Quantum Numbers - Principal, Definition, Formula, Applications, FAQs

The quantum numbers determine the behavior of electrons in the atom. They were introduced by the Danish physicist Niels Bohr in 1913 to describe the distinct energy levels of electrons. Later there was another model of the atom which Bohr's Model developed to understand the concept of quantum numbers. The quantum helps to understand the shape and orientation of thr electron where it is arranged in the space in the atom. By knowing the quantum number we can predict the exact location of the electron inside the atom.

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This Story also Contains

1. Quantum Number
2. Some Solved Examples
3. Summary

Quantum Number

Quantum numbers:

They are the set of four numbers that explain the state of an electron i.e., location, energy, type of orbital, orientation of orbital, etc. in an atom. Various quantum numbers are as follows:

1. Principal quantum number(n)
2. Azimuthal quantum number(l)
3. Magnetic quantum number(m)
4. Spin quantum number(s)

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Principal quantum number(n):

It represents the principal shell of an atom. It can have integral values except zero like 1,2,3,.... Also denoted as K,L,M,.....etc.

The maximum number of electrons in a principal shell can be 2n² where n is the principal quantum number.

This quantum number gives information about :

• Distance of electron from nucleus i.e., size of electron cloud.
• Energy of electron in any shell in unielectronic species

$\mathrm{E}_{\mathrm{n}}=-\frac{1312 \times \mathrm{Z}^2}{\mathrm{n}^2} \mathrm{~kJ} / \mathrm{mol}$

Where Z is the atomic number and n is a principal quantum number.

• In the case of multielectronic species, the energy of an electron is given by Aufbau's Principle, which we shall be studying later.

Azimuthal quantum number(l):

It gives us an idea of the three-dimensional shape of the orbitals.

The azimuthal quantum number represents the subshell or sub energy shell in an atom.

l has values from 0 to (n-1).

e.g. for n=2 ; l= 0, 1

Subshell notation for l = 0, 1, 2, ..... is s, p, d ...

A maximum number of electrons that can be accommodated in a subshell with azimuthal quantum number 'l' is given by [2(2l+1)]:

e.g. for s subshell = 2; for p subshell = 6.

Related Topics Link,

• Hydrogen Spectrum
• Atomic number mass number
• Aufbau principle
• Proton Neutron Discovery
• Shapes of Orbitals
• Protons
• Electrons

Magnetic quantum number(m):

It gives us information about the spatial orientation of the orbitals in the subshell concerning the standard set of coordinate axes.

Every value of m represents a possible orientation of the orbital

It represents the number of orbitals present in a subshell.

m has values ranging from -l to +l including zero.

For eg: for ‘s’ subshell :

1. Value of l is 0
2. m has value=0

It means that there is only one possible orientation for the s orbital

For ‘p’ subshell :

1. Value of l is 1
2. m has value= -1, 0, +1

It means that there are three possible orientations for the p-orbital

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Spin quantum number(s):

An electron in an orbital can spin in either a clockwise or anticlockwise direction. The spin quantum number has no classical analogue and any spin direction can be assigned +1/2 and the opposite spin will be automatically assigned -1/2. These values of +1/2 or -1/2 are not fixed for a particular spin direction.

Thus, an electron can have only two possible values of this quantum number, either $+\frac{1}{2}$ or $-\frac{1}{2}$ respectively.

Recommended topic video on (Quantum Numbers )

Some Solved Examples

Example.1

1. The energy of an electron in a hydrogenic ion depends on

1) (correct)Principal quantum number only

2)Principal and azimuthal quantum numbers only

3)Principal, azimuthal, magnetic quantum number only

4)All four quantum numbers

Solution

As we learn

Principal Quantum Number -

The principal quantum number determines the size and to a large extent the energy of the orbital.

The energy of hydrogenic ions depends on the principal quantum numbers only.

Hence, the answer is the option (1).

Example.2

Which of the following combination of statements is true regarding the interpretation of the atomic orbitals?

(a) An electron in an orbital of high angular momentum stays away from the nucleus than an electron in an orbital of lower angular momentum.

(b) For a given value of the principal quantum number, the size of the orbit is inversely proportional to the azimuthal quantum number.
(c) According to wave mechanics, the ground state angular momentum is equal to $\frac{h}{2 \pi}$ .

(d) The plot of $\psi V s r$ for various azimuthal quantum numbers, shows peak shifting towards a higher r-value.

1)(a), (c)

2)(b), (c)

3) (correct)(a), (d)

4)(a), (b)

Solution

An electron in an orbital of high angular momentum stays away from the nucleus than an electron in an orbital of lower angular momentum.

According to Bohr's theory, angular momentum is an integral multiple of h/2π. Hence, the ground state angular momentum is equal to h/2π. But according to wave mechanics, $\frac{h}{2 \pi}$ is wrong because it's $\frac{h}{4 \pi}$.

As we know the principal quantum number depends on size whereas the azimuthal quantum number doesn't depend on size.

Option(d) is also correct, as the plot of $\psi V s r$ shows a peak shifting towards a higher R-value.

Hence, the answer is the option (3).

Example.3

3. The electrons identified by quantum numbers n and l :

(1) n = 4, l = 1 (2) n = 4, l = 0 (3) n = 3, l = 2 (4) n = 3 , l = 1

Can be placed in order of increasing energy as :

1) (correct)$(4)<(2)<(3)<(1)$

2)$(2)<(4)<(1)<(3)$

3)$(1)<(3)<(2)<(4)$

4)$(3)<(4)<(2)<(1)$

Solution

As discussed in the concept

Principal Quantum Number (n) -

It is a positive integer with a value of n = 1,2,3.......

and

Azimuthal Quantum Number(l) -

For a given value of n, l can have n values ranging from 0 to n – 1, that is, for a given value of n, the possible values of l are: l = 0, 1, 2, ....( n –1)

(1) $n=4, l=1 \Rightarrow 4 p$

(2) $n=4, l=0 \Rightarrow 4 s$

(3) $n=3, l=2 \Rightarrow 3 d$

(4) $n=3, l=1 \Rightarrow 3 p$

Increasing the order of energy is

$3 p<4 s<3 d<4 p$

$(4)<(2)<(3)<(1)$

Alternatively,

For (1) $n+l=5 ; n=4$

(2) $n+l=4 ; n=4$

(3) $n+l=5 ; n=3$

(4) $n+l=4 ; n=3$

Lower $n+l$ means less energy and if for two subshells $n+l$

is same than lower n, lower will be the energy.

Thus correct order is (4) < (2) < (3) < (1)

Hence, the answer is the option (1).

Example.4

4. The correct set of four quantum numbers for the valence electrons of the rubidium atom (Z=37) is :

1) (correct)$5,0,0,+\frac{1}{2}$

2)$5,1,0,+\frac{1}{2}$

3)$5,1,1,+\frac{1}{2}$

4)$5,0,1,+\frac{1}{2}$

Solution

We know that -

Principal Quantum Number (n) -

It is a positive integer with a value of n = 1,2,3.......

Azimuthal Quantum Number(l) -

For a given value of n, l can have n values ranging from 0 to n – 1, that is, for a given value of n, the possible values of l are: l = 0, 1, 2, ....( n –1)

Magnetic Quantum Number (m) -

For any sub-shell (defined by ‘l ’value) $2 l+1$values of m are possible and these values are given by :

m = – l , – ( l –1), – ( l – 2)... 0,1... ( l – 2), ( l –1),l

Spin Quantum Number (s) -

It has two values +1/2 and -1/2

${ }_{37}^{R b}=>1 S^2, 2 S^2, 2 p^6, 3 S^2, 3 p^6, 4 S^2, 3 d^{10}, 4 p^6, 5 S^1$

the value of n, l, m & s for the last electrons are

$5,0,0,+\frac{1}{2}$

Hence, the answer is an option (1).

Example.5

5. The quantum number of four electrons is given below:

$\begin{aligned} & I . n=4, l=2, m_l=-2, m_s=-1 / 2 \\ & \text { II.n }=3, l=2, m_l=1, m_s=+1 / 2 \\ & \text { III.n }=4, l=1, m_l=0, m_s=+1 / 2 \\ & \text { IV.n }=3, l=1, m_l=1, m_s=-1 / 2\end{aligned}$

The correct order of their increasing energies will be :

1)$I<I I<I I I<I V$

2)$I V<I I I<I I<I$

3) (correct)$I V<I I<I I I<I$

4)$I<I I I<I I<I V$

Solution

The higher the value of ($n+l$) the higher will be the energy.

If two orbitals have the same value of (n+l), the orbital with the higher value of n will have higher energy.

So, order will be => IV < II < III < I

Hence, the answer is the option (3).

Summary

Quantum numbers are very useful in determining the properties of electrons in atoms. They define the energy levels, arrangement of atoms in the space or spatial arrangement, and spin orientation of electrons. Quantum numbers describe the number of energy levels or shells of electrons. It also determines the size and energy of the orbital. quantum numbers provide a comprehensive of an electron's state within an atom, including its energy, spatial arrangement, and its magnetic properties

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