How fast does the electron orbit the Nucleus?

Question

Kevin................ it's a "Hydrogen" Atom LOL 8-)

Answer 1

Elementary particles (including electrons) are excitations of fields spread out over space. Sometimes it's meaningful to describe a field as having a velocity and sometimes it's not. For example a sound or light wave coming from a localized source can be thought of as travelling away from the source at a definite speed, but it's also possible to have standing waves which are not moving in any sense. For example inside an organ pipe, the sound (pressure) field is oscillating at each point in space but there's truly no motion of any waveform and the energy density is constant.

Similarly, an electron in a hydrogen atom is in the form of a standing wave. The spatial dependence of the wave field is stationary or frozen. In addition, the standing wave (for the ground state which is an S state) is purely radial and has no *orbital* component whatsoever; the orbital angular momentum is zero. So the concept of orbital motion is irretrievably meaningless.

On the other hand, it's possible to calculate the average kinetic energy just as it's possible to calculate the energy density of a standing sound wave. For an electron in a single hydrogen atom in the ground state, this is the same as the ionization energy (13.6 electron volts). That's what the total energy expectation value would be if the nucleus and its field were somehow suddenly to disappear (remembering that this is *radial* kinetic energy and the orbital kinetic energy is zero).

Kinetic energy in quantum mechanics is related to the spatial variation of the wavefunction and doesn't have anything to do with motion per se, except that for a free particle, the usual relationship between kinetic energy and speed does hold approximately, so it's possible to compare the kinetic energy of the electron in the atom with the kinetic energy of an electron which actually is moving in the usual sense - whatever that's worth:

The speed at which a free electron would have to be moving to have the same energy (13.6 ev) as the kinetic energy of the electron in the H atom, would be (1/137) c, or 2.2 * 10^6 meters/second.

Answer 2

Well, it is hard to calculate in atoms that have several electrons, due to the repulsive interactions between them. For a Hidrogen atom, an electron in 1s goes around the nucleus at a speed you can calculate like this:
(GMpMe)/r^2 = Me(Vorb^2/r)
G is Cavendish constant
M is mass
P is proton
E is electron
r is radius
Vorb is orbital velocity
And r= 5,3*10^-11m, Me= 9,11*10^-31kg, Mp= 1,67*10^-27 kg
so you get Vorb= 4,58*10^-14m/s

Answer 3

Really, really fast.