Transcript No Slide Title

z = 2 c (N/V)
Assumes volume swept out is independent of whether
collisions occur (not a bad assumption in most cases)
distance
collisions
c
,z 
sec
sec
 = 1/[2(N/V)]
Some typical numbers: STP: 6.02  1023 molecules / 22.4 liters
P = 1 Torr

z = 5.45  106 sec -1 (P = 1/760 atm)
 P  1Torr  
4  10 4 cm / sec
3

7.3

10
cm / collision (P = 1/760 atm)
6
1
5.45  10 collisions / sec
Distribution of Molecular Speeds
Real gases do not have a single fixed speed. Rather molecules
have speeds that vary giving a speed distribution.
This distribution can be measured in a laboratory (done at
Columbia by Polykarp Kusch) or derived from theoretical
principles.
Molecular Beam Apparatus for Determining Molecular Speeds
Collimating slits

Box of Gas
at temperature T
Whole apparatus is evacuated to
roughly 10-6 Torr!
Detector
Synchronized rotating sectors
Maxwell-Boltzmann Speed Distribution
(N/N) =
2
4[m/(2kT)]3/2 e-[(1/2)mc / kT](c2)∆c
∆N is the number of molecules in the range c to c +∆c and
N = Total # of molecules.
∆N/N is the fraction of
molecules with speed in
the range c to c+∆c
y = e -x
1
x = (1/2)mc2/kT
0
x
For our case x = (1/2) mc2/kT = /kT, where  = K.E.
y = e -x
1
x
0
c2
parabola
c
c2
c2
or
e
 (1/2)mc
2
e
 (1/2)mc
2
/ kT
/ kT
peak
c
2
(N/N) = 4[m/(2kT)]3/2 e-[(1/2)mc /kT] (c2)∆c
[Hold ∆c constant at say c = 0.001 m/s]
Typical Boltzmann Speed
Distribution and
Its Temperature Dependence
0.0025
0o C
0.002
Fractional #
of Molecules 0.0015
(N/N)
0.001
1000o C
2000o C
0.0005
High Energy Tail
(Responsible for
Chemical reactions)
0
0
500
1000
1500
2000
2500
Speed, c (m/s )
3000
3500
4000
1) The Root Mean Square Speed:
crms = (3RT/M)1/2
If N is the total number of atoms, c1 is the speed of atom 1,
and c2 the speed of atom 2, etc.:
crms = [(1/N)(c12 + c22 +c32 + ………)]1/2
2)
3)
cmp is the value of c that gives (N/N) in the Boltzmann
distribution its largest value.
Boltzmann Speed Distribution for Nitrogen
0.0025
0.002
[2RT/M]1/2
Cmp (870 m/s) =
Fractional #
of Molecules 0.0015
(N/N)
0.001
0.0005
Cavg (980 m/s)
= [8RT/M]1/2
Crms (1065 m/s)
= [3RT/M]1/2
1000o C
0
0
500
1000
1500
2000
2500
Speed, c (m/s)
3000
3500
4000
Cleaning Up Some Details
A number of simplifying assumptions that we have made in
deriving the Kinetic Theory of Gases cause small errors in the
formulas for wall collision frequency, collision frequency (z),
mean free path (), and the meaning of c, the speed:
1)
A
ct
The assumption that all atoms move only perpendicular to the
walls of the vessel is obviously an over simplification.
2) For the collision frequency, z, the correct formula is
The (2)1/2 error here arises from the fact that we assumed only
one particle (red) was moving while the others (blue) stood still.
3) Even though the formula for wall collisions used in deriving
the pressure was incorrect, the pressure formula is correct!
This is because of offsetting errors
made in deriving the wall collision rate, I, and the momentum
change per impact, 2mc.
A final question that arises concerns which c, cavge, crms, or
cmp is the correct one to use in the formulas for wall collision
rates (I), molecule collision rates (z), mean free path ()
and pressure (p).
For p the correct form of c is crms while for I, or z considered as
independent quantities, cavge is correct.

= cavge/{(2)1/2(N/V)  2 cavge}
Bonus * Bonus * Bonus * Bonus * Bonus * Bonus
Summary of correct Kinetic Theory formulas: