Electron Rest Mass: Inverse Proportion to Nuclear Distance

[scott_alexsk]

Why is the relation between an electrons rest mass inversely proportional to its distance from the nucleus (sp)?

Thanks,
-scott

 

[Meir Achuz]

Why is the relation between an electrons rest mass inversely proportional to its distance from the nucleus (sp)?

Thanks,
-scott

An electron's rest mass is NOT inversely proportional to its distance from the nucleus.

 

[scott_alexsk]

Really, are you sure? I was just looking up relativistic contraction which was used to explain why mercury is a liquid. The site that I looked at said that because the rest mass increases, since the atom moves faster since it weighs more, the electrons are held more closely to the nueclus causing Mercury to somewhat obtain a nobel gas configuration. This is my own phrasing but the site that I looked at phrased it in the same way that my question is phrased by saying that the distance to the nucleus in inversly proportional to the electron's rest mass.

Thanks,
-scott

 

[ZapperZ]

Really, are you sure? I was just looking up relativistic contraction which was used to explain why mercury is a liquid. The site that I looked at said that because the rest mass increases, since the atom moves faster since it weighs more, the electrons are held more closely to the nueclus causing Mercury to somewhat obtain a nobel gas configuration. This is my own phrasing but the site that I looked at phrased it in the same way that my question is phrased by saying that the distance to the nucleus in inversly proportional to the electron's rest mass.

Thanks,
-scott

You can't state this without citing the source. If not, there's no way we can check if what you read is correct, or if you interpreted it correctly.

Zz.

 

[Doc Al]

I'm sure the site meant the electron's "relativistic" mass, which depends on its speed, not rest mass. An electron's rest mass is a constant. While I would not describe it using those words, relativistic effects are important in understanding the behavior of mercury.

 

[scott_alexsk]

Thanks for clearing that up Doc. Now what would your explanation be for the electron getting closer to the nucleus?

Thanks,
-scott

 

[Astronuc]

Thanks for clearing that up Doc. Now what would your explanation be for the electron getting closer to the nucleus?

What does one mean?

Mercury has the electonic configuration - [Xe].4f¹⁴.5d¹⁰.6s², which is not anywhere near noble gas configuration. On the other hand, it does have full 5d and 6s subshells, and these electrons are more bound to the atom (nucleus).

http://www.webelements.com/webelements/elements/text/Hg/econ.html

 

[Doc Al]

Now what would your explanation be for the electron getting closer to the nucleus?

I don't think I can do it justice, but here's a handwaving, semi-classical argument that might help. Since the effective mass of the electron increases (it's "relativistic" mass), and since the radius of a Bohr orbit is inversely proportional to the mass of the electron, the average radius of the "orbit" will be smaller.

Anyone up on relativistic effects in atomic physics, please chime in. (Unfortunately, the real answer is to use relativistic quantum mechanics. I'm not even sure that's practical; often relativistic effects are just approximated by a potential and non-relativistic quantum mechanics is still used.)

The only reference I could find on the web (in 10 minutes) is this:
Why is Mercury Liquid? Or, Why Do Relativistic Effects Not Get into Chemistry Textbooks?
L.J.Norby
J.Chem.Ed. v.68 p.110-113 (February 1991)

I don't have that journal available to me, so I don't know what it says.

Edit: Astronuc, please straighten me out if I'm too far off.

 

[scott_alexsk]

No Astronauc, the idea is that the relativistic effects caused by the fast speed of the mercury atoms causes somehow, according to several sources, the electrons to get closer to the nucleus and cause the atom to be more stable as it is and have less of an affinity for electrons. The question is how the increased speed effects the electrons in a way that causes it to get closer to the nucleus.

PS: Doc Al that was about the argument I have heard before.

Thanks,
-scott

 

[scott_alexsk]

Here are some links

http://www.madsci.org/posts/archives/may97/862179191.Ch.r.html"

http://antoine.frostburg.edu/chem/senese/101/inorganic/faq/why-is-mercury-liquid.shtml"

-scott

 

[Astronuc]

Scott, thanks for the references.

the idea is that the relativistic effects caused by the fast speed of the mercury atoms causes somehow, according to several sources, . . .

Mercury atoms are not traveling at relativistic velocities. They are not even vibrating at relativistic velocities. It has to do with the s-electron wave function.

There is also a discussion here - http://www.hull.ac.uk/php/chsajb/heavy_ho/5dmetals.html

All s-electrons are affected in this way since they spend appreciable time near the nucleus. The contraction of p-orbitals and particularly d and f-orbitals is somewhat less as the time spent near the nucleus decreases as the orbital angular momentum increases.

As the electrons are pulled closer to the nucleus by this effect, they are stabilized and harder to ionize.

The s-electron wave functions show a higher probability of being 'in the vicinity' of the nucleus.

I'll have to look elsewhere to see if there is a better discussion of the relativistic QM.

 

[scott_alexsk]

Thanks astronauc,

By the way does anyone know why heavier molocuels/atoms travel at faster speeds then lighter ones?

Thanks,
-scott

 

[Doc Al]

It's not the heavier atoms that move faster, it's the inner (s) electrons of the heavier atoms that must move faster to avoid falling into the nucleus.

 

[imaginary]

in heavier atoms.. we see the nucleus is heavier and has charge opposite to that of an electron.. and since all the protons plus the neutron together attract the electron.. the electron faces a larger force on it.. too overcome this gravitaional and electomagnetic interaction.. the electron must have more centrifugal force.. which is obtained by revolving faster.

 

[Astronuc]

in heavier atoms.. we see the nucleus is heavier and has charge opposite to that of an electron.. and since all the protons plus the neutron together attract the electron.. the electron faces a larger force on it.. too overcome this gravitaional and electomagnetic interaction.. the electron must have more centrifugal force.. which is obtained by revolving faster.

I believe that one will find gravitational forces between electrons and nuclei are insignificant. There is neglible attraction between neutrons (neutral, or no charge), but there is significant coulombic interaction (EM) between protons (+ charge) and electrons (- charge).

By the way does anyone know why heavier molocuels/atoms travel at faster speeds then lighter ones?

In gases, molecular speeds are governed by statistical thermodynamics - http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/disfcn.html.

Other distributions are Bose-Einstein and Fermi-Dirac. See link.
or http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/statcn.html

For a given energy (temperature) of a gas the molecular speed is lower for heavier atoms/molecules, or in liquids and solids, the vibrational freqency and amplitudes are lower for heavier atoms/molecules.

http://hyperphysics.phy-astr.gsu.edu/Hbase/kinetic/kintem.html

 

[Gokul43201]

Thanks astronauc,

By the way does anyone know why heavier molocuels/atoms travel at faster speeds then lighter ones?

By asking this question, you have shown that you really didn't understand the explanation. Go back, and read the first 2 lines in Doc Al's earlier post #8. If there's some part of this that you don't understand, ask.

 

[scott_alexsk]

Well I think I understand all but that part of it. Tell me if this interpretation is correct. Because of the high speed of the Mercury atom, the relativistic mass of the electrons increases. As a result the energy those electrons have is no longer adequate to keep them in their current energy state, so they fall back, towards the nucleus, to a lower energy state. Now if this is correct than the only missunderstanding that I need to get through is why heavier atoms travel faster than lighter ones. Gokul, I think I know what you are getting at. When I am referring to heavier atoms, I am implying a heavier rest mass, not heavier relativistic mass. I know that the mass of an atom is not only determined by its speed.

Thanks,
-scott

 

[Astronuc]

Well no according to one of the articles, the other did not go into much detail, because the rest mass of Mercury is greater it moves faster, . . .

Please provide the reference (url, or citation). The 'electrons' move faster, not the atom. It is the 'relativistic mass' of the electron which increases with speed.

The motion of the atom is more or less independent of the motion of the electrons.

I need to do some more digging around, but I think the property of mercury has more to do with the pairing and filling of the d and s electrons, particulary the 6s, than it does 'relativistic correction' (or contraction).

 

[ZapperZ]

Well I think I understand all but that part of it. Tell me if this interpretation is correct. Because of the high speed of the Mercury atom, the relativistic mass of the electrons increases. As a result the energy those electrons have is no longer adequate to keep them in their current energy state, so they fall back, towards the nucleus, to a lower energy state. Now if this is correct than the only missunderstanding that I need to get through is why heavier atoms travel faster than lighter ones. Gokul, I think I know what you are getting at. When I am referring to heavier atoms, I am implying a heavier rest mass, not heavier relativistic mass. I know that the mass of an atom is not only determined by its speed.

Thanks,
-scott

I don't kow why you keep harking to having the atoms as having "high speed". It doesn't say that in both of the links you cited. They do, however, argue about the relativistic corrections to the ELECTRON orbitals. This is not the same as the atom moving with relativistic speed. One could classically have the atoms at rest in your reference frame and the explanation would still work.

Zz.

 

[Gokul43201]

I need to do some more digging around, but I think the property of mercury has more to do with the pairing and filling of the d and s electrons, particulary the 6s, than it does 'relativistic correction' (or contraction).

While the fully filled 6s subshell is itself important, the relativistic correction to the 6s energy is quite significant and is largely responsible for the difference between Hg and its lighter relatives, Zn & Cd.

A rough estimate of this correction is not too hard to do. The number I get says that the energies in Hg are about 30% lower than the non-relativistic estimates.

 

[scott_alexsk]

You are right. Sorry it was a misunderstanding on my part. OK let me try again. Since the 6s electrons are traveling at a higher speed because they occupy a higher energy state, relativistic mass increase curbs this extra energy and causes the 6s electrons and everything bellow it to move into a lower energy state closer to the nucleus.

Thanks
-scott

 

[Hans de Vries]

Why is the relation between an electrons rest mass inversely proportional to its distance from the nucleus (sp)?

Thanks,
-scott

An hydrogen atom with the electron replaced by a muon (the electron's
heavier brother) is a factor 206.7682838 times smaller as a normal hydrogen
atom because the muon is 206.7682838 times heavier as the electron.
This has been experimentally verified.

It's because the rest frequency of the muon is higher by the same factor
that the de Broglie wavelength is shorter by this factor and that's why the
solutions of Schroedinger's equation are also smaller by this factor.
(The mass m of the electron occurs in Schroedinger's equation) Regards, Hans

 

[Gokul43201]

A rough estimate of this correction is not too hard to do. The number I get says that the energies in Hg are about 30% lower than the non-relativistic estimates.

I now think the above estimate for the relativistic correction is wrong - I didn't consider screening! If I account for screening using Slater's approximation, I get a number closer to 5%. In addition, I now think that, possibly "as important" as this relativistic correction, is the fact the the 6s electrons are very poorly screened by 5d and 4f electrons - the same phenomenon that is responsible for the Lanthanide contraction (and related effects).