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# The Distribution of Molecules over Velocities Maxwell Distribution

## 1.

MEPhI General PhysicsLecture 09

The Distribution of Molecules

over Velocities

Maxwell Distribution

## 2.

SOME MATHEMATICS: the probability distributionSmooth CHART – distribution function

f(x)

BAR CHART

Nx

Na

Area =dPx

ΔP

area =

x

0

X

x x+a

dPx f ( x)dx

N

P

i

i

N

i

1,

0

x x+dx

X

dN x

Px dPx

f ( x) lim

lim

N Ndx

x x

dx

dP f ( x)dx 1

x

## 3.

Normal distributionPk=(2/πN)1/2exp(-2(k-N/2)2/N) =>

The normal (or Gauss) distribution –

is the smooth approximation of the

Newton’s binomial formula

k

250

Parameters of Gauss distribution:

x – some random value

μ — the most probable (or

expected) value of the random (the

maximum of the distribution function)

σ — dispersion of the random value.

In case of the Eagles and Tails game: μ

200

150

100

50

k

0 2 4 6 8 10

= N/2, σ = (N/2)1/2 << N

## 4.

Gauss DistributionThe normal distribution is very often

found in nature.

Examples:

• Eagle and Tails game

• Target striking

• the deviations of experimental

results from the average (the

dispersion of results = the

experimental error)

## 5.

Statistical Entropy in Physics.Statistical Entropy in Molecular Physics: the logarithm of the number of

possible micro-realizations of a state with certain macro-parameters,

multiplied by the Boltzmann constant.

S k ln J/K

1.

2.

3.

4.

5.

In the state of thermodynamic equilibrium, the entropy of a closed

system has the maximum possible value (for a given energy).

If the system (with the help of external influence)) is derived from the

equilibrium state - its entropy can become smaller. BUT…

If a nonequilibrium system is left to itself - it relaxes into an equilibrium

state and its entropy increases

The entropy of an isolated system for any processes does not

decrease, i.e. ΔS > 0 , as the spontaneous transitions from more

probable (less ordered) to less probable (more ordered) states in

molecular systems have negligibly low probability

The entropy is the measure of disorder in molecular systems.

## 6.

Statistical Entropy in Physics.For the state of the molecular system with certain macroscopic parameters

we may introduce the definition of Statistical Entropy as the logarithm of the

number of possible micro-realizations (the statistical weight of a state Ώ) of a this state, multiplied by the Boltzmann constant.

S k ln J/К

Entropy is the additive quantity.

2

1

i

3

p p1 p2 ... pN

pi

i

1 2 ... N

N

S k ln k ln 1 ln 2 ... ln N

S Si

i 1

## 7.

Statistical Entropy and the Entropy of Ideal GasNot a strict proof, but plausible considerations.

• the number of variants of realization of a state (the statistical weight of a

state) shall be higher, if the so called phase volume, available for each

molecule (atom), is higher: Phase volume Ω1 ~Vp3~VE3/2 ~VT3/2

• the phase volume for N molecules shall be raised to the power N:

Ω ~ VNT3N/2.

For multy-atomic molecules, taking into account the possibilities of rotational

and oscillational motion, we shall substitute 3 by i : Ω

~ VNT iN/2

• As molecules are completely identical, their permutations do not change

neither the macrostate, nor the microstates of the system. Thus we have

to reduce the statistical weight of the state by the factor ~ N! (the number

of permulations for N molecules)

Ω~ VNTiN/2 /N!; S=k ln Ω =kNln(VTi/2/NC)=

= v(Rln(V/v) + cVlnT +s0)

## 8.

Statistical Entropy and the Entropy of Ideal GasΩ~ VNTiN/2 /N!; S=k ln Ω =kNln(VTi/2/NC)=

= v(Rln(V/v) + cVlnT +s0)

The statistical entropy proves to be the same physical quantity, as was

earlier defined in thermodynamics without even referring to the molecular

structure of matter and heat!

## 9.

MEPhI General PhysicsThe Distributions of Molecules

over Velocities and Energies

Maxwell and Boltzmann Distributions

That will be the Focus of the next lecture!

## 10.

Distribution of molecules over velocitiesIf gas is in thermodynamic equilibrium

state –the macroscopic parameters

(temperature, pressure) are kept stable

and the distribution of molecules over

velocities and energies remains also

stable in time and space.

This distribution was first derived in 1859

by J.C.Maxwell.

James Clerk Maxwell

1831-1879

## 11.

Distribution of molecules over velocitiesEach velocity vector can be presented as a point

in the velocity space,

As all the directions are equal – the distribution

function can not depend on the direction, but

only on the modulus of velocity

f(V)

## 12.

Distribution of molecules over velocitiesThe probability to have the x-component of the

velocity within the range between Vx and Vx+dVx

dP Vx ,Vx dVx Vx dVx

dVx dV y dVz

The probability that the end of the 3dimensional velocity vector V, will fir into

the small cube dVx dV y dVz

nearby the

velocity V can be calculated by

multiplying probabilities:

f V Vx Vy Vz dVx dV y dVz

dP(V)

From the other hand, as all the directions are equal, this probability may

depend only on the modulus of velocity

f V Vx Vy Vz

## 13.

Distribution of molecules over velocitiesNoe some mathematics:

f V Vx Vy Vz

dVx dV y dVz

ln( f (V )) ln( (Vx )) ln( (Vy )) ln( (Vz ))

We will calculate the derivative by dVx

Vx

f V V

f V Vx Vx

as

f V 1 Vx 1

f V V Vx Vx

Vx

Vx

V

2

2

2

Vx

V

Vx V y Vz

## 14.

Distribution of molecules over velocitiesThe only function which satisfies the equation:

f V 1 Vx 1

f V V Vx Vx

dVx dV y dVz

as well as the initial condition

f V Vx Vy Vz

is:

V x 2

Vx A exp(

)

2

here α must be negative! α <0

## 15.

Distribution of molecules over velocitiesV x

Vx A exp(

)

2

2

From normalization condition

: Vx dVx 1

we obtain:

Vx

A exp(

)dVx 1

2

The Poisson integral:

2

A

2

2

exp(

Х

)dХ

## 16.

Probability Distribution and Average ValuesKnowing the distribution of a random value x we may calculate its average

value <x>:

x xdPx xf ( x)dx,

Moreover, we may calculate the average for any function ψ(x):

( x) ( x)dPx ( x) f ( x)dx

THE PROPERTIES OF AVERAGES.

• Average of the sum of two values equals to the sum of their averages

<x + y> = <x> + <y>

• Average of the product of two values equals to the product of their averages ONLY

in case if those two values DO NOT depend on each other

<xy> = <x><y> only if x and y are independent variables

## 17.

Probability Distribution and Average ValuesExamples: in case of even distribution of molecules

over certain spherical volume V (balloon with radius R):

Y

X

x xdPx xf ( x)dx, = 0

<x +y> = 0;b <x y> = 0

<x2> = R2/5 >0

<r2> = < x2 +y2 +z2> = 3R2/5 > <r>2

<r> = <(x2 +y2 +z2)1/2> = 3R/4

Calculation – on the blackboard…

V

Z

## 18.

Different Kinds of Averages<r> = <(x2 +y2 +z2)1/2> = 3R/4 - average

(<r2>)1/2 = 0,61/2R > <r> - squared average

<x> = 0; <x2>1/2 = R/51/2 >0

Y

Median average rmed ; the quantity of molecules

with r<rmed equals to the quantity of molecules Z

with r>rmed

V

X

rmed = R/21/3 = 0,7937R > (<r2>)1/2 =0,7756R

> <r> = 0,75R

This all is about even distribution of molecules over space in spherical balloon.

What about the distribution of molecules over velocities and energies?

It can be spherically symmetric, but it can not be even as formally there is no

upper limit of velocity…

## 19.

Distribution of molecules over velocitiesV x

Vx A exp(

)

2

2

A

2

The average of the squared velocity equals to:

Vx 2

Vx 2 Vx dVx

2

Vx 2

Vx exp(

)dV

2

2

This integral once again can be reduced to the Poisson integral:

Vx

2

1

; (Vx ) | 1 / 2 V exp( V / 2 V )

2

x

2

x

2

x

## 20.

Distribution of molecules over velocities(Vx ) | 1 / 2 V exp( V / 2 V )

2

x

2

x

2

x

The basic assumption of thermodynamics (every degree of freedom

accumulates the same energy):

mV x

2

2

m

1

2

Vx kT

2

2

Vx

2

kT / m

2

mVx

m

Vx

exp(

)

2 kT

2kT

## 21.

Distribution of molecules over velocities2

mVx

m

Vx

exp(

)

2 kT

2kT

The distribution over absolute values of velocities:

f V Vx Vy Vz

,

mV y 2

m

Vy

exp(

)

2 kT

2kT

m

f V

2 kT

m

mVz 2

Vz

exp(

)

2 kT

2kT

3

2

2

mV

exp(

)

2kT

## 22.

Distribution of molecules over velocitiesFunction

m

f V

2 kT

3

2

2

mV

exp(

)

2kT

dVx dV y dVz

defines the probability that velocity is

within the “cubic” range:

dVx dV y dVz

dP f V dVx dVy dVz

Probability to find the absolute value of the velocity between V and V+dV

dP f V 4 V dV F (V )dV

2

F (V ) 4 V f (V ) - Maxwell’s function

2

## 23.

Maxwell’s Functionm

F V

2 kT

3

2

mV 2

4 V 2

exp

2

kT

V Vвер

F V V

2

V Vвер

mV 2

F V exp

2

kT

## 24.

Maxwell’s FunctionT

T2 1

2

m

m2 1

2

Area under the curve is always equal to 1

## 25. Stern’s experiment (1920)

• The outer cylinder is rotatingS R t

R

t

V

R 2

V

S

## 26. Lammert’s Experiment (1929)

• Two rotating discs with radial slots. One is rotatingahead of the other. The angle distance is

t1 l

V

t2 l

V l

## 27. Most probable velocity.

Most probable velocity corresponds to the maximum of the Maxwell’sfunction)

dF v

mV 2

mV 2

m 2

4

2

0

V exp

dv

kT

2 kT

2kT

3

Most probable velocity

Vвер

2kT

m

The value of the Maxwell’s function maximum:

F vвер

4

m

4 1

e 2kT

e vвер

Vвер

## 28. Average velocity.

Average velocity by deffinitionV VdP VF V dV

For Maxwell’s function:

0

3

2

2

mV

m

3

V 4

dV

V exp

2 kT 0

2kT

8kT

V

m

## 29.

Average squared velocity.Average squared velocity by definition

Vср.кв.

V2

3

V2

2

2

mV

m

4

4

dV

V exp

2 kT 0

2kT

Vср.кв.

3kT

m

## 30.

Three kinds of average velocities2kT

m

Most probable:

Vвер

Averge:

8kT

V

m

Average squared:

Vср.кв.

3kT

m

## 31.

Maxwell’s Functionm

F V

2 kT

3

2

mV 2

2

exp

4

V

2

kT

Vвер = (2kT/m)1/2

Vср = (8kT/πm)1/2

Vср.кв = (3kT/m)1/2

Vср.кв.

3kT

m

>

8kT

V

m

>

2kT

m

Vвер

>

## 32.

Example:Example: The mixture of oxygen and nitrogen (air) has the

temperature T = 300 K. What are the average velocities of two types

of molecules:

VO

2

8kT

mO

2

VO

VN

2

2

VN

2

8kT

mN

8 8,31 300

3 м

м

0

,

45

10

450

с

с

3,14 32 10 3

8 8,31 300

3 м

0

,

48

10

480 м

3

с

с

3,14 28 10

2

## 33.

Energy distribution functionm

F V

2 kT

F ( E )dE

3

2

mV

exp

2kT

2

2

4 V 2

F(V)dV = F(E)dE;

E = mV2/2;

dV = dE/(2mE)1/2

E

E dE

exp

kT

kT kT

∫F(E)dE = 1

<E> = ∫E F(E)dE = 3kT/2

F(E)

Eвер = kT/2

Eср = 3kT/2

E

## 34.

Energy distribution functionm

F V

2 kT

F ( E )dE

3

2

mV

exp

2kT

2

2

4 V 2

F(V)dV = F(E)dE;

E = mV2/2;

dV = dE/(2mE)1/2

E

E dE

exp

kT

kT kT

This is the distribution of molecules

over kinetic energies. he question is:

how the distribution will look like, if to

take into account also the potential

energy (gravity)?

That will be the topic of the next

lecture..

F(E)

<E> = 3kT/2

Eвер = kT/2

Eср = 3kT/2

E