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Kinetic Theory of Gases: Maxwell-Boltzmann Velocity Distribution, Study notes of Chemical Kinetics

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.

What you will learn

  • What are the applications of the Maxwell-Boltzmann velocity distribution in calculating average properties and fluxes?
  • What is the Maxwell-Boltzmann velocity distribution?
  • How does the mass of a gas affect its velocity distribution?
  • What is the Kinetic Theory of Gases?

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2021/2022

Uploaded on 09/12/2022

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Kinetic Theory of Gases
Kinetic Theory of Gases
Kinetic Theory: Theory that deals with prediction of
transport (µ, κ, D) and thermodynamic properties of gases
based on statistical (average) description of the translational
motion of its components (atoms & molecules).
statistical
description
velocity
distribution
function
kT
mv
/
e
kT
m
v)v(F 2
23
2
2
2
4
=
π
π
Maxwell-Boltzmann speed distribution
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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Download Kinetic Theory of Gases: Maxwell-Boltzmann Velocity Distribution and more Study notes Chemical Kinetics in PDF only on Docsity!

Kinetic Theory of GasesKinetic Theory of Gases

Kinetic

Theory:

Theory

that

deals

with

prediction

of

transport (

,^

, D) and thermodynamic properties of gases

based on statistical (average) description of the translationalmotion of its components (atoms & molecules).

statisticaldescription

velocity

distribution

function

kT mv

/

e

kT m v ) v ( F

2

2 3

2

2

2

4

  

  

=

π

π

Maxwell-Boltzmann speed distribution

Probability & Statistics ReviewProbability & Statistics Review

Suppose we have a series of observation for a value x which can take ona discrete set of values

{x

, x 1

2

, x

,…x 3

m

Suppose that in a series of N measurements we observe that

x

1

occurs N

1

times

x

2

occurs N

2

times

:^

:^

x

m

occurs N

m

times

x

1

x

2

x

3

x

4

…x

m

N

Average value of x

m i

i i

m m

x
N N
N
x
N
x N x N x N x

1 3 3 2 2 1 1

Probability & Statistics ReviewProbability & Statistics Review

The value P(x) is proportional to the length of segment

x. We

define probability density function f(x) such that P(x)=f(x)

x

Thus, the probability that an experimental outcome will be between xand x+

x is

N
) x ( N ) x ( P

The average value of x, is now given by

Let

x

0 and number of segments

x
x
xf
x
xP
x

segments all

segments all

=

<

dx ) x ( xf

x

Integral is over the whole domain of x; f(x): distribution function

Maxwell-Maxwell

-Boltzmann

Boltzmann velocity distribution

velocity distribution

kT mv

/

x

x

e

kT m

)

v ( f

2

2 1

2

2

  

  

π

vx

vy

vz

x

kT mv

/

x

x

dv

e

kT m

dv )

v ( f

x 2

2 1

2

2

  

  

π

Probability that v

x^

is between

v

x

and v

+dvx^

x

Gaussian probabilitydistribution

ExampleSiH

4

: m

SiH

×

×

kg

H

2

: m

H

×

×

kg

k= 1.

×

J/K

T = 300 K

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

e
e
e
m kT
) v ( f ) v ( f ) v (
f^

2

2

2

2 3

2

2

2

vx

vy

vz

dv

v

2

2

2

2

z

y

x^

v

v

v

v

=

speed

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

Probability of finding amolecule that has speedbetween v and v+dv

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?

Example: average kinetic energy and C

v

kT^ m

dv ) v ( F v v

0

2

2

∞ ∫

kT

v

m

K

2

R
U T
C
U
RT

kT

N

E

v

v

Av

K

Example: average flux to a plane

substrate

v

x

A

v

x ∆

t

If a particle has velocity v

x

it

will strike A if it is v

∆x

t away

Random flux to a surfaceRandom flux to a surface

v

x

A

v

x ∆

t

of collisions = gas density

×

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

The random flux

Mean free path & Gas phase collision ratesMean free path & Gas phase collision rates

×

Ar

×

H

2

×

He

MF

(cm)

(cm

2

Molecule

Example: Reaction rates of gas phase reactions that proceed atgas phase collision frequencies.Consider for example the following gas phase reactionWhat is the upper limit for k?

OH

SiH

O

SiH

k

→ 

3

2

4

B

A

B

A

AB

AB

AB

N

kN

N

N

v

Z

=

<

=

σ

Equal if theyreacteverytimethey colide

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