大学物理（热学）

1) Mean Free Path

2) Diffusion, Thermal conductivity, and Viscosity

2006-10-12

Equilibrium distribution

N

vvvv N

222

212

dn is the number of molecules xxx vvv d

vy vy + dvy

zzz vvv d

in

N

nv

N

nv

i

ii

d22

N

vnvn

222

211

ni the number of molecules of iv

near v

among N molecules, how many are near v

probability that one molecule is near v

proportional to dvx ...

xx vvf d

Meaning of dn/N

ddd),,(d

zzyyxxzyx vvfvvfvvf

N

vvvn

Isotropic assumption

2222zyxzyx vvvvvfvfvf

22 exp xx vCvf

The solution is

The probability should be normalized

N

nd1

3223 dexp

d1

xx vvCN

n

xx v

v

C

dexp

1

2

2

xx vvC dexp1 22

12

1

each normalized

d d dexp

1

),,(d

2

2223

zyxzyx

zyx

vvvvvv

N

vvvn

?m

Tkv

3 B2 Known

222

2

3),,(d

N

vvvnvv zyxwhile

zyx

zyx

zyx

vvvTk

vvvm

Tk

m

N

vvvn

ddd2

exp2

,,d

B

22223

B

For unit volume ,,d zyx vvvn

Maxwell velocity distribution

zyx

zyx vvvTk

vvvm

Tk

mn ddd

2exp

2 B

22223

B0

vvTk

mv

Tk

mnvn d42

1

exp2

)(d 2

B

223

B0

Maxwell speed distribution

For one molecule, what speed is most possible?

Most probable speed

vvvfn d4)( 20

Among N molecules, what is the speed most molecules have in unit interval?

02

1exp

d

d

B

22

Tk

mvv

v

m

Tkv B

m2

vvv

v

vN

vndexp

14

)(d 22m

223

2m

Maxwellian can be written as

The average speed

vvvvvv

v d exp1

4 22m

2

0

23

2m

m

Tk

B8

xxv x de 4 2

0

3m

m

Tk

m

Tkv BB

m 41.12

m

Tk

m

Tkv BB 59.1

8

m

Tk

m

Tkv BB

rms 73.13

xDJm

x

TJQ

thermal conductivity

Fourier’s law

Conduction

Transport phenomena

Heat transfer :

conduction convection thermal radiation

diffusion

viscosity

Diffusion

xDJ m

x

vJ y

p

Fick’s law

6

Viscosity

Microscopic theory of transport coefficients

ymvTkvmm , 2

3

2

1 , B

2

pQM , ,

)()()()(dd6

1d xxnxxntvA

For mass transport, mn

xmnxmntvAM dd6

1d

x

tvA2dd

6

1

x

vtA

MJ M

3

1

dd

d

vD3

1

TCTkn VB2

3 VDC

D

Similarly

n2

1

r = 1.12×10−10m

sm1069.1 8

m1068.2

kg1033.3

3B

325B

27

m

Tkv

Tkpn

m

Hydrogen under STP

m1067.1 7

2)2(~ r

K)(mW1071.8

, sm1042.92

25

D

sm1028.1= 25D

Experimental results:

)K(mW1068.1= 2