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Thermodynamics and Statistical Mechanics

Thermodynamics and Statistical Mechanics. Statistical Distributions. Multiple Outcomes. Distinguishable particles. Degenerate States. Suppose there are g j states that have the same energy. Boltzmann Statistics (Classical). Most Probable Distribution. Most Probable Distribution.

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Thermodynamics and Statistical Mechanics

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  1. Thermodynamics and Statistical Mechanics Statistical Distributions Thermo & Stat Mech - Spring 2006 Class 18

  2. Multiple Outcomes Distinguishable particles Thermo & Stat Mech - Spring 2006 Class 18

  3. Degenerate States Suppose there are gj states that have the same energy. Thermo & Stat Mech - Spring 2006 Class 18

  4. Boltzmann Statistics (Classical) Thermo & Stat Mech - Spring 2006 Class 18

  5. Most Probable Distribution Thermo & Stat Mech - Spring 2006 Class 18

  6. Most Probable Distribution Thermo & Stat Mech - Spring 2006 Class 18

  7. Constraints (Lagrange Multipliers) Thermo & Stat Mech - Spring 2006 Class 18

  8. Most Probable Distribution Thermo & Stat Mech - Spring 2006 Class 18

  9. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 18

  10. Quantum Statistics • Indistinguishable particles. • Bose-Einstein – Any number of particles per state. Particles with integer spin:0,1,2, etc • Fermi-Dirac – Only one particle per state: Particles with integer plus ½ spin: 1/2, 3/2, etc Thermo & Stat Mech - Spring 2006 Class 18

  11. Bose-Einstein • At energy ei there are Ni particles divided among gi states. How many ways can they be distributed? Consider Ni particles and gi – 1 barriers between states, a total of Ni + gi – 1 objects to be arranged. How many arrangements? Thermo & Stat Mech - Spring 2006 Class 18

  12. Bose-Einstein Thermo & Stat Mech - Spring 2006 Class 18

  13. Bose-Einstein Thermo & Stat Mech - Spring 2006 Class 18

  14. Bose-Einstein Thermo & Stat Mech - Spring 2006 Class 18

  15. Constraints (Lagrange Multipliers) Thermo & Stat Mech - Spring 2006 Class 18

  16. Bose-Einstein Thermo & Stat Mech - Spring 2006 Class 18

  17. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 18

  18. Fermi-Dirac • At energy ei there are Ni particles divided among gi states, but only one per state. gi³ Ni. • How many ways can the Ni occupied states be selected from the gi states? Thermo & Stat Mech - Spring 2006 Class 18

  19. Fermi-Dirac Thermo & Stat Mech - Spring 2006 Class 18

  20. Fermi-Dirac Thermo & Stat Mech - Spring 2006 Class 18

  21. Fermi-Dirac Thermo & Stat Mech - Spring 2006 Class 18

  22. Constraints (Lagrange Multipliers) Thermo & Stat Mech - Spring 2006 Class 18

  23. Fermi-Dirac Thermo & Stat Mech - Spring 2006 Class 18

  24. Distributions Thermo & Stat Mech - Spring 2006 Class 18

  25. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 18

  26. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 18

  27. Partition Function Thermo & Stat Mech - Spring 2006 Class 18

  28. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 18

  29. Ideal Gas Thermo & Stat Mech - Spring 2006 Class 18

  30. Ideal Gas Thermo & Stat Mech - Spring 2006 Class 18

  31. Gamma Function Thermo & Stat Mech - Spring 2006 Class 18

  32. Partition Function for Ideal Gas Thermo & Stat Mech - Spring 2006 Class 18

  33. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 18

  34. Ideal Gas Thermo & Stat Mech - Spring 2006 Class 18

  35. Quantum Statistics • When taken to classical limit quantum results must agree with classical. B-E and F-D must approach Boltzmann in classical limit. What is that limit? • Low particle density! Then distinguishability is not a factor. Thermo & Stat Mech - Spring 2006 Class 18

  36. Classical limit Thermo & Stat Mech - Spring 2006 Class 18

  37. Quantum Results Thermo & Stat Mech - Spring 2006 Class 18

  38. Chemical Potential Thermo & Stat Mech - Spring 2006 Class 18

  39. Three Distributions Thermo & Stat Mech - Spring 2006 Class 18

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