Quiz 5 Solutions for Physics 176
April 7, 2011
Problems That Require Writing
1. A scientist wants to study the properties of a certain equilibrium solid substance for various values of
the temperature
T
, volume
V
, and chemical potential
μ
by placing the substance in a large reservoir
for which these three parameters are held constant during a given experiment.
(a)
(5 points)
Find an appropriate thermodynamic potential
K
(
T,μ,V
) for this experimental setup.
Answer:
K
=
U

TS

μN
.
As discussed in lecture, one can identify or discover a thermodynamic potential by integrating by
parts terms in the thermodynamic identity for the thermal energy
U
until the variables of interest
appear as diﬀerentials. Here we want a function
K
(
T,μ,V
) so we proceed as follows:
dU
=
TdS

PdV
+
μdN
(1)
=
[
d
(
TS
)

SdT
]

PdV
+ [
d
(
μN
)

Ndμ
]
,
(2)
which can be rewritten as
d
(
U

TS

μN
) =
dK
=

SdT

PdV

Ndμ.
(3)
Eq. (3) tells us that the quantity
K
=
U

TS

μN
is a thermodynamic potential that is a
function of the variables
T
,
V
, and
μ
as desired.
A challenge: is it possible to ﬁnd a thermodynamic potential that is a function of the variables
T
,
P
, and
V
?
(b)
(5 points)
Show how to calculate the entropy
S
, pressure
P
, and the number of particles
N
for
this system in terms of
K
.
Answer:
The thermodynamic identity for
K
, Eq. (3) directly tells us that
S
=

±
∂K
∂T
²
V,μ
,
P
=

±
∂K
∂V
²
T,μ
,
N
=

±
∂K
∂μ
²
T,V
.
(4)
2.
(6 points)
A single helium atom with mass
m
He
is placed inside an ideal gas of neon atoms, each
with mass
m
Ne
> m
He
. If the gas is in thermodynamic equilibrium with temperature
T
, write down
in terms of
m
He
,
m
Ne
, and
T
a mathematical expression for the probability for the He atom to have a
speed between 100 m
/
s and 400 m
/
s.
Note: you do not have to evaluate your expression, just write it down.
Answer:
prob[100
≤
v
≤
400] =
Z
400
100
³
m
He
2
πkT
´
3
/
2
4
πv
2
e

βm
He
v
2
/
2
dv,
(5)
where the mass
m
He
is in kilograms, the speed
v
in meters per second, and the temperature
T
is in
kelvin if the Boltzmann factor is in the usual SI units.
This question tested whether you understand how to identify “what is the system” and “what is the
reservoir”. Since the question is asking about statistical properties about the He atom, we choose the
single He atom as the system and all Ne atoms as the reservoir. Because the He atom is the system,
1
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View Full Documentthe energies
E
s
that appear in the Boltzmann factor and in the partition function
Z
are the energies
of a noninteracting freely moving He atom:
E
s
=
E
(
v
) =
1
2
m
He
v
2
,
(6)
where the diﬀerent states of the He atom are labeled by the speed
v
. The Boltzmann factor for a
state
s
with energy
E
(
v
) is therefore
p
s
=
ce

βm
He
v
2
/
2
,
(7)
where the constant
c
is determined by normalization, that the sum of all the probabilities
p
s
must add
to 1. But this means that
c
can only depend on the mass
m
He
since the integral over the probabilities
Eq. (7) only involves
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 Spring '08
 Behringer
 Physics, Thermodynamics, Statistical Mechanics, Entropy, Eq.

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