Thermodynamics and Statistical Mechanics

1. Thermodynamics and Statistical Mechanics Partition Function Thermo & Stat Mech - Spring 2006 Class 19

2. Free Expansion of a Gas Thermo & Stat Mech - Spring 2006 Class 19

3. Free Expansion Thermo & Stat Mech - Spring 2006 Class 19

4. Isothermal Expansion Thermo & Stat Mech - Spring 2006 Class 19

5. Isothermal Expansion Reversible route between same states. đQ = đW + dU Since T is constant, dU = 0. Then, đQ = đW. Thermo & Stat Mech - Spring 2006 Class 19

6. Entropy Change The entropy of the gas increased. For the isothermal expansion, the entropy of the Reservoir decreased by the same amount. So for the system plus reservoir, DS = 0 For the free expansion, there was no reservoir. Thermo & Stat Mech - Spring 2006 Class 19

7. Statistical Approach Thermo & Stat Mech - Spring 2006 Class 19

8. Statistical Approach Thermo & Stat Mech - Spring 2006 Class 19

9. Partition Function Thermo & Stat Mech - Spring 2006 Class 19

10. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 19

11. Maxwell-Boltzmann Distribution • Correct classical limit of quantum statistics is Maxwell-Boltzmann distribution, not Boltzmann. • What is the difference? Thermo & Stat Mech - Spring 2006 Class 19

12. Maxwell-Boltzmann Probability wB and wMB yield the same distribution. Thermo & Stat Mech - Spring 2006 Class 19

13. Relation to Thermodynamics Thermo & Stat Mech - Spring 2006 Class 19

14. Relation to Thermodynamics Thermo & Stat Mech - Spring 2006 Class 19

15. Chemical Potential • dU = TdS – PdV + mdN • In this equation, m is the chemical energy per molecule, and dN is the change in the number of molecules. Thermo & Stat Mech - Spring 2006 Class 19

16. Chemical Potential Thermo & Stat Mech - Spring 2006 Class 19

17. Entropy Thermo & Stat Mech - Spring 2006 Class 19

18. Entropy Thermo & Stat Mech - Spring 2006 Class 19

19. Helmholtz Function Thermo & Stat Mech - Spring 2006 Class 19

20. Chemical Potential Thermo & Stat Mech - Spring 2006 Class 19

21. Chemical Potential Thermo & Stat Mech - Spring 2006 Class 19

22. Boltzmann Distribution Thermo & Stat Mech - Spring 2006 Class 19

23. Distributions Thermo & Stat Mech - Spring 2006 Class 19

24. Distributions Thermo & Stat Mech - Spring 2006 Class 19

25. Ideal Gas Thermo & Stat Mech - Spring 2006 Class 19

26. Ideal Gas Thermo & Stat Mech - Spring 2006 Class 19

27. Ideal Gas Thermo & Stat Mech - Spring 2006 Class 19

28. Entropy Thermo & Stat Mech - Spring 2006 Class 19

29. Math Tricks • For a system with levels that have a constant spacing (e.g. harmonic oscillator) the partition function can be evaluated easily. In that case, en = ne, so, Thermo & Stat Mech - Spring 2006 Class 19

30. Heat Capacity of Solids • Each atom has 6 degrees of freedom, so based on equipartition, each atom should have an average energy of 3kT. The energy per mole would be 3RT. The heat capacity at constant volume would be the derivative of this with respect to T, or 3R. That works at high enough temperatures, but approaches zero at low temperature. Thermo & Stat Mech - Spring 2006 Class 19

31. Heat Capacity • Einstein found a solution by treating the solid as a collection of harmonic oscillators all of the same frequency. The number of oscillators was equal to three times the number of atoms, and the frequency was chosen to fit experimental data for each solid. Your class assignment is to treat the problem as Einstein did. Thermo & Stat Mech - Spring 2006 Class 19

32. Heat Capacity Thermo & Stat Mech - Spring 2006 Class 19