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What are Quantum Numbers? Detailed Explanation

The route and mobility of an electron in an atom can be described by using quantum numbers. When all quantum numbers of the electrons in an atom are added together, the Schrodinger equation must be satisfied. They define the values of a quantum system’s preserved quantities. They can be described as a set of numerical values that provide Schrodinger wave equation solutions for hydrogen atoms.

What are Quantum numbers?

Quantum numbers are a set of numbers that describe the position and energy of an electron in an atom. Or they can also be defined as the values which are used to describe the available energy levels to the atoms and molecules. To describe the state of an electron in an atom or ion, there are four quantum numbers.

  • Principal quantum number (n): Gives information about energy level.
  • Azimuthal quantum number (l): Provides information about subshell and angular momentum.
  • Magnetic quantum number (ml): Describes the orbital of subshell and magnetic moment.
  • Spin quantum number (ms): Describes the spin of electron.

Here, before moving towards details, it is necessary to understand the way in which electrons get arranged in subshells.

What are Quantum Numbers? Detailed Explanation

What is Principal quantum number?

The symbol of principal quantum number is n and it designates the electron shell of an atom. It also tells about the maximum distance between the nucleus and electrons. Its value is always an integer and can’t be zero or negative. This thing indicates that it is impossible for an atom to have negative number of principal shells or zero shell. Its value must be equal to or greater than 1.

n = 1, 2, 3, ………

If n = 1, it indicates that electron is in lowest energy level or in ground state.

What are Quantum Numbers? Detailed Explanation

If an electron jumps to higher energy level from ground state by gaining energy, then value of n increases and if electron returns back to ground level by emitting energy, then n decreases.

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What is azimuthal quantum number?

It is also known as orbital angular momentum quantum number and is indicated by l. It describes the shape of an orbital. Its value depends upon the total number of angular nodes in an orbital. Its value is in range of 0 to (n-1), where n is principal quantum number, so clearly, azimuthal quantum number is dependent on principal quantum number.

What are Quantum Numbers? Detailed Explanation

For example, if n = 3, then possible values of l are 0, 1 and 2. Azimuthal quantum number is related to angular momentum by the following relation:

L2 = ℏ2l (l + 1)

Under this condition, when l = 0, then electron is in s- subshell. For l = 1, 2, electron will reside in p and d subshells respectively. Hence for n = 3, there are 3 possible subshells, 3s, 3p and 3d.

What is magnetic quantum number?

This quantum number is indicated by ml and it indicates number of orbitals present in a subshell. Furthermore, orientation of these atomic orbitals is also determined by it. Its value depends upon azimuthal quantum number. Since the value of ml has a range of ( – l  to + l ), so indirectly it is also dependent on “n”.

For example, if n = 4, then clearly l = 3, then the possible values of ml are ( -3, -2, -1, 0, 1, 2, 3 ). Total number of orbitals of a subshell can be found by the formula (2l + 1). As in this example, l = 3 so number of possible ml values are 7.

Relation of magnetic quantum number with angular momentum is as:

Lz = ml

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What is spin quantum number?

Electron spin quantum number is independent of previous quantum number values: n, l, and ml. The spin quantum number is indicated by ms and tells the direction of an electron in which it is spinning. Its possible values can be ( +1/2 and -1/2 ). +1/2 indicates that electron has an upward spin and it is shown by ↑. Negative ½ means spin down or downward spin and is indicated by ↓. Ability of an atom to produce magnetic field is also determined by this quantum number.

What are Quantum Numbers? Detailed Explanation

In an orbital, electrons must have opposite spins due to Pauli Exclusion Principle, that’s why an orbital can never contain more than two electrons. Spin quantum number formula is gien as:

Sz = ms

Summary

What are Quantum Numbers? Detailed Explanation

An integer that indicates the number of the shell of the electron is the principal quantum number. Its value is 1 or higher (never 0 or negative). The angular momentum quantum number is an integer that is the orbital value of the electron (e.g., s=0, p=1). L is greater than or equal to zero and less than or equal to n-1. With integer values ranging from -l to l, the magnetic quantum number is the orbital orientation. Therefore, for the orbital p, where l=1, m could have values of -1, 0 , 1. The quantum number of the spin is a half-integer value that is either -1/2 (known as ‘spin down’) or 1/2 (known as ‘spin up’).

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