Problems and Solutions on Thermodynamics and Statistical Mechanics

254 Problems d SdlLtiond on Thermodytamics d Statistical Mechanics

Because exp(-thw/rkT) decreases rapidly as t -+ 1, we have

where A = (16Nk2/h) t2exp(-[)dE.
Hence a = 7 = 1.

2084

Given the energy spectrum

EP = [(Pc)~ + m,,c 2 4 1 112 + pc as p-+ 00.

(a) Prove that an ultrarelativistic ideal fermion gas satisfies the equa-

tion of state pV = E/3, where E is the total energy.

(b) Prove that the entropy of an ideal quantum gas is given by

S= -k~[n;ln(n;)i(lfn;)ln(l+n,)]

a

where the upper (lower) signs refer to bosons (fermions).

(SUNY, Buflulo)

Solution:

(a) The number of states in the momentum interval p to p + dp is

87rV 1

Fp2dp (taking S = -). From E = cp, we obtain the number of states in

2

the energy interval E to E + dE:

87rV

c3h3

N(e)de = -e2de.

So the total energy is