Impurity atoms are held at fixed

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 The Boltzmann factor, the canonical ensemble and the partition function

Problem

N = 4.0\times10^{19}

impurity atoms are held at fixed, separate sites in a solid, so they are distinguishable and independent; each is weakly coupled to the solid, which acts as a reservoir at

T = 200\,\mathrm{K}

. Each impurity has a single ground state of energy

0

and, above it, a level of energy

\varepsilon = 2.50\times10^{-21}\,\mathrm{J}

that is doubly degenerate (two distinct states of that same energy). ((a)) Evaluate

Z_1

for one impurity and the free energy

F

of all

N

of them. ((b)) Evaluate the internal energy

U

of all

N

impurities. ((c)) Use

F = U - TS

from Chapter 7 to obtain the entropy

S

of the

N

impurities. ((d)) Check your entropy with the Gibbs formula

S = -k_B\sum_s P_s\ln P_s

of Chapter 11 applied to one impurity: quote the per-impurity entropy as a multiple of

k_B

. Take

k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}

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