Problems and Solutions on Thermodynamics and Statistical Mechanics

304 Problems d Sdutiow on Thermcdylamics d Statistical Mechanics

(e) When N >> 1, the most probable energy is the average energy

NE, M NE

2NEsinh (s)

1 + 2 COS~ (g) ’

where a = exp(E/kT).

(f) JmmdT = lmdS = S(o0) - S(0) = Nkln3.

oT

2126
Find the pressure, entropy, and specific heat at constant volume of an
ideal Boltzmann gas of indistinguishable particles in the extreme relativistic
limit, in which the energy of a particle is related to its momentum by E = cp.
Express your answer as functions of the volume V, temperature T, and
number of particle N.
(Princeton)
Solution:
Let z denote the partition function of a single particle, Z the total
partition function, p the pressure, S the entropy, U the internal energy,
and c the specific heat. We have

81rV

N