In 1913 Niels Bohr made an assumption that had no justification in the physics of the time: that an electron orbiting a nucleus can only occupy orbits whose angular momentum is a whole-number multiple of the reduced Planck constant. It was an arbitrary rule bolted onto classical mechanics, and it should not have worked. It reproduced the hydrogen spectrum to four significant figures.
This Bohr model calculator from Arb Digital runs that model for any hydrogen-like atom — any nucleus with exactly one electron left. It returns the orbital radius, the electron's speed, the binding energy of the level, and the energy and wavelength of the photon released when the electron moves between two levels. Every constant is the exact or current CODATA value, and the reduced-mass correction is included, which is what takes the answer from four correct figures to six.
What This Bohr Model Calculator Does
You give it a nuclear charge Z, a principal quantum number n, a second level, and a nuclear mass. The headline is the energy of level n in electronvolts, always negative, because a bound electron sits below the zero of a free electron at rest. Its magnitude is the ionisation energy of that level: the work needed to remove the electron entirely.
The grid covers the orbit and the transition. Orbital radius grows with n squared and shrinks with Z, so the electron in a hydrogen atom at n = 5 is twenty-five times further out than at the ground state. Electron speed falls as 1/n and rises with Z, and the tool also reports it as a fraction of the speed of light because that fraction is the fine-structure constant in disguise. Transition energy and photon wavelength describe the light emitted or absorbed when the electron moves between the two levels you entered.
The note names the spectral series. Transitions ending on n = 1 form the Lyman series and are ultraviolet; those ending on n = 2 form the visible Balmer series that gives nebulae their red glow; n = 3 gives the infrared Paschen series, and so on.
How to Use It
- Set Z to the nuclear charge, not the element's electron count. Neutral helium has Z = 2 but two electrons, so the model does not apply; singly ionised helium has Z = 2 and one electron, and it does.
- Enter the level you want described in n. The hero and the first two grid items all describe this orbit.
- Enter the other end of the jump in the second field. The transition results use both levels, and the order makes no difference to the magnitude.
- Leave the nuclear mass at 1.00728 for ordinary hydrogen. Set it to 0 if you want to compare against a textbook that uses the infinite-mass idealisation.
- Check the series name in the note. If you expected a visible line and the tool says Lyman, one of your two levels is wrong.
The Four Formulas, and a Worked Example
Radius. rn = n²a₀/Z, where a₀ is the Bohr radius. NIST gives it as 5.29177210544 × 10⁻¹¹ metres. The n² is the striking part: orbits are not evenly spaced, they spread out quadratically.
Speed. vn = Zαc/n, where α is the fine-structure constant, 7.2973525643 × 10⁻³. So the ground-state electron in hydrogen travels at almost exactly 1/137 of the speed of light — that is the entire physical meaning of α in this context.
Energy. En = −RyZ²/n², where Ry is the Rydberg energy. NIST lists it as 13.605693122990 eV. The Z² is why one-electron ions get dramatically more tightly bound as the nucleus grows.
Transition. ΔE = RyZ²|1/n&sub1;² − 1/n&sub2;²|, and the photon wavelength is λ = hc/ΔE, with hc conveniently equal to 1,239.842 eV nm.
Work hydrogen through. At Z = 1, n = 1, with the infinite-mass idealisation: r = 0.0529177 nm, v = 2.18769 × 10⁶ m/s, and E = −13.6057 eV, which is hydrogen's ionisation energy. Jump from n = 2 down to n = 1 and ΔE = 13.6057 × (1 − 0.25) = 10.2043 eV, giving λ = 1,239.842/10.2043 = 121.50 nm. That is Lyman-alpha, the ultraviolet line that dominates the hydrogen spectrum.
The measured Lyman-alpha wavelength is 121.567 nm. The 0.06 nm gap is not experimental error, and the next section is about where it comes from.
Why 121.50 Becomes 121.567 nm
The formulas above assume the nucleus is infinitely heavy, so the electron orbits a fixed point. A real proton is only about 1,836 times the electron's mass, which is heavy but not infinite. Both particles orbit their shared centre of mass, and the electron behaves as though its mass were slightly reduced.
The correction is the reduced mass μ = meM/(me + M). Every energy scales by μ/me, and every radius scales by the inverse. For hydrogen that factor is 1/(1 + 1/1836.15) = 0.9994557, so the ground-state energy becomes −13.5983 eV rather than −13.6057 eV, and Lyman-alpha lands at 121.567 nm. Five parts in ten thousand, and it accounts for the entire discrepancy.
This is not a curiosity. It is how deuterium was discovered. A deuterium nucleus is roughly twice a proton's mass, so its reduced-mass factor differs, and every deuterium line sits a fraction of a nanometre away from the corresponding hydrogen line. Harold Urey found deuterium in 1931 by spotting exactly those faint displaced companions in a hydrogen spectrum. Set the nuclear mass field to 2.01355 and you can watch the shift appear.
Hydrogen-Like Means Exactly One Electron
The most common misuse of this model is putting a neutral element's atomic number into Z and expecting a real answer. Set Z = 6 for carbon and the tool obediently returns numbers, but they describe C⁵⁺ — a carbon nucleus stripped of five of its six electrons — not carbon.
The reason is that the Bohr model has no way to represent electron-electron repulsion or screening. With two or more electrons, each one partially shields the others from the nucleus, so the effective charge each electron feels is somewhere between 1 and Z, varies by orbital, and cannot be derived from within the model. Helium's first ionisation energy is 24.59 eV; the Bohr model with Z = 2 predicts 54.4 eV, which is actually helium's second ionisation energy — the one-electron case it can handle.
The systems where it genuinely works are H, He⁺, Li²⁺, Be³⁺ and so on up the isoelectronic sequence, plus muonic and positronium-like systems if you adjust the reduced mass. For anything with multiple electrons, the practical route is the aufbau ordering our electron configuration calculator applies, which is a bookkeeping scheme rather than a solved orbit.
What the Model Gets Wrong, and Why It Survives
The Bohr model is wrong about the mechanism in almost every respect. Electrons do not follow orbits; they occupy probability distributions with no trajectory. Angular momentum in the ground state is zero, not one unit of ℏ as Bohr required. The model cannot predict line intensities, cannot handle magnetic field splitting, and takes no account of electron spin. Quantum mechanics replaced it within thirteen years.
It survives because its energy levels are right. The Schrödinger equation for a one-electron atom yields exactly the same En = −RyZ²/n², and Dirac's relativistic treatment adds only fine-structure corrections of order α², roughly one part in twenty thousand. A wrong picture that produces the right spectrum is an unusually useful thing to have, and it remains the fastest way to estimate any hydrogen-like transition.
Treat the radius and speed accordingly. They are not measurable properties of an electron. The Bohr radius is real and important, but as the scale parameter of the ground-state probability distribution — the most likely distance from the nucleus — rather than the size of a circle anything travels around.
Where This Sits Next to Our Other Tools
This page is about discrete atomic energy levels and the sharp lines they produce. For the continuous thermal spectrum of a hot body instead, the blackbody radiation calculator handles Planck, Wien and Stefan-Boltzmann. If you already have a wavelength and just want the photon's energy, the photon energy calculator is the direct route, and the wavelength calculator converts between wavelength and frequency.
On the chemistry side, the electron configuration calculator covers multi-electron atoms where this model breaks down, and the average atomic mass calculator handles isotope-weighted masses — useful when you need a nuclear mass for the reduced-mass field. For the rest-energy side of atomic physics, the mass-energy equivalence calculator converts mass to energy, and the scientific notation converter tidies up the exponents.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using a neutral element's atomic number — the model needs one electron. Z = 2 describes He⁺, not helium, and the two have very different ionisation energies.
- Dropping the minus sign on the energy — bound levels are negative by convention, and the sign is what makes ionisation energy come out positive.
- Forgetting the reduced-mass correction when comparing to measured lines — it shifts wavelengths by about five parts in ten thousand, which is far larger than modern spectroscopic precision.
- Treating the orbital radius as a measurable orbit — it is the scale of a probability distribution, not the size of a path the electron follows.
- Scaling energies linearly with Z — the dependence is Z squared, so a lithium nucleus binds its last electron nine times as tightly as hydrogen, not three.
Related Free Tools From Arb Digital
Pair this with the photon energy calculator for single-photon energies and the wavelength calculator for wavelength and frequency conversions. The blackbody radiation calculator covers continuous thermal spectra rather than discrete lines, the electron configuration calculator handles multi-electron atoms, and the average atomic mass calculator gives isotope-weighted masses. The mass-energy equivalence calculator and scientific notation converter round out the set. Everything is indexed on the free online tools hub.
Frequently Asked Questions
The radius is n squared times the Bohr radius divided by the atomic number Z. NIST gives the Bohr radius as 5.29177210544 times ten to the minus eleven metres, so hydrogen's ground state sits at about 0.0529 nanometres and its n equals 5 level is twenty-five times further out.
The speed is Z times the fine-structure constant times the speed of light, divided by n. For hydrogen's ground state that is about 2.188 million metres per second, which is roughly one one-hundred-and-thirty-seventh of the speed of light. That fraction is exactly what the fine-structure constant measures here.
Because the zero of energy is defined as a free electron at rest, infinitely far from the nucleus. A bound electron has less energy than that, so its level is negative, and the magnitude of the level equals the energy needed to ionise the atom from it.
Only for hydrogen-like ions, meaning any nucleus with exactly one remaining electron, such as singly ionised helium or doubly ionised lithium. With two or more electrons the mutual repulsion and screening cannot be represented in the model, which is why it predicts 54.4 electronvolts for helium when the real first ionisation energy is 24.59.
The nucleus is not infinitely heavy, so both particles orbit their shared centre of mass and the electron behaves as though slightly lighter. Multiplying every energy by the reduced mass divided by the electron mass shifts hydrogen's ground state from minus 13.6057 to minus 13.5983 electronvolts and moves Lyman-alpha from 121.50 to 121.567 nanometres.
The series is named for the lower of the two levels. Ending on n equals 1 gives the ultraviolet Lyman series, n equals 2 gives the visible Balmer series, n equals 3 the infrared Paschen series, then Brackett, Pfund and Humphreys for levels 4, 5 and 6.
Because its energy levels are correct. The Schrodinger equation for a one-electron atom gives the same expression, and relativistic corrections change it by only about one part in twenty thousand. The orbits themselves are not physical, but the spectrum the model predicts is, which makes it the fastest route to any hydrogen-like transition wavelength.
This tool is provided for educational and reference use. The Bohr model is a semi-classical approximation: it reproduces hydrogen-like energy levels accurately but does not describe electron behaviour, line intensities, fine structure or any multi-electron atom, so treat the orbital radius and speed as scale estimates rather than physical measurements.
