The Boltzmann factor
The isothermal atmosphere I
Pressure and density difference
due to molecules on top
Extra pressure due to molecules
Dh
within Dh: each has mass mg,
there are n(h)·A·Dh of them
(n=number density):
F
mg  nADh
Dp    
 n  mgDh
A
A
p+Dp
p
The isothermal atmosphere II
N
p
Use ideal gas law: pV = NkT  n  
V kT
mgDh
Dp  n  mgDh  
p
kT
Take limit for very small Dh:
dp
mg

 p  p  p0
dh
kT
Likewise n  n0
mgh

 e kT
mgh

 e kT
Generalisation
Note that mgh is P.E. of particle in
gravitational field
This is generally true: n  n0
e

P.E.
kT
P.E.

 e kT
is called the Boltzmann factor
Kinetic energy
Likewise, it can be shown that the probability
of finding a molecule with kinetic energy Ek is
P
Ek

 e kT
For the distribution of velocities we find
(normalising to a total probability of 1)
2
mv
3/ 2
  v 2  e  2 kT
m

P (v)  4  

 2kT 
Maxwell-Boltzmann distribution
The probability of finding a molecule with
speed v at temperature T is given by:
Diffusion and mobility
Collisions between molecules
Mean time between
collisions : tmean
Mean free path  = v · tmean
Collisional cross section s :
area in which the center of
the particle must be for
collision to take place
Collisional cross section
Chance of collision
in dx = s n0 dx
Area covered:
sn0dx
On average 1
collision per :
s n0  = 1
Classical model:
s   r1  r2 2 2
dx
unit area
Drift speed
Say that on some molecules we exert a force F
They collide but make net progress in the
direction of F
Speed picked up since last collision is on
average:
vdrift
F
  tmean
m
Ionic conductivity I
tmean / m is called mobility m.
d
Ions inside battery move with
vdrift
V
 mF  mqE  mq
d
In Dt all ions within vdrift·Dt
reach the plate
For ion density ni: ni·A·vdrift·Dt
ions are within this distance
E
+–
Ionic conductivity II
Each ion carries a charge q. So:
total charge collected in Dt is
DQ = q ·ni·A·vdrift·Dt
d
Current is charge over time:
DQ
mq 2ni A
I
 qni Avdrift 
V
Dt
d
E
+–
Resistance
Compare to Ohm’s Law:
I
mq 2ni A
R 
d
d
V
d
mq ni A
2
Note: resistance in wires etc. is
due to collisions of electrons
with ions in the wire
E
+–
Diffusion
Due to random motion molecules spread
throughout gas even without additional forces;
e.g. smell of cooking spreads through house.
Net flow depends on difference in density
throughout the room:
dn
Jx  D 
D is called diffusion coefficient
dx
Diffusion and drift
Diffusion coefficient depends on the speed v
and the mean free path : D  13  v  
Recall  = v·tmean and tmean = m ·m: D  13  v 2  m  m
Use
1 mv 2
2
 32 kT :
D  m  kT
PS225 – Thermal Physics topics
The atomic hypothesis
Heat and heat transfer
Kinetic theory
The Boltzmann factor
The First Law of Thermodynamics
Specific Heat
Entropy
Heat engines
Phase transitions