Ben Sauerwine
Practice for Qualifying Exams
Solution based on advice from Diana.
Problem Source:
CMU February 2006 Qualifying Exam
Consider an ideal gas of N atoms confined to a container of volume V and internal
surface area A.
Although you may neglect the interactions between atoms, there is
an attraction between the atoms and the walls of the container that cannot be
ignored.
A simple model for the atoms adsorbed onto the surface is to treat them as
a twodimensional classical ideal gas, where the energy of an adsorbed atom is
(
)
0
2
2
ε
ε
−
=
m
p
p
v
v
and
p
v
is the twodimensional momentum.
Do not concern yourself with the details
of the binding.
Treat
0
ε
as a known parameter.
(a)
What is the classical partition function of the adsorbed atoms if N’ atoms are
bound to the surface?
The classical partition function is given for one atom by:
(
)
∫ ∫
∞
∞
−
∞
∞
−
−
=
q
pd
d
e
Z
n
n
q
p
H
v
v
,
β
In this case, then, I have:
0
2
0
0
2
0
2
2
1
2
2
2
0
2
0
2
0
2
2
βε
β
βε
ε
β
ε
β
β
π
β
β
π
π
e
m
A
Z
dv
e
dr
m
r
dv
m
r
v
rdr
e
Ae
rdr
e
A
p
d
e
A
Z
v
m
r
m
r
n
m
p
=
=
=
=
=
=
=
∫
∫
∫
∫
∞
−
∞
−
∞
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
∞
∞
−
⎟
⎟
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 Spring '12
 Dr.Jaouni
 Thermodynamics, Statistical Mechanics, Fundamental physics concepts, Helmholtz free energy

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