Bohr Model
The Bohr model, proposed by Niels Bohr in 1913, describes the hydrogen atom as having electrons orbiting the nucleus in discrete, quantized circular orbits with specific allowed energies. Electrons can jump between orbits by absorbing or emitting photons whose energy equals the difference between the two energy levels, explaining the discrete spectral lines of hydrogen. While superseded by quantum mechanics, the Bohr model correctly predicts hydrogen's spectral series and introduced the revolutionary idea of quantized atomic energy levels.
Key Formula
Eₙ = −13.6 eV / n², rₙ = n² × a₀
LaTeX: E_n = -\frac{13.6\,\text{eV}}{n^2}, \quad r_n = n^2 a_0
| Symbol | Meaning | Unit |
|---|---|---|
| Eₙ | Energy of the nth orbit | eV |
| n | Principal quantum number (1, 2, 3, …) | dimensionless |
| 13.6 eV | Ionization energy of hydrogen | eV |
| rₙ | Radius of the nth orbit | m |
| a₀ | Bohr radius ≈ 0.529 Å = 5.29 × 10⁻¹¹ m | m |
Worked Example
Problem
Calculate the energy of the photon emitted when an electron in hydrogen transitions from n = 3 to n = 2 (the H-alpha line of the Balmer series).
Solution
Step 1: Calculate energy of n = 3 level. E₃ = −13.6 / 3² = −13.6 / 9 = −1.511 eV Step 2: Calculate energy of n = 2 level. E₂ = −13.6 / 2² = −13.6 / 4 = −3.400 eV Step 3: Calculate photon energy. ΔE = E₃ − E₂ = −1.511 − (−3.400) = 1.889 eV Step 4: Convert to wavelength. λ = hc / ΔE = (4.136×10⁻¹⁵ eV·s × 3×10⁸ m/s) / 1.889 eV λ = 1.241×10⁻⁶ / 1.889 ≈ 657 nm
Answer
Photon energy ≈ 1.89 eV; wavelength ≈ 657 nm (red light — the H-alpha spectral line).
Bohr Model: Energy Levels and Orbital Radii for Hydrogen
| Quantum Number (n) | Energy (eV) | Orbital Radius (Å) | Spectral Series |
|---|---|---|---|
| 1 | −13.60 | 0.529 | Lyman (UV) |
| 2 | −3.40 | 2.116 | Balmer (visible) |
| 3 | −1.51 | 4.761 | Paschen (IR) |
| 4 | −0.85 | 8.464 | Brackett (IR) |
| 5 | −0.54 | 13.225 | Pfund (IR) |
| ∞ | 0 | ∞ | Ionization limit |
Interactive Tools
PhET Models of the Hydrogen Atom
Compare classical and quantum models of the hydrogen atom interactively.
Open ToolKhan Academy — Bohr Model of the Hydrogen Atom
Detailed explanation with solved examples and practice problems.
Open ToolWolfram Alpha — Hydrogen Energy Levels
Compute energy levels and transition wavelengths for hydrogen.
Open Tool
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Related Terms
Energy Level
An energy level is one of the discrete, quantized values of energy that a bound quantum system (such as an electron in an atom or a molecule) is permitted to have. Unlike classical systems where energy can take any continuous value, quantum mechanics constrains bound particles to specific allowed states, each characterized by a set of quantum numbers. Transitions between energy levels result in the absorption or emission of photons with energies exactly equal to the difference between the two levels, producing the characteristic spectral lines used in atomic spectroscopy.
Emission Spectrum
An emission spectrum is the set of discrete wavelengths (spectral lines) of electromagnetic radiation emitted by an atom or molecule when its electrons transition from higher to lower energy levels, releasing photons. Each element produces a unique pattern of spectral lines that serves as its "fingerprint," allowing identification of elements in distant stars, gas clouds, and laboratory samples. The energy of each emitted photon equals exactly the energy difference between the two levels involved in the transition: E = hf = hc/λ.
Ground State
The ground state is the lowest possible energy state of a quantum mechanical system, such as an atom, molecule, or nucleus, in which all quantum numbers take their minimum allowed values consistent with the Pauli Exclusion Principle. A system in the ground state is thermodynamically stable and does not spontaneously emit radiation. The ground state energy of hydrogen is −13.6 eV, and the ground state represents the reference level from which excitation energies of higher states are measured.
Named after Danish physicist Niels Bohr (1885–1962), who proposed the model in 1913. "Bohr" is simply the scientist's surname, of Danish/Norwegian origin. The model built on Rutherford's nuclear atom and Planck's quantum hypothesis.