# Quantum Numbers

Last Updated : 16 Feb, 2024

Quantum numbers in Chemistry, are the sets of numbers that describe an electron’s orbit and movement within an atom. When the quantum numbers of all the electrons in a given atom are added together, they must satisfy the Schrodinger equation.

Quantum numbers are the set of numbers used to describe the position and energy of an electron in an atom. There are four types of quantum numbers: principal, azimuthal, magnetic, and spin. Quantum numbers represent the values of a quantum system’s conserved quantities.

## What are Quantum Numbers?

Quantum numbers are the set of constant values in the quantum approach. Quantum Numbers or Electronic quantum numbers describe an electron with numerical values that provide solutions to the Schrodinger wave equation for hydrogen atoms these numbers can define the position, energy and orientation of an electron in an atom through the set of numbers.

According to the Pauli exclusion principle, no two electrons in an atom may have the same set of quantum numbers. A half-integer or integer value is used to characterize each quantum number. The principal, azimuthal and magnetic quantum numbers are respectively related to the size, shape, and orientation of the atom.

Four quantum numbers can be used to fully describe all of the properties of a given electron in an atom; these are:

1. Principal quantum number
2. Orbital angular momentum quantum number (or Azimuthal quantum number).
3. Magnetic quantum number
4. The electron spin quantum number

## Types of Quantum Numbers

Four quantum numbers are used to fully describe all the characteristics of an electron in an atom. These quantum numbers are:

• Principal Quantum Number (n)
• Azimuthal Quantum Number (l)
• Magnetic Quantum Number (ml)
• Electron Spin Quantum Number (s)

### Principal Quantum Number (n)

The symbol ‘n’ represents the principal quantum numbers. They denote the atom’s primary electron shell. Because it describes the most likely distance between the nucleus and the electrons, a larger value of the principal quantum number implies a greater distance between the electron and the nucleus (which, in turn, implies a greater atomic size).

• The principal quantum number’s value can be any integer with a positive value equal to or greater than one. The value n=1 denotes an atom’s innermost electron shell, which corresponds to an electron’s lowest energy state (or ground state).
• As a result, the principal quantum number, n, cannot have a negative value or be equal to zero because an atom cannot have a negative value or no value for a principal shell.
• When an electron is infused with energy (excited state), the electron jumps from one principal shell to a higher shell, causing the value of n to increase.
• Similarly, as electrons lose energy, they return to lower shells, lowering the value of n. Absorption refers to the increase in the value of n for an electron, emphasizing the photons or energy absorbed by the electron.
• Similarly, a decrease in the value of n for an electron is referred to as emission, and this is where the electrons emit their energy.

### Azimuthal Quantum Number (l) – Orbital Angular Momentum Quantum Number

The azimuthal quantum number (or orbital angular momentum) describes the shape of an orbital. It is represented by the letter ‘l,’ and its value equals the total number of angular nodes in the orbital.

• A value of the azimuthal quantum number can denote either an s, p, d, or f subshell, the shapes of which vary.
• This value is determined by (and limited by) the value of the principal quantum number, i.e. the azimuthal quantum number ranges between 0 and (n-1).
• For example, if n = 3, the azimuthal quantum number can have three values: 0, 1, and 2.
• When l is set to zero, the resulting subshell is an ‘s’ subshell.
• When l=1 and l=2, the resulting subshells are ‘p’ and ‘d’ subshells, respectively (respectively).
• As a result, when n=3, the three subshells that can exist are 3s, 3p, and 3d. In another case where n = 5, the possible values of l are 0, 1, 2, 3, and 4. If l = 3, the atom contains three angular nodes.

### Magnetic Quantum Number (ml)

The magnetic quantum number determines the total number of orbitals in a subshell as well as their orientation. It is represented by the symbol ‘ml.’ This number represents the projection of the orbital’s angular momentum along a given axis.

• The magnetic quantum number is determined by the azimuthal (or orbital angular momentum) quantum number.
• For a given value of l, the value of ml falls between -l to +l. As a result, it is indirectly dependent on the value of n.
• For example, if n = 4 and l = 3 in an atom, the magnetic quantum number can be -3, -2, -1, 0, +1, +2, and +3. The total number of orbitals in a given subshell is determined by the orbital’s ‘l’ value.
• It is calculated using the formula (2l + 1). The ‘3d’ subshell (n=3, l=2), for example, has 5 orbitals (2*2 + 1). Each orbital can hold two electrons. As a result, the 3d subshell can accommodate a total of 10 electrons.

### Electron Spin Quantum Number (s)

The electron spin quantum number is independent of n, l, and ml values. The value of this number, denoted by the symbol ms, indicates the direction in which the electron is spinning.

• The ms value indicates the direction in which the electron is spinning. The electron spin quantum number can have values between +1/2 and -1/2.
• A positive value of ms denotes an upward spin on the electron, also known as spin up.
• If ms is negative, the electron in question is said to have a downward spin or spin down.
• The value of the electron spin quantum number determines whether or not the atom in question can generate a magnetic field. The value of ms can be generalized to ±½.

## Significance of Quantum Numbers

Quantum numbers are significant because they can be used to estimate an atom’s electron configuration and where its electrons are most likely to be located. The atomic radius and ionization energy of atoms, among other properties, is also determined by quantum numbers.

Each Quantum Number has its own significance which is describe in the following table:

Number

Symbol

Possible Values

Significance

Principal Quantum Number

n

1, 2, 3, 4, 5,…..

It describes the electron levels of an atom.

Azimuthal Quantum Number

l

0, 1, 2, 3,…. (n-1)

It represents the shape of the electron cloud.

Magnetic Quantum Number

ml

-l,…., -1, 0, 1,…., l

It explains the orientation of the electron cloud.

Spin Quantum Number

s

+1/2,-1/2

It tells the spin electrons can have.

## Atomic Orbital

As we know that electrons behave like waves and the position of the electron inside the atom can easily be defined with the help of the wave theory of quantum mechanics by solving the Schrodinger wave equation at the specific energy level of an atom.

These wave functions that define the position of an electron inside an atom are called atomic orbitals. These orbitals are the places that have the highest probability of finding the electron. There are four types of orbitals inside an atom

• s – orbital
• p – orbital
• d – orbital
• f – orbital

Atomic orbitals are also defined as the physical space inside the atom where the probability of finding the electron is highest.

Three other rules and principles in chemistry help us understand the position, path, orbits and energy levels of electrons in an atom, these are discussed in subsequent subsections:

### Aufbau Principle

The Aufbau principle, also called the Aufbau rule, says that electrons go into lower-energy atomic orbitals first before higher-energy ones. “Aufbau” means “building-up” in German.

#### NCERT Definition of Aufbau Principle

In the ground state of the atoms, the orbitals are filled in order of their increasing energies.

The Aufbau principle helps us figure out how electrons arrange themselves in atoms or ions. For instance, the 1s subshell gets filled up before the 2s subshell.

Here’s the order in which electrons fill up the orbitals: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, and so forth. This filling pattern applies to every atom.

For example, using the Aufbau Principle, the electronic configuration of sulfur is written as [S] = 1s2 2s2 2p6 3s2 3p4.

### Hund’s Rule of Maximum Multiplicity

Hund’s rule says that each orbital in a sublevel gets one electron before any of them get a second one. And all the electrons in these single-occupied orbitals have the same spin.

#### NCERT Definition of Hund’s Rule

Pairing of electrons in the orbitals belonging to the same subshell (p, d or f) does not take place until each orbital belonging to that subshell has got one electron each i.e., it is singly occupied.

Hund’s rule, also called the maximum multiplicity rule, can result in atoms having multiple unpaired electrons. These unpaired electrons can spin in different directions, creating magnetic moments in various directions.

Hund’s rule applies to certain molecules that have unpaired electrons.

### Pauli Exclusion Principle

The Pauli exclusion principle says that two identical particles with half-integer spins can’t be in the same quantum state within a system.

#### NCERT Definition of Pauli Exclusion Principle

No two electrons in an atom can have the same set of four quantum numbers.

OR

Only two electrons may exist in the same orbital and these electrons must have opposite spin.

In chemistry, the Pauli exclusion principle tells us that within the same atom, no two electrons can have all four of their quantum numbers identical. This means that at most, two electrons can occupy the same orbital, and they must have opposite spins.

The Pauli exclusion principle sets limits on how many electrons can be in a shell or a subshell.

## Solved Questions on Quantum Numbers

Questions 1: Find all four quantum numbers of the last electron of the Rubidium.

Solution:

Rubidium has the atomic number, Z = 37.

Electronic Configuration of Rubidium,

1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 5s1

Valence last shell electron is 5s1

Therefore,

Principal Quantum Number, n = 5,

Azimuthal Quantum Number, l = 0,

Magnetic Quantum Number, ml = 0,

Spin Quantum Number, s = +1/2

Questions 2: State the possible values of the magnetic quantum number for l = 2.

Solution:

Given that, the Azimuthal Quantum Number, l = 2

We know that,

ml = – l to + l

Therefore,

ml = -2 to +2

i.e.

m2 = -2, -1, 0, +1, +2

Questions 3: Find all four quantum numbers of the last electron of the Sodium.

Solution:

Sodium has the atomic number, Z = 11.

Electronic Configuration of Rubidium,

1s2 2s2 2p6 3s1

Valence shell last electron is 3s1

Therefore,

Principal Quantum Number, n = 3,

Azimuthal Quantum Number, l = 0,

Magnetic Quantum Number, ml = 0,

Spin Quantum Number, s = +1/2

Questions 4: State the possible values of the magnetic quantum number for l = 3.

Solution:

Given that, the Azimuthal Quantum Number, l = 3

We know that,

for l = 3,

ml = – 3 to + 3

i.e.

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

## FAQs on Quantum Numbers

### Define Quantum Numbers.

The set of numbers which are used to define the position and energy of the number of electrons in an atom are called quantum numbers.

### How many quantum numbers are there?

The four quantum numbers are:

• Principal Quantum Number (n)
• Azimuthal Quantum Number (l)
• Magnetic Quantum Number (ml)
• Electron Spin Quantum Number (s)

### Which quantum number specifies the shape of an orbital?

Azimuthal Quantum Number (l) also called Angular quantum number defines the shape of the orbital.

### Which quantum number determines the orientation of the orbital?

Magnetic Quantum Number (ml) is used to represent the orientation of the orbital in the three-dimensional space.

### How many quantum numbers are required to specify an orbital?

Three quantum numbers are required to specify the orbital of an atom which are:

• Principal Quantum Number (n)
• Azimuthal Quantum Number (l)
• Magnetic Quantum Number (ml)

### Which quantum number determines the energy of the electron?

The energy of the electron can easily be determined by using the Principal quantum number(n) and Azimuthal Quantum Number(l) of an electron.

### What is Quantum Energy?

The energy of quantum particles (i.e. very very small particles) is called quantum energy. One way to measure quantum energy is using Photon which is the smallest unit to measure light energy and energy of other electromagnetic waves.

### What is spin of an electron?

Electron spin is a quantum property of electrons. It is a shape with angular momentum. As a teaching technique, instructors compare electron spin to the planet rotating on its own axis every 24 hours. Spin-up occurs when the electron spins clockwise on its axis; spin-down occurs when the electron spins counterclockwise.

### What is Aufbau principle?

Aufbau principle is a concept in chemistry that explains how electrons fill atomic orbitals in an atom. According to this principle, electrons occupy the lowest energy orbitals available before moving to higher energy ones.

### What is Hund’s rule class 11?

Hund’s rule, often discussed in class 11 chemistry, states that electrons will occupy orbitals of the same energy level (subshell) singly before pairing up. Additionally, electrons in singly occupied orbitals will have parallel spins.

### What is Fullform of SPDF?

SPDF stands for the four sublevels or orbitals in an atom: s, p, d, and f. These letters represent different shapes and orientations of atomic orbitals where electrons are likely to be found.

• S: Sharp
• P: Principal
• D: diffuse
• F: fundamental

### Why is quantum called quantum?

Term “quantum” originates from the Latin word meaning “how much” or “quantity“. In physics, it refers to the discrete and indivisible units in which certain physical quantities, such as energy and momentum, are quantized according to quantum theory. These discrete units are fundamental to understanding the behavior of particles at the atomic and subatomic levels. Therefore, the field of quantum physics is named after the concept of quantization.

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