Is the Rydberg Equation Calculator free?

Yes, Rydberg Equation Calculator is totally free :)

Can I use the Rydberg Equation Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Rydberg Equation Calculator?

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What is the Rydberg equation, and what does each term mean?

The Rydberg equation is 1/λ = R·Z²·(1/n_f² − 1/n_i²). λ is the wavelength of the photon, R is the Rydberg constant (about 1.097 × 10⁷ m⁻¹), Z is the nuclear charge, and n_i and n_f are the principal quantum numbers of the starting and ending levels. It works for any atom or ion with exactly one electron.

Why does the calculator give 656.47 nm for Hα when my textbook says 656.3 nm?

656.47 nm is the vacuum wavelength with the reduced-mass constant R_H. Most tables quote the wavelength in air, which is 656.28 nm because air slows light by about 0.03 %. Books that use R∞ get 656.11 nm, and a rounded R such as 1.097 × 10⁷ m⁻¹ gives 656.34 nm. Switch the constant or read the air value to match your source.

What is the difference between R∞ and R_H?

R∞ assumes an infinitely heavy nucleus. A real nucleus also moves, so the electron's mass is replaced by the reduced mass, giving R_H = R∞ / (1 + mₑ/M) for hydrogen. The correction depends on the isotope: deuterium's Hα is 0.18 nm shorter than hydrogen's, which is how Harold Urey discovered deuterium in 1932.

Which hydrogen series fall in the visible range?

Only the Balmer series (transitions ending on n = 2). Hα at 656 nm is red, Hβ at 486 nm is cyan, Hγ at 434 nm and Hδ at 410 nm are violet, and the rest crowd towards the series limit at 365 nm in the near ultraviolet. Lyman lines are in the ultraviolet; Paschen, Brackett and Pfund lines are in the infrared.

Does the Rydberg equation work for helium or other atoms?

Only for one-electron species: H, D, T, He⁺, Li²⁺ and so on, where the formula is multiplied by Z². Neutral helium, sodium and every other many-electron atom need quantum defects or a full quantum calculation, because the other electrons screen the nucleus differently for each orbital.

How do I find the initial energy level from a wavelength?

Rearrange the equation: 1/n_i² = 1/n_f² − 1/(λ·R·Z²) for emission, then take the nearest whole number. For a 434 nm photon emitted onto n = 2 this gives n_i = 5.005, so the electron started in n = 5. The Photon → transition tab does this, checks the match against a tolerance and warns when several levels fit.

Rydberg Equation Calculator – Hydrogen Spectral Lines

Transition	λ vacuum	λ air	ṽ (cm⁻¹)	E (eV)	Region
3 → 2	656.47 nm	656.288 nm	15233	1.88865	Visible — red
4 → 2	486.274 nm	486.138 nm	20564.5	2.54968	Visible — cyan
5 → 2	434.173 nm	434.051 nm	23032.3	2.85564	Visible — violet
6 → 2	410.294 nm	410.178 nm	24372.8	3.02184	Visible — violet
7 → 2	397.124 nm	397.011 nm	25181.1	3.12206	Visible — violet
8 → 2	389.019 nm	388.909 nm	25705.7	3.1871	Visible — violet
9 → 2	383.651 nm	383.542 nm	26065.4	3.23169	Visible — violet
10 → 2	379.901 nm	379.794 nm	26322.6	3.26359	Ultraviolet A
11 → 2	377.174 nm	377.067 nm	26513	3.28719	Ultraviolet A
12 → 2	375.125 nm	375.019 nm	26657.7	3.30514	Ultraviolet A
∞ → 2	364.705 nm	364.601 nm	27419.4	3.39957	Ultraviolet A

n	E_n (eV)	E_n (J)	E_n (kJ/mol)	Ionization energy from n (eV)
1	−13.5983	−2.17869 × 10⁻¹⁸	−1312.04	13.5983
2	−3.39957	−5.44671 × 10⁻¹⁹	−328.009	3.39957
3	−1.51092	−2.42076 × 10⁻¹⁹	−145.782	1.51092
4	−0.849893	−1.36168 × 10⁻¹⁹	−82.0022	0.849893
5	−0.543931	−8.71474 × 10⁻²⁰	−52.4814	0.543931
6	−0.37773	−6.05191 × 10⁻²⁰	−36.4454	0.37773
7	−0.277516	−4.44630 × 10⁻²⁰	−26.7762	0.277516
8	−0.212473	−3.40420 × 10⁻²⁰	−20.5006	0.212473
9	−0.16788	−2.68974 × 10⁻²⁰	−16.198	0.16788
10	−0.135983	−2.17869 × 10⁻²⁰	−13.1204	0.135983
∞	0	0	0	0

Rydberg Equation Calculator – Spectral Lines of Hydrogen-Like Atoms

The Rydberg equation gives the wavelength of every line in the spectrum of hydrogen, and of any other atom or ion with a single electron. This Rydberg equation calculator works both ways. Give it two energy levels and it returns the wavelength, wavenumber, frequency and photon energy. Give it a photon and it finds the level the electron came from or went to, or names the line.

From Balmer to Bohr

In 1885 Johann Balmer noticed that the four visible hydrogen lines fit λ = B·n²/(n² − 4) with B = 364.56 nm. In 1888 Johannes Rydberg generalised this to 1/λ = R·(1/n₁² − 1/n₂²), which covers every series. Neither knew why it worked. In 1913 Niels Bohr derived it from quantised electron orbits, and in doing so he gave the constant in terms of other constants: R∞ = mₑe⁴/(8ε₀²h³c) = 10 973 731.568 m⁻¹. For a nucleus of charge Z the levels are E_n = −hcR·Z²/n², and the photon carries the difference between two of them.

Two worked examples

Hα (3 → 2). The bracket is 1/2² − 1/3² = 5/36, so 1/λ = 10 967 758 m⁻¹ × 5/36 = 1 523 300 m⁻¹ and λ = 656.47 nm in vacuum, a photon of 1.889 eV.

Which level did a 434 nm photon start from? For emission onto n = 2, rearrange to 1/n_i² = 1/4 − 1/(λ·R_H). This gives n_i = 5.005, so the electron fell from n = 5 and the line is Hγ. The inverse tab does this for you, checks the result against a tolerance, and warns you when several levels near a series limit fit equally well.

The hydrogen series

Lower level  Series     Range (vacuum)       Region
n = 1        Lyman      91.2 – 121.6 nm      ultraviolet
n = 2        Balmer     364.7 – 656.5 nm     visible / near-UV
n = 3        Paschen    820.6 – 1875.6 nm    near infrared
n = 4        Brackett   1.46 – 4.05 µm       infrared
n = 5        Pfund      2.28 – 7.46 µm       infrared

Each series starts with an α line and crowds towards a series limit, the wavelength of a photon that frees the electron completely. For an ion of charge Z every wavelength is divided by about Z², so He⁺ lines sit four times further into the ultraviolet.

Reduced mass and the isotope shift

The nucleus is not infinitely heavy, so the electron and nucleus orbit their common centre of mass. Replacing the electron mass with the reduced mass gives R_M = R∞/(1 + mₑ/M). For ordinary hydrogen this is 10 967 758 m⁻¹. Deuterium has a heavier nucleus, so its lines are about 0.18 nm shorter. Harold Urey found deuterium in 1932 by spotting exactly this faint companion to the Balmer lines.

Air and vacuum wavelengths

Between 200 and 2000 nm, spectroscopists quote the wavelength measured in standard air, which is shorter than the vacuum value by the refractive index of air (about 1.00028). Hα is 656.47 nm in vacuum but 656.28 nm in air, which is why textbook values often look 0.2 nm off. The calculator always shows the vacuum value first and adds the air value inside that window.

The Pickering lines

In 1896 Edward Pickering found a series of lines in the star ζ Puppis that sat between the Balmer lines and almost on top of them. For years they were blamed on hydrogen. Bohr showed they belong to He⁺ dropping to n = 4: because Z² = 4, every second Pickering line nearly matches a Balmer line, offset slightly by helium's different reduced mass.

Where it is used, and where it stops

Astronomers map star-forming gas with Hα and the solar chromosphere with the Balmer series. Radio astronomers observe recombination lines such as H109α near 5.009 GHz, between levels above n = 100. X-ray astronomers use the Lyman-α lines of highly charged ions like Fe²⁵⁺ near 7 keV to measure the temperature of hot plasma. For atoms with more than one electron, use quantum defects or a full calculation instead.

An exact formula for an idealised atom

The Rydberg equation is exact only for a point nucleus and a non-relativistic single electron. It ignores fine structure and QED shifts (about 1 part in 10⁵ for hydrogen, growing as Z²), and it does not apply to many-electron atoms.

Rydberg Equation Calculator

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