Quantum numbers

Orbitals of electrons in atoms differ in size shape and orientation. Definite energies and angular movements characterize atomic orbitals. The state of an electron in any atom is defined by certain permissible values of energy and angular momentum, which describe its location with respect to its nucleus and its energy level. These permissible states are called orbitals and are expressed by a set of four numbers 'n', 'l', 'm' and 's' called quantum numbers. These numbers serve as the signature of the electrons, uniquely describing its position in the atom. The 'n', 'l' and 'm' indicate the spatial distribution while 's' indicates the spin orientation of the electrons.

Principal quantum number

This quantum number determines the main energy shell or energy level in which the electron is present. The principal quantum number gives the average distance of the electron from the nucleus and energy associated with it.

It is denoted by the letter 'n' that can take whole number values starting from 1, 2, 3, 4, ….. . The shell with n = 1 is called first shell or 'K' shell. The shell with n = 2 is the 'L' shell and so on. The first shell is closest to the nucleus. As the value of 'n' increases, the distance from the nucleus as well as the energy of the electrons increases.

Azimuthal quantum number or angular quantum number

The Azimuthal quantum number determines the angular momentum of the electron, denoted by the letter 'l'. The value of 'l' gives the sub level or sub shell in a given principal energy shell to which the electron belongs. It can have only positive integral values from zero to (n-1) where 'n' is the principal quantum number. The various sub shell values of l are also designated by the letters s, p, d, f,…… For any main energy level, the energies of the sub shell follow the order s > p > d > f.

The different sub shells are represented by first writing the value of 'n' and then the letter designated for the value of 'l'.

To illustrate,

n = 1 l = 0 one sub shell 1s

n = 2 l = 0,1 two sub shells 2s, 2p

n = 3 l = 0,1,2 three sub shells 3s, 3p, 3d

n =4 l = 0,1,2,3 four sub shells 4s, 4p, 4d, 4f

Thus for each value of 'n' there are 'n' values of 'l'.

The value of azimuthal quantum number gives the shape of the sub shell or orbital. So it is also called as orbital quantum number.

Magnetic quantum number

The magnetic quantum number describes the behaviour of electron in a magnetic field. In the absence of external magnetic field electrons / orbitals having same values of 'n' and 'l' but different values on 'm' have the same energies. They are called degenerate orbitals. However, in the presence of an external magnetic field the orbitals vary in their energies slightly. This is because the preferred orientation of the orbital in space is a result of interaction of its own magnetic field with that of the external magnetic field.

It is denoted by the letter 'm' the values of which depends on 'l'. This quantum number can have all integral values from '-l' to '+l' including 0. Thus for given 'l' value there are (2l + 1) values of 'm'. Two orbitals in the same shell can have identical 'n' and 'l' values but they must have different fixed values of 'm'.

The number of orbitals in each sub shell are given below:

s sub shell l = 0 m = 0 only one orientation one orbital

p sub shell l = 1 m = +1,0, -1 three orientations three orbitals

d sub shell l = 2 m = +2,+1,0,-1,-2 five orientations five orbitals

Spin quantum numbers

The orientation of spin of an electron is designated by its spin quantum number 's'. The spin orientation is an intrinsic characteristic of the electron connected more with its magnetic behaviour rather than rotation of an electron about its own axis. This number can have only two values corresponding to clockwise and anticlockwise spins i.e., +½ and -½. The clockwise spin is represented by an arrow (h) pointing upwards. The anti clockwise spin is represented by an arrow (i) pointing downwards. Each orbital can accommodate a maximum of two electrons provided they have opposite spins.

Fig: 3.15 - Number of subshells and orbitals in the K,L and M shells

Problem

9.(a) What are the permissible values for l and m when n = 3?

(b) Which orbital is specified by l = 2 and n = 3?

Solution

(a) For n = 3, the permissible values for 'l' and 'm' are: l = 0, 1, 2

For 'l' = 0 m = 0 (s-orbital)

For 'l' = 1 m = +1, 0, -1 (p-orbital)

For 'l' = 2 m = +2, +1, 0, -1, -2 (d-orbital)

(b) For 'n' = 3 and 'l' = 2:

'l' = 2 means 'd' orbitals

The given orbital is '3d'.