A sealed cell of volume holds argon gas (molar mass ) at temperature and pressure

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical mechanics of ideal classical systems · Statistical mechanics of ideal classical systems

Thermal Physics, Ch The classical ideal gas from the partition function

Problem

A sealed cell of volume

V = 1.00\,\mathrm{cm^3}

holds argon gas (molar mass

M = 39.95\,\mathrm{g\,mol^{-1}}

) at temperature

T = 300\,\mathrm{K}

and pressure

p = 1.00\times10^{5}\,\mathrm{Pa}

. Treat the gas as ideal and each atom as a point particle moving freely in the box, with only its translational states counted; the momentum grid is so fine against the thermal momentum that the state sum may be replaced by an integral, giving one state per volume

h^3

of position-momentum space. Take

h = 6.626\times10^{-34}\,\mathrm{J\,s}

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

and

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

. ((a)) Find the thermal de Broglie wavelength

\lambda_T

of an argon atom at this temperature. ((b)) Find the single-particle partition function

Z_1

for one atom in this cell. ((c)) Find the mean spacing

n^{-1/3}

between atoms, where

n = N/V

is the number density. ((d)) Express the mean spacing as a multiple of

\lambda_T

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