The Maxwell distribution of speeds · Ø The Maxwell velocity distribution gives the fraction of molecules having certain velocities. Ø There are distributions for velocity components,

$Page 1: The Maxwell distribution of speeds · Ø The Maxwell velocity distribution gives the fraction of molecules having certain velocities. Ø There are distributions for velocity components,$
Experimental Physics II - Molecular-kinetic theory of gases 19

The Maxwell distribution of speeds

xv

yv

zv

dxdydz

dv

24)()( vvfvF p×=!

( )þýü

îíì-÷

øö

çèæ=

kTmvv

kTmvF

2exp

24

22

2/3

pp

0 500 1000 1500 2000 2500 30000.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3

1.2x10-3

hydrogen

prob

abilit

y de

nsity

velocity v (m/s)

oxygen

vm <v>vrms

The most probable speed:

0)(=

dvvdF m

kTvm2

=

The average speed:

mkTvp8

=ò= dvvvFv )(

mkTvrms3

=The rms speed:

kTvm 232

21 =


Experimental verification of Maxwell distribution

Classical experiments by Stern

Loven

metalvapor

(Disadvantage: detects simultaneously multiple velocities)

v

phase is shifted

Experimental Physics 4a - Molecular-kinetic theory of gases 23


Boltzmann distribution

2

31 vmnP =

0P

P

A

dh

2000 3

1 vmnP = kTvmvm 3220 ==

( ) AdhhPhPghM ×+-= )()()(

( )kTAdhhnhn )()( +-=

mAdhhnhM )()( »

kTmgdh

ndn

-= òò -=1

0

1

0

h

h

n

n kTmgdh

ndn

dhdhhdnhndhhn )()()( +»+

dhdhhdnkTA )(

-=

þýü

îíì D-=kThmgnn exp01

þýü

îíì-=

kTE

nn potentialexp01

Boltzmann distribution

vdkTEndn total 3

0 expþýü

îíì-=

Maxwell-Boltzmann distribution


At equilibrium, each elementary process should be equilibrated by its reverse process.

A

yv

yv'

v!

Boltzmann distribution: detailed balance

Two ensembles withequal total energies:


Avogadro’s number

Jean Baptiste Perrin3 2 0 3 2 2 5 3 1 2

3 1 1 0 3 3 4 3 4 4

0 3 1 3 1 4 2 2 1 3

1 1 2 2 3 0 1 3 4 3

0 2 2 1 0 2 1 3 2 4

2 1 0 0 1 1 3 1 0 0

0 2 0 0 0 0 1 2 2 0

2 1 3 3 1 0 0 0 3 0

1 0 2 1 0 0 1 0 1 0

1 1 0 2 4 1 0 1 0 1

Some thousands of readingsare required if some degree ofaccuracy is aimed at. To takean example, I have copiedbelow the numbers given by 50consecutive readings at twolevels 30 μ apart in one of theemulsions I have used:

Lower end Upper end

Annales de Chimie et de Physique 18, 1-114 (1909)

30 µm

The gamboge, which is used for a water-colour, comes from the desiccation of the latex secreted by Garcinia morella (guttier of Indo-China).

R = 0.212 µmrc = 1.207 g/cm3

rl = 1.0 g/cm3

T = 20°C n1/n2 = 2.08

þýü

îíì --=

kTgzmmnn lc )(exp0

)/ln(3))((4

21

123

nnTgzzRk lc --

-=rrp

The mean radius of the granules of the emulsion employed was found equal to 0.212 μm, by counting 11,000granules of a titrated emulsion, and to 0.213 μm by application of the law of Stokes. The difference of densitybetween the material of the granules and the inter-granular water was 0.2067 at 20°, the temperature to whichthe measurements refer.

J/K1013.1 23-´=


Ø The Maxwell velocity distribution gives the fraction

of molecules having certain velocities.

Ø There are distributions for velocity components,

for velocities and for molecular speeds.

Ø Due to occurrence of such distributions,

one may introduce different average

quantities.

ØThe Boltzmann distribution describes

variation of particle densities in an external

conservative potential field.

To remember!

Documents

The Maxwell distribution of speeds · Ø The Maxwell velocity distribution gives the fraction of molecules having certain velocities. Ø There are distributions for velocity components,