Gravitation and the Relativity of Gold

The other day I was taught about some basics of relativistic effects in chemistry. I’m not at all into relativistic quantum mechanics (yet) but I was still amused about the following statement:

In heavy atoms, the electrons approach the speed of light which makes them gain relativistic mass. This causes additional gravitational attraction between the electron and the nucleus which leads to a contraction of the electron’s orbital.

It is obvious that gravity is by far too weak to play any role on this scale. However, “obvious” is not scientific so here I’ll supply the respective calculation.

In Bohr’s picture, the electron circles around the nucleus on a stable pathway that is determined by the equilibrium of Coulomb attraction and centripetal force. The former is given by

Equation 1

where Qe is the elementary charge, ε0 the electric constant (vacuum permittivity), Zeff is the effective nuclear charge “seen” by the electron and r the nucleus–electron-distance.

The gravitational force on the other hand would be given by

Equation 2

with the gravitation constant (Newtonian constant) G and the mass of the electron and the nucleus me and mn respectively. Or — if the electron’s mass increases by Δme — an additional gravitational force of

Equation 3

emerges. Comparing this with equation 1 we find that

Equation 4

On Wikipedia (ref. 1) I found that the 1 s electron in gold (which is considered to be one of the elements with the highest relativistic effects) “travels at a speed” of 58 % c. This means its mass increases by

Equation 5

which is in agreement to the value claimed by the person mentioned above. Putting this in equation 4 we obtain

where I have used the standard atomic mass of gold of 197.0 amu (1 amu = 1.661E-27 kg) and the effective nuclear charge for the gold 1 s electron found at [2].
Using G = 6.674E-11m3 kg-1 s-2, ε0 = 8.854E-12 A s V-1 m-1, Qe = 1.602E-19 C and me, rest = 9.109E-31 kg we finally get

making it absolutely clear that gravity is completely irrelevant.

On the other hand the centripetal force is given by

Equation 6

Letting this equal the Colombian force and solving for r we find

Equation 7

If we keep everything else constant but once put me = me, rest and once me = me, rest + Δme the two results will differ by

This is — accepting this simple but useful picture of an atom — the relativistic orbital contraction for the 1 s orbital of gold.

References

[1] Relativistic quantum chemistry. (2010-10-27). In Wikipedia, The Free Encyclopedia. Retrieved 05:27, 2010-11-14, from http://en.wikipedia.org/w/index.php?title=Relativistic_quantum_chemistry&oldid=393187449
[2] Effective nuclear charges for gold. (2010). In WebElements: the periodic table on the web. Retrieved 05:28, 2010-11-14 from http://www.webelements.com/gold/orbital_properties.html

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