Statistical Mechanics - Physics at Oregon State University

A.3. SOLUTIONS FOR CHAPTER 3 245
(A) Calculate the partition function Z1(T, V ) for N=1.
(B) Calculate Z(T, V, N).
(C) Calculate p(T, V, N), S(T, V, N), and µ(T, V, N).
Ignore spin
Transform to integral
Z1(T, V ) =
Z1(T, V ) =
nx,ny,nz
√ cπ − k e B T L n2 x +n2 y +n2z x = cπ
kBT L (nx, ny, nz)
cπ
−3
1
kBT L 8
d 3 xe −x
where we extended the range of integration to all values of x,and not only one
octant.
3 ∞
kBT L
Z1(T, V ) =
4πx
c2π 0
2 dxe −x
3 3 kBT L
kBT
Z1(T, V ) =
8π = V 8π
c2π
c2π
This is allowed at low density, when the points in x-space are close together
Z(T, V, N) = 1
3N N kBT
V (8π)
N! c2π
N
F = −kBT log(Z) = −NkBT log(8π V
3 kBT
) − NkBT
N c2π
Where we used Stirling
Check:
∂F
p = −
=
∂V T,N
NkBT
V
∂F
S = −
= NkB log(8π
∂T V,N
V
N
µ =
∂F
∂N
T,V
3 kBT
) + 4NkB
c2π
= −kBT log(8π V
3 kBT
)
N c2π