Physical Chemistry , 1st ed.

Example 15.2

From the above expressions, determine the possible values ofjfor the fol-

lowing.

a.A single delectron

b.A single helectron (where 5)

Solution

a.For a delectron,2 and salways equals ^12 , so the two possible values of

jfor a single delectron are ^52 and ^32 .

b.For an helectron (which would exist as an excited state in, say, the sixth

principal quantum shell),5, so the two possible values ofjfor a single h

electron are ^121 and ^92 .

This example again shows the possible values for the jquantum number as

half-integers. For single electrons jis always a half-integer. For multiple elec-

trons,Jcan be either integers or half-integers.

Example 15.3

What are the possible values ofmjfor the delectron in Example 15.2?

Solution

For an angular momentum that follows the normal quantum-mechanical

rules for angular momenta, the possible values ofmjrange from jto j,in

integer steps. Therefore,mj    ^52 , ^32 , ^12 ,^12 ,^32 , and ^52 for the j^52 state, and

^32 , ^12 ,^12 , and ^32 for the j^32 state of the delectron. There are six possible

values ofmjfor j^52 and four possible values ofmjfor j^32 .

The point of this example is worth repeating: for an electron that has a

total angular momentum indicated by the quantum number j, the possible

values ofmjare

mj jto jin integer steps (2j 1 possible values) (15.8)

The overallenergy of an electron is dependent on the value of the jquantum

number. The mjquantum number does not affect the energy of the electron

unless the atom is in the presence of a magnetic or electric field. These state-

ments are consistent with the known effects ofand mon the energy of a

(hydrogen-like) electron.

Completely filled subshells (not shells, but subshells) contribute no overall

angular momentum to the total angular momentum of the atom. All angular

momenta, orbital and spin, are paired so that there is a net zero effect. However,

if an electron from a filled subshell is excited to a higher-energy state, this

statement no longer applies and the effect on the total angular momentum by

the partially filled subshell as well as the excited electron must be taken into

account.

However, first we ought to be able to obtain some understanding of the

electronic spectra of atoms that have a single electron in their valence subshell.

Such atoms have the electron configurations ns^1 ,np^1 ,nd^1 ,or nf^1 (where nis

some allowed value of the principal quantum number). Because of the lone

524 CHAPTER 15 Introduction to Electronic Spectroscopy and Structure