Quantum Numbers

To locate the position of an electron in the atom 4 identification number are required and these identification number are called as quantum number.

1. Principal quantum number (n)→ Shell (Orbit)
2. Azimuthal quantum number (l)→ Sub shell
3. Magnetic quantum number (m)→ Orbital
4. Spin quantum number (S) →Spin of e^–

(a) Principal Quantum Number

Given By→ Bohr’s

It represents the name, size and energy of the shell to which e^– belongs

(1) The value of n lies between 1 to ∞

i.e. n = 1,2,3,4 —— ∞ corresponding name of shells are K, L, M, N, O, —–

(2) Greater the value of n , greater is the distance from the nucleus.

r₁< r₂< r₃< r₄< r₅ ——–

(3) Greater the value of n, greater is the energy of shell

E₁<E₂<E₃<E₄ ——–

(4) Velocity of electron 

(5) The angular momentum of a revolving electron is 

where n = Principal quantum number.

(6) The number of electrons in a particular shell is equal to2n².

(b)Azimuthal quantum number/Subsidiary quantum number

Represented by ‘ ’ (Given by-Sommerfeld)

(1) It represents the shape of the subshell and orbital and orbital angular momentum

(2) Value of between 0 and (n-1)

i.e.    = 0, 1, 2 ——— (n-1)

ℓ  = 0 (s Subshell)

ℓ  = 1 (p Subshell)

ℓ= 2 (d Subshell)

ℓ  = 3 (f Subshell)

Ex.  If n =1 then ℓ = 0 ⇒1 s i.e. in n=1 shell, only one subshell ‘s’ is present.

If n =2 then  ℓ = 0, 1 ⇒ 2s, 2p i.e. in n=2 shell, two subshell ‘s’& ‘p’ are

present.

– The orbital angular momentum =  {h is called as ‘hash’}

Orbital angular momentum:           For s subshell = 0

For p subshell =  

– The number of electron in a particular subshell is equal to 2 (2ℓ +1)

for s subshell number of electrons = 2e^–

for p subshell number of electrons = 6 e^–

for d subshell number of electrons = 10 e^–

for f subshell number of electrons = 14 e^–

– Shape of the subshell:

s →   spherical

p  → dumb bell shape

d  → double dumb bell shape

f  →  complex shape

(c) Magnetic Quantum Number / Orientation Quantum Number (m) :

Given by lind : It represents the shape of different orbitals and the orientation of electron cloud (orbital)

(1) Under the influence of magnetic field each subshell is further subdivided into orbitals (The electron cloud is known as orbital)

Magnetic quantum number describesthis different distribution of electron cloud.

(2) Value of m = all integral value from – to + including zero.

i.e.   Value of m = – ℓ to +ℓ

An orbital can be represented by 3 set of quantum numbers =Ψn,ℓ,n

Ex. 1 :

Ex. 2 :

(d) Spin Quantum Number (s) : (Given  by Gold Schmid’t and Ulhenback)

It represents the direction of electron Spin around its own axis

(1) For clock wise spin/spin up (↑ ) electron → ½

(2) For anticlock wise spin/spin down (↓ ) electron → ± ½

Spin angular momentum of an e^– = 

(3) Each orbital can accommodate 2 electrons with opposite spin or spin paired.

Correct ↓↑Spin paired e                                   Wrong ↑↑Spin parallel e