Quantum NumbersJEE Main

Concept

Four quantum numbers label every electron. $n$ (shell) sets the size and energy, $l$ (subshell) the shape, $m_l$ the orientation, and $m_s$ the spin. Their allowed combinations fix exactly how many electrons each shell and subshell can hold.

Key formula

Derivation

For a given $n$ there are $n$ subshells ($l=0$ to $n-1$). Each subshell has $2l+1$ orbitals (the $m_l$ values), and each orbital holds 2 electrons of opposite spin — so a subshell holds $2(2l+1)$.

Summing $\sum_{l=0}^{n-1} 2(2l+1) = 2n^2$ gives the shell capacity: 2, 8, 18, 32 for $n=1,2,3,4$.

Scenarios to explore

• Quantum Numbers — Shells, subshells, orbitals and electron capacity.

Real-world applications

• Building electron configurations via the Aufbau principle.
• Explaining the shape of the periodic table (s, p, d, f blocks).
• Predicting spectral lines and magnetic behaviour.

JEE exam tips

• Number of orbitals in a shell $= n^2$; electrons $= 2n^2$.
• Subshell electron caps: s=2, p=6, d=10, f=14.
• No two electrons in an atom share all four quantum numbers (Pauli).

Common mistakes

• Allowing $l = n$ — the maximum is $n-1$.
• Giving $m_l$ a range wider than $-l$ to $+l$.
• Forgetting the factor of 2 from spin when counting electrons.

Exam traps to avoid

• The 3rd shell holds 18, but the 3d subshell fills only after 4s (energy order).
• $m_l$ counts orbitals, not electrons — multiply by 2 for electrons.