Bohr Model Calculator
Find electron orbit radius, energy levels, and the photon wavelength emitted or absorbed during a transition between energy levels.
Calculator verified • Last updated: August 2026
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| Level | Orbit Radius | Energy |
|---|---|---|
| ni | — | — |
| nf | — | — |
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Animated Orbit Diagram
Reading the diagramThe dashed circles show the ni and nf orbits, scaled to their relative radii (with a minimum size so a small inner orbit never gets lost behind the nucleus dot). The electron animates around the outer orbit — press play to see it in motion. A larger n means a larger, higher-energy orbit farther from the nucleus.
The Bohr Model Explained
The Bohr model describes an electron orbiting a nucleus in fixed, quantized circular paths, each corresponding to a specific energy level. Unlike a planet orbiting the sun at any distance, the electron can only exist at certain allowed radii — jumping between them by absorbing or emitting a photon of exactly the right energy.
Orbit radius:
rn: orbit radius for level n, in meters (m).
n: the principal quantum number (1, 2, 3, …).
a0: the Bohr radius, ≈ 5.29×10⁻¹¹ m.
Z: the atomic number (number of protons).
Energy level:
En: energy of level n, in electronvolts (eV).
Z: the atomic number.
n: the principal quantum number.
The negative sign reflects that the electron is bound to the nucleus — energy must be added to free it entirely.
Photon wavelength (Rydberg formula):
λ: the photon's wavelength, in meters (m).
RH: the Rydberg constant, ≈ 1.097×10⁷ m⁻¹.
Z: the atomic number.
ni, nf: the initial and final principal quantum numbers; the absolute value means the formula works the same way whichever direction the electron moves. This calculator separately checks whether n_i is above or below n_f to report emission or absorption.
This formula is named for Swedish physicist Johannes Rydberg, who found the empirical pattern behind hydrogen's spectral lines in 1888 — a quarter-century before Bohr's model supplied a physical explanation for why it worked.
Worked Example: The Balmer-Alpha Line
Using the "Balmer-alpha" preset — a hydrogen electron (Z=1) dropping from n=3 to n=2 — the calculator finds a photon wavelength of about 656 nm, which falls in the visible red part of the spectrum. This is the famous H-alpha line seen in the light from nebulae and the surface of the Sun, and it's one of the four visible lines Balmer originally used in 1885 to find the empirical pattern that later became the Rydberg formula.
Why Only Certain Orbits Are Allowed
Bohr's key insight was that the electron's angular momentum must be an integer multiple of h/2π — a quantization rule he introduced without a full theoretical justification at the time, simply because it correctly predicted hydrogen's spectral lines. This restriction directly forces the radius to scale as n², which is why orbits get farther apart (not closer together) as n increases, and why the energy levels bunch up near zero as n grows large, eventually reaching the ionization limit.
A Brief History of the Bohr Model
Niels Bohr introduced this model in 1913, combining Ernest Rutherford's nuclear atom with Max Planck's new idea of quantized energy to explain why hydrogen's line spectrum showed only specific wavelengths rather than a continuous rainbow. The model correctly predicted the hydrogen spectrum with remarkable accuracy, but it couldn't explain more complex atoms — a limitation resolved only after Erwin Schrödinger and Werner Heisenberg developed full quantum mechanics in the mid-1920s, which describes electrons as probability clouds rather than orbiting particles.
Common Bohr Model Mistakes
Applying the formulas here to multi-electron atoms (like neutral helium or oxygen) is the most common error — the Bohr model is only exact for hydrogen-like systems with a single electron. Mixing up which level is (initial) and which is (final) in the Rydberg formula is another frequent slip; this calculator handles either order automatically and reports whether the result is emission or absorption. Forgetting to square in the energy formula is a third common mistake, since it's easy to assume the scaling is linear when it's actually quadratic.
Atomic Physics Terms You Should Know
Principal Quantum Number (n) — a positive integer labeling which orbit or energy level the electron occupies.
Bohr Radius (a₀) — the radius of the smallest hydrogen orbit (n=1), about 0.529 Ångströms.
Rydberg Constant (R_H) — an empirical constant setting the scale of hydrogen's spectral line wavelengths.
Ionization Energy — the energy needed to remove the electron entirely, equal to the energy of the n=1 level with the sign flipped.
This calculator applies strictly to hydrogen-like (single-electron) systems. Multi-electron atoms require the full quantum mechanical treatment, not covered here.
Frequently Asked Questions
Why does the electron only occupy specific orbits in the Bohr model?
Niels Bohr proposed that the electron's angular momentum could only take on specific, quantized values — whole-number multiples of h/2π. That restriction is what forces the orbit radius to jump in discrete steps (proportional to n²) rather than taking any value, which in turn explains why atoms only emit and absorb light at specific wavelengths instead of a continuous spectrum.
What's the difference between emission and absorption?
Emission happens when an electron drops from a higher energy level to a lower one, releasing the energy difference as a photon of light. Absorption is the reverse — an incoming photon with exactly the right energy kicks the electron up to a higher level. This calculator shows which one applies based on whether your initial level is higher or lower than your final level.
Does the Bohr model work for atoms other than hydrogen?
Only exactly for hydrogen-like systems — a single electron orbiting a nucleus of charge Z, such as He⁺ (Z=2) or Li²⁺ (Z=3). For any atom with more than one electron, the electrons interact with each other in ways the simple Bohr model doesn't account for, so real multi-electron atoms need the more complete quantum mechanical model to describe accurately.