A cell of volume contains molecular hydrogen (molar mass ) at and

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 cell of volume

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

contains molecular hydrogen

\mathrm{H_2}

(molar mass

M = 2.016\,\mathrm{g\,mol^{-1}}

) at

T = 300\,\mathrm{K}

and

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

. Treat the gas as ideal and each molecule as a point particle moving freely in the cell, and count only its translational states (rotation and vibration are not to be included here); the state sum may be replaced by an integral with 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

\mathrm{H_2}

molecule. ((b)) Find the translational single-particle partition function

Z_1

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

n^{-1/3}

between molecules, with

n = N/V

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

\lambda_T

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