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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 2kT 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