High School Questions I Cannot Answer

It happened last year when biology got seriously cellular and molecular.

It's happening again with chemistry: some questions I cannot answer:

Why does the 4s orbital fill before the 3d orbital, anyway?

What is it with Chromium--it has only one electron in its 4s orbital, while both its predecessor Vanadium and its successor Manganese have two?

Similarly, Niobium, Molybdenum, Technetium, Ruthenium, Rhodium, Silver, Platinum, Gold, plus the whole horde of f-orbital filling elements--what's going on with them?

The periodic table in the book says that elements 110 and 111 have only one electron in the 4s orbital. Do we know this from experiment? Do we know this from quantum-mechanical calculation? Or is it just a guess from the behavior of Platinum and Gold?

I'm still further handicapped by the fact that I *never* understood the chemistry of metals at all...

Comments

[cb]

re: 4s orbitals

A real chemist can correct me, but it has always been my impression that the rules for filling orbitals are not really scientific rules based on first principles, but rather rationalizations designed to fit empirics.

We can observe Argon (?) and rationalize that the spherical shape of the s orbitals results in electrons being located closer to the nucleus (rather integrating over the likelihood of the electron) blah blah blah...but I think the truth is that we just know that it is so and we have a best story and the story then becomes the truth. But as I recall, it is a pretty close call between the two orbitals.

[cb]

re elements 110 and 111, I would count the electrons in the table and make sure they add up. That sounds fishy and one would think (again I am not a chemist) that being unable to verify through experimentation (my guess), they would list the shells in a non-controversial way. I find it very hard to believe that if you trust what yo are told as an undergrad that 7s is of lower energy than the second in 4s.

On the overall topic of the chemistry of metals -- as has been said in organic..."if you don't understand this, you will never understand metals and you might not be cut out for chemistry."

[Raghav Krishnapriyan]

The 4s orbital is simply of lower energy than 3d; it's empirical fact. They're very close though; nearly degenerate. If you look at a 4s orbital superimposed on a 3d orbital, you can sort of see this.

Chromium has only one electron in its 4s orbital because half-filled orbitals are especially stable. So rather than having 4s2 3d4, it chooses to move an electron from the 4s orbital to have two half-filled orbitals instead. This is the case for a lot of other "exceptions" as well.

The rules get a bit dodgy after Z > 100. Suffice it to say that these results come from analyzing atomic spectra, and a much more complicated model than the basic "Slater's rules" that are taught at this level is needed to correctly predict them.

[Steve]

Here's my understanding of things, as a (sort of) real chemist (graduating in December, if I can get that $!*# dissertation written):

If you look at the orbital wavefunctions, p- and d-orbitals don't have any electron density located at the nucleus (the wavefunction smoothly decays to zero there), whereas s-orbitals do. Consequently, electrons in an s-orbital more directly "feel" the nuclear charge than p- and d-orbitals do. You can think of Z-effective as a kind of weighted average of the charge that an electron experiences, added up over the whole extent of the wavefunction. The part of the s-orbital at the nucleus sees the full core, while the parts further out are "screened" by the inner electrons. (If you're really far away, for example, you don't see a nucleus of charge +Z and 2 1s electrons of charge -1, you just see some object with net charge Z-2.)

What this means is that you can have s electrons at lower energies than p or d, but it can be a tricky balance depending on the exact shape of the wavefunction (so 4s and 3d are a close call, because the 4s is on average located further away than the 3d, but there's that part close to the nucleus that the 3d doesn't have...). Usually the 4s wins, but then we get to cases like Chromium...

For Chromium, the configuration is 4s1 3d5, instead of the 4s2 3d4 that you might expect. The reason for this has to do with the quantum mechanical concept known as exchange, which doesn't get covered really until undergrad P-Chem (Chem 120 at Berkeley). Essentially, electrons are indistinguishable (you can write out all of these electron configurations, but you can't actually label a specific electron and subsequently find it at any later time), and there is an energy to be gained by having them be "similar" to each other; the idea is that in the 3d5 case, you have one electron in each d sub-band, all with spin pointing in the same direction. These electrons are not distinguishable, though, so one way to think of it is as each electron occupying each of the 5 d-orbitals 20% of the time. There is an energy gain associated with effectively increasing the volume of each electron in this fashion, and for Cr that beats out the Coulomb repulsion from putting two electrons into the 4s orbital. (Note that this doesn't beat out the 4s electron's close approach to the nucleus, though, which is why Vanadium doesn't have a 4d0 3d5 configuration.)

As for the elements further along in the periodic table, all I can say is that things get very hard very quickly. Ideas which are important here are the spreading out of exchange-able electrons as before, but also very bizarre things start happening to the shapes of orbitals due to essentially the influence of relativistic effects. A fun undergrad p-chem question is to calculate the average kinetic energy of, say, a 1s electron in Au, assuming some sort of harmonic oscillator type picture (conservation of total energy, but it flows from potential to kinetic and back again). I forget whether or not you get that the electron is travelling faster than the speed of light, but it's certainly close. The point is that all of your orbital shapes based on non-relativistic quantum mechanics have to get thrown out the window, and you have to start recalculating the wavefunctions using a more sophisticated approach. Weird things start happening as a result.

I've got no clue on the 110 and 111 elements, though. The way these things usually go is that they try and make some compound out of them in the (brief) time those elements are around and then do spectroscopy on that compound (my memory suggests that halogenated compounds are usually what's done, but I'm really rusty on this), and the types of things you would usually measure are electronic transitions and perhaps deflections in a magnetic field, which can give clues about how many unpaired electrons there are and things like that. I know there's really sophisticated theory behind this sort of thing that I just don't have a feel for, though.

Hope this is at all helpful/intelligible!

[Rosalinda]

I've always thought of this problem as one of these examples of simplication that one finds in high school textbooks. The orbitals DO fill in the order from lowest energy to highest energy.

The relative energy of the orbitals changes as electrons are added. The difference in energy between 4s and 3d orbitals is rather small, and the order shifts depending on how many electrons are in the orbitals.

Certainly the quantum-mechanical calculations could be done for elements 110 and 111. www.webelements.com states that the electron configuration is based on extrapolations from Pt and Au. Experimental confirmation for such short-lived elements can be troublesome.

The electronic structure of metals is better modeled by a band model than the discrete orbital approach favored by high school textbooks.

Try this link for a brief explanation.

hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html

[Brautigan]

huh?

[Tom Slee]

I won't (ie, can't) add to what Steve said above, except to say two things:
1. the whole idea of an orbital is an independent electron model. As soon as interactions (that's externalities on this blog I guess) get important then the orbital description of atomic and molecular electronic structure becomes an aproximation.
2. the relativistic thing is indeed important for the heavier elements. It is the main reason why lead forms two bonds while carbon and silicon (in the same period) form four.
and 3. (because no one expects the Spanish Inquisition) just a note to those who say it is just an empirical fact -- not true! The results can be calculated and explained, not just observed.

[Charles Kinbote]

There is a simple (and correct) answer to all of these questions:

They are the way they are because that's how God made them.

[Robert P.]

Rosalinda - in these sorts of examples, we are talking about the electronic structure of _individual_ metal atoms and ions, not of bulk metals, so that band structure doesn't really come in. Other than that your comment is mostly correct. Let me build upon it some - it'll give me an opportunity to tell Brad about an odd historical linkage between the fields of Theoretical Chemistry and Economics.

One must remember that the whole concept of "orbitals" in anything other than the Hydrogen atom is based upon an _approximation_. In this treatment (known as the Hartree-Fock or Self-Consistent Field Approximation) the direct, pairwise interaction between electrons is neglected; instead each electron interacts with a "smeared out" charge distribution representing all of the other electrons. You solve the Schrodinger equation for the chosen electron in the smeared-out field of the rest, getting a set of orbitals - the charge distribution associated with these orbitals then gets fed back in to define the smeared-out field with which another electron interacts, and so forth. Run around the loop a few times until you converge - that is the "Self-Consistent Field." (Actually, the mathematical problem is usually solved by converting it to a matrix diagonalization problem rather than doing direct iteration, but the end result is the same. Also, what I have described is actually a simpler version of the approximation called "Hartree" - Hartree-Fock modifies the procedure so that it respects the Pauli Exclusion principle, and thus takes into some of the "Exchange" phenomena that Steve referred to, although not all.

Since the orbitals are constructed in this self-consistent fashion, the orbital energies as well as the orbitals themselves actually depend upon which _other_ orbitals are filled, as Steve and Rosalinda have noted. In particular, 4s is slightly below 3d in K and Ca, but as you walk down Period 3 the 3d orbitals drop down below the 4s. As a result, when you ionize Ti to give the Ti^2+ ion, the electrons come out of the 4s orbital, not the 3d, because 4s is above 3d in Ti even though it was below 3d in K and Ca.

The simple orbital interpretation of atomic structure works amazingly well for a large number of phenomena. Much of its success can be attributed to a great deal of error cancellation - the observed properties are commonly things like differences between state energies, and the errors in the calculation of each largely cancel. Cr is an example of a place where it fails. The usual rationale is given by Raghav above - there is a certain degree of "stabilization" associated with a half-filled or completely filled d shells. The very language of the rationale shows, however, that we have to go beyond the picture of independent electrons moving in self consistent fields and consider the detailed *correlations* among individual electrons.

Now for the economics link. The electronic structures of the atoms ultimately come from experiment in particular, photoelectron spectroscopy, in which you measure the energy required to remove an electron from an atom to make an ion. When you look at the Hartree-Fock equations it seems, at first, that it is going to be difficult to relate this measured energy to the underlying "orbital energies", because of all the self-consistent stuff that's going on in the background. However, in 1934 a physics graduate student named Tjalling Koopmans showed that with a few additional approximations that hold in a broad range of circumstances, all the mess cancels and the observed energy is simply equal to the difference in orbital energies. Having contributed "Koopmans' Theorem" to the vocabulary of theoretical chemistry, he proceeded to abandon the field, eventually ending up with a Nobel Prize in Economics.

[RW]

The band theory of solids (thanks for the link) seems more useful than orbitals when trying to understand metals but frankly some of the explanation is simply beyond me (I did find the notion of a "fermi sea" intriguing though).

Alternatively, it's instructive to see an example of a "simple (and correct) answer" that provides neither explanation nor any hope of further understanding.

[RW]

Thanks Steve, that explanation helps.

[Ben V-L]

The reason 4s fills before 3d is electron-electron interactions, as opposed to electron-proton interactions.

It's useful to understand the "Hydrogenic atom" (speaking as a physicist, with a physicist's notion of useful). This is what you would get if you could turn off the interactions between the electrons, and it is implicitly what you are thinking of when you are bothered by a 4s orbital filling before a 3d orbital.

The Hydrogenic atom is useful because we can understand it. Each electron independently interacts with the nucleus only. In this case, the possible orbitals are exactly those of Hydrogen. The only differences from Hydrogen are a slightly stronger nuclear attraction, which is trivial and just rescales the energies associated with each orbital, and the Pauli exclusion principle, which forces the multiple electrons to pile up into the higher orbitals. In the Hydrogenic atom, the electrons would fill the 3d shell before starting on 4s. To expect this to happen is to think Hydrogenically.

But this neglects a whopping source of energy in the atom, the electron-electron repulsions. Amazingly, we can neglect this whopping source of energy and still get the atomic structure about right, even if the energy levels are off. This is a testament to the strength of the Pauli exclusion principle, which forces electrons to march the plank to the next available energy level, shifts in said energy levels be damned.

But sometimes those energy levels shift due to electron-electron interactions enough so that their order gets switches. So then the electrons end up in 4s before going into 3d. That's about it, apart from some math for determining when this should occur and some qualitative descriptions of the math.

As for metals, you need to start with the right picture, which is somehow hard to get from the textbooks. When atoms pack together into a solid, they can either decide to each hold tightly onto all their electrons, such as quartz, or they can let go of one (or more) of their electrons in exchange for having on average one electron (or more) to replace it. That is a metal. It's properties are dominated by the sea of nearly free electrons running through it. Simplistic explanations of the behavior of that sea of electrons are, in my opinion, not likely to be helpful in understanding its behavior.

[JK]

As has been said, the basic point here is about inter-electron interactions.

Raghav and Steve's explanation of Cr is correct. Note that the other "exception" in the first row transition metals is copper which has configuration 3d10 4s1. It comes between nickel (3d8 4s2) and zinc (3d10 4s2). The completion of the 3d sub-shell lowers the 3d10 4s1 energy below that of 3d9 4s2.

I may be wrong, but I sense Brad might feel that 3d "ought" to fill before 4s because he thinks quantum numbers are some how defined by reference to energy. In fact quantum numbers are more about spatial variation of a wavefuntion. If you want to go into this more deeply, that is what you need to think about.

Generally re: Robert P's remarks, and explaining effects like the CR configuration, my high school chemistry teacher used to explain that the subject is about small differences between large numbers.

[RW]

Eureka! Don't know about Brad either but energy levels is exactly what I was thinking of; thanks JK.

[R Chatel]

Interesting replies and interesting post. Did I miss something? As a humanites graduate of thirty years ago but nonetheless a regular reader of the blog and not totally economically / chemically illiterate (my father taught chemistry and was one of my high school teachers), am I missing something or just stupid when I ask--what is the point of this post/ another Sokol Affair?

[bailey]

More & more you sound like an old tenured Prof., living out your days looking for fun intellectual exercises. Remember the good old days when you wanted to make a difference in the field of Economics?

[coriolis]

As a chemist, who is returning to University for a doctorate in physics, I would have to say that Ben V-L's explanation agrees the most with my understanding of this magnetizing subject.
I don't want to be parochial, but chemists do a lot of hand waving about orbitals. and don't get me started about my girlfriend's botony class lecture on electron "shells".
There are experimental facts, there is quantum theory and there are the little mental pictures that teachers use to illustrate that actually obfuscate. The first time I calculated a wavefunction was a transcendent experience.

[Walt]

bailey: He wants to be able to answer any questions his children have when they get to college. An admirable goal, I think.

[I want to answer the questions my children have now...]

[Bruce Wilburn]

From Robert P comments

“The usual rationale is given by Raghav above - there is a certain degree of "stabilization" associated with a half-filled or completely filled d shells.”

The only problem with this rationale or theory is that it doesn’t hold true. For Cr or Mo (both d5 s1), the theory holds. However, for W, right below Cr and Mo in periodic table, the electronic configuration is d4 s2 (To simplify matters, I am ignoring the orbital numbers.)

Most of the elements refer to by Brad also have problems with their electronic configurations in that they are irregular. For example, V, Nb, and Ta are all group 5 elements with 5 valence electrons. V and Ta have same configuration d3 s2; however, Ta is d4 s1. (Valence electrons, for non-chemists, are the electrons available for bonding and reactions.)

Fe, Ru, and Os have 8 valence electrons. Both Fe and Os are d6 s2 while Ru is d7 s1.

Co, Rh, Ir have 9 valence electrons, Co and Ir are d7 s2 while Rh is d8 s1.

Ni, Pd, and Pt have 10 valence electrons. Ni is d8 s2, Pd d10 or Pt d9 s1.

I think that the best explanation is that the electronic configurations of the above elements depend on the fine points of electron-electron and nuclear-electron interactions.

Or as Steve said earlier

“As for the elements further along in the periodic table, all I can say is that things get very hard very quickly”

Finally, Tom Slee mentions

“2. the relativistic thing is indeed important for the heavier elements. It is the main reason why lead forms two bonds while carbon and silicon (in the same period) form four.”

Lead can form four covalent bonds. The best example is tetraethyl lead.

[Walt]

Oops, sorry. I meant to type "high school".

[Tom Slee]

Bruce - true enough, but it is far more common for it to form two -- eg, PbO is the most common oxide of lead, wherease Carbon is more commonly found as CO_2 than CO.

[Winslow R.]

The inability to answer chemistry questions should not concern you. What should concern you is the inability to answer economic questions.

Sorry, I just couldn't pass that one up....I'll stop now...at least for a while.

[Bruce Webb]

I am missing the premise here. And I entered Cal intending to be a Nuclear Physicist. (whereupon I encountered the difference between being smart and being Fermi).

Is there really a high school in the world that is asking questions about electron orbitals?

[Chapter 3: Atomic Structure and Electron Configuration: Problem 26c: Identify the element with the following electron configuration: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s1 4d10 ]

And if there is shouldn't they be teaching kids the three 'R's instead? Because last I observed most kids were missing the fundamentals of reading, riting and rithametic.

"High School Questions I Cannot Answer" Is this High School in Piedmont? Because if your kid is bringing these kind of questions home from Berkeley High School then there has been a major revolution in secondary education that has passed unnoticed around much of this country.

[Zack]

Bruce: I first encountered these concepts in high school. I did go to what is arguably the best private high school in Los Angeles, but I had friends at LAUSD public high schools who were also doing this sort of thing. If memory serves, it is included on the AP chemistry syllabus (AP = advanced placement = 1st-year-college equivalent courses taken in high school).

The kids who are taking these courses are not the same kids who are having problems with the three 'R's, generally speaking.

[Bruce Webb]

Zack I first encountered these concepts in High School myself. But the notion that "Why does the 4s orbital fill before the 3d orbital, anyway?" is something that some high school kid should be able to answer in anything other than the most trivial way is not obvious. Something here smacks of teaching to the test.

And not to switch the point Advanced Placement is bullshit and always was. I went to an elite public school where pretty much the majority of students were kids of doctors and lawyers who served the peons of San Jose and what was not known then as Silicon Valley from their beautiful houses in the hills above Los Gatos. Los Gatos High put most of its graduates into prestigious programs, with or without scholarships but as late as 1974 we had never even heard of Advanced Placement. We just took Calculus and won national awards for French Language knowledge and didn't blink. But somehow when I returned to college in 1981 as a Junior after four years in the Navy somehow everything had changed. High Schools had introduced AP classes and Cal Berkeley inexplicably had allowed them to grant an extra grade point for every AP class taken. A B in History scored as an 4.0 A for grade point purposes. And a A in Chemistry scored as a 5.0. Which meant that you could go through your entire time at Balboa racking up C's in English and still graduate with a 4.0.

And the results showed.

"If memory serves, it is included on the AP chemistry syllabus" Bingo. At age eight I knew how to add, divide and multiply in base six. Because for whatver reason that was what the geniuses behind "New Math" decided was what they would teach. Did that mean I could do the same thing in base seven? Or really knew anything at all about number theory? Well no but I put up impressive numbers on standardized tests of my mastery of base six multiplication.

I really doubt that if you took the broad spectrum of academic subjects that even superlative high school students needed to learn that the mechanics of filling the 4s orbital before the 3d orbital is representative of the depth of knowledge in each reach of the spectrum.

[don]

And one can be sure that the Rosevilla, CA schools aren't teaching these problems with science any more than those with evolution! Another reason to sue them! Has Larry Caldwell seen this blog?

[spencer]

Bruce you are missing the point about our schools.

We have multiple education systems.
One for kids like delongs childrens -- upper income educated parent in afflunent communities -- that are great and expose the kids to things like this.

One for middle class kids that are OK, they generally prepared the kids to go to good old state u.

A third for poor blacks in inner cities and poor white trash in the rural regions where they teach intelligent design. Maybe they have a teacher like I did in North Georgia in the 1950s -- think Deliverance -- that taught that zero times the number was the number.

Your problem is that you have been drinking the koolaide
so much you have come to believe that the entire system is like the third class of schools.

[RW]

Spencer is essentially correct although I'm not sure about the koolaide. The bulk of the variance in school outcomes can be traced to parental educational level, neighborhood and community resources; there have always been multiple tiers of schooling in the US and there has been scant change in secondary education in the last century aside from graduating more students as a percentage of total school-age population than at any other time in history (whether some of those students 'deserve' to graduate is an argument over essential criteria, a different topic). Brad Delong's question seems consistent with honors chemistry curriculum in upper tier public high schools (and consistent with curricula I have seen in some accelerated schools in less affluent communities).

That said, I've not been impressed by many of the AP programs I've seen at any school; far too reminiscent of a test cram in most cases IMHO.

[Unstable Isotope]

I guess I don't need to jump in here, since it's been explained already. I think the problem comes in because we're taught of electrons occupying "shells" when the truth is closer to electrons occupying probability clouds. The chemistry of metals gets very complex, especially heavy metals. That is why they are so versatile, they occupy many different stable stats. The simplistic answer is best "they occupy the lowest energy state."

I wouldn't worry about elements 110 and 111 myself, these are elements that existed on a very short timescale, so we don't even know if we've made the most stable conformation or not.

[Matt]

Speaking of non-linear electron densities in atomic physics...

There's a little story I once heard about the Thomas-Fermi equation. The Thomas-Fermi equation is a 'simple' non-linear differential equation for electron density in an atom-- 'simple' in quotes because, being non-linear, it can't be solved analytically. Consequently, it was one of the very first equations whose solution was obtained numerically by the new-fangled 'computers' that appeared in the early '50's. The story is that the person who did the early calculation let it run overnight, only to find himself facing a huge pile of paper filled with numerical results the next morning. The story is that he was the very first person that this ever happened to...

[Tim Worstall]

F-orbital filling elements?

Well, as an economist(sorry, B.Sc only and some layman’s interest) and one who actually makes his living dealing with the rare earths the answer is: You don’t need to know. As indeed I don’t need to and don’t except in the most vague manner. And chemists do.

I think there’s a name for this somewhere. Specialization anybody?

[But I do want my Fifteen-Year-Old to have the option of someday being a chemist...]

[Maynard Handley]

The physics behind this has been covered about as well as one can expect in this sort of forum up above. I'd like to make just one observation which is that (and school does a piss-poor job of this) one should distinguish between different types of expanation.
At the fundamental level we have underlying physics that is (at least within its domain ofrelevance) well understood, quantitative and close to exact.
Thus we know at how to write down an expression describing the behavior of a chromium nucleus along with its attendant electrons, even including QED effects. We then know how to approximate the size of QED effects, and, based on that, to write down a simpler expression which ignores them, a Schrodinger equation. We then know how to approximate the magnitude of correlation effects and write down a simpler expression (Hartree or Hartree-Fock equation), and at this point we've arrived at something we can actually solve numerically.

At this point we are faced with two issues.
The first is to interpret usefully what our model tells us. This is a far from trivial matter, even in the simplest case of interpreting gaseous spectra, and the relevance of Hartree-Fock atomic calculations to either solid metals or metal chemistry (ie metal ions or molecules involving metals) is pretty tenuous. Some of the posters above have already covered these issues.
The second issue is that of trying to make useful sense of the patterns of results we encounter. At this point various heuristics enter; but the point is that these heuristics are conceptually very different from what went before, ie QED, Schrodinger, Hartee-Fock. We know that the specific results that come out of Hartree-Fock, or alternatively out of empirical spectroscopy, consist of very finely balanced effects that can, as specifics change, point in one direction or the other. The heuristics are an attempt to regularize this but they are, in no way,"explanations". They are more rules of thumb that give one some sort of intuition about how to predict what usually happens, with no guarantee that, in this particular case, the balancing of the effects might go slightly differently giving an exception to the rule.

Hence my advice, Brad, is to be more irritated than upset at the question.
It is asking questions that make little actual sense; questions that assume that a chromium atom is very similar to a hydrogen atom, a way of viewing the world that makes very little sense.
Far more useful (for physics) would be a better explanation of the importance of angular momentum in spherically symetrical potentials and the consequences thereof. Far more useful (for chemistry) would be an explanation of the very large effects that are balanced in this situation, a list of rules-of-thumb that apply, and a throrough understanding that what one is dealing with are simply rules of thumb, not "natural laws".

(In this respect I am actually much more tolerant of the second question. While the answer also involves heuristics, it allows for bringing up the issue of exchange in a way that is not completely stupid.
As for the third question, I wish both chemistry and physics texts at this level did a whole lot less of this exceptionally stupid stretching of a model from 1930 way beyond its limits, and a whole lot more simple description of empirical effects. Who cares whether d orbitals are filled before f's or vice versa? Explain that what really matters here is that this effects the chemistry, the solid state properties [these are all metals], and the magnetic properties.)

[meno]

Brad, the real answer to your confusion is: you're thinking about it the wrong way.

Ask yourself this: Why _shouldn't_ the 4th set of s orbitals be at a lower energy level than the 3rd set of d orbitals?

I think some vague intuition that "3 is less than 4" is doing you harm. They're a different shape, so they interact differently with the other subatomic particles. Those interactions define the energy level.

[almostinfamous]

i remember the snake very well.

the snake is where you write down

1s
2s 2p
3s 3p 3d
4s 4p 4d 4f

and draw a writhing snake starting from 1s.. the shape is slightly off, but if you draw it right, you get the orbital sequence. 5 years + booze puts a lot of distance between you and atomic chemistry:)

[almostinfamous]

well if you put the 5 series down, you'll probably get a better idea.

the sequence should be 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d and so on.

[MikeB]

I remember when I first learned about the periodic table in middle school. I asked two separate teachers why some rows had 8 columns while others had much more than 8. I got two different answers, neither of which was correct. I had to hope I would learn the right answer in high school.

In high school I learned about s, p, d, and f orbitals, but my books and teachers failed to explain why these strange things existed, let alone why they filled in such an odd order. I had to hope I would learn the right answer in college.

As a physics major in college I finally learned the connection between orbitals and angular momentum, and how the quantum numbers arise in the course of solving Schrodinger's equation for the hydrogen atom, and so forth.

But I don't think I really learned about electron exchange until the first or second year of graduate school in physics.

So... starting as a middle school student, it took me eleven years and three educational institutions to learn the explanations which your child -- thanks to this blog, and the Ph.D.s that comment on it -- has learned in less than a day.

The Internet is truly the invention of the century.

(Of course, your child probably doesn't really *understand* the full meaning of these explanations. But, alas, neither do I... )

[Karlsfini]

The correct answer is spenser is right.

Brad is helping his son with his homework. I've got to get back to the homework for my kid.

I'd sue the socks off the American education system but it would just help the neocones destroy it.

The reason we have concrete and asphalt paving is that it's more difficult to tear up those materials like paving stones and use then for rebellion.

What if Galbraith turned 17 instead of 97 today? What do you think would come of him? I guess he'd be a basketball star.

[dearieme]

The correct answer was taught to me in my first freshman Physics tutorial: "Chemistry - the Science of hindsight."

[Bruce Webb]

Lots of folks way smarter than me about sub-orbitals are chiming in but I can't help but think that getting your 4s separated from your 3d means more people are knowing more about the AP tests than they need to. Science is big. History is huge. Teaching details about electron orbitals shrieks teaching to the test. Who knew that question would pop up? and wasn't it convenient that we spent the last six weeks exploring that particular corner of chemistry?

[Graydon]

"Teaching to the test" is an inappropriate bugaboo; "we are not testing the right things" is a perfectly sensible complaint, but teaching to a well designed test is precisely what a lot of very successful education systems do.

Further, being able to deal with an unforgiving mass of complex detail is a core professional skill for pretty much all professions. What specific detail it happens to be will of course vary by eventual chosen profession, but the skills for handling complexity transfer better over types of detail than they do over habits of thought.

Besides, electron shells are one of the doors-just-ajar into the idea that there's a vast, complex, fundamentally interesting field of knowledge just through here, and it Makes Sense; the awareness that it's there, the awareness that there are lots of these fields, that they do indeed Make Sense, is something any sensible education system should be presenting.

I got something very similar to the questions Brad mentions in chemistry in a middlin' suburban high school in the eighties, in Canada; I don't think it's at all inappropriate for a high school course.

[Ben V-L]

Bruce, let me offer a counterargument. I am myself critical of the trend to push high school students through advanced placement courses to advanced placement tests without paying attention the big picture of what they need to learn. In my field we see it with students who have gotten through their high school calculus, but mostly skipped trig to get there. So they can integrate a cosine, but not tell me what cosine of zero is, for example.

Okay, those are my creds. I still think the electrons filling 4s before 3d is important to cover in a higher level high school class. The main reason is that it explains the periodic table. Students know that potassium is like sodium, and so the columns on the periodic table make sense to them. They can't possibly learn the group theory that explains why there are 2*1 s orbitals and 2*3 p orbitals, and 2*5 d orbitals, but they can accept these and be taught some important concepts, i.e. that (i) atomic structure determines atomic properties, and (ii) atomic structure is mostly understood. This message gets confused when the atomic energy levels shift, and they do it early enough that it concerns common, everyday stuff.

My 4 year old is demanding my attention, so I won't have a go at editing that - hopefully it makes some sense.

[weichi]

Someone commented:

"The Thomas-Fermi equation is a 'simple' non-linear differential equation for electron density in an atom-- 'simple' in quotes because, being non-linear, it can't be solved analytically."

This is picking a nit, but it is incorrect to say that non-linear differential equations can't be solved analytically. Maybe Thomas-Fermi can't be solved analytically, but there are other non-linear differential equations that *can* be solved analytically.

For example, Dy(x) = x/y(x) with y(5)=-3 has y(x)=-sqrt(x^2 - 16) as it's soln.

[Mr. Aviles]

Hi everyone. I need to thank you for such a gratifying time reading all your comments. I am a high school chemistry teacher in a poor school in the city of Pomona. This is a school for high-risk students where the population is about 80% Hispanic, 15% white, and 5% black. Last week we started covering atomic structure, energy levels and electron configuration. Students do have a hard time dealing with these concepts, but what is important at this stage in the education process is that they can relate the reactivity of the elements based on the position of the electron in the highest energy level. Once we are able to make sense of that relationship and how elements are grouped base on those properties, then it becomes easier to teach the subject.
I do not teach the snake anymore, I want the students to use the table to figure out the electron configuration of each element.
Yes we still teach orbital and energy levels in high school, but thanks to the Internet and the amazing videos you can find on almost any subject you look for, it is easier to exemplify difficult topics.
I am a chemist graduated from CSU system who worked in the industry for 12 years and just last year made the transition to teaching. I came to this country 17 years ago from Guatemala, and I think I made the right choice about transitioning to teaching. Every teachers ultimate goal is to reach those kids who think that going to college could not be done and make them believe that everything is possible as long as they are willing to work hard to achieve their goals. The kids that go for science will have to find their own answers as they reach college, as most of you guys have done already. High school is to get them interested in science and to get partial knowledge of everything that we can cover in a short year. Hopefully the desire to learn will be awaken and a new scientist will be born.

[Gary Kerns]

It is understandable to think that all the 3 levels should be filled before proceeding to the 4 levels. However, in atomic architecture there are two types of interaction which affect the sequence of electron filling: electron-proton interactions (attraction); and electron-electron interactions (repulsion).

If the order of electron filling were to proceed according to electron-proton interactions only, the 3d levels would be filled before the 4s levels. This would be true if quantum numbers were defined only by reference to energy. In fact, quantum numbers are more about spatial variation of a wave function. As electrons fill up the levels, their envelopes of probability change. At certain electron populations the influence of electron-electron repulsive interactions becomes greater than the electron-proton attractive interactions.

Thus after filling the 3p levels, the energy levels shift due to electron-electron interactions so that the simple spatial wave function of a 4s sphere (electron-proton attractive interaction is prevalent) is easier to fill than the more complex 3d spatial wave function (electron-electron repulsive interaction is prevalent).

I think the comment posted by “Meno” (dh 9Oct05) is definitive: “The notion that 3 is less than 4 is misleading. They’re different in shape, so they interact differently with other subatomic particles. Those interactions define the energy level.”

Hope this will be of assistance,
Gary Kerns