[MikeS (OP)]

An electron orbits the nucleus of an atom presumably at relativistic speed.  As an electron has mass why does it not exhibit an increase in mass as would be expected from relativistic speed?

[Phractality]

The electron does have relativistic mass, but its orbital speed is such a small fraction of the speed of light that the difference is very small. The rest mass of an electron is 9.11×10^−31 kg, which is equivalent to .511 Mev of energy.
Ionization energies of electrons are seldom more than 100 ev. The inner orbitals of heavy atoms have the higher binding energies.

So the mass of an electron in orbit may vary a few hundredths of a percent, depending on which orbital it occupies. That is a significant fraction of the electron's rest mass, but the mass of the nucleus is thousands of times greater than that of the electrons. So the mass of an atom with varies only a few millionths of a percent because of electrons jumping between orbitals.

If you add up the rest masses of the electrons, protons and neutrons for a given atom, you get a smaller value than the mass of the atom, but the relativistic mass of the electrons is a tiny fraction of the difference.

[Bored chemist]

Binding energies for inner electrons in heavy nuclei can get to about 100,000 eV (0.1MeV)
http://hyperphysics.phy-astr.gsu.edu/hbase/tables/kxray.html#c1
The "speeds" of electrons near the nuclei can get quite close to c.
The electrons do have higher masses but the effect would be hidden by the weight of the nucleus in most cases.

[damocles]

The relativistic mass increase of inner electrons does become important for chemists trying to do calculations of chemical bonding and molecular structures or spectra. It is generally recognised that any calculation involving elements around the atomic number of iron (Z=26) or greater need to allow for relativistic effects to produce results that are at all realistic. For the usual C,H,O,N,S type calculations relativitty corrections can safely be ignored.

[MikeS (OP)]

Thanks guys, interesting.


damocles
Why does the element affect the result?  I would have thought an orbiting electron would always orbit at the same speed for any given orbital.  Presumably, it is related to mass?

[damocles]

Mike, it is a very crude picture but think of the electrons as though they were tiny particles orbiting a bohr atom.

A hydrogen electron has a charge of -1 and it is orbiting a proton with a charge of +1, so the central force is 1 unit.

The innermost electron of an iron atom is still -1, but its "sun" has a charge of +26, so its orbit will be much closer and much faster as it experiences a central force 26 times as large.

OK -- with the new quantum theory of the atom this is a lousy description, but the same underlying principles will rule.

(If you are interested in Philopsophy of science, this answer displays important differences between the ways that physicists think and the ways that chemists, like me, think).

[MikeS (OP)]

damocles
Thanks.

[Phractality]

Mike, it is a very crude picture but think of the electrons as though they were tiny particles orbiting a bohr atom.

A hydrogen electron has a charge of -1 and it is orbiting a proton with a charge of +1, so the central force is 1 unit.

The innermost electron of an iron atom is still -1, but its "sun" has a charge of +26, so its orbit will be much closer and much faster as it experiences a central force 26 times as large.

OK -- with the new quantum theory of the atom this is a lousy description, but the same underlying principles will rule.

(If you are interested in Philopsophy of science, this answer displays important differences between the ways that physicists think and the ways that chemists, like me, think).

Ignoring quantum effects, at time frames longer than a few orbits, orbitals are like clouds of smeared electric charge. If the inner orbitals are contained within the outer orbitals, you can apply Newton's shell theorem. So the attractive force for the inner orbitals comes from the total charge of the nucleus, while the outer orbitals are only attracted to the net charge of the nucleus minus the inner electrons. If the outer shell has two electrons, they are only attracted to a net positive charge of +2.

However, not all orbitals are spherical, and there may be some overlapping, so the picture is much more complex that that.

[lightarrow]

An electron orbits the nucleus of an atom presumably at relativistic speed.  As an electron has mass why does it not exhibit an increase in mass as would be expected from relativistic speed?

There is no need of relativistic mass. The atom *invariant mass* is slightly greater because of the electron's motion (because of its kinetic energy), with respect to the imaginary case of a non moving electron (at the same distance from the nucleus).
A simpler example: a spinning hard disk in a PC has rest mass slightly greater than the same when stationary.

[granpa]

http://en.wikipedia.org/wiki/Bohr_model

When Z = 1/α (Z ≈ 137), the motion becomes highly relativistic, and Z2 cancels the α2 in R; the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei

[granpa]

http://en.wikipedia.org/wiki/Old_quantum_theory#Relativistic_orbit

Arnold Sommerfeld derived the relativistic solution of atomic energy levels

This solution is same as the solution of the Dirac equation