## Atomic Energy Levels

(More about Bohr's Atomic Model for Hydrogen)

This electric force _is_ the centripetal force that holds the electron in a circular orbit,

The total energy of the system is

From setting the electric force equal to the centripetal force, we know

so the total energy is

This equation connects energy with orbital radius r. But what determines r? Bohr found that the following condition gave the desired results: An electron is allowed only to be in a state or orbit such that its angular momentum , L = mvr, is equal to an integer n multiplied by h divided by 2,

m v

_{n }r_{n}= n ( h / 2 ) , n = 1, 2, 3, . . .where h is again Planck's constant (h = 6.63 x 10

^{-34}J s). v and r now carry subscripts as v_{n}and r_{n}to indicate that they correspond to a particular value of n. n is called aquantumnumber. Restricting the angular momentum to particular, discrete values is referred to as thequantizationof angular momentum.This quantization means the energy is now restricted to particular, discrete values,

Using the centrepetal force equation and the quantization of angular momentum, we can solve for r

_{n},n = 1, 2, 3, . . .

This means the total energy is restricted to the following particular, discrete, quantized values,

n = 1, 2, 3, ...

Evaluating this gives

E

_{1}= - 13.6 evE

_{n}= E_{1}/ n^{2}We can represent this on an

energy-level diagram,Recall what negative energies mean. The hydrogen atom is a

boundstate. We must provide 13.6 eV of energy -- or do 13.6 eV of work -- to break it apart or to separate the proton and electron so they are infinitely far away from each other.Photons are emitted as the hydrogen atom makes a transition from one allowed state to another allowed state. The photon's energy is equal to the

differencein energy of those two states.

Absorption of photons is the reverse process of emission. If a photon with energyequaltothedifferencein energy of two states of an atom passes by, that photon may be absorbed and its energy will put the atom into a higher energy state. The photon's energy equals the change in energy of the atom becauseenergy is conserved. If the photon's energy is not equal to the difference in energy of two states of the atom, the photon willnotbe absorbed. This explains the line spectra observed in absorption spectra . In continuous or white light, photons of all wavelengths are present. Only those with particular energies (or wavelengths) corresponding to differences in energy will be absorbed; all others pass by untouched.

(c) Doug Davis, 1997; all rights reserved