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5/28/2004
H133 Spring 2004
1
Chapter 7
•
In this chapter we solve some of the interesting
mysteries that have popped up along our path through
thermodynamics.
±
We will make extensive use of the Boltzmann Factor that
we derived in Chapter 6.
±
We want to study the
²
MaxwellBoltzmann Distribution
o
Tells how molecular velocities are distributed in
a gas with temperature T.
²
Count Velocity States.
²
Average Energy of a Quantum System
²
Energy storage in a Gas molecule
•
In our study of a small system in contact with a large
reservoir we never stated that the two substances
needed to be different.
±
They don’t! In fact, we can think of a large container of
gas and treat
one
of the molecules as the small system
and the other molecules as the reservoir.
±
We know from our study of an ideal gas that
±
But we would like to know what the
distribution
of
velocities looks like
²
²
²
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H133 Spring 2004
2
MaxwellBoltzmann Distribution
•
So we are looking for something like this
•
We know from our study in chapter 6 that
•
Note: this is the probability that we are in
a
state with
speed v but it is not the probability that it simply
has
speed v.
±
If there are many states with speed v, the probability that
the particle will have a speed v is proportional to the
number of states with speed v.
±
The number of states with speed between vdv/2 and
v+dv/2 is given by v
2
dv
²
Rough Justification for this statement
o
o
o
o
o
T
k
mv
T
k
E
B
B
e
e
state
/
/
2
2
1
v)
speed
with
Pr(
−
−
=
∝
5/28/2004
H133 Spring 2004
3
Maxwell Boltzmann Distribution
•
So now the probability that we are in
any
state within dv
centered on v is
•
We have a term which depends on mass (m), k
B
, and
T, which has units of m
2
/s
2
. Let’s define that as a
special parameter v
p
.
•
Since v
p
is just a constant, we can put it in our
relationship (it just changes the constant of
proportionality)
•
Notice that this version is unitless…like what we would
expect for a probability.
That means the constant of
proportionality that is left to determine is also unitless.
By demanding that the total probability is equal to 1 we
can determine this remaining constant of
proportionality.
±
It is 4/
π
1/2
T
k
mv
B
dve
v
2
/
2
2
v)
around
dv
within
Pr(
−
∝
2
2
v)
around
dv
within
Pr(
−
∝
p
v
v
e
v
dv
v
v
p
p
2
/
1
2
2
/
1
2
4
)
(
)
(
v)
around
dv
within
Pr(
2
=
=
=
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This note was uploaded on 06/03/2011 for the course H 133 taught by Professor Furnstahl during the Spring '11 term at Ohio State.
 Spring '11
 Furnstahl
 Thermodynamics, Quantum Physics

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