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

[space-time]

too late for me, the questions has been answered already. as my name implies, I could have answered if I got online a little earlier. That was a good question for a 13 year old student. make sure she keeps focused on science and she has a bright career ahead.

[TheTominator]

Yes, h is Planck's constant.

Some helpful web pages I found via Google for the equations.
http://hyperphysics.phy-astr.gsu.edu/Hbase/debrog.html
http://en.wikipedia.org/wiki/Bohr_model

[quote Ted King]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.
Oh right. Sorry about that. Yes, s-orbitals have only a pair of electrons. It has been a long time since high-school AP chemistry. (In physics where I got my degree we rarely talked about s, p, d orbitals and the shell model.) The logic is the same for filling p-orbtials and the atom radius shrinking. I didn't want to touch them since it is easier conceptually to think of the spherical s-orbital.

[quote Ted King]
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?
Yes the spectra will change. The spectra measure the differences between electron energy levels since the light emitted or absorbed corresponds to quantized photons as the electrons change positions among these available energy states. Since the energy states themselves go as the atomic number squared, the distance betwee the corresponding states on different atoms should change.

If I extend the formulas presented in the Wikipedia link above, I get that the energy difference (the photon energy) between the n_i (n sub initial) and n_f (n sub final) states is
E_photon = Z^2 * R_E * [ (1/n_f)^2 - (1/n_i)^2]
where Z is the number of protons and R_E is a constant (involving the mass of the electron, speed of light, and so on).
When comparing a transitional line between two different energy levels n_i=1 and n_f=2 for example, two different atoms will have different photon energies because of the Z^2 term. The spectral lines should therefore be different. Since all of the photon energies scale with Z^2, in this model the spectra between atoms will shift by a constant factor.

Note that quantum mechanics and quantum chemistry were not my areas of expertise.

[anonymouse1]

God, I love this website!

[Ted King]

Wow, thanks TheTominator and everyone else. I guess I should have been more confident there would be someone here (or several) who could answer the question. I did find out information about the wave nature of electrons around a nucleus but I couldn't seem to find information that specifically answered the student's question. Thanks again!

[x-uri]

Man - am I late to THIS party. FWIW, as a university physics instructor, I give TheTominator's response 8.5 for 10.

The problem isn't so much with TheTominator, as it is with the hand-waving way in which Ted King invoked the de Broglie wavelength.

These topics are not beyond a bright middle school student, but if you are going to talk about wave functions, then you have to talk about what they mean. Your student asked a very good question. A better question still is "Why do the electrons appear only occupy orbitals that correspond to whole number multiples of the de Broglie wavelength?"

Try this for a less nuance-dependent (but still hand waving) explanation:

As TheTominator points out, the de Broglie wavelength is inversely proportional to the linear momentum of the electron. The faster the electron goes, the shorter its "wavelength". This result is well supported by experiment, so you do not have to ask the students to accept it without proof.

As you go up the periodic chart, the charge on the nucleus increases -- which means that the electrostatic attraction between the nucleus and any electron in any orbit will increase.

It is a simple matter to demonstrate that the linear velocity of an object orbiting under the influence of an inverse square force (like the Coulomb interaction between point charges, or the gravitational attraction between widely separated masses, etc.) increases with increasing attraction.

That is to say, you could do just a little algebra on the board to show that, as the charge on the nucleus increases, the velocity (hence the momentum) of the electron will increase, meaning that its de Broglie wavelength will decrease. Smaller wavelengths mean a shorter resonant circumference for the mysterious standing wave rule of electron orbits.

As a bonus excercise, if you are comfortable with the math, you could work out the allowed orbitals for a planet in orbit around the Sun -- and show that the the quantum mechanical result duplicates the Newtonian result (and that, for a gravitational potential acting between the Sun and a planet, the difference between adjacent allowed orbital radii is vanishingly small)

edit: scary typo

[Craig]

[quote Ted King]I did find out information about the wave nature of electrons around a nucleus but I couldn't seem to find information that specifically answered the student's question.
Ted-


First, I want to reiterate how cool it is that you are looking for an answer to your student's question. It shows a level of dedication to teaching that, unfortunately, is not present enough in our nation's schools. It is obvious that you are enthusiastic about what you do and instead of being embarrassed about not knowing the answer, you have used her question as an opportunity to learn something new yourself. Kudos.

I think that Tominator's answer is as good as any you will find using google. The reason that you are not finding a specific easy answer to her question is not because you aren't using the right keywords- it is because there isn't an easy way to explain some things. Understanding what exactly is happening at the quantum level is, quite bluntly, outside the ability of most people. This is truly the realm that only the greatest scientific minds will ever really comprehend.

The rules of Newtonian physics break down at the extreme ends of our everyday world. Go large, small, or fast enough and the things that we intuitively know don't apply. As a physics teacher, I am sure that you know this. So in order to give people a basic understanding of what is happening we use intuitive models that approximate what is really going on. Sometimes these models work well. In this case the first three periods behave pretty well within the Bohr model that Tominator used, but sometimes they don't. As pbarra1 mentioned- even the smartest people don't always agree on what or why things happen the way they do.

As Tominator said in his first post, there are other factors at play including atomic shielding, but even these effects are very difficult to precisely calculate and would require an understanding of the Schrödinger equation or Slater's empirical formulas. In addition, there are complex direct and indirect relativistic effects at play since the electrons are in accelerated motion and therefore the equations of special relatively do not rigorously apply. Also, as atomic mass increases the speed of the innermost electrons approaches the speed of light. Now you are moving into an area the requires a knowledge of relativistic Hamiltonian. You can throw spin-orbit effects into the mix as well. You can see where I am going with this- it is difficult subject matter.

Let me reiterate that in my opinion Tominator's explanation and simplifications are as good as you will probably find on the internet. It is on the level of information that is taught to physics and chemistry students at the bachelor's level. If you want to know more then you will probably need to find a quantum chemist/quantum physicist to explain it. In my experience, the problem that you will undoubtedly encounter is that the people that really understand this stuff usually have a very difficult time explaining it to those of us that don't (see my post above.) Some of the smartest people that I know sat in Dr. Shultz's class for a semester and admitted at the end that they understood about a quarter of what was taught. I understood about 1/10th of the information presented and was glad to take my C+ and move along. And I wasn't a "moron major" either.

I think the central issue here is that a 13 yo was insightful and curious enough to even ask the question. She was obviously thinking at the next level. In my opinion she should be watched closely and nurtured along if she shows an inclination to pursue the sciences. Unfortunately, women are woefully underrepresented in the the hard sciences. While some of this is undoubtedly a genetic predisposition, it has been known for years that boys in America are given preferential treatment in science classes and yet very little has been done to rectify this inequality. There are undoubtedly cultural influences as well since Asian American women do not seem to share the same disparity that is seen among other ethnic groups. This leaves me curious as to what her parents do.

In any case, there are worse ways to spend your Saturday then reading up on quantum mechanics. Admittedly- there are very few people here who would agree with that statement. Here are a couple interesting links:

http://en.wikipedia.org/wiki/Lanthanide_contraction
http://zopyros.ccqc.uga.edu/lec_top/rltvt/node28.html
http://books.google.com/books?id=3stGfWt5XT8C&printsec=frontcover




Craig

[Z]

Nothing useful to add to the conversation (a lowly non-moron major ChE here) - but just wanted to thank craig, the tominator, x-uri and others who chimed in for bringing a smile to my face. Well done. We need more of this.

[Ted King]

The only reason I mentioned de Broglie's hypothesis was to inform the students that there is a generally accepted underlying reason for the energy levels that Bohr did not originally envision. I will admit I was hand waving because I don't understand it beyond a superficial level - but my intent was only to let them in on the notion that there is a hypothesis to explain it. I only spent a minute on de Broglie's hypothesis. (I will say in my defense that I frequently admit to the students that there is a great deal of information beyond that which I possess on almost all of the topics we cover - especially in the non-biological sciences). I do that sort of thing fairly regularly - just try to give the students a small sense of the deeper complexity that underlies many of the concepts we discuss. I can see the case for that not being a desirable thing to do since the oversimplifications do involve giving the students a certain amount of misimpression. This has gotten me to wondering if I should just leave those kinds of things out since I don't have the knowledge or time (given the amount of curriculum we have to cover) to do better justice to those deeper conceptions.

[x-uri]

[quote Ted King]The only reason I mentioned de Broglie's hypothesis was to inform the students that there is a generally accepted underlying reason for the energy levels that Bohr did not originally envision. I will admit I was hand waving because I don't understand it beyond a superficial level ...
You are to be applauded for making the effort.

When I teach Astronomy, I have to try to explain a little bit of quantum theory to students who have gone through their entire academic careers without algebra (or even very much arithmetic).

The get it first when I talk about stellar spectroscopy. Then they get it again when we talk about cosmology,

What is most frustrating to me is that this latter discussion is frequently the thing that most fires the imagination of my students, and stimulates to most (and most enthusiastic) questions. However, without the math and a firm grounding in classical mechanics, there is simply no way for them to learn anything about QM.

And I don't mean that they need to know how to solve partial differential equations, they just have to be able to follow the algebra when I derive and simplify.

Teaching QM to physics students presents different challenges. Most QM classes are arranged chronologically -- outlining the historical development of the theory as a way to introduce concepts. The problem with this approach is that much of the early work in QM was done by folks like Born and Dirac, who made some astonishing leaps in developing the formalism of QM -- so it is very hard to build concept upon concept, the way that you do when teaching classical physics.

I am curious now, what is the core curriculum for this class? What topics in biology do you cover with gifted middle school students?

[Ted King]

Craig and x-uri, I very much appreciate you taking the time to give me feedback. Somebody in a later thread mentioned that after reading this thread that they felt less intelligent than they did before having read it. When I saw that I thought, "Oh, yeah." I mean, I knew there was a LOT I didn't know, but to see in more explicit terms just how much I don't understand is definitely humbling. A part of me would love it if I suddenly became blessed with enough ability to understand quantum mechanics and the relativity theories at say the level of a physics graduate student at Cal Tech. But, I don't feel too bad. I've risen to the level of my competence and I'm trying to do what I can to encourage curious young minds to come to a basic understanding of how the world works from a scientific perspective and to help keep alive the flame that a few of them have that may lead them to pursue science in college and careers. IOW, I believe for the most part (with some unfortunate exceptions) I'm trying to make the most of my limited capacities.

[quote x-uri]

I am curious now, what is the core curriculum for this class? What topics in biology do you cover with gifted middle school students?
Ours is a small school so the students have me as their science teacher for both 7th and 8th grade. Seventh grade curriculum is primarily what is called "Life Science". Eighth grade curriculum is focused more on what is called "Physical Science". But the reality is that since I have pretty much the same kids for two years, I tend to blur the lines somewhat and the net effect is that I try to teach the basics concepts of all sciences (including ecology, geology and astronomy) over the two year span.

The underlying theme I use throughout the two years is that science is about trying to spot patterns in the natural phenomena we experience and trying to develop causal explanations for those perceived patterns, following general standards of scientific methodology in hopes of assuring that we are justified in having some level of confidence in the tentative conclusions. In the physical sciences in particular - especially physics - I hope the students come to understand that the patterns are often so regular that they can expressed by mathematical equations. For example, we measure the pressure of air at a few different temperatures, record and graph the data and then a I have the students derive linear equations from the data (that only requires Algebra 1 skills to do). Then we discuss the information we have derived in terms of the kinetic energy of the air molecules as a way to look for a causal explanation for those results. I try as much as possible to reinforce the idea of finding patterns and explaining them in causal terms. The treatment is usually not as mathematical in the life sciences we do in 7th grade but we do look for patterns of form in living things and try to explain them in terms of function.

One of the underlying sub-themes of 7th grade life science is that living things create a bounded internal environment where nutrients need to pass in and be processed for the maintenance of the internal environment and reproduction of new generations of the living thing and wastes need to pass out of that internal environment. We focus first on how that happens at the cellular level with particular emphasis on the role of DNA as the master code for protein synthesis (going into some detail about the structure of DNA and transcription and translation). We look at the major organelles and their roles. About that time we also begin to look at what geologists think the early earth was like and how a simple cell-type organism might have gotten a start given that kind of environment. Then we go over what the evidence - especially fossil evidence - seems to indicate the first recognizable cell life was like and how it might have effected the earth (e.g., the role of cyanobacteria). We discuss theories about how cells might have evolved from prokaryotic to eukaryotic. Then we take a close look at Protists and talk about how organisms like them might have evolved into colonial structures and on to multicellular life. With multicellular life the fossil record seems to become much more clear, so we take an overview of that - especially focused on how animal life appears to have evolved more complex cellular organization through tissue differentiation and the development of specialized organs to meet the needs of maintaining an internal environment of a large aggregate of cells. There is also special emphasis on aerobic respiration and photosynthesis. That is the backdrop for looking at the evolution of animals following plants onto the land and evolving adaptations to the land environment. This, of course, leads to a look at the evident sequence of land vertebrate evolution from amphibians to reptiles (and dinosaurs) to birds and mammals. Along the way we cover topics like classification schemes of living things and the logic behind those schemes. That is about a half a year of curriculum in the 7th grade. Third quarter is devoted to the students doing in-depth research on a particular phylum or class of animal and then doing a group oral presentation to the class on that phylum or class - that takes about half of the quarter. The other half of third quarter and on to fourth quarter is devoted to looking specifically at human anatomy and physiology - which I hope the students can see as a natural extension of what they have found out about the structure and function of other animals. I end 7th grade with a unit on plants because that is in the spring and early summer when plants here in California are doing the most growth and development.

Oh man, I see I got carried away. I'll try to be more brief about what we do in 8th grade. The main focus is basic physics and chemistry. I do also include a unit on the rocks and minerals part of geology and that leads to a unit on mineral cycles in ecological systems and also energy flow through those systems - reintroducing photosynthesis and aerobic respiration and showing their roles in energy flows through ecosystems from the perspective of life essentially using light energy from the sun to reverse entropy in the subsystem that is a living organism; and how that energy is passed from the producers to different levels of consumers. I already in an earlier post talked about some of what we do in the chemistry part of 8th grade science. The physics topics range from the notion of energy and its "manifestations" - particularly as waves, to the relationship between the electric and magnetism sides of electromagnetism, to forces in fluids, to motion in terms of matter and forces, to simple machines (not necessarily in that order and I try to treat them as interrelated topics). I usually do astronomy at the end of the year just because students seem naturally interested in it and end-of-the-year 8th graders can be challenging to keep interested.

That's probably a lot more than you cared to know. I don't feel like this summary does justice to what I try to accomplish, though. But hopefully it gives you an idea of what we do.