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Class XII · Chapter 12

Atoms

How a gold foil experiment revealed the nucleus, and how Bohr patched classical physics just enough to explain why atoms don't collapse — and why hydrogen glows in very specific colours.

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1Alpha-Particle Scattering and Rutherford's Nuclear Model

Rutherford's team fired fast, positively-charged alpha particles at an extremely thin gold foil, expecting the older "plum pudding" model (positive charge spread evenly through the atom) to deflect them only slightly. Instead, most particles passed straight through — but a small fraction bounced back at large angles, some almost directly backward.

Rutherford's Own Words, Paraphrased It was as surprising as firing a shell at tissue paper and having it bounce back at you — the only way to explain such sharp deflections was if almost all of the atom's mass and positive charge were concentrated in an incredibly tiny, dense region: the nucleus.

This led to the nuclear model: a tiny, massive, positively-charged nucleus at the centre, with electrons orbiting around it at a comparatively vast distance — an atom is mostly empty space.

2Why Rutherford's Atom Couldn't Be Stable

Classical electromagnetism creates an immediate problem for this picture: an orbiting electron is constantly accelerating (changing direction), and accelerating charges radiate energy (Chapter 8). An electron losing energy this way should spiral into the nucleus in a fraction of a second — meaning, by classical physics, atoms shouldn't exist at all. Something in classical theory had to give, and that something was quantization.

3Atomic Spectra

Heated gases emit light only at specific, sharply-defined wavelengths — a line spectrum, not a continuous rainbow. Each element produces its own unique fingerprint of lines, and hydrogen's pattern, in particular, follows a strikingly simple mathematical rule that any successful atomic model would need to explain.

4Bohr's Postulates

Niels Bohr rescued Rutherford's model by imposing quantum rules on top of it — rules with no classical justification at the time, but which worked.

  1. Electrons occupy specific stationary orbits in which they do not radiate energy, despite classical theory predicting they should.
  2. Only orbits where the electron's angular momentum is an integer multiple of h/2π are allowed:
    Angular Momentum Quantization
    L = n (h / 2π), n = 1, 2, 3, ...
  3. An electron emits or absorbs a photon only when it jumps between orbits, with the photon's energy exactly matching the energy difference:
    Transition Condition
    hν = Ei − Ef

5Radius and Energy of Bohr Orbits

Combining these postulates with Coulomb's Law (Chapter 1) providing the centripetal force gives clean, closed-form expressions for hydrogen's orbits:

Orbit Radius
rn = n² a₀, where a₀ ≈ 0.529 Å
a₀ is the Bohr radius — the radius of the smallest (n=1, ground state) orbit. Radii grow as n².
Orbit Energy
En = −13.6 / n² eV
Negative because the electron is bound to the nucleus; n = 1 (ground state) has the most negative, lowest energy. As n → ∞, En → 0, corresponding to a free (ionised) electron.

6Hydrogen Spectral Series

Every spectral line corresponds to an electron transition between two specific orbits, and the wavelength follows the Rydberg formula:

Rydberg Formula
1/λ = R (1/nf² − 1/ni²)
R = Rydberg constant ≈ 1.097×10⁷ m⁻¹, ni > nf for emission.
  • Lyman series (transitions to n=1): ultraviolet.
  • Balmer series (transitions to n=2): visible light — the only series you can actually see, historically discovered first.
  • Paschen, Brackett, Pfund series (transitions to n=3, 4, 5): infrared.

7De Broglie's Explanation of Bohr's Quantization

Bohr's second postulate (quantized angular momentum) looked arbitrary until de Broglie's matter-wave hypothesis (Chapter 11) gave it a physical justification: an electron's orbit is only stable if it forms a standing wave around the nucleus — meaning the orbit's circumference must be an exact whole number of de Broglie wavelengths.

Standing-Wave Condition
2π rn = n λ
Substituting λ = h/mv (de Broglie) directly reproduces Bohr's angular momentum quantization, L = nh/2π — turning what looked like an arbitrary rule into a natural consequence of matter behaving as a wave.

Formula Summary

Angular Momentum
L = nh/2π
Transition Energy
hν = E_i − E_f
Orbit Radius
r_n = n²a₀
Orbit Energy
E_n = −13.6/n² eV
Rydberg Formula
1/λ = R(1/n_f²−1/n_i²)
Standing Wave Condition
2πr_n = nλ

Solved Examples

Example 1 · Radius and Energy of an Orbit

Find the radius and energy of the second Bohr orbit (n=2) of a hydrogen atom.

Solution: r₂ = n²a₀ = 4 × 0.529 Å = 2.116 Å.

E₂ = −13.6/n² = −13.6/4 = −3.4 eV.

Example 2 · Balmer Series Wavelength

Find the wavelength of the Hα line of the Balmer series (transition from n=3 to n=2).

Solution: 1/λ = R(1/2² − 1/3²) = 1.097×10⁷ × (0.25 − 0.1111) = 1.097×10⁷ × 0.1389 ≈ 1.524×10⁶ m⁻¹.

λ ≈ 1/(1.524×10⁶) ≈ 6.56×10⁻⁷ m = 656 nm — the well-known red Hα line, visible in hydrogen discharge tubes and stellar spectra alike.

Quick Check

1. According to Bohr's second postulate, the electron's angular momentum is quantized in integer multiples of:
h
✓ h/2π (correct)
2πh
h²
2. The Balmer series of hydrogen's spectrum lies in which region?
Ultraviolet
✓ Visible (correct)
Infrared
X-ray
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Dual Nature of Radiation and Matter

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Nuclei

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