Kinetic Theory
Download
Report
Transcript Kinetic Theory
Kinetic theory
We consider a dilute gas of N atoms in a box in the classical limit
a
de Broglie wave length
p k
h
h
h
p
2mEkin
2m kBT
h
L
N
1/ 3 with density n 3
L
N
2m kBT
N1/3 a L
1/ 3
h
1/ 3
N
L
n
Low density
hn
1
2m kBT
classical limit
high temperature
Ideal gas equation of state
Momentum change during
collision with wall:
t t
t t
dt F (t )dt
dt
t
t
Average perpendicular force component
acting on particle during time interval t
t t
1
av
mvx (t t ) mvx (t ) 2mvx (t ) Fxav, atom t
Fx, atom
Fx (t )dt
t t
p
dp
L
x
Average perpendicular force component acting on wall during time interval t
Fxav, wall Fxav, atom
2mvx (t ) Fxav, wall t
Time of flight t between successive collisions with right wall:
av
with Fx, wall 2mvx (t ) / t
Fxav, wall
mv
x
Lx
t
2
2 Lx
vx
Total force exerted on the wall by N particles
N
Fxtot
i 1
mi vx ,i
2
Lx
Pressure P on right wall of area A=LyLz
Fxtot
1
P
Ly Lz Ly Lz
N
i 1
mi vx ,i
2
Lx
1 N
2
mi vx ,i
V i 1
Same argument for particles flying
Fytot
1 N
2
mi vy ,i
in y-direction P
Lx Lz V i 1
tot
z
F
1 N
2
mi vz ,i
in z-direction P
Lx Ly V i 1
1
P
3V
1
3V
m v
N
i 1
N
i
mi vi
i 1
2
2
x ,i
v y ,i v z ,i
2
2
1 N
2
mi vi
Further inspection of P
3V i 1
1 N
2
m
v
i i
1
2
2
i 1
with
mi vi
and mi m i
N
2
arithmetic
average
21
PV N m v 2
3 2
1
1
2
m v m v2
2
2
Mean square
velocity
Compare with the ideal gas equation of state
PV N kBT
1
3
m v 2 k BT
2
2
Correct only in the classical limit
vrms :
v2
3kBT
m
See textbook for generalized expressions where
mi are not identical
Note: we did not derive yet the equation of state
1
3
m v 2 k BT
2
2
To do so we need to show
Calculation of statistical averages requires knowledge of the
distribution function:
f ( r , p, t ) d 3 r d 3 p :number of molecules at a given time, t, within a
volume element d3r located at r and
momentum element d3p around p=mv
Fztot
1
P
Lx Ly V
N
m v
i 1
Note that (r, p) spans the 6-dimensional µ-space where f is
defined. Later we will introduce a density function defined over
the 6N-dimensional -space. Don’t confuse!!!
2
i z ,i
With f we are able to calculate averages such as:
v2
2
3
3
v
f
(
r
,
p
,
t
)
d
r
d
p
f ( r , p, t ) d 3 r d 3 p
v2
2
3
3
v
f
(
r
,
p
)
d
r
d
p
3
f ( r , p, t ) f ( r , p) f1 ( p) f 2 ( r )
For
3
f ( r , p) d r d p
2
3
v
f
(
p
)
d
p
1
3
f1 ( p) d p
f (r ) d
f2 (r ) d 3r
2
3
r
2
3
v
f
(
p
)
d
p
1
f1 ( p) d 3 p
How to get f ?
In equilibrium, f turns out to be the
Maxwell-Boltzmann distribution function
We will encounter various approaches for its derivation throughout the
course.
Let’s start with a heuristic consideration
P(h h )
P(h ) P(h h)
(h) A h g
Mg
P(h h)
A
A
P(h h ) (h ) h g
h
Mass of the air contained in the volume element
A
h
Remember the barometric formula P=P(h)
h
air in good approximation considered as an ideal gas
M
V
n
M PP
PVnRT
P
RT
n RT
RT
M M
1/
P(h ) P(h h ) (h )
h g
P ( h h ) P ( h )
M gP(h )
h
n RT( h )
dP(h )
M g P( h )
dh
n R T(h )
h 0
In the case T=const. and with M=nNAm and R=NAkB
dP
mg
P k B T
P0
P
dP(h)
mg P(h)
dh
kB T
P(h) P0 e
mgh
kB T
and
(h) 0 e
mgh
kB T
h
dh
0
( r )d 3 r = number of particles in d3r probability of finding a particle in d3r
in accordance with our definition of f we expect
f ( r , p) d 3 p ( r ) ( x ) 0 e
mgx
kB T
0 e
E pot ( x )
kB T
p2
E pot ( r ))
Ansatz f ( r , p) f ( E
2m
d
3
p f ( E ) 0 e
3
d
p
E pot
kB T
d
dE pot
df ( E ) dE
1
0 e
dE dE pot
k BT
E pot
kB T
1
df ( E )
1
3
d
p
d
p f (E)
dE
kBT
3
1
3 df ( E )
d
p
f
(
E
)
0
dE kBT
1
3 df ( E )
d
p
f
(
E
)
0 E pot
dE kBT
df
E
df kBT
df ( E )
1
f (E) 0
dE
k BT
f e
f e
E
k BT
e
p2
E ( r )
2m
p2
E ( r )
2m
e
with
p2
2
m
E ( r )
e
:
1
k BT
f1 ( p 2 ) f 2 ( r )
velocity distribution
independent of position
Let’s calculate the Maxwell Boltzmann distribution f1(p2) with all constants
C d pe
3
p2
N
2m
C d pe
3
p2
2m
with
dx e
C 4 p dp e
2
p2
2m
0
x 2
0
1
2
4 C p dp e
2
0
2m
2m C 2m 3/ 2 C
f1 ( p )
2m
N
e
3/ 2
2m kBT
p2
2 mkBT
3/ 2
p2
2m
8m C
dp
e
0
2m
N 4m C
4m C 2m
2
p2
3/ 2
2
When asking for the probability, f(v), of finding a particle with velocity between
v v
and
v dv
f1 ( p 2 )d 3 p / N
we transform from d3p to dv
1
e
3/ 2
2m kBT
mv 2
2 k BT
m
2
4 mv d mv 4 v 2
2
k
T
B
3/ 2
e
mv 2
2 k BT
dv
f(v)
Maxwell speed distribution function
m
f (v) 4 v
2 k BT
3/ 2
2
e
mv 2
2 k BT
With, e.g., f(p) at hand we can actually derive the ideal gas equation of state
p
by calculating
3
2
3
2
2
1
1
m v2
2
2m
d
p p f ( p)
d
3
p f ( p)
1
2m
d
ppe
d
3
pe
2m
p2
2m
d
1
1
m v2
2
2m
3
d
p p 2e
3
pe
p2
p
2
2m
1
2m
2
p
dp e
4
p
dp e
0
2m
p2
2m
dpp
2
0
0
e
p2
2m
p2
2m
0
p2
2m
dp p e
4
2m
dp e
0
p2
2m
2m
1 2m 1 2m
2m
2
4
2
dpp
e
0
p2
1
1
m v2
2
2m
d
3
p p 2e
d
3
2m
pe
m 2m
2
3/ 2
3/ 2
3m
1
3/ 2
2
m
5/ 2
4
p2
2m
p
2
2m
1 3m 3
k BT
2m
2
with
21
PV N m v 2
3 2
PV NkBT