Concise Physical Chemistry

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

8 IDEAL GAS LAWS

1

0

ev

−^2

− 3 v 3

FIGURE 1.1 The probability density for velocities of ideal gas particles atT=0.

misleading because it may suggest that the most probable velocity is zero. Not so. The

particles are not standing still at any temperature above absolute zero. The peak at

v=0 arises because we don’t know which direction any particle is going, left or

right. In our ignorance, assuming a random distribution, the best bet is to guess zero.

We will always be wrong, but the sum of squares of our error over many trials will

be minimized. This is an example of the principle ofleast squares.

The Maxwell–Boltzmann distribution of molecular speeds was originally derived

assuming that particle velocities are distributed along a continuous spectrum like

Fig. 1.1. This implies thatEkincan take any value in a continuum as well. The laws of

quantum mechanics, however, deny this possibility. They require a distribution over

adiscontinuousenergyspectrumor manifold of energy levels like that in Fig. 1.2.

The connection between Figs. 1.1 and 1.2 can be seen by tilting the page 90◦to the

left. The number of particles at higher energies tails off according to a Gaussian dis-

tribution. The Maxwell–Boltzmann distribution over nondegenerate, discontinuous

energy levels is

Ni

N 0

=e−Ei/kBT

....................

................

.........

.....

..

.

E

FIGURE 1.2 A Maxwell–Boltzmann distribution over discontinuous energy levels. Particles

are not static; they exchange energy levels rapidly. The levels need not be equally spaced.