What is the Bohr model?
The Bohr model, proposed by Niels Bohr in 1913, describes the hydrogen atom as a nucleus surrounded by an electron orbiting in fixed, quantized circular paths. Each allowed orbit corresponds to a specific energy level given by Eₙ = −13.6 × Z²/n² eV. While superseded by quantum mechanics, it correctly predicts hydrogen's spectral lines. See also our Rydberg Equation Calculator.
What is a hydrogen-like atom or ion?
A hydrogen-like (or hydrogenic) atom is any ion that has only one electron, such as H, He⁺, Li²⁺, Be³⁺, or B⁴⁺. The Bohr model applies exactly to these systems because the single-electron simplification holds. The energy levels scale as Z², where Z is the atomic number.
How is the energy difference ΔE calculated?
The energy of level n is Eₙ = −13.6 × Z² / n² eV. The energy difference between two levels is ΔE = E_final − E_initial = 13.6 × Z² × (1/n₁² − 1/n₂²) eV. A positive ΔE means a photon is emitted (emission), while a negative value means a photon is absorbed.
How does the calculator find photon frequency and wavelength?
Once ΔE is known in joules, the photon frequency is ν = |ΔE| / h, where h = 6.626×10⁻³⁴ J·s (Planck's constant). The wavelength is λ = h·c / |ΔE|, where c = 3×10⁸ m/s. Wavelength is then converted to nanometres for practical use.
What are the Lyman, Balmer, and Paschen series?
These spectral series are named by the final orbit (n₁). Lyman series transitions end at n₁ = 1 and emit ultraviolet light. Balmer series transitions end at n₁ = 2 and produce visible light for hydrogen. Paschen series transitions end at n₁ = 3 and fall in the infrared region.
What is the Rydberg formula and how does it relate?
The Rydberg formula 1/λ = R∞ × Z² × (1/n₁² − 1/n₂²) directly gives the wavenumber of emitted or absorbed light, where R∞ ≈ 1.097×10⁷ m⁻¹ is the Rydberg constant. This is mathematically equivalent to ΔE = h·c/λ derived from the Bohr energy levels, providing the same result for wavelength.
Can the Bohr model be used for multi-electron atoms?
Strictly speaking, no. The Bohr model only works accurately for one-electron (hydrogen-like) ions. For multi-electron atoms, electron–electron repulsion and quantum mechanical effects make the simple Z²/n² energy formula inaccurate. Modern quantum mechanics (Schrödinger equation) is needed for those cases.
What does it mean if n₁ equals n₂?
If the initial and final quantum numbers are equal, there is no transition — the electron stays in the same orbit. ΔE = 0, meaning no photon is emitted or absorbed. A valid transition requires n₁ ≠ n₂; this calculator will flag that case and return zero values. You might also find our Calculated Uncertainty — Heisenberg's Uncertainty Principle useful.