Calculation result
Photon emission using Z = 1.00000.
Ionization from the initial level
Bohr orbit properties at n = 3
Calculation steps
1. Selected constant: Infinite nuclear-mass Rydberg constant, R = 1.09737e+7 m−1.
2. Initial level: E3 = −hcRZeff2/neff2 = -1.51174 eV.
3. Final level: E2 = -3.40142 eV.
4. Energy change: ΔE = Ef − Ei = -1.88968 eV.
5. Photon values: f = |ΔE|/h and λ = c/f.
6. Classification: Photon emission, Balmer series, Visible light.
Energy-level diagram
The dashed line marks the ionization continuum. The arrow shows the selected transition.
Energy-level table
| n | Energy (eV) | Energy (J) | Above ground (eV) | Ionization (eV) | Radius (m) | Speed (m/s) | Gap to n+1 (eV) |
|---|---|---|---|---|---|---|---|
| 1 | -13.6057 | -2.17987e-18 | 0 | 13.6057 | 5.29177e-11 | 2,187,691 | 10.2043 |
| 2 | -3.40142 | -5.44968e-19 | 10.2043 | 3.40142 | 2.11671e-10 | 1,093,846 | 1.88968 |
| 3 | -1.51174 | -2.42208e-19 | 12.0939 | 1.51174 | 4.76259e-10 | 729,230 | 0.661388 |
| 4 | -0.850356 | -1.36242e-19 | 12.7553 | 0.850356 | 8.46684e-10 | 546,923 | 0.306128 |
| 5 | -0.544228 | -8.71949e-20 | 13.0615 | 0.544228 | 1.32294e-9 | 437,538 | 0.166292 |
| 6 | -0.377936 | -6.05520e-20 | 13.2278 | 0.377936 | 1.90504e-9 | 364,615 | 0.100269 |
Calculator inputs
Formula used
The base model uses hydrogen-like energy levels. The selected Rydberg constant sets the energy scale. Optional corrections adjust nuclear mass, charge, and quantum defect.
How to use this calculator
- Select a hydrogen-like atom or enter a custom atomic number.
- Enter the initial and final principal quantum numbers.
- Choose optional physical corrections when needed.
- Select energy, wavelength, precision, and notation settings.
- Press the calculation button to update every result.
Example data
| Example | Z | Initial n | Final n | Expected series | Expected behavior |
|---|---|---|---|---|---|
| Hydrogen Balmer-alpha | 1 | 3 | 2 | Balmer | Visible emission |
| Hydrogen Lyman-alpha | 1 | 2 | 1 | Lyman | Ultraviolet emission |
| Hydrogen absorption | 1 | 2 | 4 | Balmer | Photon absorption |
| Helium ion transition | 2 | 3 | 2 | Balmer-type | Higher photon energy |
Understanding electron energy levels
Bound electron energies are negative because zero represents a free electron. Levels approach zero as the principal quantum number increases. Ionization occurs when enough energy reaches the continuum.
Emission happens when an electron moves downward. Absorption happens when it moves upward. The photon energy equals the level-energy difference.
The Bohr model works best for one-electron systems. Multi-electron atoms need more advanced quantum methods. Effective charge and quantum defect only provide approximations.
Frequently asked questions
Why are electron energies negative?
Zero energy describes a free electron at infinite distance. A bound electron has less energy, so its value is negative.
What does a positive energy change mean?
A positive ΔE means the atom absorbs energy. The electron moves to a higher energy level.
What does a negative energy change mean?
A negative ΔE means energy leaves the atom. A photon is emitted during the downward transition.
Which atoms fit this model?
Hydrogen and one-electron ions fit best. Examples include He⁺, Li²⁺, Be³⁺, and B⁴⁺.
What is the reduced-mass correction?
The nucleus is not perfectly stationary. Reduced mass slightly changes the Rydberg constant and calculated energies.
What is effective nuclear charge?
It estimates the net attraction experienced by an electron. Shielding can make it smaller than the actual atomic number.
What is a quantum defect?
It adjusts the effective principal quantum number. It is useful for approximate non-hydrogenic Rydberg states.
How is spectral series identified?
The lower transition level sets the named series. Level one is Lyman, while level two is Balmer.
Why can visible color be approximate?
Color perception varies across people and displays. Wavelength boundaries are also conventional rather than perfectly sharp.
Can this replace spectroscopy software?
No. It is an educational hydrogen-like model. Precision spectroscopy requires fine structure, relativistic effects, and experimental calibration.