2.2 Quantum Numbers

This topic explains the four quantum numbers that describe the location, energy, and behavior of electrons in an atom. It covers the principal, azimuthal, magnetic, and spin quantum numbers.

Definition

Quantum Numbers: A set of four numbers assigned to each electron in an atom to specify its exact location and energy.

Main Idea

The approach of expressing the presence of electrons around the nucleus in a digital/mathematical way gave rise to the concept of quantum numbers. A set of four numerals, called quantum numbers, are assigned to each electron present in an atom. No matter how big the electronic cloud in an atom is, the specific set of quantum numbers is enough to specify the exact location of an electron in an atom.

Core Concept

• Energy of the electron
• Shape of the orbital
• Orientation of the orbital in space
• Spin of the electron

No two electrons in an atom can have the same set of all four quantum numbers (Pauli's exclusion principle).

Detailed Explanation

1. Principal Quantum Number (n)

• This quantum number is the expression of Bohr's shell in quantum form.
• Represented by n.
• Possible values: n = 1, 2, 3, …
• Depicts the period number in the periodic table, because each period starts with a new shell.
• Represents Bohr's shell.
• If n is greater, the energy of the shell is greater, and so is its size.
• Arose from the emission spectrum of hydrogen, which was first explained by Bohr and Max Planck.
• Each spectral line in the emission spectrum is explained by the possible jumps within shells of H atoms.

2. Azimuthal Quantum Number (l)

• Denoted by l.
• Describes the shapes of orbitals of electronic shells.
• An orbital is a space around the nucleus where the probability of finding the electron is maximum.
• One shell may comprise more than one subshell or orbital.
• Determines the orbital's angular momentum of the electron.
• Specifies the subshell within a given principal energy level (n).
• Values: l = 0, 1, 2, 3, …, n-1
• s, p, d, f orbitals represent the blocks in the periodic table.
• The presence of subshells is related to the fine spectral lines in the spectrum.
• Number of subshells in a shell depends upon the shell number — bigger the shell, more subshells and more electrons can be accommodated.

3. Magnetic Quantum Number (m)

• The splitting of the fine spectral lines under the influence of a magnetic field can be explained on the basis of the magnetic quantum number.
• Also called the orientation quantum number.
• Denoted by m.
• Possible values: m = -l, …, -2, -1, 0, +1, +2, …, +l
• For s-orbital (l = 0): m = 0 (only one value)
• s-orbital remains spherically symmetrical under the influence of a magnetic field.
• For p-orbital (l = 1): m = -1, 0, +1 (three values)
• These three values show that the p-orbital splits into three degenerate orbitals: p_x, p_y, p_z, oriented on x, y, and z axes respectively.
• For d-orbital (l = 2): m = -2, -1, 0, +1, +2 (five values)
• Five orbitals oriented at different positions in space: d_xy, d_yz, d_xz, d_x²-y², d_z²
• For f-orbital (l = 3): m = -3, -2, -1, 0, +1, +2, +3 (seven values)

4. Spin Quantum Number (s)

• Maximum two electrons can reside in one orbital formed by the magnetic quantum number.
• These two electrons must have opposite spin.
• If one spins clockwise, the other's spin will be anti-clockwise.
• Goudsmit in 1925 proved that an electron also spins about its axis — this is called self-rotation and causes a spin magnetic field around it.
• To reside in one orbital, two electrons must cancel each other's spin magnetic field — they have opposite spins.
• Clockwise spin: represented by +12, symbolised by ↑
• Anticlockwise spin: represented by -12, symbolised by ↓

Table of Possible Quantum Numbers

Intermediate Level

Principal Quantum Number (n)

• Shell number.
• n = 1, 2, 3, …
• Energy and size increase with n.
• Also gives the period number.

Azimuthal Quantum Number (l)

• Subshell.
• l = 0, 1, 2, …, n-1
• s, p, d, f orbitals.
• Determines the shape of the orbital.

Magnetic Quantum Number (m)

• Orientation.
• m = -l, …, 0, …, +l
• Determines the number of orbitals in a subshell.

Spin Quantum Number (s)

• Spin.
• +12 \) or \( -12
• Two electrons in the same orbital must have opposite spins.

Number of Orbitals and Electrons

Important Points

• 1. Principal quantum number (n): Shell, energy, size, period number.
• 2. Azimuthal quantum number (l): Shape, subshell, block in periodic table.
• 3. Magnetic quantum number (m): Orientation, number of orbitals.
• 4. Spin quantum number (s): Spin, +12 or -12.
• 5. No two electrons can have the same four quantum numbers.
• 6. s: 1 orbital, 2 electrons; p: 3 orbitals, 6 electrons; d: 5 orbitals, 10 electrons; f: 7 orbitals, 14 electrons.
• 7. Emission spectra help determine electronic configuration.

Remember

• - n: Shell
• - l: Shape
• - m: Orientation
• - s: Spin
• - s orbital: 1 orbital, 2 electrons
• - p orbital: 3 orbitals, 6 electrons
• - d orbital: 5 orbitals, 10 electrons
• - f orbital: 7 orbitals, 14 electrons

Common Mistakes

• 1. Confusing n and l: n = shell; l = subshell.
• 2. **Forgetting the range of m:** m = -l to +l.
• 3. Thinking two electrons can have the same spin in the same orbital: They must have opposite spins.
• 4. Mixing up the number of orbitals: s=1, p=3, d=5, f=7.
• 5. Forgetting that quantum numbers determine electronic configuration: They are essential for writing electron configurations.

Quick Revision

• Principal quantum number (n): Shell, energy, size.
• Azimuthal quantum number (l): Shape, subshell.
• Magnetic quantum number (m): Orientation.
• Spin quantum number (s): Spin.
• No two electrons can have the same four quantum numbers.
• s: 1 orbital, 2 electrons; p: 3 orbitals, 6 electrons; d: 5 orbitals, 10 electrons; f: 7 orbitals, 14 electrons.
• Emission spectra help determine electronic configuration.

60-Second Revision

• n: Shell
• l: Shape
• m: Orientation
• s: Spin
• s orbital: 1 orbital, 2 electrons
• p orbital: 3 orbitals, 6 electrons
• d orbital: 5 orbitals, 10 electrons
• f orbital: 7 orbitals, 14 electrons

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