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**What is** **Quantum Number** ?

These are certain numbers which are used to locate electron in an atom or in other words these are address of electron. They are of four types out of which three are obtained by mathematical solution of Schrödinger wave equation.

# **4 Quantum Numbers**

(i) *Principal Quantum Number* (n):

It is obtained by mathematical solution of r-part of 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 distance of electron from the nucleus. Higher is the value of “n” away is the electron from the nucleus.

- Numbers of electrons in a shell are calculated by it as 2n
^{2}.

(ii) *Azimuthal Quantum Number *(ℓ):

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

ℓ = 0, 1, 2, 3, ……….., n-1

- It represents the sub shells or energy sub-levels

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

s stands for sharp

ℓ = 1 It means p-sub shell

(dumbbell shaped)

p stands for principal

ℓ = 2 It means d-sub shell

(sausage shaped)

d stands for diffused

ℓ = 3 It means f-sub shell

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 shapes of orbitals.
- Number of orbitals in a sub-shell can be calculated by it.

Number of orbitals in a sub-shell = 2 ℓ + 1

- 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-sub shell and for p-sub shell,

Values of m_{ℓ}_{ } = 2ℓ + 1 = 2 x 1 +1 = 3

These values of m_{ℓ}_{ } = -1,0,+1

It means that p-sub shell 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-sub shell, 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-sub shell has five orbitals with five different orientations.

orbitals d_{xy} d_{yz} d_{z}2 d_{xz} d_{x}2-_{y}2

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 clock wise spin (↑)

m_{s} = + ½ electron shows anti-clock wise spin (↓)

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Categories: atomic structure, chemistry, Structure of Atom