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Chapter 6 Boltzmann Statistics

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Boltzmann Factor & Partition Functions U R, S R U, S Huge reservoir System Changes in energy of the reservoir are very small compared to its total energy. Say the system has 5 atoms and 2 units of energy.

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Boltzmann Factor & Partition Functions Say the system has 5 atoms and 2 units of energy. Atom 1Atom 2Atom 3Atom 4Atom 5 20000 02000 00200 00020 00002 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 What is the probability of finding a particular atom with 2, 1, or 0 units of energy?

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Boltzmann Factor & Partition Functions Say the system has 10 atoms and 4 units of energy. What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy? U R, S R U, S Huge reservoir System

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Boltzmann Factor & Partition Functions Say the system has 10 atoms and 4 units of energy. What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy? U R, S R U, S Huge reservoir System

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Boltzmann Factor & Partition Functions Say the system has 10 atoms and 4 units of energy. What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy?

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Boltzmann Factor & Partition Functions Say the system has 10 atoms and 4 units of energy. What is the probability of finding a particular atom with 4, 3, 2, 1, or 0 units of energy?

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Boltzmann Factor & Partition Functions Boltzmann Factors Partition Function Boltzmann or Canonical Distribution

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Boltzmann, Entropy, & Gibbs

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Partition Functions & Hydrogen Atom What’s the energy of the electron on a hydrogen atom? Solution to Rydberg or Bohr Model can be used.

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Partition Functions & Hydrogen Atom What’s the energy of the electron on a hydrogen atom? Solution to Rydberg or Bohr Model can be used.

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Hydrogen Atom @ 300K

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Hydrogen Atom on Sun

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A System with Smaller Energies

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Average Values Say the system has 10 atoms and 4 units of energy. What is the average energy of the system if 4 atoms have n=0 3 atoms have n=1 2 atoms have n=2 1 atom has n=3 0 atoms have n=4?

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Averages Values Be careful about using the proper probability when computing averages.

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Rotation of Diatomic Molecules

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This can be calculated as an integral if dj is small compared to k B T/ (high temperature limit).

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Rotation of Diatomic Molecules A simplified partition function in the high temperature limit.

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Rotation of Diatomic Molecules HCl rotations

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Rotation of Diatomic Molecules Average energy and heat capacity.

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Rotational Partition Function For diatoms with unlike atoms For diatoms with like atoms Unlike atoms distinguishable like atoms Indistinguishable Boltzmann Factors

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Rotational Partition Function Boltzmann Factors

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Rotational Energies Unlike atoms distinguishable like atoms Indistinguishable

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Maxwell Speed Distribution vxvx vyvy vzvz v For continuous variables, we talk about probability density function or probability distribution function (pdf).

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Maxwell Speed Distribution What is Z?

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Maxwell Speed Distribution What is maximum probability speed? The Maxwell Speed Distribution

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Maxwell Speed Distribution What is the average speed?

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Maxwell Speed Distribution

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Example: Nitrogen @ 300K For Monday: (a) Show the maximum probability speed is 517 m/s. (b) Show the average speed is 476 m/s. (c) Show the rms speed is 422 m/s. (d) Calculate the probability of a molecule moving faster than 1000 m/s. (Need Maple or Mathematica)

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Nitrogen Speed Distribution

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Lecture 4. Maxwell-Boltzman Distribution

Lecture 4. Maxwell-Boltzman Distribution

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