Re: How fast do electrons in an atom travel?

I will try to simplify the quantum mechancal answer for this.  The 
electrons in an atom move about the nucleus in a cloudlike motion  
described by the electron's wave function.  Quantum mechanics can be used 
to find the rms velocity (rms v is the square root of the average of v^2.) 
of any electron as the square root of the expectation value .   
That is v=sqrt{}.  For the ground state of the hydrogen atom, 
this is given by  v=sqrt{2B/m}, where B is the binding energy.  This QM 
answer is the same as the simple Bohr model would give, even though the 
Bohr model does not give the correct picture of how the electron moves 
about the nucleus.  
Numerically, for hydrogen, v=sqrt{2 X 13.6/511,000}c= 0.0073 c.

The innermost and fastest electrons in heavier atoms will be given by Z 
times the Hydrogen result.  The outer electrons will move more slowly as 
you go to the outer shells, and the slowest electrons in any atom will 
move at about the hydrogen speed.  So that the rms electron velocity in 
uranium will vary from about v=0.007 c to about v= 0.7 c.  The high 
velocity of the fastest electrons in heavy atoms would have to be 
corrected for relativity, but would still be about 0.7 c for uranium.  

If by "bound to other atoms" you mean covalent molecular binding, the 
calculation gets much more complicated.  The fastest, innermost electrons 
would not be affected.  The slower, outer electrons would have (I believe) 
only a slight change, probably a slight increase in their rms speed.