Blundell Thermal Physics · Problem 20.9

The partition function

Physics — Statistical Mechanics · paramagnet-isothermal-entropy

Problem

The energy of a paramagnet can be written as U=−m⋅BU = -\boldsymbol{m} \cdot \boldsymbol{B}. Writing TS=U+FTS = U + F, show that if B\boldsymbol{B} is varied isothermally then T δS=−B⋅δm.(20.61)T\,\delta S = -\boldsymbol{B} \cdot \delta \boldsymbol{m}. \tag{20.61} [Hint: use m=−(∂F/∂B)T\boldsymbol{m} = -(\partial F/\partial \boldsymbol{B})_T.] Show that this is consistent with δU=T δS−m⋅δB\delta U = T\,\delta S - \boldsymbol{m} \cdot \delta \boldsymbol{B} (as in eqn 17.29).

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