How fast does an electron orbit the nucleus of an atom?
Electrons don’t orbit the nucleus. The wavelength of an electron is larger than the area that it might be said to be “contained” within, which means that it isn’t in any specific place at any given time. (This is a small pet peeve of mine: The Bohr model of the atom (the one with the electrons zooming around in rings) was corrected very shortly after it was introduced, yet it still persists.)
However, for some calculations it’s useful to calculate the velocity of an electron (usually one that isn’t attached to a specific atom). At those times, the velocity can vary wildly (much like that of light–what we think of as “the speed of light” is really the speed of light in a vacuum–photons slow down in anything that isn’t a vacuum, and have been slowed to a crawl under very special conditions).
Interestingly, in the Bohr model the velocity of an electron in the ground state of the Hydrogen atom is alpha * c which is the the fine structure constant (~1/137) times the speed of light. So in the Bohr model this electron would “orbit” at about 0.0073 c.
Which is a small pet peeve of mine. Photons always travel at c (at least, so far as has yet been determined). Whenever you read about light travelling slower through a medium, it is the global average velocity of a group of photons (due to absorption and re-emission) that is being referred to. But individual photons have thus far always been observed to travel at c.
Electrons don’t have definte postions in the electron shell but obviuosly they do have momentum and therefore velocity.