A Pirani gauge measures the pressure in a vacuum system from the heat that the gas…

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 Pirani gauge measures the pressure in a vacuum system from the heat that the gas conducts away from an electrically heated wire to the surrounding wall. Model the gas between wire and wall as a flat slab of thickness

L = 2.0\,\mathrm{mm}

, with the wire surface at

340\,\mathrm{K}

and the wall at

300\,\mathrm{K}

; ignore radiation and convection. The gas is nitrogen: molecular mass

m = 4.65\times10^{-26}\,\mathrm{kg}

, hard-sphere diameter

d = 0.37\,\mathrm{nm}

, and

f = 5

active degrees of freedom, so the energy per unit volume is

\tfrac52 nk_BT

. Take the gas properties at

300\,\mathrm{K}

throughout. ((a)) Using

\kappa = \tfrac13\bar v\lambda\left(\tfrac52 nk_B\right)

and

\lambda = 1/(\sqrt2\,n\sigma)

, show that the estimate for

\kappa

contains no number density, and evaluate it. ((b)) Estimate the heat flux across the gap. ((c)) Find the pressure at which the mean free path equals the gap width

L

. ((d)) Explain in two or three sentences why the gauge can only read the pressure below the value found in (c), and why the reason is the same as the reason the viscosity of a dilute gas does not depend on its pressure.

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