Physics 53
Thermal Physics 2
I have concluded that all of man's troubles have one cause, that he
cannot sit still in a room by himself.
— Pascal
Average energy of a molecule
Toward the end of the 19th century great insight was gained into the meaning of
thermodynamics by application of statistical methods. A leading figure in this work was
Ludwig Boltzmann. We quote here without proof one his most important findings:
In a system at equilibrium at temperature
T
, the probability that a particular
particle will have energy
E
is proportional to
e
−
E
/
kT
.
In this,
k
is Boltzmann’s constant, and
T
is the Kelvin temperature.
From this statement about the probabilities, one can derive an important theorem about
the average energy of a molecule:
Equipartition of energy
For a system in thermal equilibrium at
temperature
T
, each degree of freedom
contributes energy
1
2
kT
to the average
energy of a molecule.
Here a
degree of freedom
is an independent part of the energy of a molecule. For
example, the motion of the CM of the molecule is described by
K
CM
=
1
2
m
(
v
x
2
+
v
y
2
+
v
z
2
)
,
which has three independent parts for the motion in the
x
,
y
, or
z
directions. These
represent three degrees of freedom.
To be able to count the degrees of freedom we must make a model of the molecule. We
will treat the atoms as point masses, and imagine that the bonds between atoms are like
stiff springs connecting these masses.
PHY 53
1
Thermal Physics 2
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A monatomic gas (such as helium) will consist simply of point masses, each with three
degrees of freedom from the three terms in
K
CM
as above.
A diatomic molecule will have four additional degrees of freedom besides these three:
•
The molecule can rotate about two independent axes passing through the CM and
perpendicular to the line between the atoms. This gives two degrees of freedom.
Rotation about the line between pointlike atoms gives no degree of freedom because the moment of
inertia about that axis is zero and hence there is no rotational energy.
•
The atoms can vibrate back and forth along the line between them. The kinetic
energy of this vibration gives one degree of freedom.
•
The potential energy of the “spring” in this vibration gives another degree of
freedom.
Since it has three degrees of freedom, a monatomic gas molecule should, if equipartition
of energy holds true, have average energy
3
2
kT
, while a diatomic molecule which has
seven degrees of freedom should have average energy
7
2
kT
. We will see later that these
predictions can be tested by experimental measurements of specific heats.
Thermal expansion
The increase in average energy of the molecules with increasing temperature is
responsible for many familiar phenomena in our everyday experience. One of these is
the fact that most solid or liquid substances expand when heated.
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 Spring '07
 Mueller
 Physics, Thermodynamics, Energy, TA, Ludwig Boltzmann, Stefan

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