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Chapter 6 Statistical Thermodynamics

09/19/2001. Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993). Combinatorial Analysis. Consider a system of N particles that are allowed to occupy r states.Microstate --- The description of the system that provides the state of each particle.Number of possible micros

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Chapter 6 Statistical Thermodynamics

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    1. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Chapter 6 Statistical Thermodynamics Notes on Thermodynamics in Materials Science by Robert T. DeHoff (McGraw-Hill, 1993).

    2. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Combinatorial Analysis Consider a system of N particles that are allowed to occupy r states. Microstate --- The description of the system that provides the state of each particle. Number of possible microstates =

    3. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    4. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Assumptions Consider all particles to be identical. The net value of a macroscopic property depends on the number of particles (ni) in each state (i). Exchanging the specific identity of the particles in a state does not change the value of the property. On average the fraction of time each particle spends in any energy state is the same.

    5. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Probability of Macrostates Hypothesis Fraction of time in a macrostate = probability of that macrostate.

    6. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Probability of Macrostates

    7. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    8. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    9. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    10. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    11. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    12. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    13. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Boltzman Hypothesis where S is the entropy. W is the # of microstates in a macrostate. The Boltzman constant, k = R/NO. NO is Avogardos number. R is the ideal gas constant.

    14. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    15. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Find Conditions for Equilibrium Find an expression for change in entropy of the system. Determine the constraints. Apply the constraints and the extremum criterion:

    16. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Find an expression for dS(ni) Substitute for W:

    17. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Find an expression for dS(ni)

    18. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Isolation Constraints

    19. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Constrained Maximum Entropy Apply Lagrange multipliers to constraints & add to condition for entropy maximum.

    20. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    21. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Constrained Maximum Entropy

    22. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Constrained Maximum Entropy

    23. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    24. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    25. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    26. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    27. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Monatomic Gas Model Assumptions: All particles are identical. Volume = lx x ly x lz Energy of the system is not quantized & is equal to S kinetic energies of the particles.

    28. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    29. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    30. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993) Einsteins Model of a Crystal Consider a simple cubic crystal --- 6 nearest neighbors, 1 atom & 3 bonds per unit cell. Hypothesis --- Energy of crystal is the sum of the energies of its bonds. The atoms vibrate around equilibrium positions as if bound by vibrating springs. Only certain vibrational frequencies are allowed in coupled springs. The energies (ei) of the bonds are proportional to their vibrational frequencies (n). ei = (i + 1/2) hn where h = Plancks constant. The adjustable parameter n is set by assuming an Einstein temperature: qE = hn/k

    31. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    32. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

    33. 09/19/2001 Notes from R.T. DeHoff, Thermodynamics in Materials Science (McGraw-Hill, 1993)

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