Complete the below:
Q: 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]
(c) Find the low and high temperature limits (T → 0 and T → ∞) of S.