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The Boltzmann factor

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

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The Boltzmann factor

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  1. The Boltzmann factor

  2. 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): The isothermal atmosphere I 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 = sn0 dx • On average 1 collision per l: sn0l = 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 d • tmean / m is called mobilitym. • 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 E + –

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

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

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