How to Use Bohr Model Calculator
The Bohr Model Calculator takes an atomic number Z and a principal quantum number n, returning the orbit radius, electron velocity, total energy, and any selected transition’s frequency and wavelength.
- Enter Z - The Bohr Model Calculator accepts hydrogen-like ions (one electron only). Enter 1 for hydrogen, 2 for He⁺, 3 for Li²⁺, and so on.
- Enter n - The principal quantum number n = 1, 2, 3, … selects the orbit. n = 1 is the ground state; higher n means a larger, higher-energy orbit.
- Set the transition (optional) - Pick a transition n₁ → n₂ to see the emitted or absorbed photon’s frequency and wavelength. The Bohr Model Calculator automatically identifies which spectral series the transition belongs to (Lyman, Balmer, Paschen, etc.).
- Read the full results panel - Separate result cards display orbit radius, electron speed, binding energy, photon frequency, and vacuum wavelength, making it easy to copy individual values or compare across different n levels.
Formula & Theory - Bohr Model Calculator
The Bohr Model Calculator uses Bohr’s quantization of angular momentum:
r_n = (a₀ / Z) · n² (orbit radius)
v_n = (Z · α · c) / n (electron speed)
E_n = − (Z² · 13.6 eV) / n² (total energy)
ΔE = E_n₁ − E_n₂ (photon energy)
f = ΔE / h (photon frequency)
λ = c / f (vacuum wavelength)| Symbol | Meaning |
|---|---|
| a₀ | Bohr radius (5.29e−11 m) |
| α | Fine structure constant (~1/137) |
| Z | Nuclear charge |
| n | Principal quantum number |
| h | Planck constant |
| c | Speed of light |
These formulas reproduce the gross structure of hydrogen-like spectra (Lyman, Balmer, Paschen, Brackett, Pfund series).
Hydrogen Spectral Series
The Bohr Model Calculator identifies each transition by its target level n₁:
| Series | Lower level (n₁) | Spectral region |
|---|---|---|
| Lyman | 1 | Ultraviolet |
| Balmer | 2 | Visible / near-UV |
| Paschen | 3 | Near-infrared |
| Brackett | 4 | Infrared |
| Pfund | 5 | Infrared |
For example, the Balmer-α (Hα) line at n = 3 → 2 falls at 656 nm — the bright red line visible in a hydrogen discharge tube.
Assumptions and Limits
The Bohr model assumes circular orbits, instantaneous Coulomb attraction, and infinite nuclear mass. To improve accuracy for hydrogen-like ions, use the reduced electron mass; for multi-electron atoms, switch to quantum-chemistry calculations.
Use Cases for Bohr Model Calculator
The Bohr Model Calculator is useful when you need a quick, transparent calculation for introductory atomic physics. Common uses include:
- Hydrogen-like spectra - Predict line positions for H, He⁺, Li²⁺, and similar one-electron systems with exact Bohr-model energies.
- Plasma diagnostics - Estimate transition energies for highly ionized species in laboratory or astrophysical plasmas.
- Physics homework - Verify problem-set answers using the same formula your textbook uses, and check that units cancel correctly.
- Conceptual demos - Show how orbit radius grows with n² and shrinks with increasing Z, making the atom smaller for heavier ions.
- Rydberg state studies - Explore very large n (n > 50) Rydberg atoms, where classical-quantum correspondence becomes apparent.
- First-principles estimates - Use the Bohr model as a quick sanity check before running a full quantum-chemistry simulation.
For high-precision spectroscopy or multi-electron atoms, switch to quantum mechanics or use spectroscopic databases such as NIST. The Bohr Model Calculator covers the essential first-order picture that underpins every introductory atomic-physics course.