Thermodynamics, Statistical Physics, and Quantum Mechanics

THERMODYNAMICS AND STATISTICAL PHYSICS 239

We then obtain, from (S.4.105.2),

The cross derivatives of (S.4.105.3) are equal so

The heat capacity at constant magnetic field is given by

from which we obtain

By again exchanging the order of differentiation in (S.4.105.6) and using
the result found in (S.4.105.4), we have

Replacing M by in (S.4.105.7) yields the desired

b) For an adiabatic process, the entropy S is constant. Writing
we compose the differential

and by (S.4.105.4) and (S.4.105.5),