Bohr Model Calculator — Orbits by the Numbers

Niels Bohr’s 1913 atom quantised the electron’s orbits by fiat and by genius together: only certain radii, energies and speeds are allowed, indexed by the integer n. Enter n, and this calculator returns the orbit’s size, the electron’s energy and its speed — the numbers that first explained hydrogen’s spectrum.

The Atom That Shouldn't Have Worked

Classical physics condemned the orbiting electron to spiral into the nucleus in a trillionth of a second, radiating continuously. Bohr's 1913 answer was to forbid it by postulate: angular momentum comes in ħ-sized units, which quantises the orbit radii to r = n²a₀ (a₀ = 0.529 Å, the Bohr radius) and the energies to E = −13.6 ÷ n² eV, and electrons radiate only in jumps between them. The postulates were outrageous and the numbers were perfect: the model reproduced hydrogen's entire spectrum (the Rydberg page's constant fell out of fundamental constants), predicted the ionisation energy, and sized the atom correctly. Quantum mechanics later replaced the orbits with orbitals and kept every one of these numbers — the model is wrong as a picture and exact as an energy ledger, which is why it is still taught, and why this calculator computes its outputs without apology and its caveats without omission.

Worked Example: n = 2, the Balmer Launchpad

The defaults set n = 2: radius = 4 × 0.529 = 2.117 Å, energy = −13.6 ÷ 4 = −3.40 eV, speed = 2.188×10⁶ ÷ 2 = 1.094×10⁶ m/s — about 0.36% of light speed, still non-relativistic enough for the model to trust itself. From this level the Balmer series departs: a fall to n = 2 from n = 3 releases the difference (1.89 eV, the hydrogen-levels page's headline) as the red H-alpha photon of the Rydberg page. Climb instead to n = 10 and the calculator shows the atom swelling to 52.9 Å across with the electron crawling at 219 km/s and bound by just 0.136 eV — a Rydberg atom, a laboratory giant so loosely held that passing fields tear it, and the bridge between Bohr's atom and the classical world his correspondence principle promised at large n.

Why the Numbers Survive the Model

Quantum mechanics' hydrogen atom has no orbits, yet its energy levels are Bohr's to the precision this page prints — because Bohr, guided by the correspondence principle, had effectively quantised the action of the true system, and the true system's energies care about exactly that. The radius formula survives transfigured (the most probable radius of the 1s orbital is exactly a₀; the mean is 1.5 a₀ — close enough that "the Bohr radius" remains atomic physics' unit of length, enshrined in the constants tables). Even the speed survives as the virial theorem's average. The calculator's three outputs are thus not antiques: they are the ground floor of the modern theory, wearing the 1913 scaffolding that first reached them — the correct way to remember a model that was a ladder, kicked away by the very theory it let us climb to.

The Pickering Series: Bohr's Public Victory

The model's making was a courtroom win: astronomers had found a stellar series (Pickering's, in ζ Puppis) whose lines fit Rydberg's form with half-integer quantum numbers — an embarrassment — until Bohr pointed out they belonged not to hydrogen but to ionised helium, He⁺, whose Z = 2 shifts the ladder exactly so, half-integers and all, and predicted the series' remaining lines' positions to spectroscopic precision. Fowler, the leading spectroscopist, contested the precision, demanding rigour about the nuclear mass — and the correction Bohr supplied (the reduced mass again) improved the agreement further, converting the chief sceptic in print. The calculator's hydrogen-only inputs carry that history quietly: its constants are the reduced-mass-corrected tradition, and the half-integer affair is why physicists of 1913 began, reluctantly, to believe. A model's authority is earned exactly this way — by resolving an anomaly it was not built for, in public, against a hostile expert — and the episode remains the template for how a calculator page of this hub hopes to earn trust: print the constants, show the audit trail, and let the arithmetic argue.

Moseley's Law: The Model Counts the Elements

Within months of Bohr's paper, Henry Moseley used its Z-scaling on X-ray spectra to assign every element its atomic number — the nuclear charge in units of e — and the periodic table's order snapped from atomic-weight approximation into exact integer law, with gaps correctly predicting undiscovered elements. The Bohr formula with Z inserted is Moseley's law's skeleton: characteristic X-ray energies scale as (Z − 1)² × 13.6 eV × (1 − 1 ÷ 4) for the K-alpha line, a computation this calculator's structure performs mentally per element. Moseley was killed at Gallipoli in 1915, aged 27, and Britain thereafter kept its scientists from the infantry — the policy footnote to the model's fastest triumph.

Related Tools

More from the Physics Calculators hub: the Hydrogen Energy Levels Calculator — the levels and the transitions between them — and the Rydberg Equation Calculator, for the photons the jumps emit.

Frequently Asked Questions

Where does 13.6 eV come from?

From the constants: E₁ = −m_ee⁴ ÷ (8ε₀²h²) evaluates to −2.179×10⁻¹⁸ J = −13.598 eV — Bohr's derivation of the Rydberg constant from first principles was the result that converted the sceptics, because spectroscopy had measured the number for decades without any theory beneath it. The calculator uses the rounded −13.6 eV (infinite-nuclear-mass value, essentially); the reduced-mass refinement shaves it to 13.598 for hydrogen's real nucleus. A web calculator printing 13.6 and naming the refinement is striking the same bargain the textbooks strike.

Why can't the electron sit between levels?

In the model, because angular momentum is quantised (mvr = nħ) — only the radii satisfying it exist as stationary states; in the modern theory, because the Schrödinger equation's boundary conditions admit only discrete bound-state energies, the same arithmetic in deeper dress. Between levels there is simply no state to occupy — an electron "in between" is not a physical configuration but a superposition notation. The discreteness that scandalised 1913 is the discreteness every spectrum displays, lines instead of rainbows, and this calculator's integer-only input enforces it at the keyboard.

Does the model work for helium?

For He⁺ (one electron, Z = 2) excellently, with Z² scaling: energies −54.4 ÷ n² eV, radii quartered — run the mental adjustment on this page's n = 1 output. For neutral helium (two electrons) it fails instructively: the electrons' mutual repulsion, which the model ignores, shifts every level, and the failure drove a decade of crisis until the new quantum mechanics and the exclusion principle resolved it. The calculator confines itself to hydrogen (Z = 1) as labelled; the Z-scaling for hydrogen-like ions is the reader's one-multiplication extension, with the boundary of the model's competence stated in this FAQ rather than discovered in an error.

What is alpha, the fine-structure constant?

α = e² ÷ (4πε₀ħc) ≈ 1 ÷ 137.036 — the electromagnetic coupling's dimensionless strength, which falls out of this model as the ground-state speed in units of c (the calculator's note points at it: v₁ = αc). It governs the fine structure that splits spectral lines (Bohr's model, extended by Sommerfeld's ellipses, first approximated it), the electron's anomalous magnetic moment, and by its eerie dimensionlessness has attracted a century of numerology the physics community endures. That a three-input calculator's note can point from a school model to α in one sentence is the model's density of consequence in miniature.

How big is a Rydberg atom at n = 100?

Radius = 10,000 × 0.529 Å = 0.529 µm — bacterium-sized, visible in principle to a microscope, with the electron bound by 1.36 meV and orbiting (in the picture) at 21.9 km/s. Such atoms are manufactured routinely (laser excitation) for quantum sensing and computing experiments, their hugeness making them exquisitely field-sensitive. Enter n = 100 in the calculator and watch the three trends of the note racing — r up quadratically, E vanishing, v crawling — to the edge of the classical world, which is where Bohr always said his model should hand over.

What is the model's angular momentum at n?

L = nħ exactly — the postulate the whole structure hangs on (mvr with r = n²a₀ and v = v₁ ÷ n returns nħ, a consistency the reader can verify from this page's three outputs in one line of arithmetic: at n = 2, m × 1.094×10⁶ × 2.117×10⁻¹⁰ = 2.11×10⁻³⁴ = 2ħ). Quantum mechanics revises the value (the ground state's orbital angular momentum is 0, not ħ — the model's clearest quantitative error in its home territory), yet the quantisation of angular momentum in ħ units survives as the theory's backbone. The calculator's Bohr figures are the postulate's consequences, displayed so the reader can audit the postulate from them.

Who was Bohr?

Niels Bohr (1885–1962), Danish physicist whose 1913 trilogy of papers founded the quantum atom while he was a postdoc in Manchester, and whose Copenhagen institute became quantum mechanics' debating chamber through the 1920s (the Bohr–Einstein debates are its scripture). He won the Nobel in 1922, midwifed the complementarity interpretation, escaped occupied Denmark by boat and air in 1943, and lobbied — futilely, honourably — for international control of the atom he had taught the world to picture. This page's n runs on his postulates.