Ions like He⁺, Li²⁺ and U⁹¹⁺ that have only one electron behave just like hydrogen but with a stronger nucleus: energies grow with Z², orbits shrink with Z and electrons move faster.
How to use the Hydrogen-Like Atom Calculator
- Atomic number Z
- Principal quantum number n
- Nuclear mass (u, 0 = infinite)
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the atomic number, the level n and the nuclear mass (for the reduced-mass correction).
Inputs explained
| Field | What to enter |
|---|---|
| Atomic number Z | A number |
| Principal quantum number n | A number |
| Nuclear mass (u, 0 = infinite) | A number |
What each result means
- Energy E_n = -13.605693×[Atomic number Z]^2÷[Principal quantum number n]^2×[Reduced mass ratio μ÷mₑ]
- Orbit radius r_n = 5.29177210903e-11×[Principal quantum number n]^2÷[Atomic number Z]÷[Reduced mass ratio μ÷mₑ]
- Electron speed = [Atomic number Z]×0.0072973525693×299792458÷[Principal quantum number n]
- Ionisation energy from this level = 13.605693×[Atomic number Z]^2÷[Principal quantum number n]^2×[Reduced mass ratio μ÷mₑ]
Example values
How the main result changes when Atomic number Z changes and everything else stays at the example values above:
| Atomic number Z | Energy E_n (eV) |
|---|---|
| 1 | -13.6038 |
| 1.5 | -30.6086 |
| 2 | -54.4153 |
| 2.5 | -85.0239 |
| 3 | -122.4345 |
| 4 | -217.6613 |
Formula
E_n = −13.6057 eV·Z²/n²·(μ/mₑ), r_n = a₀·n²/Z·(mₑ/μ), v_n = Z·α·c/n
Worked example
Inputs — Atomic number Z: 2; Principal quantum number n: 1; Nuclear mass (u, 0 = infinite): 4.0026.
Results — Reduced mass ratio μ/mₑ = 0.99986296; Energy E_n = -54.415314 eV; Orbit radius r_n = 26.462487 pm; Electron speed = 4,375.382527 km/s; Ionisation energy from this level = 54.415314 eV.
Key terms
- Bohr radius
- a₀ = 52.9 pm.
Frequently asked questions
What is a hydrogen-like ion?
Any atom stripped to a single electron.
Why does energy grow with Z²?
Stronger nuclear attraction and smaller orbits.
What is the reduced-mass correction?
The nucleus also moves slightly, lowering energies by ~0.05% for hydrogen.
Are relativistic effects included?
No – they matter for heavy ions.