The Boltzmann factor

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

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

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

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)

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 On average 1 collision per l: s n0 l = 1 Classical model: Area covered: sn0dx 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:

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 + –

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 + –

Resistance Compare to Ohm’s Law: Note: resistance in wires etc. is due to collisions of electrons with ions in the wire d 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: D is called diffusion coefficient

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

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