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Chapter 6: Entropy and the Boltzmann Law

Chapter 6: Entropy and the Boltzmann Law. S = k ℓ n W. This eqn links macroscopic property entropy and microscopic term multiplicity. k = Boltzmann constant (Appendix A) States tend toward maximum W and therefore maximum S. Eqn 6.1 S = k log W. S = -k Σ p i ℓn p i.

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Chapter 6: Entropy and the Boltzmann Law

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  1. Chapter 6: Entropy and the Boltzmann Law

  2. S = k ℓn W • This eqn links macroscopic property entropy and microscopic term multiplicity. • k = Boltzmann constant (Appendix A) • States tend toward maximum W and therefore maximum S. • Eqn 6.1 S = k log W

  3. S = -k Σ pi ℓn pi • W = N!/π ni! For N objects which can have the outcome i. (Eqn 1.18) • For a die, i = 1-6 and for a coin, i = H or T • Use x! = (x/e)x for each factorial term. • Also NN = πNni since N = Σni • pi = ni/N (Eqn 1.1) • This results in W = π [ N/ni ]ni = π [1/pini]

  4. Entropy and Probability (2) • This results in W = π [ N/ni ]ni = π [1/pini] • Plug W into ℓn W = - Σ ni ℓn pi • Divide by N to get (1/N) ℓn W = - Σ pi ℓn pi (Eqn 6.4) • Compare Eqn 6.1 (using ℓn instead of log) and Eqn 6.4 to get S/k = - Σ pi ℓn pi (Eqn 6.2)

  5. Entropy and Probability or Multiplicity • Entropy is an extensive property; the value of entropy depends on the system size (mass, # mol). So ST = ΣSi • But recall that pT = π pi (Eqn 1.6) • And WT = π Wi (p.34) • We confirm that the ℓn relationship between S and p or W is correct.

  6. Lessons from Quantum Mechanics • Energy is quantized • Translational (Examples 2.2, 2.3 and Ch 11) • Vibrational (Ch 11) • Rotational (Ch 11) • Electronic (Ch 11)

  7. Flat Distribution (Ex 6.1, Prob 1-3) • Consider a system that has 4 outcomes (dipole pointing to S, E, N, W) and each outcome is equally probable. • We expect that the most probable distribution is when the ni values are equal where i = S, E, N, W. This is called a flat distribution (Figure 6.1 d). What is the most probable result (max W) when a coin is tossed 50 times?

  8. Generalize Ex. 6.1 • Maximum S is associated with flat distribution. This is because each outcome is equally probable. • I.e. there are no constaints on the outcomes. • Note that a Lagrange multiplier is used to solve this problem.

  9. Now Include Constraints. • The model is the die with 6 possible outcomes (t = 6); pi is the probability of rolling the ith outcome. • Define g = 1 = Σpi (Eqn 6.10) • The function is S which we want to maximize. • Constraint: average score is <ε> = Σpi εi • Use Lagrange multipliers α for g and β for <ε> to find set of pi*

  10. Boltzmann Distribution Law • pi* = [exp (-βεi)]/Σ [exp (-βεi)]; definitely not flat. • These are the values that maximize S when the avg score is <ε>. • The denominator is called q = partition function (Ch 10) • Ex. 6.3 and 6.4, Prob 6.10

  11. Omit pp 89-99

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