Problems and Solutions on Thermodynamics and Statistical Mechanics

(Ann) #1
Statistical Physics 333

2145
A classical gas of N point particles occupies volume V at tempera-
ture T. The particles interact pairwise, d(r;j) being the potential between
particles i and j, r,j = Ir, - rjl. Suppose this is a “hard sphere” potential


(a) Compute the constant volume specific heat as a function of tem-
V
perature and specific volume v = -.
N


PV

(b) The virial expansion for the equation of state is an expansion of


  • in inverse powers of V:
    RT


L1+-+T+... A2(T).
RT V

Compute the virial coefficient Al.
(Princeton)

Solution
For the canonical distribution, the partition function is

z=-.- ’ ’ /e-”dqdp
N! h3N
=’(-> 27rm 3N/2 .Q,
N! Ph2

where qi represents the coordinates and p; the momentum of the ith parti-
cle. Defining the function f;j = exp[-Pd(r;j)] - 1 with f;, = 0 for r;j > a,
we can write
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