The Foundations of Chemistry

QUANTUM NUMBERS

The solutions of the Schrödinger and Dirac equations for hydrogen atoms give wave func-

tions, , that describe the various states available to hydrogen’s single electron. Each of

these possible states is described by four quantum numbers. We can use these quantum

numbers to designate the electronic arrangements in all atoms, their so-called electron

configurations.These quantum numbers play important roles in describing the energy

levels of electrons and the shapes of the orbitals that describe distributions of electrons

in space. The interpretation will become clearer when we discuss atomic orbitals in the

following section. For now, let’s say that

an atomic orbitalis a region of space in which the probability of finding an elec-

tron is high.

We define each quantum number and describe the range of values it may take.

1.The principal quantum number,n, describes the main energy level,or shell, an

electron occupies. It may be any positive integer:

n1, 2, 3, 4,...

2.The angular momentum quantum number,
, designates the shape of the region

in space that an electron occupies. Within a shell (defined by the value of n, the

principal quantum number) different sublevels or subshells are possible, each with

a characteristic shape. The angular momentum quantum number designates a

sublevel,or specific shapeof atomic orbital that an electron may occupy. This number,


, may take integral values from 0 up to and including (n
1):

0, 1, 2,... , (n
1)

Thus, the maximum value of 
is (n
1). We give a letter notation to each value

of 
. Each letter corresponds to a different sublevel (subshell).

0, 1, 2, 3,... , (n
1)

spdf

In the first shell, the maximum value of 
is zero, which tells us that there is only

an ssubshell and no psubshell. In the second shell, the permissible values of 
are

0 and 1, which tells us that there are only sand psubshells.

3.The magnetic quantum number,m
, designates the specific orbital within a

subshell. Orbitals within a given subshell differ in their orientations in space, but

not in their energies. Within each subshell, m
may take any integral values from



through zero up to and including 
:

m
(

),...,0,...,(
)

The maximum value of m
depends on the value of 
. For example, when 
1,

which designates the psubshell, there are three permissible values of m
: 
1, 0, and

1. Thus, three distinct regions of space, called atomic orbitals, are associated

with a psubshell. We refer to these orbitals as the px, py, and pzorbitals (see Sec-

tion 5-16).

5-15

The s, p, d, fdesignations arise from
the characteristics of spectral emission
lines produced by electrons occupying
the orbitals: s(sharp), p(principal), d
(diffuse), and f(fundamental).

208 CHAPTER 5: The Structure of Atoms

See the Saunders Interactive
General Chemistry CD-ROM,
Screen 7.11, Shells, Subshells, and
Orbitals.