Concise Physical Chemistry

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

A DIGRESSION ON “SPACE” 9

whereNiis the number of particles at the level having energyEi. In this expression,

N 0 is the number at the lowest energy, usually designated zeroE 0 =0 in the absence

of a reason to do otherwise.^4 Energy, being a scalar, is proportional to the square

of thespeedof an ensemble of molecules. Thepopulationof the energy levels in

Fig. 1.2 drops off rapidly at higher energies.

The termdegeneracyis used when two or more levels exist at the same en-

ergy, which sometimes happens under the laws of quantum mechanics. Now the

number of particles at levelEiis multiplied by the number of levelsgihaving that

energy

Ni

N 0

=gie−Ei/kBT

The degeneracy is always an integer and it is usually small. Also, fromEkin=^32 RT

for one mole, we can find theexpectation value〈εkin〉of the kinetic energy per

representative or average particle

〈εkin〉=

3

2

R

NA

T

This leads to the important constant

R

NA

=kB=

8. 3145

6. 022 × 1023

= 1. 381 × 10 −^23 JK−^1

and

〈εkin〉=^32 kBT

wherekB, the universal gas constantper particle, is called theBoltzmann constant.It

should be evident thatkBTmust have the units of energy becauseNi/N 0 is a unitless

(pure) number, ln(Ni/N 0 )=−Ei/kBT, which is also unitless, hence the units of

kBTmust be the same asEi. We are taking advantage of the fact that ify=ex, then

lny=x.

1.9 A DIGRESSION ON “SPACE”

The terms density and probability density were used in Section 1.8. These are different

but analogous uses of the word density. In the first case, density was used in the usual

sense of weight or mass per unit volume,m/V. In the second case, theprobability

densityis defined as the probability in a specified space. Any variable measured

(^4) Like all energies, this zero point is arbitrary.