Quantum Number

Quantum Number

An atom contains number of orbits and orbitals. These are distinguished from one another on the basis of their size, shape and orientation in space. The parameters are given in terms of different numbers called quantum numbers. Quantum numbers may be defined as a set of four numbers with the help of which we can get complete information about all the electrons in an atom. Quantum number tells us the complete address of the electron (i.e. location, energy, the type of orbital occupied and orientation of that orbital) in an atom.
Quantum Number is of four types-
1. Principal Quantum Number (n)
2. Azimuthal Quantum Number (l)
3. Magnetic Quantum Number (m)
4. Spin Quantum Number (s)

Principal Quantum Number (n)

It is denoted by 'n'.
The values of 'n' can never be zero but from 1 to n.
n = 1 K shell
n = 2 L shell
n = 3 M shell
n = 4 N shell
'n' represents the orbit number (i.e. size of the atom) as well as energy of the orbit to which an electron belongs.
n ∝ R & E
As the value of 'n' increases, the energy of the electron increases and thus, the electron is less tightly held with nucleus. Angular momentum can be calculated using principal quantum number: mvr = nh/2π

Azimuthal Quantum Number (l)

i. It is denoted by 'l'and is represent the sub-shell (i.e. orbital)
ii. The value of 'l' depeds upon the value of 'n'.
iii. The values of 'l' are from 0 to (n − 1)
l = 0 for s-sub-shell, l = 1 for p-sub-shell, l = 2 for d-sub-shell and l = 3 for f-sub-shell
iv. For a given value of n, total values of 'l' are 'n'.
v. The values of 'l' signify the shape and energy level of sub-shells in a major energy shell.
vi. The number of electrons in a particular sub-shell is equal to 2(2l + 1)
vii. If the value of 'n' is same then the order of energy of various sub-shells of a shell will be s < p < d < f
4s < 4p < 4d < 4f
viii. If the value of 'l' is same but the value of 'n' is different then the order of energy of various sub-shells of a shell will be 1s < 2s < 3s < 4s < 5s < 6s
3d < 4d < 5d < 6d etc.
ix. The orbital angula momentum = √l(l+1)ℏ

Magnetic Quantum Number (m)

i. It is denoted by 'm'.
ii. It represents the orientation of orbitals.
iii. Its value depends upon the value of 'l'
m = − l to + l
iv. The positive values of magnetic quantum number 'm' represent the angular momentum component of the orbital in the direction of the applied magnetic field whereas the negative values of 'm' account for the angular momentum component of orbital in the opposite direction of applied magnetic field. v. Total values of 'm' for a given value of 'n' = n²
vi. Total values of 'm' for a given value of 'l' = (2l + 1)

Spin Quantum Number

i. It is denoted by 's'.
ii. It describe the orientation of the electron spin (rotation) in the space.
iii. The value of 'm' are +1/2(clockwise rotation) and −1/2(anticlockwise rotation)
iv. Magnitude of spin quantum number of an electron cannot be changed.
v. The spin may lie in 2s+1 = 2 orientation.
vi. Each type of subatomic particle has fixed spin quantum numbers like 0, 1/2, 1, 3/2, … etc.
vii. The spin value of an electron, proton and neutron is 1/2.
viii. The particles having half integral value (1/2, 3/2 …) of spin are called fermions.
ix. The particles having integral value (0, 1, 2..) of spin are called bosons.
x. Spin multiplicity of an atom is √s(s+1)
xi. Spin angular momentum of an electron is √s(s+1)h/2π

Which of the following set of quantum number is not possible-
a. n = 2, l = 0, m = −1, s = −1/2
b. n = 3, l = 2, m = −1, s = +;1/2
c. n = 2, l = 1, m = −1, s = −1/2
d. n = 2, l = 3, m = −1, s = +;1/2

Hints: n = 2 and l = 3 means 2f which is not possible.

What is the maximum number of electrons, which can have following quantum numbers
n = 3, l = 1, m = -1 ?
a. 2
b. 6
c. 8
d. 4

Hints: Any orbital can have a maximum of two electrons with spin quantum number ±1/2.

Which of the following quantum numbers can distinguish between two electrons present in the same orbital ?
a. Azimuthal quantum number
b. Principal quantum number
c. Magnetic quantum number
d. Spin quantum number

Hints: Spin quantum number as its value is different (i.e. +1/2 and −1/2) in the same orbital.

Pauli's Exclusion Principle

No two electrons in an atom can have the same set of all the numbers.
Example-
Quantum numbers of 5th and 6th electrons of nitrogen-
5th electron-
n = 2
l = 1
m = +1
s = +1/2
6th electron-
n = 2
l = 1
m = 0
s = +1/2
we have all the same quantum numbers except 'm' for 5th and 6th electrons of nitrogen.

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