A molecule trapped at a fixed site has a single ground state of energy and an excited…

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

A molecule trapped at a fixed site has a single ground state of energy

0

and an excited level of energy

\varepsilon = 3.00\times10^{-21}\,\mathrm{J}

that consists of three distinct states of the same energy (a threefold degenerate level). It is weakly coupled to a reservoir at

T = 250\,\mathrm{K}

. Take

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

and give numerical answers to three significant figures. ((a)) Evaluate the partition function

Z

of one molecule, summing over states. ((b)) Find the probability that the molecule is in its ground state, and the probability that it is somewhere in the excited level. ((c)) Find the mean energy

U

of one molecule, in joules. ((d)) Two such molecules are held at two distinct fixed sites and do not interact. Write down the partition function of the pair and evaluate it, and give the mean energy of the pair in joules.

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