Quantum Numbers

Quantum numbers in Chemistry, are the sets of numbers that describe an electron’s trajectory 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.

Let’s learn about all the quantum numbers in detail in this article.

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 

The four-quantum number totally defines an electron in an atom or provides complete knowledge of an electron in an atom. The four quantum numbers are:

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

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 (m_l)

The magnetic quantum number determines the total number of orbitals in a subshell as well as their orientation. It is represented by the symbol ‘m_l.’ 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 m_l falls between -l and +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 m_l values. The value of this number, denoted by the symbol m_s, indicates the direction in which the electron is spinning. 

• The m_s 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 m_s denotes an upward spin on the electron, also known as spin up. 
• If m_s 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 m_s 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:

• Principal Quantum Number (n) describes the electron levels of an atom.
• Azimuthal Quantum Number (l) represents the shape of the electron cloud. 
• Magnetic Quantum Number (m_l) explains the orientation of the electron cloud.
• Spin Quantum Number (s) 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.

Read, More

• Electronic Configuration of Elements
• Filling of Orbitals in Atom
• Shapes of Atomic Orbitals

Solved Examples on Quantum Numbers

Example 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

Vlence last shell electron is 5s¹

Therefore, 

Principal Quantum Number, n = 5,

Azimuthal Quantum Number, l = 0,

Magnetic Quantum Number, m_l = 0,

Spin Quantum Number, s = +1/2

Example 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, 

m_l = – l to + l 

Therefore, 

m_l = -2 to +2

i.e. 

m₂ = -2, -1, 0, +1, +2

Example 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

Vlence shell last electron is 3s¹

Therefore, 

Principal Quantum Number, n = 3,

Azimuthal Quantum Number, l = 0,

Magnetic Quantum Number, m_l = 0, 

Spin Quantum Number, s = +1/2

Example 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,

m_l = – 3 to + 3

i.e.

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

FAQs on Quantum Numbers

Question 1: Define Quantum Numbers.

Answer:

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.

Question 2: How many quantum numbers are there?

Answer:

The four quantum numbers are,

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

Question 3: Which quantum number specifies the shape of an orbital?

Answer:

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

Question 4: Which quantum number determines the orientation of the orbital?

Answer:

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

Question 5: How many quantum numbers are required to specify an orbital?

Answer:

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 (m_l)

Question 6: Which quantum number determines the energy of the electron?

Answer:

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

Question 7: What is Quantum Energy?

Answer:

The enegry 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. 

Question 8: What is the spin of an electron?

Answer:

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.