Physical Chemistry Third Edition

942 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

22.5 The Equilibrium Populations of

Molecular States

So far in this chapter we have studied the quantum-mechanical states of isolated atoms

or molecules. In a dilute gas, the molecules do not significantly interfere with each other,

and we can apply these states to the individual molecules. However, all molecular states

will not be occupied by the same numbers of molecules in a dilute gas at equilibrium.

TheBoltzmann probability distributiongives the probability of a molecular state of

energyεin a system at thermal equilibrium:

(Probability of a state of energyE)∝e−E/kBT (22.5-1)

wherekBis Boltzmann’s constant, equal to 1. 3807 × 10 −^23 JK−^1 , andTis the absolute

(Kelvin) temperature.

Each state in an energy level has the same energy so it will have the same population.

Ifgis the degeneracy of the level,

(Population of energy level of energyE)∝ge−E/kBT (22.5-2)

In Chapter 9 there is a derivation of the Boltzmann probability distribution for classical

dilute gases. There is a derivation of this probability distribution for a quantum dilute

gas in Part 4. For now, we introduce it without derivation. The important fact about the

Boltzmann probability distribution is that states of energy much larger thankBTare

quite improbable.

To a good approximation, the energy of a molecule is a sum of four different

energies

EtotEtr+Evib+Erot+Eel (22.5-3)

The probabilities of each of these energies is independent of the others, so that the

probability of an energy level is the product of four Boltzmann factors:

(Probability)∝gtre−Etr/kBTgvibe−Evib/kBTgrote−Erot/kBTgele−Eel/kBT (22.5-4)

The translational levels are very closely spaced. The electronic energy levels

are typically very widely spaced compared withkBT, and all energy levels except

the ground level are typically almost completely unpopulated at room temperature.

EXAMPLE22.10

Calculate the ratio of the population of one of the states of the first excited electronic level

of the Cl 2 molecule to that of the ground state at 298 K. The energy of the first excited level

is 2.128 eV above the ground state.