Quantum Numbers

The numbers used for completely specifying each electron of atom are known as Quantum Numbers. Four such quantum numbers are found to be necessary for describing completely an electron.

1. PRINCIPAL' QUANTUM NUMBER (n)

This number gives the average distance of the electron from the nucleus and hence gives the shell number (i.e. principal energy level) to which the electron belongs. Thus it gives an idea of the position, of the electron around the nucleus. The energy of the electron depends upon this number. Higher the principal quantum number, greater is its distance from the nucleus, greater is its size and also higher is its energy. It takes the value from 1 to infinity. But only values from 1 to 7 have so far been established for atoms of the known elements. These are designated either as 1, 2, 3, 4, 5, 6, and 7 or as K, L, M, N, 0, P and Q respectively. The maximum number of electrons in ‘n’ principal quantum number is given by 2n².

2. AZIMUTHAL QUANTUM NUMBER (l )

This number represents the sub-level to which the electron belongs and also determines the shape of the orbital. Azimuthal quantum number is also known as Subsidiary or Angular quantum number. It is represented by letter ‘l’.

The value of ‘l’ depends upon the value of principal quantum number ‘n’. For a particular value of ‘n’, the ‘l’ can take the values from 0 to n-1. Each value of ‘l’ represents one particular sub-shell. For example,

For n = 1, n –1 = 0; so, l = 0

For n = 2, n –1 = 1; so, l = 0,1

For n = 3, n –1 = 2; so, l = 0,1, 2

For n = 4, n –1 = 0; so, l = 0,1, 2, 3

The various sub-shells represented by Azimuthal quantum number ‘l’ are s, p, d and f i.e. l = 0, denotes ‘s’ sub-shell, l = 1, denotes ‘p’ sub-shell, l = 2, denotes ‘d’ sub-shell and l = 3, denotes ‘f’ sub-shell. Azimuthal quantum number also gives the angular moment of electron in a particular sub -shell. The angular moment of electron in a particular sub-shell is given by l( l+1) h/2π.

3. MAGNETIC QUANTUM NUMBER(m) 

This quantum number gives the number of orbitals present in a particular sub-shell and orientation of the orbital in the space. Like ‘l, the value of ‘m’ is dependent upon the value of ‘l’ and can take only integral value from to – l to + l through zero. In general, for each value of ‘l’ there will be 2l + 1 values of m.

(a). For l = 0 (s sub-shell); m = 0. Hence there is only one orbital in the ‘s’ sub-shell.

(b). For l = 1 (p sub-shell), m = – 1, 0 and +1. Hence there are three orbitals in the ‘p’ sub-shell. The three corresponding orbitals are written as px, py and pz.

(c). For l = 2 (d sub-shell), m = –2, –1, 0, +1 and +2. Hence d sub-shell have five orbitals represented by dxy, dyz, dzx, dx2-y2 and dz2.

(d). For l = 3 ( f sub-shell), m = –3, –2, –1, 0, +1, +2 and +3. Hence there are seven orbitals present in the ‘f’ sub-shell.

SPIN QUANTUM NUMBER (s)

This quantum number arises due to the spinning of the electron about its own axis. The spin can be clockwise represented by +½ S or anti-clockwise represented by –½ S. The +½ S value indicates clockwise spin (generally represented by an arrow pointing upwards, i.e.↑) and –½ S indicates anti-clockwise spin (generally represented by an arrow pointing downwards, i.e.↓).

Thus, the four quantum numbers gives the complete address of an electron in an atom. So, they are also known as Postal Index numbers of an electron.