Modern Thermodynamics

- Chapter 1

39

simulations of gas dynamics it is found that any initial velocity distribution evolves into the Maxwell

distribution very quickly, in a time it takes a molecule to undergo few collisions, which in most cases is

less than 10-8s.

The Maxwell Speed Distribution

The average velocity of a molecule is clearly zero because every direction of velocity and its opposite are

equally probable (but the average of the square of the velocity is not zero). However, the average speed,

which depends only on the magnitude of the velocity, is not zero. From the Maxwell velocity distribution

(1.6.13) we can obtain the probability distribution for molecular speed, i.e., the probability that a

molecule will have a speed in the range (v, v+dv) regardless of direction. This can be done by summing

or integrating P(v) over all the directions in which the velocity of a fixed magnitude can point. Note that

P(v) is independent of the direction of v. In spherical coordinates, since the volume element is

v2

sin θdθdϕdv , the probability is written as

P(v)v2

sin θdθdϕdv . The integral over all possible

directions is:

P(v)v2

sinθdθdϕdv

=0

2π

∫

θ=0ϕ

π

∫

=

4πP(v)v2dv

(1.6.14)

The quantity 4πP(v)v2 is the probability density for the molecular speed. We shall denote it by f(v). With

this notation, the probability distribution for molecular speeds can be written explicitly as:

f(v)dv =

4π

m

2πkBT

3/2

e−βv

2

v2dv

β =

m

2kBT

(1.6.15)

Because the molar mass M=mNA and R=kBNA, the above expression can also be written as:

f(v)dv = 4π

M

2πRT

3/ 2

e−

βv2

v2dv

β =

M

2RT

(1.6.16)

The shape of the function f(v) is shown in the Fig. 1.10(b). This graphs show that, at a given temperature,

there are a few molecules with very low speeds and a few with large speeds; we can also see that f(v)