Hydrogen Atom Energy Levels Calculator
Find the energy of a hydrogen atom level in eV and joules, plus the ionisation energy, from n and Z.
⚛️ What is the Hydrogen Atom Energy Levels Calculator?
The hydrogen atom energy levels calculator gives the energy of an electron in a chosen level of a hydrogen atom, using the famous result E = minus 13.6 times Z squared over n squared electronvolts. Enter the level number and it returns the energy in electronvolts and joules, along with the energy needed to ionise the atom from that level.
These energy levels are one of the great triumphs of early quantum theory. Bohr derived them in 1913, and they explain why hydrogen emits and absorbs light only at specific wavelengths. Students use the calculator to check homework and to see how tightly bound each level is, while the numbers underpin spectroscopy, astrophysics, and laser physics. Knowing that the ground state sits at minus 13.6 electronvolts, and that this equals hydrogen's ionisation energy, ties together a lot of atomic physics in one figure.
The negative sign is the point people most often question. Energy is measured relative to a free electron at rest, which is defined as zero. A bound electron has less energy than that, so its level is negative, and the deeper the level the more negative and more stable it is. As the level number n rises, the energy climbs towards zero and the levels bunch together, converging on the ionisation limit. For a single-electron ion heavier than hydrogen, the stronger nuclear charge deepens every level by a factor of Z squared.
This tool is useful because it turns two whole numbers into the level energy in the units you need and immediately gives the ionisation energy, with the working shown so the minus-13.6-over-n-squared scaling is clear at a glance.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Ground state of hydrogen
Example 2 - Second level of hydrogen
Example 3 - Ground state of He+ (Z = 2)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the energy level formula for hydrogen?
The energy of level n is E = -13.6 times Z squared divided by n squared, in electronvolts, where Z is the atomic number (1 for hydrogen). The 13.6 eV is the Rydberg energy. The ground state (n = 1) is -13.6 eV and higher levels are less negative, approaching zero as n grows.
Why are hydrogen energy levels negative?
The negative sign means the electron is bound to the nucleus: energy must be added to free it. Zero energy is defined as the electron at rest infinitely far away. A more negative value means a more tightly bound, lower-energy state, which is why the ground state at -13.6 eV is the most stable.
What is the energy of the ground state of hydrogen?
The ground state (n = 1) of hydrogen has an energy of -13.6 eV, equal to about -2.18 x 10^-18 joules. This is the lowest and most stable level, and its magnitude, 13.6 eV, is the energy needed to ionise a hydrogen atom from the ground state.
What is the ionisation energy of hydrogen?
It is 13.6 eV, the energy needed to remove the electron from the ground state (n = 1) to infinity. From a higher level n the ionisation energy is smaller, equal to 13.6 divided by n squared. This calculator shows the ionisation energy from whatever level you enter.
How do you calculate the energy of the n = 2 level?
Substitute n = 2 and Z = 1 into E = -13.6 Z squared / n squared: E = -13.6 / 4 = -3.4 eV. The second level is four times less negative than the ground state, so the electron is more loosely bound there and closer to being free.
Why do the energy levels get closer together?
Because the energy depends on 1 over n squared, the gaps shrink rapidly as n increases: the jump from n = 1 to n = 2 is 10.2 eV, but from n = 3 to n = 4 it is only 0.66 eV. The levels crowd together and converge on 0 eV, the ionisation limit, at high n.
How does the energy change for hydrogen-like ions?
For a single-electron ion of charge Z, the energy scales as Z squared: E = -13.6 Z squared / n squared. So the ground state of He+ (Z = 2) is -13.6 x 4 = -54.4 eV, four times deeper than hydrogen. The stronger nuclear pull binds the electron much more tightly.
How do energy levels relate to spectral lines?
A photon is emitted or absorbed when the electron moves between levels, and its energy equals the difference between them. The transition energy is 13.6 Z squared times (1/n1 squared minus 1/n2 squared). This is the energy form of the Rydberg formula, which converts directly to a wavelength.
Is this formula exact?
It is very accurate for hydrogen and hydrogen-like ions and is the standard result taught in introductory quantum mechanics. Tiny corrections, from fine structure, the Lamb shift, and hyperfine effects, shift the levels by small fractions of an electronvolt, but the -13.6 / n squared formula captures the dominant structure.