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Class 12 · Chapter 12

Atoms

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Atoms Formula Sheet

8 formulas across 3 topics in Atoms.

1 min read

Updated 2026-07-09 · v1.0.0

Rutherford Scattering & the Nuclear Model

Rutherford's gold-foil experiment revealed that almost all of an atom's mass and all of its positive charge is concentrated in a tiny central nucleus — the scattering data itself is what pins down the nucleus's size.

r₀ = 2kZe² / (mv²)

Distance of closest approach

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Energy conservation: the alpha particle's initial kinetic energy converts entirely into electrostatic potential energy at the point of closest approach

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
r₀distance of closest approachm[L]
Zatomic number of the target nucleusdimensionless[M⁰L⁰T⁰]
m, vmass and initial speed of the incoming alpha particlekg, m/s[M], [LT⁻¹]

Valid when

  • Occurs only for a head-on (zero impact parameter) collision — this gives the SMALLEST possible r₀, an upper bound on the nucleus's size
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Worth remembering

  • Most alpha particles passed straight through the foil undeflected (proving atoms are mostly empty space), while a small fraction bounced back at large angles (proving a small, dense, positively charged nucleus)

Bohr's Model of the Hydrogen Atom

Bohr rescued the nuclear model from classical collapse by simply postulating that angular momentum is quantised — everything else (orbit radius, energy, spectral lines) follows from that single assumption.

mvr = nh/2π

Bohr's angular momentum quantisation postulate

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
nprincipal quantum number (n = 1, 2, 3, ...)dimensionless (integer)[M⁰L⁰T⁰]

Valid when

  • The foundational postulate of the entire Bohr model — later justified by de Broglie's matter-wave picture (2πr = nλ)
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r(n) = n² r₁ / Z , r₁ = 0.529 Å (n=1, Z=1)

Radius of the nth Bohr orbit

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Combining the angular momentum quantisation condition with the Coulomb force providing centripetal force

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
r(n)radius of the nth orbitm (or Å)[L]

Valid when

  • Orbit radius grows as n², so higher orbits are dramatically larger — the atom's size is set almost entirely by its outermost electron's n
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v(n) = Ze² / (2ε₀nh) = (Z/n) v₁ , v₁ ≈ 2.18 × 10⁶ m/s

Velocity of the electron in the nth Bohr orbit

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
v(n)orbital speed in the nth orbitm/s[LT⁻¹]

Valid when

  • Velocity DECREASES with increasing n (unlike radius, which increases) — outer electrons move more slowly

Common mistakes

  • Radius scales as n² while velocity scales as 1/n — opposite trends that are easy to mix up under exam pressure
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E(n) = −13.6 Z²/n² eV

Total energy of the electron in the nth Bohr orbit

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Sum of kinetic energy (½mv²) and Coulomb potential energy (−kZe²/r), evaluated using the quantised radius formula

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
E(n)total energy of the electron in orbit neV[ML²T⁻²]

Valid when

  • Negative sign indicates a BOUND state — the electron needs +13.6 Z²/n² eV of energy supplied to escape entirely (n → ∞)

Common mistakes

  • Ground state energy of hydrogen is exactly −13.6 eV — one of the most memorised numbers in the syllabus; don't forget the negative sign when computing transition energies
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hν = E(n2) − E(n1) = 13.6 Z² (1/n1² − 1/n2²) eV

Energy of a photon emitted/absorbed in an electron transition

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Conservation of energy: the photon carries away exactly the energy difference between the two orbits

Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
n1, n2principal quantum numbers of the lower and higher orbit (n2 > n1)dimensionless (integer)[M⁰L⁰T⁰]

Valid when

  • Emission when the electron falls from n2 to n1 (higher to lower); absorption for the reverse transition
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The number of distinct spectral lines emitted when an electron de-excites from level n to level 1 is n(n−1)/2 — a very common counting question that pairs with the energy transition formula.

Spectral Series of Hydrogen

Every possible electron transition in hydrogen falls into one of a few named series, depending only on which orbit the electron lands in — the Rydberg formula generates every observed spectral line from one equation.

1/λ = RZ² (1/n1² − 1/n2²)

Rydberg formula for spectral lines

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Variables used in this formula, with units and dimensions
SymbolMeaningUnitDimension
RRydberg constantR ≈ 1.097 × 10⁷ m⁻¹m⁻¹[L⁻¹]

Valid when

  • Lyman series: n1=1 (UV); Balmer series: n1=2 (visible); Paschen series: n1=3 (IR) — n2 > n1 in every case

Common mistakes

  • Only the Balmer series has lines in the visible range — Lyman is entirely UV and Paschen (and beyond) is entirely infrared, a frequently tested fact
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Series limit (shortest λ, n2 → ∞): 1/λ = RZ²/n1²

Series limit of a spectral series

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Setting n2 → ∞ in the Rydberg formula — the electron transitions from just barely bound (ionisation threshold) down to n1

Valid when

  • Gives the shortest wavelength (highest energy) line possible in that series — the point where lines converge and stop
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Previous chapter11. Dual Nature of Radiation & MatterNext chapter 13. Nuclei
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