1549380323-Statistical Mechanics Theory and Molecular Simulation

212 Canonical ensemble

c. Show that the second virial coefficient in the low density limit becomes

B 2 (T) =− 2 π

∫∞

0

dr r^2 f(r)

wheref(r) = e−βu(r)−1.

4.6. An ideal gas ofNparticles of massmat temperatureTis in a cylindrical

container with radiusaand lengthL. The container rotates about its cylin-

drical axis (taken to be thezaxis) with angular velocityω. In addition, the

gas is subject to a uniform gravitational field of strengthg. Therefore, the

Hamiltonian for the gas is

H=

∑N

i=1

h(ri,pi),

whereh(r,p) is the Hamiltonian for a single particle

h(r,p) =

p^2

2 m

−ω(r×p)z+mgz.

Here, (r×p)zis thez-component of the cross product betweenrandp.

a. Show, in general, that when the Hamiltonian is separable in this manner,

the canonical partition functionQ(N,V,T) is expressible as

Q(N,V,T) =

1

N!

[q(V,T)]N,

where

q(V,T) =

1

h^3

∫

dp

∫

D(V)

dre−βh(r,p).

b. Show, in general, that the chemical potentialμ(N,V,T) is given by

μ(N,V,T) =kTln

[

Q(N− 1 ,V,T)

Q(N,V,T)

]

,

whereQ(N− 1 ,V,T) is the partition function for an (N−1)-particle

system.

c. Calculate the partition function for this ideal gas.

d. Calculate the Helmholtz free energy of the gas.

e. Calculate the total internal energy of the gas.

f. Calculate the heat capacity of the gas.

∗g. What is the equation of state of the gas?