Definition:

Quantum numbers are a set of four values that are used to describe the position and energy of the electron in an atom.

Alternatively, quantum numbers are the numbers used to indicate the size, shape, three-dimensional direction, variation of the energy level, and the direction of rotation of an electron around its own axis.

Four quantum numbers are required to fully describe the position of electrons in an atom or in an atomic orbital. They are:

- Principal quantum number, n
- Subsidiary or Azimuthal quantum number,
*l* - Magnetic quantum number, m
- Spin quantum number, s

Relation among four quantum numbers:

- “n” can have values of 1, 2, 3, 4 ………..(only integers)
- for a given “n” value,
*l*can have values starting from 0 to (n-1), i.e a total of n values. - for a given values of
*l*, there can be total of (2*l*+1) values of “m” ranging from*-l*to*+l* - Spin quantum number, “s” does not dependent on other quantum numbers. It can have two values + 1/2 (clockwise) and -1/2 (anti-clockwise)
- For each value of “m”, there will be two values (+1/2 or -1/2) of s. So, there can be two electrons within an orbital with opposite spin.

For Helium:

He(2) = 1s^{2}

Four quantum numbers are given below:

n = 1,

*l* = 0 ( electron exists in s orbital, for s orbital *l* = 0)

m = 0 ( because m = – *l* to +*l* through 0)

s = -1/2 ( every orbital contains 2 electrons but their spin should be opposite. If you draw box, it will be more clear)

For N (7) = 1s^{2}2s^{2}2p_{x}^{1}2p_{y}^{1}2p_{z}^{1} ( well understood by box system)

Here, n = 2

*l* = 1 ( please see the value of *l *for p orbital)

m = -1

s = +1/2

For O (8) =1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{1}2p_{z}^{1} ( In this case 8th electron of O pairs)

n = 2

*l* = 1 ( because electron at p orbital)

m = +1

s = -1/2 ( 8^{th} electron pairs with 5^{th} electron and spin quantum number for 5^{th} electron is =1/2)

For F(9) = 1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{2}2p_{z}^{1}

n =2

l = 1

m = 0

s = -1/2

For Ne(10) = 1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{2}2p_{z}^{2 }

Here,

n =2

*l* = 1

m = -1

s = -1/2 ( pairing occurs)

For Na(11)

Electronic configuration (EC) of Na(11) = 1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{2}2p_{z}^{2} 3s^{1} ( We can also write this configuration as [Ne] 3s^{1} )

**(A)**

(B)

Here,

n = 3

*l *= 0 ( for n =3, *l* should have 3 values such as 0, 1, and 2. Since, the11^{th} electron of Na occupies “s” orbital, the value of *l *should be 0)

m = 0 ( when* l* = 0, m = 0)

s = +1/2 ( according to **A**, clockwise/ upward direction) or -1/2 (according to **B**, anti-clockwise direction)

For Mg(12)

EC of Mg(12) = 1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{2}2p_{z}^{2} 3s^{2 }

Here, n = 3

l = 0 ( because electron is in s orbital)

m = 0

s = -1/2 ( pairing occurs, anti-clockwise direction)

Question: Find out the values of four quantum numbers for the 10^{th} electron of Na(11).

Hints: See the BOX

For Al(13)

EC of Al(13) = 1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{2}2p_{z}^{2} 3s^{2} 3p_{x}^{1} or [Ne] 3s^{2} 3p_{x}^{1}

By box system

Here,

n = 3

*l* = 1 ( ask yourself why *l* value is 1)

m = +1 (see the relation between *l* and m)

s = +1/2

**For Si(14)**

EC of Si(14) = [Ne] 3s^{2}3p_{x}^{1}3p_{y}^{1}3p_{z}

Here,

n = 3

*l *= 1

m = 0

s = +1/2

For P(15)

EC of P(15)= [Ne] 3s^{2}3p_{x}^{1}3p_{y}^{1}3p_{z}^{1}

Here,

n = 3

*l* = 1

m = -1

s = +1/2

**For S (16)**

EC of S(16) = [Ne] 3s^{2}3p_{x}^{2}3p_{y}^{1}3p_{z}^{1}

Here,

n = 3

l = 1

m = +1 ( for p_{x}, *l* = +1, for p_{y}, *l* = 0, and for p_{z}, *l* = -1)

s = -1/2 ( due to pairing)

For Chlorine, Cl

EC of Cl(17) = [Ne] 3s^{2}3p_{x}^{2}3p_{y}^{2}3p_{z}^{1}

Here,

n = 3

*l* = 1

m = 0 ( read the explanation for S)

s = -1/2 ( due to pairing)

**For Ar(18)**

EC of Ar(18) = [Ne] 3s^{2}3p_{x}^{2}3p_{y}^{2}3p_{z}^{2}

Here,

n = 3

*l* = 1

m = -1

s = -1/2

**For K(19)**

EC of K(19) = 1s^{2}2s^{2}2p_{x}^{2}2p_{y}^{2}2p_{z}^{2} 3s^{2} 3p_{x}^{2 }3p_{y}^{2}3p_{z}^{2} 4s^{1} or [Ar] 4s^{1}

Here,

n = 4

l = 0 ( for n = 4, *l* has four value such as 0, 1, 2 and 3. For “s” orbital *l* = 0)

m = 0 ( because* l* = 0)

s = +1/2