1. The Boltzmann factor

2. The isothermal atmosphere I
Pressure and density difference due to molecules on top
Extra pressure due to molecules within Dh: each has mass mg, there are n(h)·A·Dh of them (n=number density):
p+ p D D h p

3. The isothermal atmosphere II
Use ideal gas law: pV = NkT 
Take limit for very small Dh:
Likewise

4. Generalisation
Note that mgh is P.E. of particle in gravitational field
This is generally true:
is called the Boltzmann factor

5. Kinetic energy
Likewise, it can be shown that the probability of finding a molecule with kinetic energy Ek is
For the distribution of velocities we find (normalising to a total probability of 1)

6. Maxwell-Boltzmann distribution
The probability of finding a molecule with speed v at temperature T is given by:

7. Diffusion and mobility

8. 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

9. Collisional cross section
Chance of collision in dx = s n0 dx
On average 1 collision per l: s n0 l = 1
Classical model:
Area covered:
sn0dx dx
unit area

10. 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:

11. Ionic conductivity I tmean / m is called mobility m.
Ions inside battery move with
In Dt all ions within vdrift·Dt reach the plate
For ion density ni: ni·A·vdrift·Dt ions are within this distance d E + –

12. Ionic conductivity II
Each ion carries a charge q. So: total charge collected in Dt is DQ = q ·ni·A·vdrift·Dt
Current is charge over time: d E + –

13. Resistance Compare to Ohm’s Law:
Note: resistance in wires etc. is due to collisions of electrons with ions in the wire d E + –

14. 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:
D is called diffusion coefficient

15. Diffusion and drift
Diffusion coefficient depends on the speed v and the mean free path l:
Recall  = v·tmean and tmean = m ·m:
Use

16. 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