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 6.1S = k log W

3. S = -k Σ p i ℓn p i W = N!/π n i ! 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 N N = πN ni since N = Σn i p i = n i /N(Eqn 1.1) This results in W = π [ N/n i ] ni = π [1/p i ni ]

4. Entropy and Probability (2) This results in W = π [ N/n i ] ni = π [1/p i ni ] Plug W into ℓn W = - Σ n i ℓn p i Divide by N to get (1/N) ℓn W = - Σ p i ℓn p i (Eqn 6.4) Compare Eqn 6.1 (using ℓn instead of log) and Eqn 6.4 to get S/k = - Σ p i ℓn p i (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 S T = Σ S i But recall that p T = π p i (Eqn 1.6) And W T = π W i (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 n i 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); p i is the probability of rolling the i th outcome. Define g = 1 = Σ p i (Eqn 6.10) The function is S which we want to maximize. Constraint: average score is = Σ p i ε i Use Lagrange multipliers α for g and β for to find set of p i *

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

11. Omit pp 89-99