Homework problems #1
1. Helmholtz free energy of liquids can by approximated by
F(T,V,N) = −kT ln



h2
2πmkT
−3N/2

4
3
π
√
2
V
N
1/3
−σ
3


N


,
where h, m, k and σ are constants. Determine the equation of state of this gas p = p(T,V), the difference
of specific heats cp −cV , and show that both formulae correspond to ideal gas for V/N ≫ σ.
2. Computer problem: The number of microstates of the system of N classical noninteracting particles with
energy lower than E in a volume V is given by
Ω(E) =
2πmE
h2
3
2 N
VN
N!Γ(3
2N +1)
.
Let us study two such systems in a thermal interaction with total energy 2E. We shall study fluctuations
in these systems, within which one of the system has energy E +∆E and the second one E −∆E. Plot the
number of microstates of combined system as a function of ∆E for N = 10, 102, 104. (It is advisable to
express the energy in units of ε = h2/(2πm) an number of states in units of γ = VN/(N!Γ(3
2N +1)).) Let
us assume that ∆E changes discountinuously by the value of ε/100 and determine an error introduced by
assuming that the entropy is calculated only from the states corresponding to the equilibrium. (calculate
for N = 10, 102, 104, 105).
The solution should be submitted not later than on April 2nd.
1