Concise Physical Chemistry

c01 JWBS043-Rogers September 13, 2010 11:20 Printer Name: Yet to Come

THE MAXWELL–BOLTZMANN DISTRIBUTION 7

particles colliding with the wall of a constraining container, supposing that the wall

has a hole in it. Only a few particles escape the container because the hole is small.

Escape probability is determined by how fast the particle is moving. Fast particles

collide with the walls of the container more often than do slow ones.

By a standard derivation (Exercise 1.2), one finds

pV=^13 NAmu ̄^2 x

whereu ̄xis the average speed of an ensemble consisting of one mole of an ideal

gas. Notice that becausepV=RThas the units J K−^1 mol−^1 K=Jmol−^1 ,pVis

amolar energy. Increasing the temperature of a gas requires an input of energy. We

usually write the kinetic energyEkinof a single moving mass such as a baseball

asEkin=^12 mv^2 , wherevis its speed andmis its mass. Consider a hypothetical

one-dimensionalx-space along which point particles can move without interference.

If the kinetic energy of molecular particles follows the same kind of law as more

massive particles, we obtain

1

2 mu ̄

2

x=E ̄kin

whereE ̄kinis the averagekinetic energybecause kinetic energy is the only kind an

ensemble of point particles can have. Substitute 2E ̄kinformu ̄^2 xin

pV=^23 NAE ̄kin

butpValso equalsRTfor one mole of a gas, so

pV=RT=^13 NAmu ̄^2 x

This enables us to calculateu ̄xat any specified temperature. The calculation gives

high speeds. For example, nitrogen molecules move at about 400 m s−^1 (meters per

second) at room temperature and hydrogen molecules move at an astonishing speed

of nearly 2000 m s−^1. There are different ways of calculating averages (mean, mode,

root mean square), which give slightly different results for molecular speeds.

1.8 THE MAXWELL–BOLTZMANN DISTRIBUTION

All particles of a confined gas do not move with the same velocity even ifTis

constant. Rather, they move with a velocityprobability densityρvwhich is randomly

distributed aboutv=0 and which follows the familiar Gaussian distributione−v

2

.

The probability density function drops off at large values of±vbecause the prob-

ability of finding particles with velocities very much different from the mean is small.

The curve is symmetrical because, picking an arbitrary axis, the particle may be going

either to the left or to the right, having a velocityvor –v. The peak atv=0 is somewhat