Quantum Numbers: Definition, Types, Rules and Examples

Quantum numbers are a set of numbers used in quantum mechanics to describe the state of an electron in an atom. They provide important information about the energy level, subshell, orbital and spin of an electron. Every electron in an atom is described by four quantum numbers.

What Are Quantum Numbers?

In the modern quantum mechanical model of the atom, electrons do not move around the nucleus in fixed circular paths. Instead, an electron is described by a wave function and has a certain probability of being found in a particular region of space. Quantum numbers are used to describe the properties and state of an electron.

There are four quantum numbers:

1. Principal quantum number
2. Azimuthal quantum number
3. Magnetic quantum number
4. Spin quantum number

Four Quantum Numbers at a Glance

Quantum Number	Symbol	Describes	Allowed Values
Principal	n	Shell or energy level	1, 2, 3, 4, ...
Azimuthal	l	Subshell and orbital shape	0 to n - 1
Magnetic	ml	Orientation of orbital	-l to +l
Spin	ms	Spin of electron	+1/2 or -1/2

1. Principal Quantum Number

The principal quantum number is represented by n. It indicates the principal shell or main energy level in which an electron is present.

The possible values of the principal quantum number are:

n = 1, 2, 3, 4, 5, ...

The shells are commonly represented by the following letters:

Principal Quantum Number	Shell
n = 1	K shell
n = 2	L shell
n = 3	M shell
n = 4	N shell

As the value of n increases, the electron generally belongs to a shell farther from the nucleus and has a higher principal energy level.

Example

For an electron present in the fourth shell:

n = 4

2. Azimuthal Quantum Number

The azimuthal quantum number is represented by l. It is also called the angular momentum quantum number. It determines the subshell and is related to the shape of the orbital.

For a particular value of n, the possible values of l are:

l = 0 to n - 1

The values of l correspond to subshells as follows:

l	Subshell	General Shape
0	s	Spherical
1	p	Dumbbell-shaped
2	d	Complex shapes
3	f	More complex shapes

Example

For the third shell, n = 3. Therefore:

l = 0, 1, 2

Thus, the third shell contains the 3s, 3p and 3d subshells.

3. Magnetic Quantum Number

The magnetic quantum number is represented by m_l. It describes the orientation of an orbital with respect to a chosen axis.

For a particular value of l, the possible values of the magnetic quantum number are:

m_l = -l, ..., 0, ..., +l

Therefore, the total number of possible values of m_l is:

Number of orbitals = 2l + 1

For s Subshell

For an s subshell:

l = 0

Therefore:

m_l = 0

Thus, an s subshell contains 1 orbital.

For p Subshell

For a p subshell:

l = 1

Therefore:

m_l = -1, 0, +1

Thus, a p subshell contains 3 orbitals.

For d Subshell

For a d subshell:

l = 2

Therefore:

m_l = -2, -1, 0, +1, +2

Thus, a d subshell contains 5 orbitals.

For f Subshell

For an f subshell:

l = 3

Therefore:

m_l = -3, -2, -1, 0, +1, +2, +3

Thus, an f subshell contains 7 orbitals.

Number of Orbitals in Different Subshells

Subshell	l	Number of Orbitals	Maximum Number of Electrons
s	0	1	2
p	1	3	6
d	2	5	10
f	3	7	14

4. Spin Quantum Number

The spin quantum number is represented by m_s. It describes the spin state of an electron.

An electron can have one of two possible spin quantum numbers:

m_s = +1/2

or

m_s = -1/2

An orbital can accommodate a maximum of two electrons. If two electrons occupy the same orbital, they must have opposite spins.

Relationship Between Quantum Numbers

The four quantum numbers are related to each other. The principal quantum number determines the possible values of the azimuthal quantum number. The azimuthal quantum number determines the possible values of the magnetic quantum number. The spin quantum number describes the spin state of the electron.

The sequence can be remembered as:

Shell → Subshell → Orbital → Spin

n → l → m_l → m_s

Example: Quantum Numbers of a 3p Electron

Consider an electron present in the 3p subshell.

Since the electron is in the third shell:

n = 3

Since it belongs to the p subshell:

l = 1

For l = 1, the possible values of m_l are:

m_l = -1, 0, +1

The possible spin values are:

m_s = +1/2 or -1/2

Therefore, one possible set of quantum numbers for a 3p electron is:

(3, 1, 0, +1/2)

Another possible set is:

(3, 1, +1, -1/2)

Example: Quantum Numbers of a 4d Electron

For a 4d electron:

n = 4

Since d corresponds to l = 2:

l = 2

Therefore:

m_l = -2, -1, 0, +1, +2

The spin quantum number can be:

m_s = +1/2 or -1/2

Hence, one possible set is:

(4, 2, -2, +1/2)

Quantum Numbers and Pauli Exclusion Principle

The Pauli exclusion principle states that no two electrons in the same atom can have the same set of all four quantum numbers.

For example, two electrons can occupy the same orbital because they may have the same values of n, l and m_l, but their spin quantum numbers must be different.

Thus, if one electron has:

(2, 0, 0, +1/2)

the other electron in the same orbital must have:

(2, 0, 0, -1/2)

Important Rules of Quantum Numbers

• The principal quantum number n can have values 1, 2, 3, 4, ...
• The azimuthal quantum number l ranges from 0 to n - 1.
• The magnetic quantum number m_l ranges from -l to +l.
• The spin quantum number m_s can only be +1/2 or -1/2.
• The number of orbitals in a subshell is given by 2l + 1.
• Each orbital can contain a maximum of two electrons.
• No two electrons in an atom can have identical values of all four quantum numbers.

Common Mistakes in Quantum Numbers

• For n = 1, l can only be 0.
• For n = 2, l can be 0 or 1, but not 2.
• For l = 1, m_l cannot be +2.
• The spin quantum number cannot be 0.
• An s subshell has one orbital, not two.
• A p subshell has three orbitals and can accommodate six electrons.
• A d subshell has five orbitals and can accommodate ten electrons.
• An f subshell has seven orbitals and can accommodate fourteen electrons.

Quick Revision Table

Quantity	Symbol	Values
Principal quantum number	n	1, 2, 3, ...
Azimuthal quantum number	l	0 to n - 1
Magnetic quantum number	ml	-l to +l
Spin quantum number	ms	+1/2 or -1/2

Conclusion

Quantum numbers are fundamental to understanding the electronic structure of atoms. The four quantum numbers provide information about an electron's shell, subshell, orbital orientation and spin.

The most important relationships to remember are:

l = 0 to n - 1

m_l = -l to +l

m_s = +1/2 or -1/2

Understanding these rules makes it easier to solve questions related to electronic configuration, orbitals, atomic structure and the Pauli exclusion principle.

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