Statistical Mechanics - Physics at Oregon State University

152 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS.
very general statement for such variables is easy to prove in classical statistical
mechanics. Suppose we have one component c of the 6N-dimensional vector X
in phase space which appears in the Hamiltonian in a quadratic way:
H( X) = H ′ + αc 2
(7.41)
where H ′ does not depend on c. The partition function then has the form
Z = Z ′
∞
αc2 − k dce B T
′ πkBT
= Z
α
(7.42)
−∞
where Z ′ does not contain any reference to the component c anymore. The
internal energy U is related to Z by U = − ∂
∂β log(Z) and hence we find
U = U ′ + 1
2 kBT (7.43)
Hence the energy associated with a quadratic variable is simply 1
2 kBT .
Diatomic gas.
For an ideal mono-atomic gas we have kinetic energy only, and since this
involves 3N components, the internal energy of an ideal gas is 3
2 NkBT . For an
ideal diatomic gas the situation is more complicated. At relatively low temperatures
only the motion of the center of mass plays a role and the internal energy
is again 3
2 NkBT . At intermediate temperatures the molecules will rotate freely,
adding two quadratic variables to the Hamiltonian, e.g. the two rotational
momenta. Hence the energy is 5
2 NkBT . At high temperatures the atoms are
able to vibrate. This now adds two quadratic coordinates to the Hamiltonian,
one momentum variable and one variable representing the distance between the
atoms. Hence the energy is 7
2 NkBT . At very high temperatures the molecules
will start to break apart and we lose the vibrational bond energy associated
with the position coordinate. The two atoms together are now described by six
momentum coordinates and the energy becomes 3NkBT .
7.8 Effects of the potential energy.
The ideal gas is an exceptional system, since it contains no potential energy term
in the Hamiltonian. As a result D is proportional to V N and the only volume
dependence of the entropy is a term NkB log(V ). Because
∂S
p
∂V = U,N T we
find pV = NkBT . Hence the ideal gas law holds for any system for which
the Hamiltonian does not depend on the generalized coordinates, but only on
the momenta! A second case occurs when we assume that each molecule has
a finite volume b. The integration over each spatial coordinate then excludes
a volume (N − 1)b and D is proportional to (V − Nb) N for large values of N,
where (N − 1)b ≈ Nb. This gives rise to the factor V − Nb in the van der Waals
equation of state.