Chemistry - A Molecular Science

Chapter 2 Quantum Theory

values of m

(-1, 0, +1) and, consequently, three orl

bitals. These three orbitals are the 3p

orbitals. In the

l = 2 sublevel, m

= -2, -1, 0, +1, and +2 for the five orbitals in the 3d l

sublevel. Thus, in the n = 3 level, ther

e are three sublevels and nine orbitals.

The wave equation developed by

Schrödinger required only the n,

l and m

quantum l

numbers, but many details of the emission sp

ectra of the atoms could not be explained

using only this set of quantum numbers. In

addition, silver atoms that are passed through

a magnetic field divided into two beams. All

of these problems could be explained only if

the electron had magnetic properties, which required a new quantum number, the electron spin.

Electron spin is a quantum mechanical effect

that is as much a property of the

electron as is its charge. However, a simple macroscopic analogy is often used to describe this quantum mechanical effect. In the analog

y (Figure 2.9), electron spin results from the

rotation of the electron about its axis (anal

ogous to the 24 hour rotation of the earth about

its axis). The rate of rotation is the same for all electrons, but there are two different directions of rotation, which are distinguished by the spin quantum number,

m

. s

ms

, the spin quantum number,

has are only two possible values:

ms

= +

1 /^2

and

ms

= -

1 /^2

.

All magnetic properties are attributed to electron spin

, so it is often represented as the spin

of the electron about its own axis in a clockwise (

ms

= +

1 /^2

) or counterclockwise (

ms

= -

1 /^2

)

direction because a magnetic field is produced by a charge moving in a circular path. Thus, the ‘spinning charge’ can be thought to generat

e a magnetic field as indicated by the

arrows in Figure 2.9. The arrow representing t

he magnetic field is often used to represent

the m

quantum number of the electron. s

↑ (‘up spin’) is used to indicate an electron with m

(^) s
= +
1 /^2
and
↓ (‘down spin’) indicates an electron with
m
= -s
1 /^2
.
Example 2.5
Identify the incorrect quantum number and change it to an acceptable value.
m
n
l^
ml
(^) s
a)
0 1 0 ½
n cannot be 0. Any integer above 0 would be acceptable.
b)
3 3 3 -½
l cannot equal n, so n must be increased or
l decreased.
Decreasing
l is not acceptable because m
cannot be l
greater than
l. n must be increased to 4 or above.
c)
8 0 -1
½
ml
must lie between +
l and –
l, so either
l must be increased
to 1 or above, or m
must be reduced to 0. l
d)
6 5 0 0
ms
must be +
½ or -
½^

S N

N S

m=+ =

s

m=- =

s

12

12

Figure 2.9 Classical anal

og of electron spin

The ‘spinning’ electron generates a magnetic field represented by the vertical arrows. The two possible values of m

are often referred to as the ‘up-spin’ (ms

= s

(^1) +
/) and the ‘down-spin’ (m^2
= -s
1 /^2
) due to the direction of
the arrows. The north and south poles of the corresponding permanent magnet are shown next to each spinning electron.
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