# Hydrogen Energy Levels Calculator

Created by Miłosz Panfil, PhD
Reviewed by Bogna Szyk and Adena Benn
Last updated: Jun 05, 2023

The Hydrogen energy levels calculator helps you to compute the energy level of hydrogen atoms and hydrogen-like atoms and ions. You will also find answers to questions like:

• What is the energy level of hydrogen?
• How to calculate the ionization energy of hydrogen?
• ...and more.

## Hydrogen energy levels

The hydrogen atom is made of an electron and a proton. The electron has a negative electric charge, while the proton has a positive charge. This means that there is an attractive force between them, called the Coulomb force (see Coulomb's law calculator). The mass of the electron is much smaller, around one thousand times smaller, than the mass of the proton. Because of this, we can think about the electron as orbiting the proton.

The energy of the electron comes from balancing the attractive Coulomb force and the centrifugal force of circular motion. The law of quantum mechanics predicts that the energy of the electron in such a situation can take only certain values. We call these values of energy the energy levels. There is the lowest energy level, where the electron is as close to the proton as the Heisenberg uncertainty principle allows. The higher the energy level, the further from the proton the electron is. The energy level also tells us how tightly the electron and proton are bound. The lower the energy, the tighter the binding.

## Hydrogen energy levels calculator

The energy of the electron in the hydrogen atom equals

E = - m c² alpha² Z² /(2n²),

where:

• E – Energy of the electron;
• m – Mass of the electron;
• c – Speed of light;
• alpha = 1/137 – Fine structure constant;
• Z – Atomic number (Z = 1 for the hydrogen); and
• n – Energy level.

The energy is measured in electronvolts, 1 eV = 1.6 × 10-16 J.

The most important number is n, which sets the energy level. Value n = 1 means the energy is of the lowest energy level. Note that the energy is negative. This means that a proton and electron will likely form a hydrogen atom when they come close to each other.

In our calculator, you can modify the atomic number; the default value is Z = 1, which corresponds to the hydrogen. You can read the Hydrogen-like atoms section below to learn more about the meaning of Z and how you can use this energy levels formula for other atoms and ions.

## How many energy levels does Hydrogen have

There is an infinite number of energy levels. However, if you play a bit with the hydrogen energy level calculator, you might notice that when n is large, the difference in energies between neighboring levels decreases. This means that the high energy levels are becoming hard to distinguish from each other. Their discrete nature, due to quantum physics, fades away. The last, though infinite, energy level has energy equal to 0. Beyond that energy, electrons and protons do not form a bound state anymore and are just two independent particles.

You may be interested in exploring the role of a hydrogen atom in determining the degree of unsaturation of an organic molecule using the degree of unsaturation calculator.

## How to calculate the ionization energy of Hydrogen

What is ionization energy? Ionization energy is the energy required to rip one electron from the atom. In other words, it is the energy needed to overcome the binding energy. The minimal energy to separate the electron from the proton is the energy difference between the energy when the electron is free and the electron's energy level. If the electron is on the first level, then this energy is exactly equal to 13.6 eV.

## Hydrogen-like atoms

The formula for the energy levels works not only for hydrogen but also for other ions like He⁺, Li²⁺, Be³⁺ , and B⁴⁺ , which have a single electron. In these hydrogen-like atoms, the number of protons might be larger than one, which leads to a stronger Coulomb attraction and lowers energy levels. To see this effect, you can easily change the number of protons, usually called atomic number Z, in our calculator.

You can explore another quantum mechanical phenomenon in our De Broglie wavelength calculator.

Miłosz Panfil, PhD
Energy level
Atomic number
Energy
eV
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