Past exam of the chemistry A course of the University of Cambridge / 2024 / Paper 2 / A8 / d

Following the restricted model in the preceding part, its electronic factor for $N$ independent magnetic molecules is $Q_{\mathrm{mag}}=(2\cosh x{)}^N$. Its field-dependent Helmholtz free energy is thereforewhere $A_{\mathrm{ind}}$ denotes terms independent of the field. Equivalently, the change from zero field is $A(B)-A(0)=-N{k}_{\mathrm{B}}T\ln \cosh x$. Retaining the excited-state offset simply adds $N\Delta$ to the restricted free energy.

If $N$ instead counts all molecules in thermal equilibrium, the correct electronic expression isThe ordinary translational indistinguishability factor is field independent and does not alter either magnetic derivative.

Differentiation of the restricted free energy gives the magnetizationIn the weak-field limit $|\mu B|\ll k_{\mathrm{B}}T$, $\tanh x\simeq x$, soThe printed coefficient is inconsistent with the stated moment $\mu =2{\mu }_{\mathrm{B}}$. The requested expression without the factor four would follow from $\mu ={\mu }_{\mathrm{B}}$ in this restricted model. This discrepancy is present in the original PDF, rather than introduced by its transcription.

For completeness, differentiating the full thermal free energy givesIts weak-field form isAt $k_{\mathrm{B}}T\gg \Delta$ this approaches $2N{\mu }_{\mathrm{B}}^{2}B/(V{k}_{\mathrm{B}}T)$, so restoring the two nonmagnetic ground states does not recover the printed coefficient either. The inverse-temperature Curie law form is valid, but the population factor and moment squared must be retained consistently.