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What are quantum numbers? Explain briefly the various quantum numbers which completely specify an electron in an atom. 

A set of numbers used to provide a complete description of an electron in an atom is called quantum numbers. There are four quantum numbers required for a complete characterisation of various electrons in an atom. They are:

(i) Principal quantum number (n)
(ii) Azimuthal quantum number (l)
(iii) Magnetic quantum number (m)
(iv) Spin quantum number (s)

(i) Principal quantum number (n): The principal quantum number determines the main energy level or shell to which an electron belongs. It can have all positive integral values such as 1,2, 3,4. etc. The various values of n are also designated by K, L, M, N and so on.
n       =   1 2 3 4 5 
Orbits =  K L M N O
The Principal quantum number gives an idea of
(a) the size of the orbital and the distance of the electron from the nucleus.
(b) the energy of the electron. 
As the value of n increases, the electron gets farther away from the nucleus and its energy increases.
The maximum number of electrons in a shell is equal to 2n² where n is the principal quantum number.
n               =                1           2          3          4
Maximum                    2x1²      2 x 2²    2 x 3²     2 x 4^2
electrons                    (2n²) = 2      = 8      = 18     = 32

(ii) Azimuthal quantum number: Azimutal quantum number:
(a) denotes the sub-shell to which electron belongs,
(b) specifies the angular momentum of the electron and
(c) signifies the shape of the orbital.

It can have all possible whole number values from l=0 to l=n-1, so that the total values of l are equal to the value of principal quantum number, n. Each value of l represents a particular subshell or sub level.
lf n= 1 (K shell) l = 0, (1s subshell) [one value]
n = 2 (L shell) ; l = 0, 1 (2s and 2p subshell respectively) [two values]
n = 3 (M shell) ; l = 0, 1, 2, (3s, 3p and 3d subshell respectively) [three values]
n = 4 (N shell) 1 = 0, 1, 2, 3 (4s, 4p, 4d, 4f subshell, respectively) [four values]
Thus, once n is fixed, lower the value of l, lower will be the energy.
e.g. for M shell (n = 3), 3s < 3p < 3d
4s = 4p = 4d= 4f.
Designation of Azimuthal quantum number (l) by symbols:

Maximum number of electrons that can be occupied by a subshell is 2(2l + 1) or 4l + 2.

(iii) Magnetic quantum number (m). An electron due to its angular motion around the nucleus produces a magnetic field which can interact with the external magnetic field. Thus, under the influence of the external magnetic field, the electrons in a given sub-shell orient themselves in certain specific regions of space around the nucleus. These regions of space are called orbitals. This quantum number determines the number of the preferred orientations of electrons in a subshell in space around the nucleus. For a given value of l, m can have values from - l to + l including zero (-l to zero to +l) and in all (2l + 1) values. For example.
If l = 0 ; m = 0
l = I ; m = - 1, 0 + 1
l = 2 ; m -2,- 1,0 ,+1, +2
l = 3 ;m =-3,-2,- 1,0, +1, +2, +3
It means that s-subshell (l = 0) has one orientation of electron i.e. s orbital:
p-subshell (l = 1) has three orbitalsp_x,p_y,p_z d-subshell (l = 2) has five orientations i.e. has five orbitals,
f-subshell (l = 3) has seven orientations of electrons i.e. f-subshell has seven orbitals.

(iv) Spin quantum number (s). Electron revolves around the nucleus and at the same time it rotates about its own axis. An electron may have two possible spins -clockwise or anticlockwise. Therefore quantum s can have one of the two possible values +¹/₂and -¹/₂ depending upon the direction of the spin. Instead of numbers + ¹/₂ and -¹/₂ , the directions of spin are designated by arrows pointing up and down (↑ or ↓) respectively.