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Hydrogen Energy Levels Calculator

Enter the principal quantum number n to find the energy level.

How it is calculated

The tool uses the Bohr-model formula En = −13.6 ÷ n², with E in electronvolts (eV) and n the principal quantum number. The energy in joules comes from multiplying by 1.602176634 × 10−19 J per eV. The binding (ionization) energy is the absolute value of En, the energy needed to free the electron from that level. The value 13.6 eV is a rounded constant, so results are accurate to about three significant figures, even though four decimals are shown. Only whole numbers n = 1, 2, 3, … are accepted; zero, negatives and fractions such as 2.5 give no result. The model applies to hydrogen-like one-electron atoms only when scaled by Z², which this tool does not do.

Worked example

For the third level, n = 3. Then n² = 9, so E = −13.6 ÷ 9 = -1.5111 eV. In joules, −1.5111 × 1.602176634 × 10−19 = −2.4211 × 10⁻¹⁹ J. The binding energy is the absolute value, 1.5111 eV: that is how much energy would ionize an electron sitting in n = 3.

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Common mistakes

  • Entering the wavelength, the transition or the number of levels instead of n. The tool takes one level and does not compute energy differences between two levels.
  • Dropping the minus sign when comparing: the energy is negative for a bound electron, and only the binding energy is positive.
  • Typing a non-integer such as 2.5 or 0, which is rejected, since n must be a whole number of at least 1.
  • Applying the result to helium ions or other atoms; the formula here is for hydrogen (Z = 1) and would need a Z² factor otherwise.

About this calculator

The hydrogen energy levels calculator computes the energy of a hydrogen atom's electron in orbital n using the Bohr model formula Eₙ = −13.6 eV / n², where n is the principal quantum number (1, 2, 3, …). The negative sign reflects that the electron is bound to the nucleus; n = 1 is the ground state at −13.6 eV, and as n increases the levels get closer together and approach zero, the point at which the electron is no longer bound (ionization).

The energy difference between two levels, ΔE = 13.6 eV × (1/n₁² − 1/n₂²), predicts the exact wavelengths of hydrogen's spectral lines — the Lyman series (transitions to n = 1, ultraviolet), Balmer series (to n = 2, visible light), and Paschen series (to n = 3, infrared) all follow directly from this formula. Physics and chemistry students, and anyone studying atomic spectroscopy use this calculator to find a given level's energy in electronvolts or joules and check it against the Bohr model before moving on to the full quantum-mechanical treatment.

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Comments (1)

  • Editorial team

    Because the tool returns one level at a time, you can still reach spectral lines by running it twice. Level 2 gives -3.4000 eV and level 3 gives -1.5111 eV; the gap between those two values is the energy of the photon released when an electron drops from the third level to the second, which falls in visible light. The 13.6 constant is rounded, so keep extra digits while subtracting. Which transition series are you working on in your course?

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