A vessel holds argon in equilibrium at and

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 vessel holds argon in equilibrium at T=350 KT = 350\,\mathrm{K} and p=1.5 Pap = 1.5\,\mathrm{Pa}. A hole of area A=0.20 mm2A = 0.20\,\mathrm{mm^{2}} (diameter about 0.5 mm0.5\,\mathrm{mm}) is opened in the wall into a vacuum. Argon atoms have mass m=6.63×10−26 kgm = 6.63\times10^{-26}\,\mathrm{kg} and hard-sphere diameter d=0.34 nmd = 0.34\,\mathrm{nm}. Inside the vessel the velocity directions are isotropic, so the fraction of molecules whose velocity lies within the polar angle range θ\theta to θ+dθ\theta+\mathrm{d}\theta measured from the outward normal to the hole is 12sin⁡θ dθ\tfrac12\sin\theta\,\mathrm{d}\theta. The speed distribution f(v)f(v) is normalised, ∫0∞f(v) dv=1\int_0^{\infty}f(v)\,\mathrm{d}v = 1, with ∫0∞vf(v) dv=vˉ\int_0^{\infty}v f(v)\,\mathrm{d}v = \bar v. You may use ∫0π/2cos⁡θsin⁡θ dθ=12\int_0^{\pi/2}\cos\theta\sin\theta\,\mathrm{d}\theta = \tfrac12. ((a)) By counting the molecules of speed vv and direction θ\theta that reach the hole in a time dt\mathrm{d}t, derive the effusion flux Φ=14nvˉ\Phi = \tfrac14 n\bar v, where nn is the number density. ((b)) Evaluate Φ\Phi for this gas. ((c)) How many atoms leave the vessel per second at this instant? ((d)) Compute the mean free path and check that the hole is small enough for the calculation to describe effusion.

Browse the problem library