No Slide Title notes... · See also: Atkins (8th Ed.) ... A pair of molecules will collide whenever the centres of the two molecules come within a distance d (the collision diameter)

Chemical Thermodynamics : Georg Duesberg

Chapter 2

Kinetic gas theory

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Kinetic Molecular Theory of Gases

Maxwell

(1831-1879)

Boltzmann

(1844-1906)

macroscopic

(gas cylinder)

microscopic

(atoms/molecules)


Physical properties of gases can be described by motion of

individual gas atoms/molecules

Assumptions:

1) each macroscopic and microscopic

particle in motion holds an kinetic

energy according to Newton’s law

2) They undergo elastic collisions

3) They are large in number and are

randomly distributed

4) They can be treated as points of

mass (diameter<< mean free path)


-v v

Δt

2vm

Δtime

ΔvelocitymassForce

1) According to Newton's law of action–reaction, the force on

the wall is equal in magnitude to this value, but oppositely

directed.

2.) Elastic collision with wall: vafter = -vbefore

Kinetic Molecular Theory of Gases: Assumptions

3. Avogardo Number – Brownian motion

4. Gases are composed of atoms/molecules which are

separated from each other by a distance l much more than their

own diameter d

d = 10-10 m

L = 10-3 m….. few m

molecules are mass points with

negligible volume

Kinetic Molecular Theory of Gases: Assumptions


L

reactionF

Collisions of the gas molecules with a wall

As a result of a collision with the wall

the momentum of a molecule changes by

Small volume, v=LA, adjacent to

wall where L is less than the mean

free path


Pressure = Forcetotal/Area P=F/A

• Ftotal = F1 collision x number of collisions in a particular time interval

Assume that in a time Δt every molecule (atom) in the original

volume, v=LA, within the range of velocities

will collide with the wall.


All molecules within a distance xt

with x 0 can reach the wall on the

right in an interval t.

tvL x

This means that Δt is given by:



The “reaction force” of a molecule on the wall is the

negative of the average rate of change in the

momentum of gas molecules in the volume v that

collide with the wall in the time Δt.

The total force on the wall is the sum of the average rate of

momentum change for all molecules in the volume v=LA that

collide with the wall

Here we have divided by 2 since only ½ of the molecules in our

volume have a positive velocity toward the wall

dt

mvd

dt

dpF

)2(

with

L

vm

L

vmF

xx

x

22

2

2


L

We do the sum by noting that the total number of

molecules in the volume V is (N/ V)xLA

Remembering Pascal’s law

dividing by A yields the pressure

everywhere.

V=LA

N/ V = density

AFP



L

Kinetic theory: go from 1 to 3 dimensions

Velocity squared of a molecule: 2222

zyx vvvv

The average of a sum is equal to

the sum of averages…

All the directions of motion

(x, y, z) are equally

probable.

Remember homogeneous

and isotropic! Equipartition principle


Kinetic theory

Combing these results yields

From the ideal gas law

And with c = <v>

m

kTcv

32

Relation between the absolute temperature and average kinetic

energy of a molecule.

m

kT

Nm

NkTv

332

22

3

1vm

V

Nvm

V

NP x

2

3

1vNmNkTPV


vrms of a molecule is “thermal speed”:

The absolute temperature is a measure of the average kinetic

energy of a molecule.

Example:

What is the thermal speed of hydrogen molecules at 800K?

Kinetic theory


Chapter 1 : Slide 13

This is roughly correlated to the speed of sound in those media –

as a results your voice has a higher timbre in He…

Gas Temperature (°C) Speed in m/s

Air 0 331.5

Air 20 344

Hydrogen 0 1270

Carbon dioxide 0 258

Helium 20 927

Water vapor 35 402

m

kTvsound

m

kTvrms

3

γ = the adiabatic constant,

characteristic of the specific

gas


Kinetic theory

The average kinetic energy per molecule is

kTvm2

32

1 2

m

kTv

m

kTv

33 22

kTkE transkin 23


Measurement of molecular speeds

# of molecules striking various locations along drum is

directly related to speed distribution inside gas

What is the distribution of molecular speeds?


Histogram of measured speed distributions

N is total number of atoms (molecules)

∆N is number in a particular speed range v + dv


Maxwell-Boltzmann Distribution

Number of molecules having

speeds in an interval of width

v around v. It is proportional

to v, the total number of

molecules, N, and to the height

of the distribution curve.

P(v)

P(v)


IG-09

Distribution of velocities in a gas

, ,x y z x y z

transKkTP v v v dv dv dve

2

2 2 2 2

1

2trans

x y z

K mv

v v v v

2 2 2

2 2 2

1

2

1 1 1

2 2 2

x y z

x y z

m v v v

kTx y z

mv mv mv

kT kT kTx y z

P dv dv dv

dv dv dv

e

e e e

P is the probability, at temperature T , of finding a molecule with

velocity in the range (vx+dvx, vy+dvy, vz+dvz)


IG-09 19

Velocity probability distribution in polar coordinates

2 2 2

2

1

2

1

22 4

x y zm v v v

kTx y z

mv

kT

P dv dv dv

v dv

e

e

Volume element (dvx)(dvy )(dvz) in rectangular coordinates is

(4v2 dv) in polar coordinates a spherical shell of radius v and

thickness dv

Because kinetic energy depends on v,

the volume of this velocity space is

proportional to the number of ways of

obtaining a particular kinetic energy

(within a small range), i.e., all points

in this thin spherical shell of fixed

thickness Δv correspond to the same

kinetic energy. The greater the radius

of the shell, the more points it

encloses.


Velocity components of molecule (vx, vy, vz). N molecules

represented by N points in velocity space. Volume of space between

v and v + Δv is ~4v2 Δv.

However, all else is not equal. The

factor

accounts for the decreased likelihood

that a molecule will have a given

speed.

Molecular Speeds

kTmv 2exp 2

We have 4v2 in the distribution function. This says that, all else

being equal, we expect more molecules to have speeds with a range

between v and v + Δv, where Δv is fixed, the larger the value of v.


1P v dv

12

2

m ma

kT kT

2

2

mkT

kT

m

2

13

1

2

( ) xmv

kTxP v const e

1

3

22

1x x

kTP v dv const

m

32

2

mconst

kT

Normisation factor for the 3D case

See also: Atkins (8th Ed.) Justification 21.2

Normalisation factor

and

aeax

2With Standard

Integral:

This integral is the fraction of molecules with speeds lying

between the limits v1 and v2


3 2

2 24 exp 22

mP v v mv kT

kT

molecular mass (kg) Boltzmann’s constant

1.3810-23 J/K

temperature in K

(not ºC!!!)


Distribution of speeds in helium gas

Fraction of helium atoms, at 293 K, with speeds between 500

m.s-1 and 600 m.s-1

3 2

2 24 exp 22

mP v v mv kT

kT


Average, most probable and rms speed

•Root mean square speed: (rms) 2

rmsv v

2 21 1 3

2 2 2rmsmv mv kT

3rms

kTv

m

( )0mp

P vv

dv

0

8 8 3( ) 0.92

3rms

kT kTv P v dv v

m m

20.82mp rms

kTv v

m

•Average speed:

•Most probable speed: vmp The most probable speed, is that for which P(v)

has a maximum. By differentiating our distribution function and setting it = 0


P(v) = 4v2 (M/2RT)3/2e-Ek/RT


<v> = (8RT/πM)1/2

vrms = (3RT/M)½

vmp = (2RT/M)1/2 3 2

2 24 exp 22

mP v v mv kT

kT


Maxwell-Boltzmann Speed Distribution vs Temperature for Helium

20.82mp rms

kTv v

m


Maxwell-Boltzmann Speed Distribution vs mass

20.82mp rms

kTv v

m


Chapter 1 : Slide 29

A container filled with N molecules of oxygen gas

is maintained at 300K . What fraction of the

molecules has speeds in the range 599-601 m/s?

The molar mass M of oxygen is 32g/mol.

The fraction in that interval is ,

where , .

From

smv /2

N

vvN

N

dvvNf

)()(0

601

599

smv /6000

3106.2 f

Solution:

Sample calculation

3 2

2 24 exp 22

mP v v mv kT

kT


Summary of Kinetic Theory

Physical meaning of the absolute

temperature is a measure of the

average kinetic energy of a

molecule.

From this we can express

the pressure of an ideal

gas as

Chemical Thermodynamics : Georg Duesberg Chemical Thermodynamics : Georg Duesberg

Intermolecular Collisions in Hard-sphere Gases

Quantitative picture of the events that take place in a

collection of gaseous molecules.

Frequency of collisions?

Distance between successive collisions?

Rate of collisions per unit volume?

Definition:

A pair of molecules will collide whenever the centres of the two

molecules come within a distance d (the collision diameter)

of one another.

No collision. Collision occurs.

d


The Collision Cylinder

Stationary particles inside the collision tube.

2d d

Imagine one particle flying through stationary (frozen) particles.

Within the area , the collision cross section, it will

have collisions. The volume of the cylinder is give by:

2d

2dtvtvLV

L


Collision Frequency

To determine the collision frequency Z we have to consider the

relative speed of the colliding particles.

vvvvrel

22

12

2

2

1

2

1

12

8

RTv

rel

The reduced mass µ of two identical particles is m/2 and therefore

2112

11

1

MMwith

21

8RT

Mv


V

N

V

NNd

1

Collision Frequency Z

V

Nd

mV

Nv

t

tNvz

drel 22

1

1

8kT22

21

8kT2 2

mvv

rel

To determine the collision frequency Z we determine the total number

of molecules that have a collision in the time interval ∆t.

The number of centres Nd are the volume of the collision tube (with

their relative velocities) multiplied with their density minus 1.

2dtvtvLV relrel


Collision Frequency

kT

pvz

V

Nvz

2

2

1

1

kT

p

V

Nwith

The Collision Density

22

1

111

4

21

V

N

m

kT

V

NzZ

We define the collision density as the total rate of collisions per unit volume,

and therefore multiply with the density Nd .The factor ½ stems from the fact

that only AB and not BA collisions or counted

Typical Numbers: Z=5x1034 s-1m-3 for N2 at RT, 1Bar


The Mean Free Path

p

kT

p

kT

v

v

vzwithz

v

kT

p

kT

p

707.02

2

21

1

1

The Mean Free Path is inverse proportional with pressure.

The mean free path - the average distance traveled between successive collisions:

velocity divided by collision frequency.


Diameter of Molecules, D 2 Å = 2 x10-10 m

Collision Cross-section: 1 x 10-19 m

Mean Free Path at Atmospheric Pressure:

m0.3or m1031041.110

3001038.1

21 7

195

23

p

kT

At 1 Torr, 200 mm;

at 1 mTorr, 20 cm

The Mean Free Path : Examples

2d


Collisions flux with Walls and Surfaces Zw

• Rate at which molecules collide with a wall

of area A

dxvfvtAV

NZ xw )(

0

0

2

12

axewith ax

A = area

t = interwall

vm

kT

m

kT

kT

m

dvevkT

mdxvfv x

kT

mv

xx

x

4

1

2

2

2

2)( 2

00

2

“Bolzmann for 1D”


Collisions Flux with Walls and Surfaces

• Rate at which molecules collide with a wall

of area A 2

1

24

1

kTm

p

V

NvZw

It is also called the impingement rate (molecules cm-2 sec-1)

shows up in a large number of calculations and is

important in MBE, ALD, CVD…

kT

p

V

N

vm

kT

4

1

2



Effusion

2

1

000

24

1

kTm

pA

V

NvAAZw

• Rate at which molecules

pass through a small hole

of area Ao, r

A gas under pressure goes (escapes) from one compartment of a

container to another by passing through a small opening.


The Effusion Equation

Graham’s Law - estimate the ratio of the effusion rates for two different gases.

Effusion rate of gas 1 r1.

2

1

1

01,1

2

r

kTm

pAAZ ow

Effusion rate of gas 2 r2.

2

1

2

02,2

2

r

kTm

pAAZ ow

2

1

2

1

2

1

2

0

2

1

1

0

1

2

22

r

m

m

kTm

pA

kTm

pA

r

Effusion Ratio


General Chemistry: Chapter 6

Graham’s Law

• Only for gases at low pressure (natural escape, not a jet).

• Tiny orifice (no collisions)

• Does not apply to diffusion.

A

B

B

A

Brms

Arms

M

M

3RT/M

3RT/M

)(v

)(v

Bofeffusionofrate

Aofeffusionofrate

• Ratio used can be:

– Rate of effusion (as above)

– Molecular speeds

– Effusion times

– Distances traveled by molecules

– Amounts of gas effused.


Kinetic Theory of Energy Transport

Specific heat Velocity Mean free path

vCk3

1Thermal conductivity:

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No Slide Title notes... · See also: Atkins (8th Ed.) ... A pair of molecules will collide whenever the centres of the two molecules come within a distance d (the collision diameter)