Physical Chemistry , 1st ed.

spin angular momentum, with s, an orbital that has 0.) The eigenvalue

equations are therefore

Sˆ^2 s(s1)^2  (12.1)

Szms (12.2)

The values of the allowed quantum numbers sand msare more restricted

than for and m. All electrons have a value ofs^12 . The value ofs, it turns

out, is a characteristic of a type of subatomic particle, and all electrons have

the same value for their squantum number. For the possible values of the z

component of the spin, there is a similar relationship to the possible values of

mand :msgoes from sto sin integral steps, so mscan equal ^12 or ^12 .

Thus, there is only one possible value ofsfor electrons, and two possible val-

ues for ms.

Spin also has no classical counterpart. Nothing in classical mechanics pre-

dicts or explains the existence of a property we call spin. Even quantum me-

chanics, at first, did not provide any justification for spin. It wasn’t until 1928

when Paul A. M. Dirac incorporated relativity theory into the Schrödinger

equation that spin appeared as a natural theoretical prediction of quantum

mechanics. The incorporation of relativity into quantum mechanics was one

of the final major advances in the development of the theory of quantum me-

chanics. Among other things, it led to the prediction of antimatter, whose ex-

istence was verified experimentally by Carl Anderson (with the discovery of the

positron) in 1932.

Example 12.1

What is the value, in Js, of the spin of an electron? Compare this to the value

of the angular momentum for an electron in sand porbitals of an H-like

atom.

Solution

The value of the spin angular momentum of an electron is determined by us-

ing equation 12.1. We must recognize that the operator is the square of the

total spin, and to find the value for spin we will have to take a square root.

We g e t

spins(s 1 )^2 





1

2



1

2





(^1) 

6.



626

 2

10 ^34



Js



2



9.133  10 ^35 Js

The angular momentum of an electron in an sorbital is zero, since 0 for

an electron in an sorbital. In a porbital,1, so the angular momentum is

(+ 1) 1  2 

6.626

2

10 ^34 Js

1.491     10 ^34 J s

which is almost twice as great as the spin. The magnitude of the spin angu-

lar momentum is not much smaller than the angular momentum of an elec-

tron in its orbit. Its effects, therefore, cannot be ignored.

The existence of an intrinsic angular momentum requires some additional

specificity when referring to angular momenta of electrons. One must now

372 CHAPTER 12 Atoms and Molecules