Kinetic Theory: pressure
q Model: gas molecules with mass m
enclosed in a cube LxLxL
q Momentum of a molecule moving
in x direction:
q “pressure” in x direction produced
by one molecule
Fx mv 2x mv2x
px = 2 = 3 =
L
L
V
Px = mvx
q For N molecules
q Momentum change on recoil:
DPx = Px 2 - Px1 =
q For random motion we have
= m(-v x ) - mv x = 2m1v x
1
v x2 = v y2 = v z2 = v 2
3
q Force produced on the wall:
†
N
2
2
mvx = nmvx
V
px =
q Total pressure
dPx DPx 2mx v x mv x
=
=
=
dt
Dt 2L /v x
L
†
Dt = time between consecutive
1
2
p = nmv 2 = n E k = nkT
3
3
3
E k = kT
2
Fx =
wall collisions
†
†
†
Kinetic Theory: velocity distribution
q fraction of molecules with velocity
between vx and vx+dvx
dn
= f (v x ) dv x = Ae
n vx
= Ae
-
mv x2
2kT
E
- kx
kT
q Obtain Maxwell distribution
3
dv x =
q From here we calculate
mv 2
m - 2kTx
dv x =
e
dv x
2pkT
†
ß Average velocity
•
v=
constant A is determined by normalization
†
†
dv x dv y dv z = 4 pv 2 dv
8kT
pm
ß Root mean square velocity
•
v2 =
†
Úv
2
f (v)4 pv 2 dv =
0
3kT
m
ß Most probable velocity
q Rewrite in spherical coordinates
†
2
Ú vf (v)4pv dv =
0
q Total distribution
dn
= f (v x ) f (v y ) f (v z ) dv x dv y dv z =
n v
= f (v)dv x dv y dv z
2
mv
4 Ê m ˆ 2 2 - 2kT
f (v ) =
v
e
Á
˜
p Ë 2kT ¯
df (v )
2kT
=
dv
m
†
†
1
Maxwellian velocity distribution
Maxwell: T dependence
2
Maxwell: mass dependence
Kinetic Theory: impingement rate
q Model: infinite plane
perpendicular to x direction
q Need only x component of f(v)
q Engineering formulas (note units)
ß For molecules with molecular
mass M; pressure in torr
mv 2
F = 3.5.10 22
m - 2kTx
f (v x ) =
e
2pkT
ß for air (M=29 g) at room
temperature; pressure in torr
q Number of molecules striking a
surface / cm2s
†
†
•
m
F = Ú v x dn x = n
2pkT
°
F=
•
Úv e
x
-
†
mv x2
2kT
p È molecules˘
Í
˙
MT Î cm 2 s ˚
dv x
È molecules˘
F = 3.8.10 20 pÍ
Î cm 2 s ˙˚
°
1
1
8kT
p
nv = n
=
†
4
4
pm
2pmkT
†
3
Kinetic Theory: mean free path
q volume swept by a moving
molecule with diameter d in time
Dt
DV = pd 2vDt
l=
q for molecular density n
†
pd 2vDt =
l = vDt =
†
†
1
n
q mean free path
1
pd 2 n
5.10-3
[cm]
p
q Other properties of interest
ß viscosity
†
q taking into account movement of
molecules and Maxwell
†
distribution
1 1
l=
2 pd 2 n
q Engineering formula for air at
room temperature: p in torr
†
È Ns
˘
1
1
h = nmvl = r vl Í 2 = 10 p˙
Îm
˚
3
3
1
h = nmvl more rigorous result
2
ß diffusion coefficient
1
h
D = vl = [cm 2 /s]
3
r
h
more rigorous result
D = 1.34
r
†
†
†
4