Bohr Model Calculator - Orbital Radius, Energy, Wavelength

A Bohr model calculator solves the radius, energy, velocity, and period of any electron orbit in a hydrogen-like ion, then turns a chosen transition between two orbits into a photon wavelength and energy. The Bohr model calculator is built for chemistry and physics coursework on atomic structure, especially hydrogen, helium ion, and lithium ion problems that ask for a single numerical answer without launching a full quantum solver.

• • Hydrogen homework: Compute the n=1, 2, 3 orbits and the Lyman and Balmer series photon wavelengths for a chemistry or physics problem set.
• • Hydrogen-like ions: Re-run the same formulas for He+, Li2+, and Be3+ by changing the atomic number Z while keeping the model assumptions.
• • Spectroscopy check: Match a textbook spectral line to the named series (Lyman, Balmer, Paschen) using the nᵢ→n_f transition you measured.
• • Concept review: Show how radius scales with n², energy with Z²/n², and velocity with Z/n to make the textbook algebra concrete.

The Bohr model treats the electron as a particle in a flat Coulomb potential that orbits a stationary nucleus, then quantizes its angular momentum. The model gives exact predictions for one-electron systems and matches the Rydberg series that real hydrogen and hydrogen-like ions produce, which is why it stays useful long after the original 1913 paper.

For multi-electron atoms the model breaks down because electron-electron repulsion shifts the energy levels, but for any one-electron species the numbers are still the right answer. That is the scope this Bohr model calculator covers: any Z from 1 to 10 paired with principal quantum numbers from 1 to 10.

For the classical mechanics side of the same physics curriculum, the Kinematics Motion Calculator covers displacement, velocity, and acceleration problems that often sit alongside atomic structure homework.

The calculator uses the standard Bohr model expressions, with the Bohr radius a_0, the Rydberg constant R_∞, and the Hartree energy E_h taken from the 2018 NIST CODATA review. Each output is a direct evaluation of the formula that matches the printed textbook values.

• Z: Atomic number of the hydrogen-like ion. Z=1 gives neutral hydrogen.
• n: Principal quantum number of the orbit whose radius and energy you want.
• n_i: Upper level for the photon transition. Must be greater than n_f.
• n_f: Lower level for the photon transition. Determines the spectral series.

All four input fields feed into a single pure function. Atomic number Z and orbit n are used for the orbital properties, while the upper and lower transition levels feed the Rydberg formula for the photon wavelength.

According to NIST CODATA 2018, the Bohr radius is 5.29177210903e-11 m and the Rydberg constant is 1.0973731568160e7 m⁻¹; the Hartree energy used for the orbital electron energy is 27.211386245988 eV. Those three constants fix the scale of every output the calculator shows.

According to NIST CODATA 2018 - Bohr radius, the Bohr radius is 5.29177210903e-11 m, with a relative standard uncertainty of 1.7e-10.

According to NIST CODATA 2018 - Rydberg constant, the Rydberg constant R_infinity is 1.0973731568160e7 m^-1, which sets the scale of the spectral lines the calculator reproduces.

If you want to translate the orbital electron energy between electronvolts and joules, the Work–Energy–Power Calculator handles the energy and power side of the same intro-physics toolkit.

Four ideas make the Bohr model predictable: a fixed unit of length, quantized angular momentum, hydrogen-like scaling with Z, and a transition rule that turns energy differences into photons.

These four ideas let the same formulas describe hydrogen, deuterium, and He+ without re-deriving the physics. The Z² factor in energy and the 1/Z factor in radius are the easiest clues when you need to predict an answer before computing it.

Keep in mind that the Bohr model assumes one electron and a point nucleus. Real hydrogen has a small reduced-mass correction and a finite nuclear-size shift, both of which are beyond the simple model and not included here.

The Bohr model is often taught in the same modern-physics unit as relativity, and the Gravitational Time Dilation Calculator covers the static gravity piece of that unit for a Schwarzschild mass.

Enter the atomic number, the orbit you want to inspect, and the two levels of the transition you want to study. The result panel updates as you type.

1. 1 Choose the atomic number: Set Z from 1 (hydrogen) up to 10 for any one-electron species the Bohr model covers.
2. 2 Pick the orbit n: Use the principal quantum number of the orbit whose radius, energy, and velocity you want to see.
3. 3 Set the upper transition level: Enter n_i between 2 and 10. The orbit the electron drops from.
4. 4 Set the lower transition level: Enter n_f between 1 and n_i - 1. The orbit the electron lands in.
5. 5 Read the photon wavelength: The first result is the wavelength of the photon emitted for the nᵢ→n_f transition.
6. 6 Check the spectral series: Match the named series label to a textbook line list (Lyman, Balmer, Paschen, Brackett, Pfund, or Humphreys).

For the textbook Balmer-alpha line, leave Z=1, set n=2 to inspect the upper level, and use n_i=3 with n_f=2. The wavelength result should read about 656 nm and the spectral series label should show Balmer (visible). If you flip to n_f=1 the same calculation produces the Lyman-alpha line at 121.5 nm.

When the Coulomb force behind the Bohr orbit is replaced by a classic spring or weight problem, the Forces & Newton's Laws Calculator covers the F = ma view of mass, force, and acceleration.

The calculator is most useful when a chemistry or physics problem asks for a single numerical Bohr-model answer and you want it without rebuilding the formulas by hand.

• • Direct Bohr formulas: Uses the standard r_n, E_n, and Rydberg expressions with NIST CODATA constants, so the answer matches a printed textbook to four or more significant digits.
• • Covers hydrogen-like ions: Z up to 10 means the same tool handles He+, Li2+, Be3+, and the rest of the one-electron series without re-deriving the algebra.
• • Two problems in one place: Computes both the static orbit properties (radius, energy, velocity, period) and the photon from a chosen transition.
• • Series identification: Labels the transition as Lyman, Balmer, Paschen, Brackett, Pfund, or Humphreys, so a measured wavelength can be matched to a line list quickly.
• • Worked-example friendly: Defaults are set to the textbook hydrogen Lyman-alpha problem, so the result panel can be cross-checked against a sample solution.

For one-electron problems the Bohr numbers are the right ones to quote, and the calculator delivers them in seconds without setting up a spreadsheet.

For multi-electron atoms, transitions in solids, or fine-structure effects, the Bohr model stops being the right tool and the result should be cross-checked with a fuller quantum model.

Three input factors change every result, and the spectral line that the transition lands in depends on which lower level you choose. The next two caveats describe where the model is reliable and where it stops being exact.

• • The Bohr model assumes one electron. For neutral helium, lithium, or any multi-electron atom the model overestimates the binding energy because it ignores electron-electron repulsion.
• • Spin, relativistic corrections, and fine structure are not included. The 2p and 2s levels are degenerate in the Bohr model, and real hydrogen shows a 4.5 GHz fine-structure splitting between them.
• • The nucleus is treated as a point mass at the center. The reduced-mass correction and the finite nuclear size shift the levels by parts per million, which matters in precision spectroscopy.

If you need precision beyond the simple Bohr model, the next step is the Dirac fine-structure formula or a full Schrödinger-solution comparison.

For the nᵢ→n_f transition the calculator assumes emission (an electron dropping down). Entering n_f greater than n_i produces a sign flip and the page labels the result as absorption.

As published by Wikipedia - Bohr model, the orbital radius scales as n^2 a_0 and the energy levels are E_n proportional to -Z^2 / n^2, which is the pair of formulas the calculator uses.

The pH scale is the closest hydrogen-related concept in general chemistry, and the pH pOH Calculator converts pH, pOH, and hydrogen-ion concentration for the same first-year course.