In 1913, Niels Bohr solved the problem that electron orbits didn't follow classical physics. So he postulated that for a hydrogen atom:

- The electron moves in circular orbits around the nucleus with the motion described by classical physics
- The electron has only a fixed set of allowed orbits. These possible orbits have these possible values
*nh/(2pi),*where n must be an integer (h is Planck's constant) - An electron can pass from one allowed orbit to another when energy is absorbed or emitted

The allowed states for an electron are numbered: *n = 1, 2, 3, 4, ... *These integral numbers, which arise from point 2, are called **quantum numbers.**

The Bohr Theory predicts the radii of the allowed orbits

## r_{n} = n^{2}a_{0}, where n = 1,2,3... and a_{0} = 53 pm

When a free electron is attracted to the nucleus and confined to the orbit *n*, the electron energy can be described with this equation.

## E_{n} = -R_{H}/n^{2}

RH is a constant that equals 2.179 * 10^-18 J

### Example Problem:

Is it likely that there is an energy level for the hydrogen atom, E_{n} = -100 * 10^{-20} J?

In order to prove that the answer is *no, * we have to show that n wouldn't be an integer.

n^{2} = -R_{H}/n^{2}

= -2.179 * 10^{-18} J/ -1.00 * 10^{-20} J

= 2.179 x 10^{2} = 217.9

n = sqrt(217.9)
= 14.76

## 14.76 isn't an integer, so it's NOT an allowed energy level.

Normally, the electron in a hydrogen atom is found in the orbit closest to the nucleus (n = 1). This is the lowest energy, or the **ground state. **When the electron gains a quantum of energy, it moves to a higher level (n = 2, 3, and so on) and the atom is in an **excited state.** When a electron moves from a higher to a lower numbered orbit, a unique quantity of energy is emitted- the difference between the two levels.

change in Energy = E_{f} - E_{i} = -R_{H}/n_{f}^{2} - -R_{H}/n_{i}^{2} = R_{H}(1/n_{i}^{2} - 1/n_{f}^{2})

= 2.179 * 10^{-18} J(1/n_{i}^{2} - 1/n_{f}^{2})