Electron waves, electron momentum and atomic size
Electric potential energy
Electric potential energy of an electron at distance r from a hydrogen nucleus = -e2/4or = ke2/r
This is NEGATIVE. You have to put energy into the atom to remove the electron from it. The electron
is held in an electrostatic potential well in just the same way as a stone on the Earth’s surface is held
in a gravitational potential well of the Earth.
Electron wavelength and momentum
Electron wavelength =  = h/mv
Electron momentum = mv
So for a large wavelength we must have a small momentum
deBroglie standing waves
deBroglie standing waves fitting into the ‘box’ (atom).
The smaller the atom the smaller the wavelength of the standing waves that could fit into it.
It follows that if the diameter of the atom is smaller the wavelength of the electron wave must also
be smaller and since = h/mv the momentum of the electron must be larger.
Electron kinetic energy
The kinetic energy of the electron EK = ½ mv2 = ½(mv)2/m = ½ [(electron momentum)2/m]
A larger momentum will therefore also mean a larger kinetic energy.
Electron total energy
Electron total energy = electron kinetic energy + electron potential energy
Minimum size for an atom
At some critical radius (rC ) the kinetic energy is equal to the potential energy and so the total energy
of the electron is zero. This means that it is just about enough energy to escape from the potential
well of the atom.
At this value: ½(mv)2/m = e2/4or and so ½(h/)2/m = e2/4orC
But rC = /4
And so ½(h/4rC)2/m = e2/4orC
Therefore: Minimum or critical radius (rC) = h2o/8me2
Using h= 6.63x10-34 Js, o as 8.84x10-12 Fm-1 , m = 9x10-31 kg and e = 1.6x10-19 C we have:
rC = (6.63x10-34)2xx8.84x10-12/8x9x10-31x(1.6x10-19)2 = 6.62x10-11 m = 0.0662 nm.
This is then the smallest possible radius for a hydrogen atom. If we try to make the atom smaller we
have to ‘squeeze’ the electron wave. A shorter electron wave means a greater kinetic energy and so
the electron will break free from the atom.