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Consider an ideal 2-state paramagnet

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Consider an ideal 2-state paramagnet. The entropy is given by:

S = k [N ln N - Nu ln Nu - Nd ln Nd],

where Nu and Nd are related to total number N, internal energy U, magnetic field B and magnetic moment μ by

Nu = (N - U/μB)/2, Nd = (N + U/μB)/2

(a) Use the relation T-1 = (∂S/∂U)N,B to derive U/μB = -N tanh x (where x = μB/kT)

(b) Show that S can be written as

S = Nk [ln (2 cosh x) - x tanh x]