Reduced Mass, Isotopes, and Exotic Bohr Atoms
For masses m_1 and m_2, relative motion uses reduced mass mu = m_1 m_2/(m_1 + m_2). Hydrogenic radii scale as 1/(mu Z) and energies and Rydberg constants scale as mu Z^2 for fixed charge magnitudes.
Why this shows up in the exam
Comparing hydrogen and deuterium spectra · Solving positronium questions · Finding muonic-atom radii and transition energies
Learn the idea
Bohr formulas survive for two-body Coulomb systems when electron mass is replaced by reduced mass. The nucleus is not perfectly fixed: both bodies orbit their centre of mass. A heavier partner makes the motion closer to the fixed-nucleus picture; positronium and muonic atoms show much larger changes.
🧠 Memory hook: Use the mass that moves relatively: the reduced mass.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- mu = m₁ m₂/(m₁ + m₂) — reduced mass of the two-body system
- r_n proportional to n²/(mu Z) — radius scaling for a Coulombic one-particle ion
- E_n proportional to -mu Z²/n² — level-energy scaling with reduced mass
- R_species = R_infinity (mu/m_e) — reduced-mass correction for an electronic hydrogen-like atom
How to approach it
- 1Identify the two orbiting masses and charges
- 2Calculate reduced mass
- 3Replace m by mu before using radius or energy scaling
Common slip-ups that cost marks
- •Using electron mass unchanged for positronium
- •Using total mass instead of reduced mass
- •Forgetting that heavier reduced mass shrinks radius but increases binding
🌟 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.
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