Bohr Model Energy Level Calculator
Enter the nuclear charge Z and principal quantum number n to compute the energy level of a hydrogen-like atom's electron by the Bohr model Eₙ = −13.6·Z²/n² (eV).
Input Data
Results
At a glance:Enter the nuclear charge Z and principal quantum number n to compute the energy level of a hydrogen-like atom's electron by the Bohr model Eₙ = −13.6·Z²/n² (eV).
Formula
Eₙ = −13.6 · Z² / n² (eV).
$$E_n = -13.6 \cdot \dfrac{Z^2}{n^2}\ \text{eV}$$How to Use
- Enter the nuclear charge Z and principal quantum number n.
- The calculator returns the energy level Eₙ.
Hydrogen atom (Z=1) energy levels by principal quantum number
| Principal quantum number n | Energy Eₙ (eV) | Note |
|---|---|---|
| 1 | −13.60 | Ground state (most stable) |
| 2 | −3.40 | First excited state |
| 3 | −1.51 | Second excited state |
| 4 | −0.85 | Third excited state |
All energies are negative (bound states); larger n means higher level and smaller spacing.
FAQ
What is the Bohr-model energy-level formula?
For a hydrogen-like atom, Eₙ = −13.6·Z²/n² (eV), where Z is the nuclear charge and n is the principal quantum number. Negative energy means the electron is bound to the nucleus.
Why is the energy negative?
The minus sign means the electron is in a bound state, with the zero of energy set at the electron fully escaping the nucleus (n→∞). More negative means a lower level, tighter binding, and greater stability.
How do I compute the photon energy of an electron transition?
Photon energy equals the difference between two levels ΔE = E_high − E_low. For hydrogen n=2→n=1, ΔE = −3.4 − (−13.6) = 10.2 eV, corresponding to a Lyman-series line.
Does the Bohr model apply to all atoms?
No. Strictly it applies only to hydrogen-like atoms (single-electron systems, e.g. H, He⁺, Li²⁺). Multi-electron atoms need quantum mechanics because of electron-electron repulsion; the Bohr model is only qualitative there.
How do I find the ionization energy?
Ionization energy is the energy to move the electron from a level to infinity (n→∞, E=0), equal to the absolute value |Eₙ|. For hydrogen ground state, |−13.6| = 13.6 eV.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.