A rigid glass bulb of volume contains helium-4 gas (atomic mass )

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Kinetic theory and transport in gases · Kinetic theory and transport in gases

Thermal Physics, Ch Collisions, mean free path and transport in gases

Problem

A rigid glass bulb of volume 0.50 L0.50\,\mathrm{L} contains helium-4 gas (atomic mass 4.00 u4.00\,\mathrm{u}). A beam experiment shows that the most probable speed of the atoms in the bulb is 1120 m s−11120\,\mathrm{m\,s^{-1}}. The bulb has a pinhole of area 1.0 μm21.0\,\mathrm{\mu m^2} opening into a vacuum; the pinhole is much smaller than the mean free path, so the gas leaks by effusion, and the temperature of the gas is held constant throughout. Take 1 u=1.661×10−27 kg1\,\mathrm{u} = 1.661\times10^{-27}\,\mathrm{kg}. ((a)) Find the temperature of the gas. ((b)) Find the mean speed of the atoms. ((c)) The number of atoms in the bulb obeys dN/dt=−14nvˉA\mathrm{d}N/\mathrm{d}t = -\tfrac14 n\bar v A, so the pressure falls as p=p0e−t/t0p = p_0 e^{-t/t_0}. Find the time constant t0t_0. ((d)) The bulb is refilled with a mixture of helium-3 (3.02 u3.02\,\mathrm{u}) and helium-4 (4.00 u4.00\,\mathrm{u}) containing equal numbers of each. Find the ratio of the rate at which helium-3 atoms escape to the rate at which helium-4 atoms escape.

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