Tag Archives: chemistry

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

Causality

(i) Homunculus reading a book "Organic Chemistry". He: "It says, acetone is a planar molecule because the central carbon atom is sp2 hybridized." (ii) Homunculus: "Maybe I can generalize this finding." (iii) A car is going fast onto a huge rock. "There is no obstacle in my way because I'm not braking."

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(c) M.Klammler 2010, Creative Commons Share Alike.

What You See in the Mirror

(i) Homunculus with stereoidal hair in front of a mirror: "Aahw, how did my hair get like that?" (ii) He breakes a bond to an ethyl substitute and inverts the stereocenter. (iii) Homunculus shown from the front: "Now wait. Crap, I think I got it wrong again."

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(c) M.Klammler 2010, Creative Commons Share Alike.

Another Pitfall

I was never fully satisfied by the usage of dimensions like m6mol-2s-1 (or worse) for rate constants in chemical kinetics. I used to think there should be a way to express this by a dimension free coefficient and use mole fractions (xi) instead.

After many pages of paper written I have to admit that this is not possible. It is due to the fact that kinetics and thermodynamics are not fully interchangeable and the concept of molar fractions shows to be rather useless in kinetics. An obvious example is given by a container filled with two gases to react. The rate is – assuming constant temperature – determined only by the chance of two molecules colliding in a certain amount of time. This probability is not affected whatsoever by adding a third gas to the container. In terms of molecular fractions this couldn’t be expressed because increasing the overall amount of substance will decrease the molar fraction of the components. Hence, there is no way beside using concentrations even if they are not as “nice” as fractions could be…

A minute ago I mailed an article [1] to a professor of mine in which I discussed the topic in more detail. It is written in German but I’ll still attach it to this post. However, I just got aware that I didn’t mention the case of a second order reaction between molecules of the same substance in that work. I hope that it still might be useful to some of you somehow under certain kind of circumstances.

[1] Einheiten von Reaktionsgeschwindigkeitskonstanten

Solvent Drunk Ideas on a Common Afternoon

A couple of things that are quite fun to try on a dragging afternoon in the lab:

  • Explain how a laboratory centrifuge works without using the term “centrifugal force”.
  • Give a proper explanation why and how a cooling mixture of dry ice (CO2(s)) in isopropyl alcohol works pretty good while dry ice in water gives a rather poor coolant.
  • Determine if (or under which circumstances) it is a good idea to externally warm a fractionating column including a Vigreux condenser with a fan.
  • Think of what is going on when the temperature of an oil bath of a distilling apparatus is increased while it is already above the boiling point of the liquid inside.
  • Give proper explanation of the processes linked to a rotary evaporator. (Pressure is of special interest.)
  • Pour ether on your desk and set it on fire.