Bohr Model Energy Level Calculator

Q: Which ions does this work for?

This calculator applies to hydrogen-like ions with a single electron, such as H, He+, Li2+, Be3+, and similar one-electron species. It does not apply to multi-electron atoms.

Q: How do I know if it’s emission or absorption?

If the initial level n_initial is higher than the final level n_final, the electron drops to a lower level and emits a photon (emission). If n_initial is lower than n_final, the electron is excited to a higher level and absorbs a photon (absorption).

Q: Why doesn’t this match real hydrogen exactly?

The Bohr model is an idealized one-electron model that neglects effects like reduced mass corrections, fine structure, and other quantum-mechanical details. Real hydrogen and hydrogen-like ions can show small shifts relative to the ideal Bohr predictions.

Compute energy levels, photon ΔE, wavelength λ, and frequency ν for hydrogen-like ions using the Bohr model. Visualize the transition on an energy-level diagram and a wavelength spectrum gauge.

Background

In the Bohr model, a one-electron (hydrogen-like) ion has quantized energies E_n given by E_n = −(Z²·R_H) / n², where Z is the atomic number, n is the principal quantum number (1, 2, 3, …), and R_H ≈ 2.18×10⁻¹⁸ J. A transition between levels n_initial and n_final emits or absorbs a photon of energy |ΔE|, related to wavelength by ΔE = h·ν = h·c/λ.

Enter transition details

Ion (hydrogen-like)

We assume an ideal hydrogen-like ion: a single electron bound to a nucleus with charge +Z.

Energy levels (Bohr n)

Use positive integers (1, 2, 3, …). For emission, n_initial > n_final; for absorption, n_initial < n_final.

Options:

Round to sensible significant figures

Step-by-step working

Energy-level diagram

Wavelength spectrum gauge

Quick picks:

Chips prefill n-values and ion, then run the calculation.

Result:

No results yet. Enter levels and click Calculate.

How this calculator works

We treat the system as a hydrogen-like ion with a single electron and nuclear charge +Z.
Energy levels follow E_n = −(Z²·R_H) / n², where R_H ≈ 2.18×10⁻¹⁸ J.
The photon energy is |ΔE| = |E_final − E_initial|. We convert between energy, frequency, and wavelength via ΔE = h·ν = h·c/λ.
We also identify the series (e.g., Lyman, Balmer, Paschen) from the lower n, and classify the photon as UV, visible, or IR from λ.

Formula & Equations Used

Bohr energy levels (J): E_n = −(Z²·R_H) / n²

Energy difference: ΔE = E_final − E_initial

Photon relations: |ΔE| = h·ν = h·c/λ

Wavelength: λ = h·c / |ΔE|

Frequency: ν = |ΔE| / h

Energy in eV: E (eV) = E (J) / (1.602×10⁻¹⁹ J·eV⁻¹)

Example Problems & Step-by-Step Solutions

Example 1 — H Balmer-α (3 → 2)

For hydrogen (Z = 1): E_n = −R_H/n². So E₃ ≈ −2.42×10⁻¹⁹ J and E₂ ≈ −5.45×10⁻¹⁹ J. ΔE = E₂ − E₃ ≈ −3.03×10⁻¹⁹ J, so |ΔE| ≈ 3.03×10⁻¹⁹ J. Using λ = h·c/|ΔE| gives λ ≈ 656 nm (red, Balmer-α).

Example 2 — H Lyman-α (2 → 1)

E₂ ≈ −5.45×10⁻¹⁹ J, E₁ ≈ −2.18×10⁻¹⁸ J. ΔE = E₁ − E₂ ≈ −1.64×10⁻¹⁸ J, so |ΔE| ≈ 1.64×10⁻¹⁸ J. Then λ = h·c/|ΔE| ≈ 121.6 nm (ultraviolet, Lyman-α).