Statistical Mechanics - Physics at Oregon State University

154 CHAPTER 7. CLASSICAL STATISTICAL MECHANICS.
where f(N, x) is a function related to g(N, x).
If there are only Coulomb interactions, γ = −1, and we find p = T 4 f(N, V T 3 ).
In general we can write the equation in the form
pV
T
3
3
−
= V T γ −
f(N, V T γ (7.51)
The value of γ can therefore be determined from an experiment in which
we vary p and T in such a way that pV/T is constant. That implies V γ /T 3
is constant and a plot of V versus T gives γ. This description is valid for a
gas of Na + and Cl − ions at low density. The forces between ions can only be
approximated by simple Coulomb forces when the ions are far apart. At small
distances one has to include the effects of induced dipoles. Also, for a gas of
inert atoms we expect to have van der Waals interactions, in which case we have
γ = −6.
7.9 Problems for chapter 7
Problem 1.
In the presence of a magnetic induction B the classical Hamiltonian representing
the energy of the system, including the interaction energy with the
external field, follows from the Hamiltonian H(p1, · · · , pN , r1, · · · , rN ) without
a field by minimal substitution. The momenta pi are replaced by pi − e
c A(ri)
where A is the vector potential defined by B = ∇ × A.
A. Calculate the free energy G(T, B, N, V ) = U − T S − M · B
B. Prove van Leeuwen’s theorem that diamagnetism does not exist in classical
mechanics by showing that M( B) = 0 for all values of the induction B.
Problem 2.
Consider a classical system of identical rod-like molecules which have a permanent
electric dipole. The direction of each molecule is given by the angular
coordinates θi, φi. The Hamiltonian is given by
H(pi, ri, pθi, pφi, θi, φi) =
N
i=1
2 p i 1
+
2m 2 I[ ˙ θ 2 i + ˙ φ 2 i sin 2
(θi)] − dE cos(θi)
where I is the moment of inertia of the molecule and d the electric dipole
moment. The electric field E is applied along the z-axis.
A. Calculate the free energy G(T, E, N, V ) = U − T S − P · E.
B. Calculate the polarization P (T, E, N, V ).