One Electron Energy Calculator

Calculate Energy

Formula Used

The calculator uses the Bohr energy equation for a one electron atom or ion.

E_n = -K × Z² / n²

Here, K is usually 2.18 × 10^-18 joule. Z is the nuclear charge. n is the principal quantum number.

The transition energy is found from the absolute difference between two levels.

ΔE = |E_f - E_i|

Electronvolts are calculated by dividing joules by 1.602176634 × 10^-19. Frequency uses f = ΔE / h. Wavelength uses λ = c / f.

How To Use This Calculator

Enter the nuclear charge Z. Use 1 for hydrogen.
Enter the initial quantum number.
Enter the final quantum number.
Keep 2.18 and -18 for the standard energy constant.
Select decimal places for the displayed result.
Press the calculate button.
Review energy, frequency, wavelength, and transition type.
Use the CSV or PDF button to save the result.

Example Data Table

System	Z	Initial n	Final n	Transition Energy	Energy eV	Wavelength	Type
Hydrogen	1	2	1	1.635E-18 J	10.20 eV	121.57 nm	Emission
Hydrogen	1	3	2	3.028E-19 J	1.89 eV	656.70 nm	Emission
Helium ion	2	2	1	6.540E-18 J	40.82 eV	30.39 nm	Emission

Understanding One Electron Energy

One electron atoms are simple but powerful models. They include hydrogen and ions with only one bound electron. The calculator uses the Rydberg energy idea to estimate each allowed level. It starts from the constant 2.18 × 10^-18 joule. That value is the ground state energy scale for hydrogen. A negative sign means the electron is bound to the nucleus.

Why The Level Matters

The principal quantum number, shown as n, controls the size of the orbit model. Larger n values place the electron farther from the nucleus. The energy then moves closer to zero. A level with n equal to one is lowest. A level with n equal to two is higher. The difference between two levels gives the photon energy for a jump.

Role Of Nuclear Charge

For hydrogen like ions, nuclear charge also matters. The symbol Z counts the positive charge of the nucleus. Hydrogen has Z equal to one. Helium ion has Z equal to two. The energy grows with Z squared. This makes tightly bound ions much more energetic than hydrogen.

Practical Output Values

The tool gives level energy in joules and electronvolts. It also calculates transition energy. When the electron falls to a lower level, light is emitted. When it moves upward, energy is absorbed. The photon frequency comes from dividing transition energy by Planck constant. The wavelength comes from dividing light speed by frequency. These values help connect atomic structure with spectra.

Use In Study

This calculator is useful for homework, lab checks, and quick comparisons. It shows every important step in a compact format. You can test many levels and ion charges. You can also export the results for notes. The CSV file is good for spreadsheets. The PDF button is useful for reports. Always remember that this is a Bohr style result. It works best for one electron systems.

Limits And Care

Real atoms can be more complex. Electron spin, fine structure, shielding, and external fields can shift measured lines. This page does not replace a full quantum calculation. It gives a fast educational estimate. Use positive quantum numbers only. Review the sign convention before comparing sources. Bound levels are negative, while released photon energy is positive.

FAQs

What does 2.18 × 10^-18 mean?

It is the approximate Rydberg energy constant in joules. It gives the hydrogen ground state energy scale used in the Bohr model.

Why is level energy negative?

A negative value means the electron is bound. Zero energy represents a free electron far away from the nucleus.

What is Z in this calculator?

Z is nuclear charge. Hydrogen has Z equal to one. A helium ion with one electron has Z equal to two.

Can I calculate absorption?

Yes. Set the final quantum number higher than the initial number. The result will show an absorption transition.

Can I calculate emission?

Yes. Set the final quantum number lower than the initial number. The electron releases a photon during that change.

Why do electronvolts appear?

Electronvolts are convenient for atomic energy. Small joule values become easier to read when converted into electronvolts.

Does this work for multi electron atoms?

No. It is best for hydrogen like systems with one electron. Multi electron atoms need shielding and quantum corrections.

What happens when both levels match?

The transition energy becomes zero. No photon frequency or wavelength is produced because the electron does not change level.