Entropy derivations for 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]
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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]
(c) Find the low and high temperature limits (T → 0 and T → ∞) of S.
I have no idea how to derive this at all. I am unsure as to how to do parts a, b, or c. HELP!!!
(a)
S = k [N ln N - Nu ln Nu - Nd ln Nd],
T-1 = (∂S/∂U)N,B
1/T = (∂S/∂U)N,B = k ∂/∂U [ - Nu ln Nu - Nd ln Nd],
Note: Since N is a constant, ...
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