Physical Chemistry Third Edition

916 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

22.1 The Translational States of Atoms

The energy eigenfunction of a hydrogen atom is given in Eq. (17.1-17) as a product of

the center-of-mass factorψcand the relative factorψ:

Ψψc(xc,yc,zc)ψ(x,y,z) (22.1-1)

wherexc,yc, andzcare the Cartesian coordinates of the center of mass of the nucleus

and electron, andx,y, andzare the Cartesian coordinates of the electron relative to the

nucleus. We now relabel the center-of-mass factorψcasψtrand call it thetranslational

factor. The relative motion was equivalent to the motion of a fictitious particle with

a mass equal to the reduced mass of the electron and nucleus. If the reduced mass is

replaced by the electronic mass, the motion is the same as a the motion of an electron

around a stationary nucleus (a good approximation). We now relabel the relative factor

ψasψeland call it theelectronic factor.

The energy of a hydrogen atom is given by Eq. (17.1-20)

EtotalEc+ErelEtr+Eel (22.1-2)

where we now relabel the center-of-mass energyEcas the translational energyEtr, and

relabel the relative energyErelas the electronic energyEel.

For multielectron atoms, we treated the electronic motion with the assumption that

the nucleus is stationary. This is a good approximation because the nucleus is much

more massive than the electrons and moves much more slowly, allowing the electrons

to follow it to a new position almost as though it had always been there. We can still

study the motion of the nucleus. If an atom is not confined in a container its center of

mass obeys the time-independent Schrödinger equation of a free particle, Eqs. (15.3-40)

and (17.1-18). The solution of the Schrödinger equation for this translational motion is

given by Eq. (15.3-41). The translational energy is given by Eq. (15.3-42), and is not

quantized.

If an atom is confined in a rectangular box its center of mass cannot move completely

up to the walls of the box because of the electrons in the atom. However, if the box is

much larger than the size of an atom it will be an excellent approximation to apply the

formulas that apply to a particle of zero size to the translation of an atom in a box. The

translational energy eigenfunctions would be represented by the normalized version of

Eq. (15.3-21):

ψtr

√

8

abc

sin

(nxπx

a

)

sin

(nyπy

b

)

sin

(nzπz

c

)

(22.1-3)

wherea,b, andcare the dimensions of the box andnx,ny, andnzare three quantum

numbers, which we now call thetranslational quantum numbers. The translational

energy is given by Eq. (15.3-22):

Etr

h^2

8 M

(

n^2 x

a^2

+

n^2 y

b^2

+

n^2 z

c^2

)

(22.1-4)

whereMis the total mass of the atom.

The electronic energy levels of any atom other than the hydrogen atom cannot be

represented by any simple formula, and we will usually rely on experimental data for

the energy eigenvalues. The electronic energy levels are very widely spaced compared

with translational energy levels.