Thermodynamics, Statistical Physics, and Quantum Mechanics

100

We integratefrom 0 to since we only consider the atomsmoving toward
the wall. On the other hand, bydefinition, theaverage velocity is given
by

Comparing(S.4.14.2) and(S.4.14.3), we seethat

this energy For a Maxwellian distributionwe have
whereCis a normalizingconstant:

The numerator isthe totalenergy of the atoms leaving thecontainer per
second, and thedenominator is thetotal number of atomsleaving the con-
tainer persecond. Define From part(a), we canexpress this
integral in terms of theaverage velocity Then we have

We know that (since it is anormalizing factor, seeProblem
4.13), and

SOLUTIONS

This result applies for any type of distribution function We onlycon-
sider a flow from theinside to theoutside of the container. Since thehole
is small, we canassume that thedistributionfunction of theatoms inside
the containerdoes notchange appreciably.

b) The averagekineticenergy of the atoms leavingthe containershould be
somewhat higherthan the averagekinetic energy inthe containerbecause
faster atomsstrike the wall more often than theonesmoving moreslowly.
So, the faster atomsleave thecontainer at a higherrate. Let us compute