Chapter 6 Boltzmann Statistics

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.

Boltzmann Factor & Partition Functions Say the system has 5 atoms and 2 units of energy. Atom 1Atom 2Atom 3Atom 4Atom What is the probability of finding a particular atom with 2, 1, or 0 units of energy?

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

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

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?

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?

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

Boltzmann, Entropy, & Gibbs

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

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

Hydrogen 300K

Hydrogen Atom on Sun

A System with Smaller Energies

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?

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

Rotation of Diatomic Molecules

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

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

Rotation of Diatomic Molecules HCl rotations

Rotation of Diatomic Molecules Average energy and heat capacity.

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

Rotational Partition Function Boltzmann Factors

Rotational Energies Unlike atoms distinguishable like atoms Indistinguishable

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

Maxwell Speed Distribution What is Z?

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

Maxwell Speed Distribution What is the average speed?

Maxwell Speed Distribution

Example: 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)

Nitrogen Speed Distribution