Rydberg Formula Calculator
Find the wavelength of a hydrogen spectral line from its two energy levels using the Rydberg formula.
🌈 What is the Rydberg Formula Calculator?
The Rydberg formula calculator finds the wavelength of light emitted or absorbed when an electron jumps between two energy levels in a hydrogen atom. Give it the lower and upper quantum numbers and it returns the spectral line wavelength, names the series it belongs to, and gives the photon's energy and frequency.
The formula sits at the heart of atomic spectroscopy. Chemistry and physics students use it to predict the coloured lines a hydrogen discharge tube produces and to identify the famous red H-alpha line. Astronomers rely on the same lines, seen in starlight, to measure the composition and motion of distant objects. The tool turns a pair of integers into a precise wavelength, making the abstract idea of quantised energy levels concrete.
The key idea is that the lower level n1 defines the spectral series. Transitions ending at n1 = 1 form the Lyman series in the ultraviolet, those ending at n1 = 2 form the visible Balmer series, and n1 = 3 gives the infrared Paschen series. The upper level n2 must always be larger, because a photon is released as the electron falls from a higher orbit to a lower one. A common slip is swapping the two numbers, which makes the formula return a negative value.
This calculator is useful because it handles the reciprocal arithmetic and the tiny Rydberg constant for you, and it works for hydrogen-like ions too through the atomic number Z, where wavelengths scale as one over Z squared. Enter the levels and read the wavelength with the working shown, so you can see exactly how each term contributes.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Balmer H-alpha (the red line)
Example 2 - Lyman-alpha (ultraviolet)
Example 3 - Paschen line (infrared)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the Rydberg formula?
The Rydberg formula gives the wavelengths of spectral lines: 1 / lambda = R Z squared times (1/n1 squared minus 1/n2 squared), where R is the Rydberg constant (about 1.097 x 10^7 per metre), Z is the atomic number, and n1 and n2 are the lower and upper energy levels of the electron transition.
How do you calculate a spectral line wavelength?
Put the two energy levels into the formula, compute 1 over lambda, then take the reciprocal. For the Balmer H-alpha line (n1 = 2, n2 = 3) in hydrogen: 1/lambda = R (1/4 minus 1/9) = R x 0.1389, giving lambda = about 656 nanometres, the familiar red line.
What are the hydrogen spectral series?
Each series is defined by the lower level n1: Lyman (n1 = 1) in the ultraviolet, Balmer (n1 = 2) in the visible, Paschen (n1 = 3), Brackett (n1 = 4), and Pfund (n1 = 5) in the infrared. The upper level n2 can be any integer larger than n1, giving a set of lines within each series.
What is the Balmer series?
The Balmer series is the set of hydrogen lines that fall to the n1 = 2 level, and four of them lie in the visible spectrum: H-alpha (red, 656 nm), H-beta (blue-green, 486 nm), H-gamma, and H-delta. It is the series you see in a hydrogen discharge tube and in stellar spectra.
What is the Rydberg constant?
The Rydberg constant R is about 1.0968 x 10^7 per metre for hydrogen. It sets the scale of all hydrogen spectral wavelengths. A closely related value, the Rydberg energy of 13.6 eV, is the ionisation energy of hydrogen from its ground state, since energy and inverse wavelength are proportional.
Why must n2 be greater than n1?
Because a photon is emitted when the electron drops from a higher orbit (n2) to a lower one (n1), so n2 is always the larger number. If you reverse them the formula gives a negative result. For absorption the roles swap, but the same pair of levels is involved and the wavelength is identical.
Can the Rydberg formula be used for other elements?
It works exactly for hydrogen-like ions, those with a single electron such as He+ (Z = 2) or Li2+ (Z = 3), where the wavelength scales as 1 over Z squared. For multi-electron atoms it is only approximate, because electron-electron interactions and shielding shift the levels.
How is wavelength related to the photon energy?
The energy of the emitted photon is E = h c over lambda, so a shorter wavelength means a more energetic photon. The Lyman series (ultraviolet) has the highest energies because it ends at the tightly bound n1 = 1 level, while the infrared Paschen series has the lowest.
Who discovered the Rydberg formula?
Johannes Rydberg published the general formula in 1888, building on Johann Balmer's 1885 empirical fit to hydrogen's visible lines. It was purely empirical until 1913, when Niels Bohr derived it from his quantised atomic model, explaining why the integers n1 and n2 appear.