Rydberg Equation Calculator

You can find the line spectrum of hydrogen-like atoms with this Rydberg equation calculator.

Transition wavelength656.1123 nmn3 → n2 releases 1.88968 eV — visible light

Photon energy1.88968 eV3.028e-19 J

Energy of the lower level-3.401423 eVlevel n2 — negative because the electron is BOUND, and zero is a free electron at rest, infinitely far away

Energy of the upper level-1.511744 eVlevel n3 — closer to zero, so less tightly bound

Ionisation energy from the lower level3.401423 eVwhat it costs to free the electron completely — 13.60569 eV from the ground state of hydrogen

Orbit radius at the lower level211.6709 pma₀n²/Z — the radius grows as n SQUARED, so highly excited atoms are enormous

Electron speed at the lower level1.09385e+6 m/sαcZ/n — in hydrogen's ground state that is α, about 1/137 of the speed of light. THAT is what the fine-structure constant is

Frequency456.9225 THz

Wavenumber15,241.2938 cm⁻¹spectroscopists quote this rather than wavelength because it is proportional to energy

SeriesBalmer (visible)all transitions down to n2 in hydrogen

Series limit364.5068 nmthe shortest wavelength in this series, from n = ∞ — the lines crowd together as they approach it

Scaling with Zenergies go as Z²this hydrogen-like ion with Z = 1 has 1× hydrogen's energies and 1× its radii — which is why heavy ions emit X-rays where hydrogen emits visible light

Where the Bohr model failsanything with two electronsit works only for one electron around a nucleus. It gets the hydrogen energies exactly right and then fails completely for helium, because it has no account of electron–electron repulsion or of spin

Reduced-mass correction0.05446% longerthe nucleus is not infinitely heavy, so both orbit their common centre. Everything above uses R∞, the infinite-mass idealisation — for hydrogen the real lines sit about 0.05% further to the red

Corrected wavelength in vacuum656.4696 nmusing R_H rather than R∞ — this is the figure a spectrometer in vacuum actually measures

And in air656.2878 nmair slows light by about 277 parts per million, so air and vacuum wavelengths differ in the fourth figure. The familiar 656.28 nm for H-α is the AIR value — quoting it against a vacuum calculation is a standard way to appear 0.03% wrong

The formula

E_n = −13.606 Z²/n² eV; 1/λ = RZ²(1/n₁² − 1/n₂²)

Negative energies mean bound

Every level here is negative because the zero of energy is a free electron at rest infinitely far from the nucleus. An electron in an atom has less energy than that, so its energy is negative, and the ionisation energy is simply how far it has to climb to reach zero. Levels get closer together as n rises because the depth goes as 1/n².

Radii go as n²/Z while energies go as Z²/n². A hydrogen-like ion of a heavy element is therefore both far smaller and far more tightly bound — which is why such ions emit X-rays where hydrogen emits visible light.

The fine-structure constant, physically

The electron's speed in the ground state of hydrogen is αc, about 1/137 of the speed of light. That is what α actually is in the Bohr picture, and it is also why the model works as well as it does: hydrogen is only mildly relativistic, so a non-relativistic model gets the energies right to about a part in 10⁵.

Its limits are immediate

The Bohr model reproduces hydrogen's spectrum exactly and then fails completely for helium. It has no account of electron–electron repulsion, no spin, and no explanation of why some transitions are strong and others forbidden. It survives as a teaching device because its energies happen to be right, not because its picture of orbits is.

One correction is worth knowing: the nucleus is not infinitely heavy, so both particles orbit a common centre and the reduced mass shifts every line by about 0.05% in hydrogen. That shift is far larger than spectroscopic precision, and comparing it between hydrogen and its heavier isotope is how deuterium was discovered.