1. Chapter 6 Boltzmann Statistics

2. 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.

3. 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?

4. 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

5. 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

6. 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?

7. 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?

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

9. Boltzmann, Entropy, & Gibbs

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

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

15. Hydrogen Atom @ 300K

16. Hydrogen Atom on Sun

18. A System with Smaller Energies

21. 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?

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

23. Rotation of Diatomic Molecules

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

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

27. Rotation of Diatomic Molecules HCl rotations

28. Rotation of Diatomic Molecules Average energy and heat capacity.

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

31. Rotational Partition Function Boltzmann Factors

32. Rotational Energies Unlike atoms distinguishable like atoms Indistinguishable

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

34. Maxwell Speed Distribution What is Z?

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

36. Maxwell Speed Distribution What is the average speed?

37. Maxwell Speed Distribution

38. 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)

39. Nitrogen Speed Distribution