The Bohr Atom

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

1. The electron moves in circular orbits around the nucleus with the motion described by classical physics
2. 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)
3. 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 = n²a₀, where n = 1,2,3... and a₀ = 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/n²

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.

n² = -R_H/n²
= -2.179 * 10^-18 J/ -1.00 * 10^-20 J
= 2.179 x 10² = 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_f² - -R_H/n_i² = R_H(1/n_i² - 1/n_f²)

= 2.179 * 10^-18 J(1/n_i² - 1/n_f²)