Orbital Speed, Acceleration, and Period
Combining Coulomb centripetal force with angular-momentum quantization gives v_n proportional to Z/n and r_n proportional to n^2/Z. Therefore period 2 pi r_n/v_n scales as n^3/Z^2 and acceleration v_n^2/r_n scales as Z^3/n^4.
Why this shows up in the exam
Comparing orbital periods · Finding revolution frequency · Scaling electron acceleration across ions
Learn the idea
Bohr-orbit speed scales as Z/n, while revolution frequency scales as Z squared over n cubed. A stronger nucleus drives a faster, tighter orbit. A higher orbit is larger and slower, so one revolution takes much longer.
🧠 Memory hook: Higher n: wider, slower, and much longer to circle.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_n = Z alpha c/n — electron speed in the nth hydrogen-like Bohr orbit
- T_n proportional to n³/Z² — orbital period scaling
- f_n proportional to Z²/n³ — revolution-frequency scaling
- a_n proportional to Z³/n⁴ — centripetal-acceleration scaling
How to approach it
- 1Separate orbital motion from radiative transition
- 2Write the n and Z scaling
- 3Check powers by combining r and v if unsure
Common slip-ups that cost marks
- •Confusing revolution frequency with emitted-photon frequency
- •Using v proportional to n
- •Forgetting Z when comparing ions
🌟 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.