Thermodynamics, Statistical Physics, and Quantum Mechanics

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d) The mean square speed of the atoms may befoundimmediately by

recalling the equipartition theorem (see Problem 4.42) and using the fact

that there is energy per degree of freedom. So

For completeness, though, the integral may be shown:

4.14 Slowly Leaking Box (Moscow Phys-Tech, Stony

Brook (a,b))

a) The number of atoms per unit volume moving in the direction normal
to the wall (in spherical coordinates) is

where is the azimuth angle, is the polar angle, is the number density
of atoms, and is the speed distribution function(Maxwellian). To
determine the number of atoms striking the area of the hole A on the wall
per time dt,we have tomultiply(S.4.14.1) by dt (only the atoms
within adistance dt reach thewall). To obtain the total atomic flow
rateRthrough thehole, we have tointegrate thefollowingexpression:

THERMODYNAMICS AND STATISTICAL PHYSICS