Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physic8^371

Hence

nvzP(v,)dv, = n (K)’”exp (--) MU;

27rM 2kT

where n is the molecular number density.

2181

A gas in a container consists of molecules of mass m. The gas has a

well defined temperature T. What is

(a) the most probable speed of a molecule?

(b) the average speed of the molecules?

(c) the average velocity of the molecues?

Solution:

The Maxwell velocity distribution is given by

dw = (-) m 3/2 exp[-mv2/2kT]dv,dv,dv,.

2skT

(a) Let f(v) = v2 exp(-mv2/2kT). The most probable speed is given

by df (v) = 0, as

u=($) 112.

(b) The average speed is

(c) The average velocity V = (is,, isy, is,) is given by

5, = (n1/27rkT)~/’ v, exp[-mv2/2kT]dv,dv,dv, = 0 ,.

01

and isy = is, = 0. Thus V = 0.