Bohr Model Calculator – Atomic Structure & Electron Properties
Calculate electron energy levels, orbital radius, velocity, and wavelength for any atom using the Bohr atomic model.
Enter the atomic number, principal quantum number, and optional orbital and magnetic quantum numbers to explore electron properties.
Bohr Model Calculator – Atomic Structure & Electron Properties
Calculate electron energy levels, orbital radius, velocity, and wavelength for any atom using the Bohr atomic model.
About the Bohr Model Calculator
The Bohr model, introduced by Danish physicist Niels Bohr in 1913, was the first quantum mechanical description of atomic structure to successfully explain the hydrogen spectrum. Although it has since been superseded by the more rigorous quantum mechanical model, the Bohr model remains an essential teaching tool and provides accurate results for hydrogen-like ions where a single electron orbits a nucleus of atomic number Z.
At the heart of the Bohr model are two postulates. First, electrons orbit the nucleus only in certain allowed circular orbits, called stationary states, in which they do not radiate energy. Second, electrons can jump between these orbits by absorbing or emitting a photon whose energy equals the difference between the two energy levels. These two ideas introduced the concept of quantized energy states into atomic physics and laid the groundwork for modern quantum mechanics.
The energy of the nth level for a hydrogen-like atom is given by E_n = −13.6 × Z² / n² eV, where Z is the atomic number and n is the principal quantum number (n = 1, 2, 3, …). The negative sign indicates that the electron is bound to the nucleus; a less negative energy means a higher, less tightly bound orbit. The ground state of hydrogen (Z = 1, n = 1) has an energy of −13.6 eV, while the first excited state (n = 2) has an energy of −3.4 eV.
The orbital radius scales as r_n = a₀ × n² / Z, where a₀ = 5.292 × 10⁻¹¹ m is the Bohr radius, the most probable distance of the electron from the proton in the ground state of hydrogen. For higher shells the radius grows rapidly with n², meaning excited electrons occupy much larger orbits. The electron velocity in each orbit decreases with n as v_n = α × c × Z / n, where α ≈ 1/137 is the fine-structure constant and c is the speed of light.
Beyond energy and radius, the Bohr model also allows calculation of the electron's de Broglie wavelength λ = h / (m_e × v), the orbital period T = 2π r / v, and the allowed orbital (l) and magnetic (m) quantum numbers that describe the shape and orientation of the orbit within the fuller quantum mechanical picture.
This calculator implements all of these relationships and is useful for students studying atomic physics, spectroscopy, quantum chemistry, and related fields. Enter the atomic number Z (the number of protons) and the principal quantum number n to get immediate results for energy, radius, velocity, and wavelength. The optional orbital quantum number l and magnetic quantum number m further specify the quantum state within a given shell.
Bohr model examples
Worked examples showing how to apply the Bohr model to real atomic configurations.
| tool.bohr-model-calculator.examples.colInput | Result | Explanation |
|---|---|---|
| Z = 1, n = 1 (Hydrogen ground state) | E = −13.60 eV, r = 5.29 × 10⁻¹¹ m | The electron sits in the lowest energy orbit at the Bohr radius. This is the most stable state of hydrogen. |
| Z = 1, n = 2 (Hydrogen first excited state) | E = −3.40 eV, r = 2.12 × 10⁻¹⁰ m | The electron has absorbed 10.2 eV from the ground state. The orbital radius is four times larger than in n = 1. |
| Z = 2, n = 1 (Helium, hydrogen-like) | E = −54.40 eV, r = 2.65 × 10⁻¹¹ m | Doubling Z quadruples the binding energy and halves the orbital radius compared to hydrogen at the same n. |
| Z = 1, n = 3 (Hydrogen second excited state) | E = −1.51 eV, r = 4.76 × 10⁻¹⁰ m | The third shell is nine times larger than the first. Transitions from n = 3 produce the Paschen series in the infrared. |
How to use the Bohr model calculator
- Enter the Atomic Number (Z) — the number of protons in the nucleus. For hydrogen enter 1, for helium enter 2, and so on.
- Enter the Principal Quantum Number (n) — the shell number. Use n = 1 for the ground state, n = 2 for the first excited state, etc.
- Optionally enter the Orbital Quantum Number (l, from 0 to n−1) and Magnetic Quantum Number (m, from −l to +l) to specify a sub-state.
- Click Calculate to instantly see energy level, orbital radius, electron velocity, de Broglie wavelength, and orbital period.
- Click Reset to clear all inputs and start a new calculation.
Bohr model calculator FAQ
What is the Bohr model of the atom?
The Bohr model is a planetary model of the atom proposed by Niels Bohr in 1913. It states that electrons orbit the nucleus in fixed circular paths called shells, each with a discrete energy, and that electrons only emit or absorb radiation when jumping between these allowed orbits. Although later replaced by quantum mechanics for multi-electron atoms, it remains accurate for hydrogen-like (one-electron) ions.
What does the principal quantum number n mean?
The principal quantum number n (1, 2, 3, …) specifies the electron's shell and determines both its energy and its average distance from the nucleus. As n increases the energy becomes less negative (less tightly bound) and the orbital radius grows as n². In the ground state n = 1 gives the lowest energy and smallest orbit.
Why is the energy negative in the Bohr model?
The energy is defined relative to the ionisation limit, where the electron is at infinite distance from the nucleus with zero kinetic energy. A bound electron has lower energy than a free one, so bound-state energies are negative. The ground-state energy of hydrogen is −13.6 eV, meaning 13.6 eV of energy must be supplied to ionise a ground-state hydrogen atom.
Is the Bohr model accurate for multi-electron atoms?
The Bohr model is only strictly accurate for hydrogen-like ions — atoms or ions with a single electron, such as H, He⁺, Li²⁺, and so on. For multi-electron atoms electron–electron repulsion and exchange interactions require the full quantum mechanical treatment. However, the Bohr model still gives useful estimates and is an excellent pedagogical starting point.
What is the Bohr radius?
The Bohr radius (a₀ ≈ 5.292 × 10⁻¹¹ m or 0.529 Å) is the most probable distance between the electron and the proton in the ground state of hydrogen. It sets the natural length scale for atomic distances. The orbital radius for any shell is r_n = a₀ × n² / Z.
How do the quantum numbers l and m relate to the Bohr model?
In the original Bohr model only n is used. The orbital quantum number l (0 to n−1) and magnetic quantum number m (−l to +l) come from the extension of Bohr's ideas by Sommerfeld and later from full wave mechanics. They describe the shape and orientation of the orbital, refine the energy in the presence of magnetic fields (Zeeman effect), and allow specification of a unique quantum state.