Problems and Solutions on Thermodynamics and Statistical Mechanics

202 PmMems d Sdutiom on Thermodynamics d Statistical Mechunica

system, as a function of temperature. Do not attempt to evaluate these
expressions.

(c) By simplifying your expressions in (b), derive an expression for the
specific heat C(T) that is valid at very low temperatures. In what range of
temperatures is your expression valid?

(d) By simplifying your answer to (b), derive a high temperature ap-

proximation to the specific heat C(T). What is the range of validity of your

approximat ion?

(Prince ton)

Solution:

(a) For a classical rotator, one has

E=- 1 1

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z = /e-pEdpadp,dOdp = p 8r21 ,

a 1

(E) = --lnz = - = kT.

ap @

Thus C(T) = k.

(b) In quantum statistical mechanics,

@h h2

(c) In the limit of low temperatures, - >> 1, or -- >> kT, so only

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