Tag Archives: relativity

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.


[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

Relativistic Aspects

Nuclear fusion of hydrogen brings comfort and artistic inspiration. (George Herbert McCord: 'Florida Sunrise' 1880)

Nuclear fusion of hydrogen brings comfort and artistic inspiration. (George Herbert McCord: 'Florida Sunrise' 1880)

Oxyhydrogen combustion in the ignited second stage propelles a Saturn V with the Apollo 11 vehicle moonwards.

Oxyhydrogen combustion in the ignited second stage propelles a Saturn V with the Apollo 11 vehicle moonwards.

The Equivalence of mass and energy surely ranks among one of the things most frequently misunderstood. Especially when talking about chemical issues, things are often not pronounced correctly. Today it appeared that — for the first time in my life — I heard a chemist mentioning the mass defect within the discussion of a chemical reaction. Unfortunately it was, frankly spoken, wrong. In the next few lines I’d like to discuss some basic applications of Einstein’s famous formula E = m c². Energy and mass are related by the square of the speed of light (somewhat about 3.00E8 m/s).

Considering the helium atom (2 protons, 2 neutrons and 2 electrons) and deuterium (1 proton, 1 neutron and 1 electron) we find that the first has a mass of 4.003E-3 kg/mol while the latter’s is 2.014E-3 kg/mol. (That is 4.028E-3 kg for 2 mole.) Where did the difference of 2.5E-5 kg/mol go? If we use Einstein’s formula we find that the observed mass defect is related to a loss of energy of 2.3E12 J/mol. This is almost exactly the energy that is released by the fusion of two moles of deuterium to one mole helium. The origin of this energy is the stronger (i.e. less energetic) nuclear bonding in the Helium core. We do the same thing with the mass of Uranium 235 and its respective fission products to easily calculate the energy of the reactions going on inside a nuclear reactor.

Next, let us have a look at the oxyhydrogen or “knallgas” reaction where hydrogen and oxygen gas are combusted to yield water. The H2 molecule (hydrogen gas) has a mass of 2.016E-3 kg/mol and O2 3.200E-2 kg/mol. From experiments we know that burning two moles of hydrogen gas and one mole of oxygen gas to one mole of liquid water releases 2.858E5 J. This is seven orders of magnitude bellow the value we calculated for the fusion of helium before. However, we can use Einstein’s formula again and calculate the mass defect as Δm = ΔE/c² This tells us that — after condensing and cooling down to initial temperature — the mole of water is 3.18E-12 kg lighter than was the mixture of gas before we ignited the explosion. Now obviously (within our today’s instruments) there is no way to make this tiny mass accessible for measurement. Anyhow, this is no reason to deny its existence.

But of course the equivalence of mass and energy is not limited to bonding energies. When accelerating particles close to the speed of light an increase of mass is observed that is equivalent to the (relativistic!) kinetic energy of the particle. Hence an photon traveling at the speed of light gets heavy enough to kick out electrons from a charged metal plate. This is known as the photoelectric effect and was first explained by Albert Einstein who was honored with the Nobel price for that. But again: This is not an exotic effect but appears to any object that is accelerated. You can cheat on your next deal buying exactly one kilogram of gold at the market place and reselling 1.000000000000015 kg of gold in a bus traveling at a speed of 130 km/h. (Unfortunately, the energy costs for accelerating the bus are several orders of magnitude higher. But never mind.)

Today I heard a chemist talking about a chemical reaction of isolated molecules in the gas phase. Will their mass decrease after they reacted? We have to be carefully. First, the product is formed in an excited state. Until they give off energy via emission of electromagnetic waves or interaction with other molecules they store the energy gained from the reaction within vibrations, rotations or velocity. In that case, they are energetically equal to the species they were formed from and hence there is no mass defect. But it will appear clear as day as soon as they fall back to lower energetic states.