A row of identical two-state units ("spins") is fixed in place; each unit shows…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Microstates, multiplicity and the statistical entropy

Problem

A row of

20

identical two-state units ("spins") is fixed in place; each unit shows either up or down and, in this chapter's model, carries no energy, so every arrangement of the row has the same energy. The row is isolated, so every microstate is equally likely. Separately, a grain of solid is held at constant volume at

300\ \mathrm{K}

, and is large enough that its temperature does not change measurably when the small quantity of heat below enters it. ((a)) How many microstates belong to the macrostate "exactly

13

up"? ((b)) What is the probability of that macrostate? ((c)) Chapter 10 treats the number of up units in the row as a binomial variable with

p = 1/2

. How many standard deviations above the mean is

13

? ((d)) The grain absorbs

1.00\times10^{-20}\ \mathrm{J}

of heat at

300\ \mathrm{K}

. Using the Chapter 6 result for the entropy change when heat enters at a fixed temperature, together with

S = k_B\ln\Omega

, by what factor is the multiplicity of the grain multiplied?

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