Series Limits, Longest and Shortest Wavelengths
For a series ending at n_f, the longest wavelength uses n_i = n_f + 1, while the shortest wavelength uses n_i approaching infinity. The limit wavenumber is R Z^2/n_f^2.
Why this shows up in the exam
Comparing Lyman and Balmer limits · Finding longest Paschen wavelength · Locating ultraviolet, visible, and infrared series
Learn the idea
Within a series, the first line is longest and the series limit is shortest. The smallest energy drop into a fixed lower level comes from the next level and makes the longest wave. Starting from infinity gives the largest drop and shortest wave.
🧠 Memory hook: Next rung gives longest; infinity gives shortest.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- 1/lambda_max = R Z² (1/n_f² - 1/(n_f+1)²) — longest wavelength, first line of a series
- 1/lambda_min = R Z²/n_f² — shortest wavelength, series limit
- nu_limit = c R Z²/n_f² — series-limit frequency
How to approach it
- 1Fix n_f from the series
- 2Choose n_i = n_f + 1 or infinity
- 3Invert only after calculating the wavenumber
Common slip-ups that cost marks
- •Calling the first line the shortest
- •Substituting n_i = 0 for the limit
- •Comparing wavelengths by comparing wavenumbers in the same direction
🌟 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.