The Solution

Shortly after Rutherford, Niels Bohr devised a model for the atom. He assumed (because it gave the right answer) that the angular momentum could only have certain values. Although we haven't discussed angular momentum, the symbol for it is l and the equation for it is

l = mvr (4)

He didn't know why the angular momentum was "quantized,'' but if he made that assumption, then he could calculate the orbits in which the electrons exist. Later, the idea of particle/wave duality (ask if you want to know) explained why electrons can only exist in these specific orbits. So, by making a specific assumption, Bohr devised a technique for calculating the allowable orbits of the electrons. When an electron is in an orbit, it has a specific speed and is at a specific radius, so it has a specific kinetic (KE) and potential (PE) energy. The total energy $({\rm E = KE + PE})$ of the electron did not change while it was in this orbit. Furthermore, the electrons could not exist between these specified orbits!