This chapter moves from the atom's outer electron structure to its innermost nucleus, covering two genuinely separate topics that happen to share a chapter — how electrons are arranged and how nuclei decay.
Bohr's model of the atom took Rutherford's nuclear picture and fixed its fatal flaw — classically, an orbiting electron should continuously radiate energy and spiral into the nucleus within a fraction of a second, yet atoms are stable. Bohr simply postulated that electrons can only occupy certain fixed orbits, and the resulting model correctly predicted the hydrogen spectrum's exact wavelengths, one of the first great successes of early quantum theory. The nuclear physics half of the chapter then shifts entirely to a different question: why some nuclei are unstable, and the precise mathematical law governing how they decay.
Core formulae for atomic and nuclear physics
| Quantity | Formula |
|---|---|
| Bohr radius of nth orbit | rn = n²r₁, r₁ = 0.529 Å |
| Energy of nth orbit (hydrogen) | En = −13.6/n² eV |
| Rydberg formula | 1/λ = R(1/n₁² − 1/n₂²) |
| Radioactive decay law | N = N₀e−λt |
| Half-life | t½ = 0.693/λ |
| Mean life | τ = 1/λ |
| Mass-energy equivalence | E = mc² |
| Binding energy | BE = Δmc² |
| Nuclear radius | R = R₀A1/3 |
Topics covered
- Rutherford's alpha-scattering experiment — and what it revealed about the atom's structure — a small, dense, positively charged nucleus
- Bohr's postulates — quantised angular momentum, and stable orbits with no radiation
- Energy levels and the hydrogen spectrum — the Lyman, Balmer and Paschen series, and their origin as electron transitions between orbits
- Nuclear structure — protons, neutrons, mass number and atomic number, and the notation used to describe a nucleus
- Nuclear size and density — and why nuclear density is essentially the same for every nucleus, regardless of size
- Mass defect and binding energy — why a nucleus weighs less than the sum of its separate parts, and where that missing mass goes
- Binding energy per nucleon curve — and what it reveals about nuclear stability, fission and fusion
- Radioactivity — alpha, beta and gamma decay, and the decay law with half-life and mean life
- Nuclear fission and fusion — the two opposite processes that release energy from the binding energy curve
Why this chapter matters
In NEET and JEE, Bohr model energy-level and spectral-line numericals, plus radioactive decay law problems involving half-life, are among the most consistently repeated question types across years. Binding energy per nucleon and mass defect calculations are equally reliable, often combined with a conceptual question about nuclear stability.
For Class 12 boards, expect a derivation of the Bohr model's energy expression and a numerical on radioactive decay using the half-life relation.
Mistakes students make in this chapter
- Forgetting hydrogen spectrum energy levels are negative. En = −13.6/n² eV is negative because the electron is bound to the nucleus; a transition to a higher n increases (makes less negative) the energy, absorbing a photon, while a transition to lower n releases one.
- Confusing half-life with mean life. They're related but not equal — mean life (τ = 1/λ) is always slightly longer than half-life (t½ = 0.693/λ), and using one where the problem specifies the other gives a wrong numerical answer.
- Assuming the amount of radioactive material remaining after two half-lives is zero. Radioactive decay is exponential, not linear — after each half-life exactly half of whatever remains decays, so a small nonzero amount always mathematically remains, however long you wait.
- Believing binding energy per nucleon increases indefinitely with mass number. It actually rises, peaks around iron (mass number near 56), and then decreases for heavier nuclei — this peak is precisely why both fission of heavy nuclei and fusion of light nuclei release energy.
- Mixing up mass defect and binding energy. Mass defect is the missing mass itself (Δm); binding energy is the energy equivalent of that missing mass (Δmc²) — they're directly related but numerically different quantities with different units.
Worked example: the red line in the hydrogen spectrum
- Write the Bohr energy levels. The energy is negative because the electron is bound, and it rises towards zero as n grows:Eₙ = −13.6 / n² eV
- Evaluate both levels.E₃ = −13.6/9 = −1.51 eVE₂ = −13.6/4 = −3.40 eV
- Take the difference. The photon carries away exactly the energy the electron loses — subtract in the order that leaves a positive number:ΔE = E₃ − E₂ = −1.51 − (−3.40) = 1.89 eV
- Convert energy to wavelength with the same hc = 1240 eV·nm shortcut:λ = 1240 / 1.89 ≈ 656 nm
- Identify the line. Every transition that ends at n = 2 belongs to the Balmer series, and 656 nm sits at the red end of visible light:n = 3 → 2 · Balmer series · H-alpha · red
What the notes contain
The PDF derives Bohr's energy level expression from first principles, works through hydrogen spectral series calculations for the Lyman, Balmer and Paschen series, and covers mass defect, binding energy per nucleon, and the radioactive decay law with several solved half-life and activity numericals.
How to use this: build the concept from these notes first, then do one quick NCERT pass before the exam for exact wording and diagram labels, and spend whatever time is left on previous year questions.
Questions
Why are the energy levels in Bohr's model negative?
The negative sign reflects that the electron is bound to the nucleus — zero energy corresponds to a free electron infinitely far away, and any bound state necessarily has lower (negative) energy than that reference. The full derivation showing why is worked through in the PDF.
What's the difference between half-life and mean life?
Half-life is the time for exactly half of a radioactive sample to decay. Mean life is the average lifetime of all the nuclei in the sample, and it's always somewhat longer than the half-life, related by t<sub>½</sub> = 0.693τ. Both are derived and applied in solved numericals.
Why does the binding energy per nucleon curve peak around iron?
Nuclei near iron have the most tightly bound nucleons on average, making iron the most stable element by this measure. This is exactly why fusion of light nuclei (moving toward the peak) and fission of heavy nuclei (also moving toward the peak) both release energy, while fusing elements heavier than iron would require energy input rather than releasing it.
Does this cover the Rydberg formula for hydrogen spectral lines?
Yes, with the derivation connecting it to Bohr's energy levels and worked examples calculating wavelengths for specific transitions in the Lyman and Balmer series.
All other chapters
Every chapter of Class 11 and Class 12 Physics, Chemistry and Biology — each with its own notes and PDF.