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What are quantum numbers?

Class 11 ChemistryNCERT chapter: Structure of Atomइसे हिंदी में पढ़ें

Exam-ready answer

Quantum numbers are a set of four numbers that describe an electron in an atom completely. The principal quantum number (n) gives the shell, size and energy; the azimuthal quantum number (l) gives the subshell and shape of the orbital; the magnetic quantum number (ml) gives its orientation in space; and the spin quantum number (ms) gives the electron's spin, +½ or −½.

Explanation

Solving the Schrödinger equation for the hydrogen atom gives many allowed wave functions called atomic orbitals. Each orbital is labelled by three quantum numbers: n, l and ml. The principal quantum number n can be 1, 2, 3 and so on. It fixes the shell (K, L, M, N…), the size of the orbital and, for hydrogen, its energy. As n increases, the orbital becomes larger and the electron sits farther from the nucleus.

The azimuthal (orbital angular momentum) quantum number l can take values from 0 to n − 1. It tells the subshell and the shape: l = 0 is s (spherical), l = 1 is p (dumb-bell), l = 2 is d and l = 3 is f. The magnetic quantum number ml takes every integer value from −l to +l, so a subshell has (2l + 1) orbitals: one s, three p, five d and seven f orbitals. A shell n has n² orbitals in total.

The first three numbers describe the orbital; the fourth describes the electron itself. An electron spins on its own axis, and the spin quantum number ms can be +½ or −½ for the two possible spin directions. By Pauli's exclusion principle, no two electrons in an atom can have the same set of all four quantum numbers. So one orbital holds at most two electrons, and they must have opposite spins.

Points to remember

  1. Principal quantum number (n = 1, 2, 3…): identifies the shell and decides the size and (largely) the energy of the orbital.
  2. Azimuthal quantum number (l = 0 to n − 1): identifies the subshell and the shape of the orbital (s, p, d, f for l = 0, 1, 2, 3).
  3. Magnetic quantum number (ml = −l to +l): gives the orientation of the orbital; a subshell has 2l + 1 orbitals.
  4. Spin quantum number (ms = +½ or −½): gives the spin orientation of the electron.
  5. Pauli exclusion principle: no two electrons in an atom can have all four quantum numbers the same; an orbital holds at most 2 electrons.

Exam tip

Make a small table with each quantum number, its symbol, its allowed values and what it tells (shell, shape, orientation, spin). Add the rule l = 0 to n − 1 and ml = −l to +l; examiners often ask for allowed sets of values.

Questions students also ask

How many orbitals and electrons can a shell hold?
A shell with principal quantum number n has n² orbitals. Since each orbital holds a maximum of two electrons with opposite spins, the shell can hold 2n² electrons. For example, the L shell (n = 2) has 4 orbitals and up to 8 electrons.
Write the quantum numbers of a 3p orbital.
For a 3p orbital, n = 3 and l = 1 (because p means l = 1). The magnetic quantum number ml can be −1, 0 or +1, which gives the three 3p orbitals. An electron in any of them can have ms = +½ or −½.
Why is a 2d orbital not possible?
For n = 2, the azimuthal quantum number l can only be 0 or 1, because l ranges from 0 to n − 1. A d orbital needs l = 2, which is not allowed when n = 2. So the 2d orbital does not exist; d orbitals start from n = 3.

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