and Gibbs free energy, as defined in this

Chapter 5, Problem 17P

(choose chapter or problem)

The enthalpy and Gibbs free energy, as defined in this section, give special treatment to mechanical (compression-expansion) work, -P dV. Analogous quantities can be defined for other kinds of work, for instance, magnetic work. Consider the situation shown in below Figure 5.7, where a long solenoid (N turns, total length L) surrounds a magnetic specimen (perhaps a paramagnetic solid). If the magnetic field inside the specimen is \(\vec{B}\) and its total magnetic moment is \(\vec{M}\), then we define an auxiliary field \(\vec{\mathcal{H}}\) (often called simply the magnetic field) by the relation

\(\vec{\mathcal{H}} \equiv \frac{1}{\mu_{0}} \vec{B}-\frac{\vec{M}}{V}\)

where \(\mu_{0}\) is the “permeability of free space,” \(4\pi\times10^{-7}\mathrm{\ N}/\mathrm{A}^2\). Assuming cylindrical symmetry, all vectors must point either left or right, so we can drop the \(\longrightarrow\) symbols and agree that rightward is positive, leftward negative. From Ampere’s law, one can also show that when the current in the wire is I, the \(\mathcal{H}\) field inside the solenoid is NI/L, whether or not the specimen is present.

(a) Imagine making an infinitesimal change in the current in the wire, resulting in infinitesimal changes in B, M, and \(\mathcal{H}\). Use Faraday’s law to show that the work required (from the power supply) to accomplish this change is \(W_{\text {total }}=V \mathcal{H} d B\). (Neglect the resistance of the wire.)

(b) Rewrite the result of part (a) in terms of \(\mathcal{H}\) and M, then subtract off the work that would be required even if the specimen were not present. If we define W, the work done on the system to be what’s left, show that \(W=\mu_{0} \mathcal{H} d M\).

(c) What is the thermodynamic identity for this system? (Include magnetic work but not, mechanical work or particle flow.)

(d) How would you define analogues of the enthalpy and Gibbs free energy for a magnetic system? (The Helmholtz free energy is defined in the same way as for a mechanical system.) Derive the thermodynamic identities for each of these quantities, and discuss their interpretations.

Figure 5.7: A long solenoid, surrounding a magnetic specimen, connected to a power supply that can change the current, performing magnetic work.