This topic follows the consequences of Bohr's postulates. Combining Coulomb attraction with angular-momentum quantization gives \(r_n=a_0n^2\), so atomic radii are no longer arbitrary. Substituting the allowed radii into the mechanical energy gives \(E_n=-E_0/n^2\).
Now the Hydrogen spectrum becomes quantitative: a transition from \(m\) to \(n\) emits a photon with \(hc/\lambda=E_m-E_n\), which reproduces the Rydberg formula. The empirical integers in spectroscopy are now explained as energy-level labels.
Hydrogen levels become a ladder of bound states. The ground state is the most tightly bound, and higher \(n\) levels approach the ionization limit \(E=0\).
Spectral lines arise when the atom moves between two rungs of this ladder.
Combining \(mvr=n\hbar\) with \(mv^2=\kappa e^2/r\) gives the allowed radii:
Substituting these radii into \(E=-\kappa e^2/(2r)\) gives the energy spectrum:
The radius grows as \(n^2\), while the energy approaches zero from below as \(n\) increases.
For a transition from \(m\) to \(n\), the photon energy is
What was empirical in Rydberg's formula becomes a consequence of energy quantization.
The success is real but narrow. Bohr explains Hydrogen-like spectra and the Rydberg constant, but not the general structure of quantum states, spin, multi-electron atoms or measurement. The model is a bridge toward wave mechanics.
To use the Bohr model in an exercise, keep the calculation chain visible. The Coulomb force supplies the centripetal force, while angular momentum quantization selects only certain radii:
Solving these two relations gives the allowed radii and energies. The spectral line then comes from an energy difference, not from a continuous orbital frequency:
This is enough scaffolding for short exercises on spectral series; detailed historical development and longer numerical examples belong in the book.
This page is designed to support short guided exercises on: Energy levels, orbital radii and how the Bohr model explains Hydrogen spectral series.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.
- Focus: Energy levels, orbital radii and how the Bohr model explains Hydrogen spectral series.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[r_n=a_0n^2,\qquad a_0=0.529\ \text{\AA}\]
- Equation: \[E_n=-\frac{E_0}{n^2},\qquad E_0=13.6\ \mathrm{eV}\]
- Equation: \[hf=\frac{hc}{\lambda}=E_m-E_n.\]
- Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
- Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.