# Photon Energy & Hydrogen Spectrum Calculator - Wavelength to eV, Rydberg Formula, Photoelectric Effect

> Convert wavelength, frequency and photon energy (eV, J, kJ/mol), compute hydrogen spectral lines, photoelectric stopping potential and de Broglie wavelength.

- URL: https://qilingtools.com/en/math/spectroscopy
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Chemistry](https://qilingtools.com/en/tags/chemistry), [Physics](https://qilingtools.com/en/tags/physics)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Photon energy is E = hν = hc/λ, with the exact SI constants h = 6.62607015 × 10⁻³⁴ J·s and c = 299792458 m/s; a handy shortcut is λ(nm) ≈ 1239.84 / E(eV). Green light at 500 nm has ν = 6.00 × 10¹⁴ Hz, a wavenumber of 20000 cm⁻¹ and E = 3.97 × 10⁻¹⁹ J = 2.48 eV, or 239.3 kJ/mol per mole of photons. The tool has four parts. (1) A photon converter between nm, μm, Å, m, Hz, THz, cm⁻¹, J, eV and kJ/mol that names the region of the electromagnetic spectrum and shows an approximate colour for visible light. (2) Hydrogen spectra: the Rydberg formula 1/λ = R·Z²(1/n₁² − 1/n₂²) for any transition in hydrogen, deuterium or hydrogen-like ions such as He⁺ and Li²⁺, with the series named (Lyman, Balmer, Paschen, Brackett, Pfund, Humphreys), level energies Eₙ = −13.6 Z²/n² eV, ionization energy (13.6 eV for H, 54.4 eV for He⁺), Bohr radius, a clickable energy-level diagram and the visible emission lines (Hα 656.3, Hβ 486.1, Hγ 434.0, Hδ 410.2 nm). It uses the reduced-mass R_H = 1.09678 × 10⁷ m⁻¹ and reports both vacuum and air wavelengths (Hα: 656.47 nm in vacuum, 656.28 nm in air); the Lyman limit is 91.2 nm. (3) The photoelectric effect: from a metal’s work function (Cs 2.14, K 2.30, Na 2.36, Zn 4.33, Cu 4.65, Pt 5.65 eV…) it finds the threshold wavelength, maximum kinetic energy and stopping potential, with a K_max–ν graph. (4) The de Broglie wavelength λ = h/p of an electron, proton, neutron, alpha particle or any mass, from speed, kinetic energy or accelerating voltage, with relativistic correction.

## How to use

1. Choose a panel: photon energy, hydrogen spectrum, photoelectric effect or de Broglie wavelength
2. Enter a wavelength or energy, pick n₁ and n₂ (or click an arrow on the energy-level diagram), choose a metal, or enter a particle’s speed or accelerating voltage
3. Read the converted values, line wavelengths and level energies, stopping potential or matter wavelength, with the full working shown

## Features

- Converts between 10 units of wavelength, frequency, wavenumber and photon energy, with copy buttons
- Names the spectral region from gamma rays to radio and shows the colour of visible light
- Rydberg formula for any transition in H, D, He⁺, Li²⁺, Be³⁺, B⁴⁺ and C⁵⁺
- Reduced-mass correction with both vacuum and air wavelengths
- Clickable energy-level diagram, visible emission spectrum and a table of each series’ lines
- Level energies, ionization energy, Bohr radius and electron speed
- Photoelectric effect with 17 metal work functions, threshold wavelength, K_max, stopping potential and graph
- de Broglie wavelength from speed, kinetic energy or voltage, relativistically correct

## FAQ

### How do I convert wavelength to photon energy?

Use E = hc/λ. Writing hc as 1239.84 eV·nm gives E(eV) = 1239.84 / λ(nm), so 500 nm is 2.48 eV and 1 eV corresponds to 1239.8 nm. Multiply eV by 1.602 × 10⁻¹⁹ for joules, and joules by the Avogadro constant divided by 1000 for kJ/mol: a 500 nm photon carries 3.97 × 10⁻¹⁹ J, or 239.3 kJ/mol.

### Why does the Rydberg formula give 656.47 nm for Hα instead of 656.3 nm?

The Rydberg formula gives the wavelength in vacuum. With R_H = 1.09678 × 10⁷ m⁻¹ the n = 3 → 2 line is 1/λ = R_H(1/4 − 1/9), λ = 656.47 nm. The 656.3 nm in tables and textbooks is measured in air, whose refractive index of about 1.00028 shortens the wavelength by about 0.18 nm; the tool shows both. Using R∞ = 1.0974 × 10⁷ m⁻¹ (an infinitely heavy nucleus) would give 656.11 nm instead.

### What are the Lyman, Balmer and Paschen series?

They group hydrogen lines by the lower level n₁. Transitions ending on n = 1 form the Lyman series (ultraviolet, 121.6 to 91.2 nm), those ending on n = 2 the Balmer series (visible and near UV, 656.3 to 364.6 nm), n = 3 Paschen, n = 4 Brackett, n = 5 Pfund and n = 6 Humphreys, all in the infrared. Within a series the wavelength shortens as n₂ grows and converges to the series limit at n₂ → ∞.

### How is the stopping potential calculated in the photoelectric effect?

Einstein’s equation gives K_max = hν − W, and the stopping potential is V₀ = K_max / e, which has the same number in volts as K_max in eV. Light of 400 nm (3.10 eV) on sodium (W ≈ 2.36 eV) gives K_max ≈ 0.74 eV and V₀ ≈ 0.74 V. If hν < W, i.e. the wavelength is longer than the threshold λ₀ = hc/W, no electrons are emitted however intense the light.

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