hw05
ChE 132C due
Feb. 16
, 2011
1)
The Boltzmann distribution gives the probability density for the velocity of a molecule as
2
( )
exp[
/(2
)]
EQ
B
f
vC
m
v
k
T
where m is the mass, k
B
is Boltzmann’s constant, and T is temperature. [answers given
for parts (a), (b), and (c) for review]
(a)
What value of C
EQ
that normalizes the distribution ? [this is a bit difficult if you
try to do the integral – use the form of a Gaussian to help you!]
(b)
What is the standard deviation in terms of
m, k
B
,
and
T
?
(c)
What is the average absolute velocity, E[
v
] = <
v
> = ?
(d)
The velocities in different orthogonal directions are independent, so the joint
distribution for velocity in three dimensional space is
32
2
2
( ,
,
)
exp[
(
) /(2
)]
xyz
E
Q
x
y
z
B
fvvv
C
mv v v
kT
which is the product of separate distributions for v
x
, v
y
, and v
z
.
Derive the
MaxwellBoltzmann distribution, i.e. the marginal distribution
f
S
(
s
) for the speed
s =
(v
x
2
+v
y
2
+v
z
2
)
1/2
.
You may find it helpful to change to spherical coordinates in
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 Spring '11
 PETERS
 Mole, Probability distribution, Probability theory, probability density function, CEQ exp

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