OT: "Deep" question for chemistry or physics majors - long

[Ted King]

Usually I can find answers to my science questions with searches on the internet, but there is one that came up recently that I haven't been able to hunt down. A little background first: I'm a middle school science teacher at a school for gifted kids - so, of course, the school uses all Macs for computers :-). I teach all the basic sciences even though my degree is in biology. At the middle school level my lack of having a lot of college coursework in physics and chemistry isn't usually a problem, but since I'm dealing with gifted kids I do like to take them as far as they are able to go and occasionally that leads me out of my comfort zone. Last week, we were having a discussion where one of the students asked a question that definitely was outside my comfort zone.

We had gone over the history of the development of atomic theory - all the way from Democritus to Dalton, Thomson, Rutherford and Bohr. As part of the explanation for why electrons are only at the energy levels they are I talked a bit about de Broglie's wave theory of matter - especially with respect to electrons and that only complete wavelengths of electrons around the nucleus were possible, hence, electrons could "inhabit" only certain energy levels. Then later, when I was having the kids try to analyze and explain the relationship of the number of protons in the nucleus, size of atoms, electronegativity and first ionization energy across the periods and down the groups on the Periodic Table, one of the students hit me with a question that stumped me. What she asked was that if the atoms were getting smaller across the period, yet all the valence electrons were in the same energy level, and if the electrons must be only at energy levels because they must have "complete" wavelengths, then how could the atoms be getting smaller and still have the "proper" wavelength for that energy level?

I thought that was an exceptionally astute question for a 13 year old. I admitted to her that I didn't know. I told her I'd try to find out but that my initial speculation was that as the greater number of protons are added to the nucleus across the period, increasing the attraction on the electrons, maybe the wavelength of the electrons at the energy levels shifts (same number of waves but with shorter wavelengths?) so that they can still "fit" full waves around the nucleus even though they are moving closer to the nucleus. I tried to make it clear to her that that was a pretty uninformed hunch on my part, though. That got me wondering, in addition, if a possible shifting of wavelength of the electrons in an energy level as they got "pulled" closer to the nucleus across a period has something to do with differences in the emission and absorption spectra of different elements (something we also discuss).

Anybody know the answer? Or know where I can find the answer? Or is the question itself misplaced?

[ztirffritz]

I'm sure that physicists and chemists are a little on the geeky side so they probably hang out on the internet somewhere. I'd suggest finding that place and asking there. There might be a few people on the forum who could answer that question, but I doubt it. I'm curious about the answer though. I remember asking the same question in my physics class (though I was older than your student) and received about the same answer. Through college it was never answered. I just figured that the answer to that question was the part of difference between being an engineer and being a physicist/chemist.

[pbarra1]

Well - cant help u with the wavelength part - I think that a quantum level view of the atom at that age level is inappropriate, but that is just my opinion. Atoms shrink as u move across the table in a particular period because the nuclear positivity is increasing as u add more protons (+ charges) to the nucleus. Greater nuclear positivity leads to more vigorous attractions for the electrons (- charges) which "shrinks" the atom a tiny bit. There is a lot more to the wavelength part u were talking about - some of it is still under debate in the journals.

[Ted King]

ztirffritz, I figured it would be a long-shot for somebody on the this forum to know the answer to the question, but I thought it would be worth and try. But that is why I also asked if anyone knew of good place to try to find the answer - I don't know of any web sites where I would be able to ask that question. I thought maybe somebody here might know of a web site where those kinds of questions might be answered.

pbarra1, we don't really delve much into quantum mechanics - just enough to have the kids have some idea of why there are energy levels for the electrons. Whenever possible, if I think the kids can understand it, I try to give them the explanation for - or have them try to figure out for themselves - why the "facts" of science seem to be what they are. These are gifted kids so they can understand quite a lot. They work in discussion groups to identify the patterns of number of protons in the nucleus, first ionization energy and electronegativity as you go across the period. Then the groups discuss and try to explain why those patterns come about - especially trying to relate all of those factors together in the explanation. Many groups are able to come up with, on their own, the "opposites attract" explanation you gave, but as I circulate around there are a few groups where I have to drop a couple of hints and they usually fit the pieces together from there. One of the primary purposes of all that is so when we get to discussions about atomic bonding that they can apply what they understand about first ionization energy and electronegativity to grasp why there is a continuum from pure covalent bonding to polar covalent to "so-polar-covalent" that we call it ionic bonding. And then they can relate that to things like why alkali metals ionically bond with halogens but non-metals tend to covalently bond with each other. That is how I transition from the unit on the Periodic Table to the unit on chemical reactions. I do believe most of the students understand it.

[DharmaDog]

My wife has a PhD in Chemistry, specializing in nanoparticles and is a professor at Texas A&M. I'll see if she knows, but she's under the weather right now.

[pbarra1]

I am glad that there are more teachers out there providing opportunities for students to create their own knowledge and understanding as opposed to just being receptacles for teachers to pour information into. Keep up the good work.

[TheTominator]

You were on track with the wavelength of the electron getting smaller with increasing atomic number.

If you focus on just the innermost spherical s-shell of the electron cloud around an atomic nucleus, as you move up the periodic table, the shell is kept closer in toward the center. As you increase the atomic number and as you add electrons into a partially full s-shell, the new electrons occupy locations at the same distance as their peers in the s-shell, but the entire s-shell is now a smaller radius. It is a smaller radius because it is in a deeper potential energy well.

In the Bohr model, the energy required to remove an electron in a particular shell from an atom goes as the square of the atomic number, Z, over the square of the index, n, of the shell.
E_n = Z^2 *R_e / n^2
This is the depth of the energy well.

For the innermost s-shell, the index n=1. So for n=1,
E is proportional to Z^2.
An atom with a more protons will have its innermost s-electrons in a lower energy state (a deeper well) than an atom with less protons.

This is relevant because the probability wave function gets "tighter" with a deeper energy well. The DeBroglie wavelength for the electron is
lambda = h/p (where h = constant and p = momentum)
A higher (magnitude) energy electron has a higher momentum and thus a smaller wavelength. A smaller wavelength means the average radius for the electron around the atom is smaller. The shell shrinks as you add protons to the nucleus.

Now examine how these shells are populated. The innermost s-shell is mostly empty. Add an electron (and a nuclear proton). That electron is at the same energy state as its peers, but the s-shell itself decreases in radius a little. Add another electron and the same thing happens. The atom decreases in size as you go right along a row of the periodic table until the shell fills up.

There are other effects such as electron screening where the electrostatic repulsion from an s-state will counter the pull of the nucleus on a p-state. But that is beyond the scope of the initial question.


Edit:
Just in case anybody was wondering, I wasn't a "Moron Major" in college.

[Craig]

Chemist here. The answer can be summed up this way:







The reason that I know this is because this man told me so.



http://chem.berkeley.edu/people/faculty/...rauss.html

Now if you don't understand that then I am guessing 13 year olds won't either. I always had a lot of respect for teachers who just flat out said, "I don't know. Why don't you come by later and we can look it up together." Don't feel bad, I didn't get it then and still don't get it.


Craig

[Filliam H. Muffman]

You might want to look at The Mechanical Universe series for a little more background in basic physics for that age. It covers emissions briefly.
http://en.wikipedia.org/wiki/The_Mechanical_Universe

Looks like TheTominator covered it pretty well. It is one of those topics that gets really complicated the more you dig into it.

[Ted King]

TheTominator, I believe I followed your excellent explanation. "h" is Planck's constant, right? I think the only thing that isn't too clear is the bit about the s-shells (s-orbital?) - I thought s-orbitals were limited to one electron pair. But the details of that don't seem to effect the underlying logic of what you are saying. I'm not sure how I'll be able to formulate an answer to my student's question without losing too much of the important nuance; this may be one of those times where I'll just have to say "you'll have to wait until you have more background before you can have your excellent question answered correctly".

If you don't mind continuing to share your knowledge, does the greater energy of the electrons as their wavelength (and, hence, the radius of the atom) decreases have an effect on the emission and absorption spectra of atoms of elements across the period? It seems like that would explain part of the difference beyond just the addition of more electrons as you go across the period. For example, is part of the difference in the emission spectrum of lithium and beryllium due to the smaller radius caused by the attraction of the extra proton as well as the additional electron?