“n plus l, then n”
The order in which orbitals fill
Lower n + l fills first. If two are equal, the one with the smaller n goes first. That single rule puts 4s before 3d without memorising the diagonal chart.
Chemistry, Class 11 and 12 · Gujarat Board, CBSE, GUJCET, JEE and NEET
Atomic Structure formulas
Atomic structure is where chemistry borrows from physics. The Bohr results and the quantum numbers carry the marks, and the rules for filling orbitals are asked in every board paper in one form or another.
19 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Waves and photons
| What | Formula | Symbols and notes |
|---|---|---|
| Wave relation | c = νλ | c = 3 × 10⁸ m s−1 |
| Photon energy | E = hν = hc/λ | h = 6.626 × 10−34 J s |
| Photoelectric effect | hν = hν₀ + ½mv² | ν₀ is the threshold frequency |
| de Broglie wavelength | λ = h/mv | every moving particle has one |
| Heisenberg uncertainty | Δx · Δp ≥ h/4π | position and momentum cannot both be exact |
The Bohr atom
| What | Formula | Symbols and notes |
|---|---|---|
| Energy of the nth level | En = −13.6Z²/n² eV | for hydrogen-like species only |
| Radius of the nth orbit | rn = 0.529n²/Z Å | grows as n² |
| Angular momentum | mvr = nh/2π | this quantisation is Bohr's postulate |
| Rydberg equation | 1/λ = R(1/n₁² − 1/n₂²) | R = 1.097 × 10⁷ m−1 |
| Number of spectral lines | n(n − 1)/2 | from the nth level down to the ground state |
Quantum numbers and filling
| What | Formula | Symbols and notes |
|---|---|---|
| Principal, n | shell number, 1, 2, 3, … | decides size and most of the energy |
| Azimuthal, l | 0 to n − 1, shapes s, p, d, f | decides the shape of the orbital |
| Magnetic, m | −l to +l | gives 2l + 1 orbitals in a subshell |
| Spin, s | +½ or −½ | two electrons per orbital, opposite spins |
| Electrons in a shell | 2n² | 2, 8, 18, 32 for the first four shells |
| Electrons in a subshell | 2(2l + 1) | s 2, p 6, d 10, f 14 |
| Aufbau order by (n + l) | lower n + l fills first; on a tie, lower n | this is why 4s fills before 3d |
| Hund's rule | fill singly before pairing | all with parallel spins |
| Pauli exclusion | no two electrons share all four quantum numbers | hence two per orbital |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
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Memory hooks
How to remember this chapter.
“Bus seats”
Hund's rule
People take empty seats on a bus before sitting next to someone. Electrons fill each orbital of a subshell singly before any pairs up.
“Half and full are happiest”
Chromium and copper
Cr is 3d⁵4s¹ and Cu is 3d¹⁰4s¹, not the expected 3d⁴4s² and 3d⁹4s², because half-filled and fully filled subshells are extra stable.
Where marks go
The mistakes that cost the most.
- Using the Bohr formulas for a multi-electron atom. They apply to hydrogen and hydrogen-like ions only.
- Writing the ground state of chromium and copper as the Aufbau order predicts.
- Forgetting that 4s fills before 3d but empties first when the ion forms.
- Giving l a value equal to n. It runs from 0 to n − 1.
- Treating the uncertainty principle as a limit of the instrument. It is a property of nature.
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