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An overview of the Kinetic Theory of Gases, focusing on the Maxwell-Boltzmann velocity distribution. The theory deals with the prediction of transport properties and thermodynamic properties of gases based on the statistical description of the translational motion of their components (atoms and molecules). the Maxwell-Boltzmann speed distribution function and its application to calculate the probability of finding molecules with certain speeds, average properties, and fluxes.
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Typology: Study notes
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kT mv
/
e
kT m v ) v ( F
2
2 3
2
2
2
4
−
=
π
π
2
m
1
1
2
2
m
m
1
2
3
4
m
∑
m i
i i
m m
1 3 3 2 2 1 1
segments all
segments all
=
<
dx ) x ( xf
x
Maxwell-Maxwell
-Boltzmann
Boltzmann velocity distribution
velocity distribution
kT mv
/
x
x
e
kT m
)
v ( f
2
2 1
2
2
−
π
x
kT mv
/
x
x
dv
e
kT m
dv )
v ( f
x 2
2 1
2
2
−
π
x^
x
x
4
SiH
2
H
Maxwell-Maxwell
-Boltzmann
Boltzmann speed distribution
speed distribution
!
The three velocity components are independent of each other
mv^ kT
mvkT
mvkT
/
z
y
x
z
y
x
2
2
2
2 3
2
2
2
−
−
−
2
2
2
2
z
y
x^
v
v
v
v
=
dv) v ( F
dv
dv
dv ) v ( f ) v ( f ) v (
f^
z
y
x
z
y
x^
=
How many molecules are there in a shell withvolume 4
π
v
2 dv?
Transform into cylindrical coordinates
dv
e
kT m
v
dv ) v ( F
kT mv
/
2
2 3
2
2
2
4
−
=
π
π
Maxwell-Maxwell
-Boltzmann
Boltzmann speed distribution
speed distribution
Why is Maxwell-Why is Maxwell
-Boltzmann
Boltzmann speed distribution useful?
speed distribution useful?
How do you use it?How do you use it?
v
kT^ m
dv ) v ( F v v
0
2
2
∞ ∫
kT
v
m
K
2
kT
N
v
v
Av
K
x
x ∆
x
∆x
Random flux to a surfaceRandom flux to a surface
x
x ∆
cylinder volume = NAv
∆x
t
Total # of collisions =
x
x
x^
dv v)
v ( f
t
NA
∞ ∫^0
∆
x
kT mv
x
/
w
dv
e v
m kT
N
t
Z A
F
Flux
x 2
0
2 1
2
2
−
∞ ∫
=
=
=
π
∆
a
dx
e
ax
(^12)
0
2
=
∞ ∫
−
kT^ m
N
kT m
kT m
N
F
/
π
π
8
4
2 1 2
2
2 1
=
=
4
<
=
v
N
F
Mean free path & Gas phase collision ratesMean free path & Gas phase collision rates
2
MF
2
OH
SiH
O
SiH
k
→
3
2
4
B
A
B
A
AB
AB
AB
N
kN
N
N
v
Z
=
<
=
σ
Gas phase collision rates & transport coefficientsGas phase collision rates & transport coefficients
s /
cm
.
k
s /
cm
,
~
v
cm
~
v k ) d d (
m
m
m
m
with
kT
v
max
AB
AB
AB
AB
max
B
A
AB
B
A
B
A
AB
3
10
2
15 10
5 1
000
50
10
(^123)
8
−
− ×
=
∴
<
×
<
=
=
=
=
< σ
σ
σ
μ
πμ
σ
λ
N
mf
2
1
=
<
=
v
D
mf λ 1 3
P T ~
D
/^ 2 3
N
C
v
V
mf
<
=
λ
κ
1 3
