Maximum number of electrons in an electron shell (principal quantum number n) For a given principal quantum number n, what is the maximum number of electrons that can occupy that shell?

Introduction / Context:
Electronic configurations in atoms are organized by quantum numbers. The principal quantum number n defines a shell, which contains subshells with orbital quantum number l = 0, 1, …, n − 1. Counting the available quantum states (including spin) yields a well-known capacity formula for each shell.

Given Data / Assumptions:

• Allowed l values: 0 to n − 1.
• Magnetic quantum numbers m_l: for each l, there are (2 l + 1) orbitals.
• Each spatial orbital holds 2 electrons (spin up and spin down).

Concept / Approach:

Total orbitals in shell n: sum over l of (2 l + 1) = 1 + 3 + 5 + … + (2 n − 1) = n^2. Multiplying by spin multiplicity 2 gives the maximum number of electrons as 2 n^2. This simple counting matches observed periodic table structure (capacity 2, 8, 18, 32 … for n = 1, 2, 3, 4, respectively), though actual filling follows the aufbau order due to subshell energies.

Step-by-Step Solution:

Verification / Alternative check:

Check n = 2: 2 n^2 = 8 → 2s (2) + 2p (6) = 8. Check n = 3: 18 → 3s(2) + 3p(6) + 3d(10) = 18.

Why Other Options Are Wrong:

n^2 omits spin; 2 n is far too small; n^3 and 4 n have no basis in quantum state counting.

Common Pitfalls:

Confusing shell capacity with the aufbau filling order (4s often fills before 3d), which is an energy ordering effect, not a capacity change.

Final Answer:

2 n^2 electrons