One mole of argon

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 grand canonical ensemble, ensemble equivalence and fluctuations

Problem

One mole of argon, treated as a monatomic ideal gas with

U = \tfrac32Nk_BT

, is sealed in a rigid vessel of fixed volume in thermal contact with a bath at

T = 300\,\mathrm{K}

. Energy crosses the vessel wall freely; atoms cannot. Take

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

and

N_A = 6.022\times10^{23}\,\mathrm{mol^{-1}}

. ((a)) Write the first-order expansion of the bath entropy

S_R(U_0 - E_s, N_0 - N_s)

for a system that could exchange both energy and particles, state which reservoir derivative supplies the

1/T

and which supplies the

-\mu/T

, and say why for this sealed vessel the particle term is absent so that the Boltzmann factor applies. ((b)) Starting from

Z = \sum_s e^{-\beta E_s}

, show that

\left(\partial^2\ln Z/\partial\beta^2\right)_V = \langle E^2\rangle - \langle E\rangle^2

, and hence that

\sigma_E^2 = k_BT^2C_V

. ((c)) Evaluate

C_V

U

and

\sigma_E

for the argon, and show that the relative width is

\sigma_E/U = \sqrt{2/3N}

by comparing your number with that formula.

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