Spring 2006
PHY 203: Introduction to Statistical Mechanics
Midterm Exam
1. Adsorption onto a flat surface:
Consider a gas of N atoms in contact with a stationary flat solid surface. There
is an interaction potential between the surface and an atom which depends only on
the vertical coordinate, V (z). It is repulsive for small z, then becomes attractive for
intermediate z, and finally tends to 0 as z → ∞. The potential is such that an atom
from the gas can become trapped at the surface. Suppose there is ONE quantum state
in the vertical direction and call its energy −ε0 with respect to V = 0.
Assume that the gas atoms do not interact with each other either in the adsorbed state
or when they are free. Further assume that the particles can be treated classically both
in the 3D state and for the x-y motion in the 2D state.
(a) What are the basic thermodynamic equations defining equilibrium between the
two states? (hint: think about what quantity determines whether a given atom
goes to the 3D or 2D gas) [2 points]
(b) Find the single particle partition function for the two-dimensional gas. From that
form the full partition function for the 2D subsystem, the free energy, and finally
the chemical potential µ2D . [6 points]
(c) Do the same for the 3D ideal gas, finding classical results. [6 points]
(d) Combine the results (a)-(c) to find an expression for the number of adsorbed
atoms, Nad in terms of the pressure in the 3D gas, Pgas . Interpret physically. [6
points]
2. Assuming that the entropy S and the statistical number Ω of a physical system are
related through an arbitrary functional form
S = f (Ω) ,
show that the additive character of S and the multiplicative character of Ω necessarily
require the function f (Ω) be of the form
S = k ln(Ω)
– continued on back page !! –
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3. Consider a gas of N particles of mass m confined to a volume V . The Helmholtz free
energy of the gas is
N
h3
N2
F (T, V, N ) = N kT ln
−
a ,
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
!
∂y
·
∂z
z
!
∂z
·
∂x
x
!
= −1
y
2. In problem 3c, Maxwell’s relations might be useful.
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