Problems and Solutions on Thermodynamics and Statistical Mechanics

Stati~tical Phpics 395

where N is the particle number density in the beam and A is the cross
section of the beam. Hence

Jrnvz2.v,e- 3 T dv,

J v,e-+dv,

Po = N rn" 2 = 2NkTo =^3 x N/m2 ,

which is much less than the pressure from the background gas.

2199

A quantity of argon gas (molecular weight 40) is contained in a chamber

at To = 300 K.

(a) Calculate the most probable molecular velocity.

A small hole is drilled in the wall of the chamber and the gas is allowed

to effuse into a region of lower pressure.

(b) Calculate the most probable velocity of the molecules which escape

through the hole.

The pressures of the chamber and the region outside the hole are ad-

justed so as to sustain a hydrodynamic flow of gas through the hole, such

that viscous effects, turbulence, and heat exchange with the wall of the hole

may be neglected. During this expansion the gas is cooled to a temperature

of 30 K.

(c) Calculate the velocity of sound c at the lower temperature.

(d) Calculate the average flow velocity V at the lower temperature, and

compare the distribution of velocities with the original distribution in the

chamber.

(UC, Berkeley)

Solution:

(a) The Maxwell distribution is given by

or