A row of two-state units is fixed in place; each shows up or down and carries no energy
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles
Thermal Physics, Ch Microstates, multiplicity and the statistical entropy
Problem
A row of 24 two-state units is fixed in place; each shows up or down and carries no energy, so all arrangements have the same energy and, the row being isolated, all are equally likely. Separately, a silver crystal is held at constant volume at 200K and is large enough that its temperature does not change measurably when the heat considered below enters it.
((a)) Find the multiplicity of the macrostate with exactly 15 up, and the multiplicity of the most probable macrostate, 12 up.
((b)) Evaluate lnΩ for the 12-up macrostate exactly, and compare it with the estimate obtained by applying Stirling's approximation lnN!≈NlnN−N (Chapter 10) to the three factorials in the binomial coefficient.
((c)) How much heat, delivered to the silver crystal at 200K, multiplies the crystal's multiplicity by exactly 10.0? Use the Chapter 6 result for the entropy change when heat enters at fixed temperature, together with S=kBlnΩ.