Statistical Mechanics - Physics at Oregon State University

74 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
4.2 Boltzmann gas again.
An ideal gas is defined as a system of (1) free, (2) non-interacting particles in
the (3) classical regime. Restriction (1) means that there are no external forces,
beyond those needed to confine the particles in a box of volume V. Restriction (2)
means that there are no internal forces. Restriction (3) means that we can ignore
the quantum effects related to restrictions in occupation numbers. This means
that we use the Maxwell-Boltzmann distribution function. Restrictions (1) and
(3) are easy to relax. Most of solid state physics is based on such calculations,
where electrons can only singly occupy quantum states in a periodic potential.
Restriction (2) is much harder to relax, and many discussions in physics focus
on these correlation effects.
Ideal gas again.
In this section we derive the results for the ideal gas again, but now from
the independent particle point of view. This implicitly resolves the debate of
the factor N! The energy levels of the particles in the box are the same as we
have used before. The ideal gas follows the Boltzmann distribution function
and hence we find
N =
fMB(ɛo) =
orb
orb
e µ−ɛo
k B T = e µ
k B T Z1
We again introduce the quantum concentration nQ(T ) and solve for µ :
MkBT
nQ(T ) =
2π2 3
2
n
µ = kBT log
nQ(T )
(4.25)
(4.26)
(4.27)
The requirement µ ≪ −kBT is indeed equivalent to n ≪ nQ(T ). The internal
energy is also easy to obtain:
U =
orb
ɛoe µ−ɛo
µ
kB T k = e B T 2 ∂
kBT
∂T
2 NkBT
U =
Z1
∂Z1
∂T N,V
orb
ɛo − k e B T (4.28)
= 3
2 NkBT (4.29)
as we had before. To get to the last step we used that Z1 is proportional to T 3
2 .
Always the same strategy.
In many cases one uses the formula for N in the same way we did here.
From it we solved for µ(T, N, V ) and we also obtained U(N,T,V) accordingly.