The quantum numbers are nothing but the details that are required to locate an electron in an atom. In an atom a large number of electron orbitals are permissible. An orbital of smaller size means there is more chance of finding the electron near the nucleus. These orbitals are designated by a set of numbers known as quantum numbers.

**Electronic configuration and quantum numbers**

**Quantum Numbers**

The quantum numbers are nothing but the details
that are required to locate an electron in an atom. In an atom a large number
of electron orbitals are permissible. An orbital of smaller size means there is
more chance of finding the electron near the nucleus. These orbitals are
designated by a set of numbers known as quantum numbers. In order to specify
energy, size, shape and orientation of the electron orbital, three quantum
numbers are required these are discussed below.

**1.** **The principal quantum
number (n)**

The electrons inside an atom are arranged in
different energy levels called electron shells or orbits. Each shell is
characterized by a quantum

number
called principal quantum number. This is represented by the letter 'n' and 'n'
can have values, 1,2,3,4 etc. The first level is also known as K level. Second
as L level, third as M level, fourth as N level and so on. The first or K level
is the orbit nearest to the nucleus and next one is second or L level and so
on.

**2.** **The subsidiary or
azimuthal quantum number ( l)**

According to Sommerfield, the electron in any particular energy level
could have circular path or a variety of elliptical paths about the nucleus
resulting in slight differences in orbital shapes with slightly differing
energies due to the differences in the attraction exerted by the nucleus on the
electron. This concept gave rise to the idea of the existence of sub-energy
levels in each of the principal energy levels of the atom. This is denoted by
the letter '*l*' and have values from 0
to n-1.

Thus, if
n=1, *l*=0 only one value (one level
only) *s* level.

n=2, *l*=0
and 1 ( 2 values or 2 sub- levels) *s* and *p*
level.

n=3, *l*=0, 1 and 2 (3 values or 3 sub-levels) *s, p* and *d* level.

n=4, *l*=0, 1, 2 and 3 (4 values or 4
sub-levels) *s, p ,d* and *f*
level.

**3.** **Magnetic quantum
number (m)**

In a strong magnetic field a sub-shell is
resolved into different orientations in space. These orientations called
orbitals have slight differences in energy. This explains the appearance of
additional lines in atomic spectra produced when atoms emit light in magnetic
field. Each orbitals is designated by a magnetic quantum number m and its
values depends on the value of '*l*' .
The values are -' *l*' through zero to
+' *l*' and thus there are (2*l*+1) values.

Thus when *l*=0, m= 0 (only one value or one
orbital)

*l*=1, m= -1,
0, +1 (3 values or 3 orbitals)

*l*=2, m= -2,
-1, 0, +1, +2 (5 values or 5 orbitals)

*l*=3, m=
-3,-2, -1, 0, +1, +2, +3 (7 values or 7 orbitals).

The three
quantum numbers labeling an atomic orbital can be used equally well to label
electron in the orbital. However, a fourth quantum

number, the spin quantum number, (*s*)
is necessary to describe an electron completely.

**4.** **Spin quantum number ( s)**

The electron in the atom rotates not only around the nucleus but also
around its own axis and two opposite directions of rotation are possible (clock
wise and anticlock wise). Therefore the spin quantum number can have only two
values +1/2 or -1/2. For each values of m including zero, there will be two
values for *s.*

To sum up, the four quantum numbers provide the
following informations:

1.
n identifies the shell, determines the size of
the orbital and also to a large extent the energy of the orbit.

2.
There are n subshells in the n^{th}
shell. *l* identifies the subshell and
determines the shape of the orbital. There are (2*l*+1) orbitals of each type in a subshell i.e., one s orbital (*l*=0), three p orbitals (*l*=1), and five d orbitals (*l*=2) per subshell. To some extent *l* also determines the energy of the
orbital in a multi-electron atom.

3.
m* _{l}*
designates the orientation of the orbital. For a given value of

4.
m_{s} refers to orientation of the spin
of the electron.

*Example Problem*

*Example 1*

**What is the total
number of orbitals associated with the principal quantum number n=3 ?**

*Solution*

For n = 3, the possible values of *l* are 0,1 and 2. Thus, there is one 3s
orbital (n = 3, *l* = 0 and m* _{l}* = 0); there are three p
orbitals (n = 3,

Therefore,
the total number of orbitals is 1+3+5 = 9.

*Example 2*

**Using s,
p, d, f notations, describe the orbital with the following quantum numbers**

** (a) n=2, l
= 1 (b) n = 4, l = 0 (c) n = 5, l = 3 (d) n = 3, l = 2.**

*Solution*

n l orbital

(a) 2 1 2p

(b) 4 0 4s

(c) 5 3 5f

(d) 3 2 3d

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