Bohr Model Calculator

Calculate the energy levels, orbital radii, and photon wavelengths for hydrogen-like atoms using the Bohr model of the atom.

ENERGY LEVEL
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Orbital Radius
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Electron Velocity
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Photon Wavelength
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Photon Energy
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The Bohr Model

The Bohr model, proposed by Niels Bohr in 1913, was a revolutionary step in understanding atomic structure. It describes electrons orbiting the nucleus in discrete circular orbits with quantized angular momentum. Unlike classical physics, which predicted that orbiting electrons would continuously radiate energy and spiral into the nucleus, the Bohr model postulates stable orbits where electrons do not radiate.

While the Bohr model has been superseded by quantum mechanics for accurate calculations, it remains remarkably useful for understanding hydrogen-like atoms (atoms with one electron). It correctly predicts the spectral lines of hydrogen and provides an intuitive picture of atomic energy levels, making it a cornerstone of introductory physics and chemistry education.

Key Formulas

En = -13.6 × Z² / n² eV
rn = 0.0529 × n² / Z nm
1/λ = RH × Z² × (1/n'² - 1/n²)

Where n is the principal quantum number, Z is the atomic number, RH is the Rydberg constant (1.097 x 107 m-1), and λ is the photon wavelength for transitions between levels.

Spectral Series

SeriesLower LevelRegionWavelength Range
Lymann=1Ultraviolet91-122 nm
Balmern=2Visible365-656 nm
Paschenn=3Near IR820-1875 nm
Brackettn=4IR1458-4051 nm

Frequently Asked Questions

Why does the Bohr model only work for hydrogen?

The Bohr model works exactly only for one-electron systems because it does not account for electron-electron repulsion in multi-electron atoms. With two or more electrons, the interactions become too complex for the simple Bohr picture. Modern quantum mechanics uses the Schrodinger equation with approximate methods to handle multi-electron atoms accurately.

What is the ground state energy of hydrogen?

The ground state (n=1) energy of hydrogen is -13.6 eV. This means 13.6 eV of energy is needed to completely remove the electron from a hydrogen atom (ionization energy). The negative sign indicates a bound state. As n increases, the energy approaches zero (free electron). This value was one of the great successes of the Bohr model, matching experimental measurements precisely.

How does the Bohr model relate to quantum mechanics?

The Bohr model can be seen as a semiclassical approximation. The full quantum mechanical treatment using the Schrodinger equation reproduces the Bohr energy levels exactly for hydrogen, but additionally provides orbital shapes (s, p, d, f) and predicts fine structure splitting that the Bohr model cannot. The Bohr model's quantization of angular momentum corresponds to the quantum numbers in the full theory.