Spring 2004
PHY 203: Introduction to Statistical Mechanics
Midterm Exam
1. The zipper problem:
A zipper has N links; each link has a state in which it is closed with energy 0 and a state in
which it is open with energy . The zipper can only unzip, however, from the left end: the
link number s can only be open if all links to the left (numbers 1, 2, . . . , s − 1) are open.
(a) Find the partition function of the zipper. (You should sum the series.) [10 points]
(b) In the limit of low temperature, kT , find the average number of open links. The
model is a very simplified model of the unwinding of two-stranded DNA molecules.
[10 points]
2. Consider a system consisting of two particles, each of which can be in any one of three
quantum states with energies 0, > 0, and 3. The system is in contact with a heat reservoir
at temperature T . Find the partition function Q and heat capacity CV of the system if
(a) the particles are distinguishable [20 points]
(b) the particles are identical bosons [20 points]
(c) the particles are identical fermions [20 points]
Discuss the heat capacity in the high and low temperature limits for all three cases.
3. Consider a gas of N particles of mass m confined to a volume V . The Helmholtz free energy
of the gas is
#
"
N2
N
h3
−
a
F (T, V, N ) = N kT ln
e (V − N b) (2πmkT )3/2
V
,
where T is the temperature, a and b are positive constants and e = 2.718 . . ..
(a) Show that this gas obeys the van der Waals equation of state,
a
P + 2 (v − b) = kT
v
with: v =
V
N
[10 points]
(b) Derive the relation between temperature and volume of this gas when it undergoes an
adiabatic process [5 points].
(c) Find the change in temperature ∆T when the gas makes a free expansion from V to
V + ∆V [5 points].
Hints:
1. For problems 3, use the following relation:
∂x
∂y
·
z
∂y
∂z
·
x
∂z
∂x
2. In problem 3c, Maxwell’s relations might be useful.
1
= −1
y