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I'm trying to have a light on these questions:

  1. At what speed do the electrons orbits nucleus of an atom?
  2. Does the speed of electron vary at different shells or levels?
  3. Is that true what I heard,that the electrons of low level orbits moves in higher speed approaching c? if so,gaining speed will literally brings up its mass (by Relativity),then those electron would gain higher mass,i don't know,may be getting closer to nucleus and till growing in speed,gaining mass,getting closer.... Will it result in something like an electron falling into nucleus or so?
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First of all it is important that you do not mistake the electrons for objects that are moving nicely on perfect circular orbits. A correct answer can only be obtained trough quantum mechanical calculations.

Perhaps take a look at this older post: How is the energy of an electron-shell related to the speed of electrons in that shell?

Or

How fast do electrons travel in an atomic orbital?

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