Tossing 100 coins starting from 60 heads and 40 tails (with Wi = 1.4 x 10^28 microstates) eventually lands on the most likely result, 50 heads and 50 tails (with Wf = 1.0 x 10^29 microstates). Using ΔS = k(ln Wf - ln Wi) with Boltzmann's constant k = 1.38 x 10^-23 J/K, what is the change in entropy?

1. Boltzmann's formula gives entropy as S = k ln W, so the change between two states is ΔS = k(ln Wf - ln Wi) = k ln(Wf/Wi), where the subscript i stands for the initial 60 heads and 40 tails state and the subscript f stands for the final 50 heads and 50 tails state. 2. From Table 15.4, Wi = 1.4 x 10^28 microstates for 60 heads and 40 tails, and Wf = 1.0 x 10^29 microstates for 50 heads and 50 tails. 3. Find the ratio of these two microstate counts: Wf/Wi = (1.0 x 10^29)/(1.4 x 10^28) = 7.14. 4. Take the natural logarithm of this ratio: ln(7.14) = 1.97. 5. Multiply by Boltzmann's constant, substituting units: ΔS = (1.38 x 10^-23 J/K)(1.97) = 2.7 x 10^-23 J/K. 6. Option B, 1.38 x 10^-23 J/K, is only Boltzmann's constant itself, without multiplying by the logarithm of the microstate ratio. 7. Option C, 2.7 x 10^-22 J/K, is off by a factor of ten from the correctly computed value, an easy exponent slip to make. 8. Option A, 9.33 x 10^-23 J/K, borrows the numeric value 9.33 from a different worked example covered earlier (the entropy change for heat transfer between two reservoirs), which is not the figure that applies to this coin-toss calculation. _Source: OpenStax College Physics (CC BY 4.0), Ch 15 "Thermodynamics", section 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation_