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Chapter 19 Statistical thermodynamics: the concepts

Chapter 19 Statistical thermodynamics: the concepts. Macroscopic World. T, P, S, H, U, G, A. Statistical Thermodynamics Kinetics Dynamics. How to translate mic into mac?. { r i},{ p i},{ M i},{ E i} …. Microscopic World. The job description. Brute force approach does not work!.

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Chapter 19 Statistical thermodynamics: the concepts

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  1. Chapter 19 Statistical thermodynamics: the concepts Macroscopic World T, P, S, H, U, G, A... Statistical Thermodynamics KineticsDynamics How to translate mic into mac? {ri},{pi},{Mi},{Ei}… Microscopic World

  2. The job description

  3. Brute force approach does not work! • 1 mol 100000000000000000000000 particles • ~100000000000000000000000 equations needed to be solved!!! Bad news: We cannot afford it! Good news: We do not need that detailed description! How? ………

  4. Thanks to them James Clark Maxwell Ludwig Boltzmann Josiah Willard Gibbs Tsung-Dao Lee Arrhennius Enrico Fermi Ising Landau Paul Dirac Bose Einstein Cheng-Ning Yang Van Hove Langevin Bardeen Mott Anderson …

  5. THE magic word Statistical We, the observers, are macroscopic. We only need average of microscopic information. Spatial and temporal average

  6. Macroscopic World Still lots of challenges (opportunities) herein! T, P, S, H, U, G, A... {ri},{pi},{Mi},{Ei}... Microscopic World

  7. Contents The distribution of molecular states 19.1 Configuration and weights 19.2 The molecular partition function The internal energy and the entropy 19.3 The internal energy 19.4 The statistical entropy The canonical partition function 19.5 The canonical ensemble 19.6 The thermodynamic information in partition function 19.7 Independent molecules

  8. Assignment for Chapter 19 Exercises: • 19.1(a), 19.2(b), 19.4(a) Problems: • 19.6(a), 19.9(b), 19.11(b), 19.15(a) • 19.3, 19.7, 19.14, 19.18, 19.22

  9. E E E E The distribution of molecular states These particles might be distinguishable .............. Distribution = Population pattern

  10. E E E Enormous possibilities! E E E E E E E E E E E ..............

  11. E E E E E Distinguishable particles

  12. Principle of equal a priori probabilities • All possibilities for the distribution of energy are equally probable. An assumption and a good assumption.

  13. E E E They are equally probable E E E E E E E E E E E ..............

  14. E E E E E They are equally probable

  15. {5,0,0,...} Configuration and weights The numbers of particles in the states

  16. {3,2,0,...}

  17. {3,2,0,...} One configuration may have large number of instantaneous configurations

  18. {N-2,2,0,...} How many instantaneous configurations? N(N-1)/2

  19. {3,4,5,6} E 18!/3!/4!/5!/6!

  20. Configuration and weights W is huge! 20 particles: {1,0,3,5,10,1} W=931000000 How about 10000 particles with {2000,3000,4000,1000}?

  21. Stirling’s approximation:

  22. W Wmax There is an overwhelming configuration {ni} {ni}max

  23. Equilibrium configuration The dominating configuration is the configuration with largest weight. The dominating configuration is what we actually observe. All other configurations are regarded as fluctuation. Constant total energy : Constant total number of molecules:

  24. Find the distribution with largest lnW

  25. Lagrange’s method of undetermined multipliers z=f(x,y) with g(x,y)=c1, h(x,y)=c2 To find the maximum of z with constraints g and h, we May use

  26. Boltzmann distribution Boltzmann constant

  27. E The molecular partition function(nondegenerate case)

  28. E The molecular partition function(degenerate case)

  29. Angular momentum

  30. The Rotational Energy Levels (Ch 16) Around a fixed-axis Around a fixed-point (Spherical Rotors)

  31. E E5 E4 E3 E2 E1 Example: Linear Molecules (rigid rotor)

  32. E=ε,g=2 q=? E=0,g=1 E E5 E4 E3 E2 q=? E1 Exercises

  33. The physical interpretation of the molecular partition function E, T E, ∞ E, T=0 q is an indication of the average number of states that are thermally accessible to a molecule at the temperature of the system.

  34. Example: Uniform ladder of energy levels (e.g., harmonic vibrator)

  35. E=ε,g=1 E=0,g=1

  36. E=ε,g=1 E=0,g=1

  37. E=ε,g=1 E=0,g=1

  38. E=ε,g=1 E=0,g=1

  39. E=ε,g=1 E=ε,g=1 E=0,g=1 E=0,g=1

  40. E=ε,g=1 E=0,g=1 E=ε,g=1 E=0,g=1

  41. Approximations and factorizations • Exact, analytical partition functions are rare. • Various kinds of approximations are employed: dense energy levels independent states (factorization of q) …

  42. Dense energy levels One dimensional box:

  43. Independent states (factorization of q) Three-dimensional box: (Translational partition function) Thermal wavelength

  44. Why q, the molecular partition function, so important? • It contains all information needed to calculate the thermodynamic properties of a system of independent particles (e.g., U, S, H, G, A, p, Cp, Cv …) • It is a kind of “thermal wavefunction”. (Remember the wavefunction in quantum mechanics which contains all information about a system we can possibly acquire)

  45. Find the internal energy U from q Total energy of the system: At T=0, U=U(0)

  46. E=ε,g=1 E=0,g=1 A two-level system U=?

  47. E=ε,g=1 E=0,g=1 A two-level system W=? Exercise

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