## Energy Levels of Hydrogen

There's an easy way to think about electron energy, and there's a hard (but more mathematically correct) way. Let's do this the easy way.

Each electron in an atom has a certain amount of energy. Shells closer to the nucleus have the lowest energies ... that means n=1 has the lowest energy, and higher n means higher energy. There is a limit though, and eventually if an electron gains enough energy, it will exit the atom altogether.

Each electron in an atom has a certain amount of energy. Shells closer to the nucleus have the lowest energies ... that means n=1 has the lowest energy, and higher n means higher energy. There is a limit though, and eventually if an electron gains enough energy, it will exit the atom altogether.

## Hydrogen - the simplest atom

Hydrogen is the easiest atom to predict electron energy for, because

*it only has one electron*. Suppose an electron is excited from n=1 (on the left) to n=3 (on the right):Q: How much energy did the electron have to gain to move like this? What wavelength of light could cause this to happen?

A: For hydrogen, we have two formulas we can use ... one for energy, and one for wavelength:

A: For hydrogen, we have two formulas we can use ... one for energy, and one for wavelength:

where R is Rydberg's constant (1.097 x 10^-7 m^-1), ni is the INITIAL shell number and nf is the FINAL shell number. h is still Planck's constant (6.626 x 10^-34 Js) and c is still the speed of light (3.0 x 10^8 m/s).

And so we can calculate wavelength OR energy of the photon from it too:

And so we can calculate wavelength OR energy of the photon from it too:

And so for this electron transition to occur, we need a photon with at least 1.938 x 10^-18 J ... which means its wavelength has to be 102.6 nm or shorter. The answers here are NEGATIVE because the photon is being ABSORBED by the electron...the photon is losing energy (the electron is gaining energy).

## Sample Question

Q: How much energy is RELEASED when an electron falls from n=6 to n=2 in a hydrogen atom? What wavelength is a photon with that amount of energy?

A: E = 7.572 x 10^-20 J.

wavelength = 2.63 x 10^-6 m = 2630 nm

A: E = 7.572 x 10^-20 J.

wavelength = 2.63 x 10^-6 m = 2630 nm