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What are Quantum Numbers?

In simple words, quantum numbers are certain numbers which are used to locate electrons in an atom.

or

In other words, we can say that quantum numbers are addresses of electrons. They are of four types out of which three are obtained by mathematical solution of the Schrödinger wave equation.    

4 Quantum Numbers

(i)   Principal Quantum Number (n):

It is obtained by mathematical solution of r-part of the Schrödinger wave equation.

Its values are:

n = 1, 2, 3, ………..,∞

• It represents the shells or orbits or energy levels

When n = 1

It means 1^st shell or K-shell

n = 2 

It means 2^nd shell or L-shell

n = 3 

It means 3^rd shell or M-shell

• It also indicates energy. Higher is the value of “n” more is the energy.
• It also indicates the distance of the electron from the nucleus. Higher is the value of “n” away is the electron from the nucleus.

• The numbers of electrons in a shell are calculated by it as 2n².

(ii)  Azimuthal Quantum Number (ℓ):

It is obtained by the mathematical solution of θ-part of the Schrödinger wave equation. It is also called a secondary quantum number. Its values are:

   ℓ = 0, 1, 2, 3, ……….., n-1

• It represents the subshells or  energy sub-levels

When ℓ = 0    It means s-sub shell (spherical shaped) 

s stands for sharp  

ℓ = 1

It means p-subshell (dumbbell-shaped)

p  stands for principal

 ℓ = 2

 It means d-sub shell

(sausage-shaped) 

d stands for diffused

ℓ = 3

It means f-subshell                 

f  stands for fundamental

• It also indicates energy up to some extent. Higher is the value of “ℓ” more is the energy.
• It also represents the shapes of orbitals.
• The number of orbitals in a sub-shell can be calculated by it.
• Number of orbitals in a sub-shell = 2 ℓ + 1

• The number of electrons in a sub-shell can be calculated by it.

Number of electrons in a sub-shell = 2(2 ℓ + 1)

(iii) Magnetic Quantum Number (m_ℓ ):

It is obtained by mathematical solution of Ø-part of Schrödinger wave equation. Its values are:

Values of     m_ℓ  = – ℓ , …,0 ,……..,+ ℓ

The number of values of      m_ℓ   = 2 ℓ + 1

• It represents orbitals and their orientations

e.g ( i)   if     ℓ = 1    It means p-subshell and for p-subshell,

Values of m_ℓ  = 2ℓ + 1 = 2 x 1 +1 = 3

These values of m_ℓ  = -1,0,+1

It means that the p-subshell has three orbitals with three different orientations.

  orbitals            p_x         p_y         p_z

  m_ℓ                    +1        -1         0

e.g(ii)  if  ℓ  = 2   It means d-sub shell

and for d-subshell, number of values of m_ℓ   = 2ℓ + 1 = 2 x 2 +1 = 5

these values of m_ℓ are = -2, -1, 0, +1, +2

It means that d-subshell has five orbitals with five different orientations.

orbitals  

 d_xy   d_yz   d_z2   d_xz       d_x2-_y2

m_ℓ      -2     -1    0      +1        +2

Spin Quantum Number (m_s):             

It describes the spin of electrons. Its values are

   m_s = + ½ , – ½

When  m_s = – ½   electron shows clockwise spin (↑)       

m_s = + ½   electron shows anti-clockwise spin (↓)    

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