AtomsNEET Physics · Class 12 · NCERT Chapter 12

Introduction

How do you study something a billion times smaller than a hair? You throw something at it and watch what bounces. Rutherford did exactly that with alpha particles in 1911 and discovered the nucleus. Bohr (1913) built a quantum model of the simplest atom (hydrogen) that explained its line spectrum. Wave mechanics took over later, but Bohr's formulas still pass NEET reliably.

Expect 1 to 2 NEET questions every year. Common asks: Bohr radius and velocity, energy of n-th orbit, ionisation energy, Rydberg formula, hydrogen series identification, and the angular momentum quantisation L = n h-bar.

Rutherford alpha scattering

Geiger and Marsden (1909) fired alpha particles at a thin gold foil. Three observations:

• Most alphas passed straight through.
• A small fraction scattered at large angles.
• Roughly 1 in 8000 backscattered (deflected by more than 90°).

This was impossible if the atom's positive charge was spread out (Thomson's plum pudding model). Rutherford concluded that the positive charge and nearly all the mass are concentrated in a tiny dense nucleus, with electrons orbiting at large distances. Atom is mostly empty space.

Rutherford fired alpha particles at a thin foil. Most went through; a few bounced back, proving a tiny dense nucleus. Closest head-on approach: r_0 = k · 2 Z e² / KE.

Distance of closest approach

For a head-on alpha, all kinetic energy converts to electrostatic PE at the closest distance r_0:

For 5 MeV alphas on gold (Z = 79): r_0 ≈ 45 fm. The nucleus is much smaller, so the alpha never actually touches it (only the strong force at 1-2 fm matters).

Limitations of Rutherford's model

A revolving electron is accelerated, so by classical electromagnetism it must radiate energy and spiral into the nucleus in less than 10⁻⁸ seconds. It does not. Also, atoms emit only specific wavelengths (line spectra), not a continuous range. Rutherford's model could not explain either fact.

Bohr postulates (1913)

1. Electrons move only in certain stationary circular orbits without radiating energy.
2. The angular momentum is quantised: $L=mvr=n\hslash$ with n = 1, 2, 3, …
3. When an electron jumps from a higher orbit to a lower one, it emits a photon of energy$h\nu =E_{n_2}-E_{n_1}$.

Postulates 1 and 3 are radical breaks from classical physics. Postulate 2 (the quantisation rule) later emerged naturally from de Broglie's standing-wave picture.

Bohr radius and velocity in n-th orbit

Combine Coulomb attraction with the angular momentum rule and Newton's second law for circular motion:

Bohr's n-th orbit: radius scales as n²/Z, velocity as Z/n, energy as -Z²/n². For ground state of hydrogen (n = 1, Z = 1), r = 0.529 Å, E = -13.6 eV.

Energy levels

Total energy = KE + PE. Using the orbit relations:

Negative E means bound. n = 1 is the ground state. n = ∞ corresponds to a free (just unbound) electron with E = 0. KE_n = +13.6 Z² / n² eV; PE_n = -27.2 Z² / n² eV; total = -KE = E_n.

Click two levels to see the photon emitted on the transition. ΔE = E_high - E_low → λ = 1240 / ΔE (nm).

Hydrogen spectrum

Electron transitions between levels emit (or absorb) photons. Five named series:

• Lyman (n_1 = 1, ultraviolet)
• Balmer (n_1 = 2, visible: H-α 656 nm, H-β 486 nm, H-γ 434 nm, H-δ 410 nm)
• Paschen (n_1 = 3, infrared)
• Brackett (n_1 = 4, IR)
• Pfund (n_1 = 5, far IR)

Click any series to see its lines. Lyman is UV, Balmer is visible (the only series we can see directly with our eyes), Paschen and onwards are IR.

Rydberg formula

Combining the energy formula with E = h c / λ:

Series limit (n_2 → ∞): $1/\lambda =R{Z}^2/n_1^2$.

Pick n_1 (the lower level) and n_2 (the higher level). The Rydberg formula gives the wavelength of the photon emitted when the electron jumps from n_2 down to n_1.

Excitation and ionisation

Excitation: the electron jumps from a lower to a higher level by absorbing a photon (or by collision with another particle of enough energy). The minimum energy to lift it from ground (n = 1) to n = 2 is the first excitation energy. For hydrogen: 10.2 eV.

Ionisation: the electron is freed completely (n → ∞). Ionisation energy = -E_1.

For hydrogen: 13.6 eV. For He+: 54.4 eV. For Li²+: 122.4 eV.

Excitation lifts the electron to a higher level. Ionisation removes it (n → ∞). For hydrogen, ionisation energy = 13.6 eV; first excitation (1 → 2) = 10.2 eV.

de Broglie picture of Bohr orbits

de Broglie (1924) interpreted the Bohr orbits as standing waves of an electron. The orbit circumference must fit a whole number of wavelengths:

That is exactly Bohr's quantisation rule. The electron forms a stable matter-wave pattern only on the permitted orbits.

de Broglie said: an electron in orbit is a standing wave. For the wave to fit, the orbit circumference must be a whole number of wavelengths: 2 π r = n λ. Plug λ = h / m v: m v r = n ℏ. That is exactly Bohr's postulate.

Limitations of Bohr's model

• Works only for hydrogen and hydrogen-like ions (one electron). Fails for helium and beyond.
• Cannot explain the relative intensities of spectral lines.
• Cannot explain fine structure, Zeeman effect, or hyperfine splitting.
• Mixes classical orbits with quantum rules in an ad-hoc way; replaced by full quantum mechanics later.

Bonus: spectral lines counter

When an electron in level n returns to ground, it can emit any combination of the lines connecting n down to 1. Total possible transitions:

When an electron de-excites from level n, it can take any path back. Total possible spectral lines: n(n - 1) / 2.

Worked NEET problems

Summary cheat sheet

• Closest approach: $r_0=k(2Z{e}^2)/E_k$.
• Bohr quantisation: $L=n\hslash$.
• Bohr radius: $r_n=n^2{a}_0/Z,a_0=0.529\text{A˚}$.
• Velocity: $v_n=(Z/n)(c/137)$.
• Energy: $E_n=-13.6Z^2/n^2\text{eV}$, KE = -E, PE = 2E.
• Rydberg: $1/\lambda =R{Z}^2(1/n_1^2-1/n_2^2)$.
• Series: Lyman (UV, n_1 = 1), Balmer (visible, n_1 = 2), Paschen, Brackett, Pfund (IR).
• Ionisation H: 13.6 eV. First excitation: 10.2 eV.
• de Broglie: $2\pi r=n\lambda \iff L=n\hslash$.
• Lines from n: n(n - 1)/2.

Next: try the interactive widgets for Bohr orbits, hydrogen spectrum and ionisation, or work through the 30 NEET PYQs with full solutions. To time yourself, take the free 10-question mock test.