大学物理(热学)
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大学物理(热学)
1) Mean Free Path
2) Diffusion, Thermal conductivity, and Viscosity
2006-10-12
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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
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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
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Isotropic assumption
2222zyxzyx vvvvvfvfvf
22 exp xx vCvf
The solution is
The probability should be normalized
N
nd1
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3223 dexp
d1
xx vvCN
n
xx v
v
C
dexp
1
2
2
xx vvC dexp1 22
12
1
each normalized
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d d dexp
1
),,(d
2
2223
zyxzyx
zyx
vvvvvv
N
vvvn
?m
Tkv
3 B2 Known
222
2
3),,(d
N
vvvnvv zyxwhile
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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
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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?
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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
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The average speed
vvvvvv
v d exp1
4 22m
2
0
23
2m
m
Tk
B8
xxv x de 4 2
0
3m
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m
Tk
m
Tkv BB
m 41.12
m
Tk
m
Tkv BB 59.1
8
m
Tk
m
Tkv BB
rms 73.13
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xDJm
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x
TJQ
thermal conductivity
Fourier’s law
Conduction
Transport phenomena
Heat transfer :
conduction convection thermal radiation
diffusion
viscosity
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Diffusion
xDJ m
x
vJ y
p
Fick’s law
6
Viscosity
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Microscopic theory of transport coefficients
ymvTkvmm , 2
3
2
1 , B
2
pQM , ,
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)()()()(dd6
1d xxnxxntvA
For mass transport, mn
xmnxmntvAM dd6
1d
x
tvA2dd
6
1
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x
vtA
MJ M
3
1
dd
d
vD3
1
TCTkn VB2
3 VDC
D
Similarly
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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
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K)(mW1071.8
, sm1042.92
25
D
sm1028.1= 25D
Experimental results:
)K(mW1068.1= 2