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Bohr Model Energy Calculator

Find the energy of a Bohr atom level.

Result

-6.0444 eV

How it is calculated

The energy of the electron in a hydrogen-like atom is E = −13.6 × Z² / n² in electron-volts. Z is the nuclear charge (atomic number) and n is the principal quantum number or level. The constant 13.6 eV is the ground-state binding energy of hydrogen, used here as a rounded value. The result is negative because the electron is bound: zero is the energy of a free electron, and the closer the value is to zero the weaker the binding. The level is rounded to the nearest whole number, and it must be at least 1; Z must be at least 1. The formula is valid only for atoms or ions with a single electron, such as H, He⁺ and Li²⁺.

Worked example

Consider the He⁺ ion, which has Z = 2 and one electron, in level n = 3. Z squared is 4 and n squared is 9, so E = −13.6 × 4 / 9 = −54.4 / 9 = −6.0444. The tool shows -6.0444 eV. For comparison, level 1 of the same ion has −13.6 × 4 = −54.4 eV, so lifting the electron from level 1 to level 3 takes about 48.36 eV.

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Common mistakes

  • Using the formula for atoms with several electrons, such as neutral helium or carbon. The Bohr model with 13.6 × Z² applies only when one electron is present, so ignore electron-electron repulsion at your peril; for neutral He the result would be badly wrong.
  • Entering the number of electrons or the mass number instead of the atomic number Z. For He⁺ the value is 2 (protons), not 1 (electrons) and not 4 (mass number).
  • Entering a non-integer level and expecting an interpolated answer. The level is rounded to a whole number, so 2.6 is treated as 3.
  • Forgetting the sign. The negative result is correct; if you need the energy to ionise the electron from that level, take the absolute value.
  • Treating the value as the energy of a photon. A photon energy is the difference between two levels, for example E(3) − E(2), not the energy of one level.

About this calculator

This calculator finds the energy of an electron orbit in a hydrogen-like atom (one electron orbiting a nucleus of charge Z) using the Bohr model formula E_n = −13.6·Z²/n² electron-volts, where n is the principal quantum number (1, 2, 3…) labeling the orbit and Z is the atomic number. Proposed by Niels Bohr in 1913, the model quantizes electron orbits into discrete allowed energy levels, and the energy differences between levels correspond exactly to the wavelengths of light absorbed or emitted when an electron jumps between them — explaining the discrete spectral lines of hydrogen's Lyman, Balmer, and Paschen series.

Chemistry and physics students use this as the standard first model for understanding atomic structure and quantized energy, before moving on to the full quantum mechanical treatment; and it remains a useful teaching and estimation tool for atomic emission spectra even though later quantum mechanics superseded it as a complete description.

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