Explore Bohr levels with flexible hydrogenic options today. Compute photons, wavelengths, and series names easily. Validate labs, homework, and spectroscopy designs in minutes anywhere.
| Scenario | Inputs | Key output |
|---|---|---|
| Ground state energy | Z=1, n=1 | E₁ ≈ −13.606 eV |
| First excited level | Z=1, n=2 | E₂ ≈ −3.401 eV |
| Balmer-α transition | Z=1, nᵢ=3 → nᶠ=2 | λ ≈ 656 nm (visible red) |
| Lyman-α transition | Z=1, nᵢ=2 → nᶠ=1 | λ ≈ 122 nm (ultraviolet) |
In the Bohr model for a hydrogen-like ion, the bound-state energy for level n is:
En = −ER · Z² / n²
For a transition between levels nᵢ and nᶠ, the photon energy is:
ΔE = Ef − Ei, and |ΔE| = h·f = h·c/λ
A common line estimate uses the Rydberg relation: 1/λ = R · Z² · (1/nlow² − 1/nhigh²).
Hydrogen’s simplest bound-state picture comes from the Bohr model, where the electron occupies discrete orbits labeled by the principal quantum number n. Each n corresponds to a fixed energy En, so the atom can only absorb or emit photons whose energies match differences between two levels.
In a hydrogen-like system, energies scale as −Z²/n². Doubling n makes the magnitude of En four times smaller, so higher levels cluster close to zero. This spacing trend explains why high-n states are easier to ionize and why spectral lines compress toward series limits.
The ionization threshold is defined at E = 0, where the electron is no longer bound. For hydrogen, the ground-state binding energy is about 13.6 eV, which is the energy needed to remove the electron completely. As n increases, En approaches zero, representing states near the continuum.
A transition from ni to nf produces a photon with |ΔE| = h·f = h·c/λ. Emission typically occurs when ni > nf, releasing energy as light. Absorption occurs when the atom gains energy and moves upward.
The lower level sets the spectral series: Lyman (n=1) lies in ultraviolet, Balmer (n=2) appears in visible and near‑UV, and Paschen (n=3) is infrared. This calculator labels the series and reports wavelength in nanometers for quick region identification.
Results depend on the constants used, especially the Rydberg value and the energy conversion between eV and joules. Small differences can appear between a wavelength computed from |ΔE| and one computed from the Rydberg relation. Both are useful cross-checks for coursework and lab reports.
In spectroscopy experiments, measured wavelengths can be mapped to n transitions by scanning plausible (ni, nf) pairs and matching λ. For plasma and discharge tubes, the same approach helps identify hydrogenic ions where Z > 1, shifting lines by Z².
A classic reference is Balmer‑α: ni=3 → nf=2 yields λ near 656 nm. Lyman‑α: 2 → 1 is near 121.6 nm. These values are widely used for calibration checks and for validating the expected spectral series limits.
The zero reference is the free electron at infinity. Bound states have lower energy, so En is negative. The magnitude indicates how much energy is required to ionize that level.
Z is the nuclear charge seen by the electron. Hydrogen uses Z=1. Hydrogen‑like ions, such as He⁺, use Z=2, which increases binding energies and shifts wavelengths by a factor of Z².
One wavelength comes from |ΔE| and fundamental constants. The other uses the Rydberg line formula. Slight differences can occur due to constant selection, reduced‑mass effects, and rounding.
Most introductory problems use n up to 20. Larger n values are allowed for completeness, but level spacing becomes tiny and results become sensitive to rounding and to physical effects not included here.
If ΔE = Ef − Ei is negative, the atom emits a photon. If ΔE is positive, the atom must absorb a photon to make the transition upward.
No. It uses the basic hydrogenic energy scaling. Fine structure, Lamb shift, hyperfine splitting, and external field effects require more advanced quantum models and additional inputs.
Choose a likely series by region (UV, visible, IR), then test ni and nf combinations until the calculated wavelength matches your measurement within uncertainty.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.