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# Entropy derivations for an ideal 2-state paramagnet

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 &#956; by

Nu = (N - U/&#956;B)/2, Nd = (N + U/&#956;B)/2

(a) Use the relation T-1 = (&#8706;S/&#8706;U)N,B to derive U/&#956;B = -N tanh x (where x = &#956;B/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 &#956; by

Nu = (N - U/&#956;B)/2, Nd = (N + U/&#956;B)/2

(a) Use the relation T-1 = (&#8706;S/&#8706;U)N,B to derive U/&#956;B = -N tanh x (where x = &#956;B/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 &#8594; 0 and T &#8594; &#8734;) 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 = (&#8706;S/&#8706;U)N,B
1/T = (&#8706;S/&#8706;U)N,B = k &#8706;/&#8706;U [ - Nu ln Nu - Nd ln Nd],

Note: Since N is a constant, ...

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