Bohr Radius and Orbit Scaling
For an infinitely heavy hydrogen-like nucleus, the radius of the nth Bohr orbit is a_0 n^2/Z. With finite nuclear mass, replace electron mass by reduced mass, producing a small isotope correction.
Why this shows up in the exam
Comparing H, He+, and Li2+ radii · Finding orbit number from radius · Estimating sizes of exotic Bohr atoms
Learn the idea
Hydrogen-like orbit size grows as n squared and shrinks with nuclear charge. Higher orbits are wider because the matter wave fits more loops, while a stronger nucleus pulls the same orbit inward.
🧠 Memory hook: n squared expands; Z contracts.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- r_n = a₀ n²/Z — radius for a hydrogen-like ion in the infinite-nuclear-mass approximation
- r₂/r₁ = (n₂² Z₁)/(n₁² Z₂) — safe ratio form for two hydrogen-like orbits
How to approach it
- 1Write n and Z for each species
- 2Use the ratio before inserting a₀
- 3Check that larger Z gives a smaller radius at fixed n
Common slip-ups that cost marks
- •Using r proportional to n
- •Putting Z squared in the radius formula
- •Ignoring different orbit numbers in a ratio
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In hydrogen, an electron transitions from n = 2 to n = 1. Using E_n = -13.6/n^2 eV, find the emitted photon energy.
More from Atoms and Nuclei
Hydrogen spectrum and spectral series
Understand the origin, calculation, and interpretation of spectral lines, series limits, wavelength and frequency relations, and transitions in hydrogen and hydrogen-like ions.
Atomic models and Bohr theory
Learn the historical development of atomic models, especially the Bohr model, and how it explains the structure, radii, and energy levels of hydrogen and hydrogen-like atoms.
Nuclear fission, fusion, and energy production
Understand the processes of nuclear fission and fusion, their energy yields, the role of binding energy, and applications like nuclear reactors and solar energy.
Mass defect, binding energy, and mass-energy equivalence
Learn how mass defect leads to nuclear binding energy, the use of E=mc², and how to calculate binding energy per nucleon and related quantities.
Nuclear size, radius, and density
Explore how nuclear size is measured, the formula for nuclear radius, its dependence on mass number, and the concept of nuclear density.
Nuclear reactions and conservation laws
Study the equations for nuclear reactions, including conservation of mass number, atomic number, and other physical quantities.