Resonance and Structure: a world in-between

An example of theoretical structures in science is drawing molecules using a capital letter for an atom and a dash for a bond, so water is H-O-H. What does it mean when that doesn’t always work? What does this failure imply about our notion of “structures”?

The much-studied example is a molecule called “benzene.” We can think of it as a ring of six carbon atoms with alternating single and double bonds. But is the first double bond between carbon atoms number 1 & 2, or is the first of the double bonds between carbons number 2 & 3? It makes a difference when we try to attach a new atom to the ring. Where will the newly attached atom end up?

It is perhaps easier to visualize if we mark carbon #1 by attaching something to it. So, say that carbon #1 has X attached. Then, if we are going to attach Y to another carbon of the ring, will it end up at carbon #2 or carbon #3?  It depends on where the double bonds are located. 

So, which way should we draw the molecule? Should the alternating double bonds start after carbon #1 or after carbon #2?

It turns out that, upon trying to add “Y” to benzene, we get half of each possible outcome. It is as if benzene can be both ways at once.

But how can that be? That doesn’t fit our scheme for drawing the structure of molecules wherein our sense of what is “real” tells us that it must be one way or the other; it can’t “for real” be both ways at the same time. Our scheme for drawing structures seems to be missing something. We can’t use it to predict our results (we can’t tell where “Y” will end up). So, what should we do?

And for that matter, what does that tell us about the physical Universe, that it doesn’t seem to always conform to our way of thinking of it as existing in fixed structures?

That was one of the great dilemmas confronting August Kekule (pronounced Kuh-COO-lay) back in the 1850’s. Kekule is credited for being one of the pioneering chemists who transformed alchemy into scientific chemistry (although, granted, no one has heard of him; the physicists get all the glory). Today’s chemistry students learn that when Kekule was starting out his career, his prospective father-in-law told him he should get a more respectable job (much like today when people would say the same to video game programmers or to rock-n-roll musicians . . . until they became rich).

And even today, organic chemists cannot get life insurance. They keep accidentally blowing themselves up too often to escape the notice of the actuaries.

Anyway, what was Kekule’s solution? (Perhaps I should add that many chemicals, including much of the body, have benzene at their core. It is not a rare or trivial problem).

And, oh yeah, Kekule was the guy who invented writing chemical structures in the first place (1857). He was the guy who figured out that atoms could form bonds at all, and he represented that with his drawings of their structures.

But his scheme didn’t work for benzene.

Kekule’s solution

Kekule’s answer was that benzene must flip back and forth between the two ways that we could draw it (with the double bonds between 1 & 2 or between 2& 3), in order to give the results of making equal numbers of each outcome. And this has come to be called “resonance structures.” 

But now take a moment to consider what this must mean. Is the world really flipping back and forth like that? When we hear the phrase “It is what it is,” should the speaker be adding, “Well, not really. The world actually flips all around a lot.”

Or is it that the problem is with how we think of our structures? There exists the way that we can draw chemical structures, and there exists the way that the world works. And are they the same?

It suggests a maxim by Descartes. “We do not describe the world we see; we see the world we can describe.” (Yes, Descartes could be an interesting character, even though in past posts I have been agreeing with how Newton was very critical of him).

In other words. there might be some drawbacks to describing the world only mathematically (or only as per how we can draw it) if, when we do, the world seems to end up flipping all around. That might be a problem for someone such as Tegmark who argues that the physical Universe literally “is” its mathematical structures (instead of saying that we use mathematics to measure the world, and we measure it as we are able).

Also, I suppose it could be argued that there might be an entire new universe created every time that benzene flips back and forth—that way, benzene could be one way in each universe—and we are living in just one of those universes. But historically, the chemists didn’t think of that idea. 

In any case, Kekule’s answer is now known to be “wrong”—it fails to give reproducible results in more complex situations—but we still use it, anyway. It is one of those things like how it is often easier to use Ptolemy’s description of planetary motion for predicting eclipses, even though Ptolemy fails for more advanced applications. Or it is like how it is easier to think of Bohr’s model of the atom than to use quantum mechanics. So we use the easiest methodology that still works for an application. And likewise, Kekule’s resonating structures are greatly used today, even though we know that they are “wrong.” (An old theory can work even if superseded by a newer theory because what was reproducible under the old theory is still reproducible).

To give some scope of what that means, I’ll relate how in my first year of organic chemistry I had a new inexperienced teacher, and he gave a final exam where we were to write out every possible resonating structure for a complicated reaction, and six hours later we students were still working on it—writing furiously as fast as we could—when the next class came in to use the room for their final exam.

You see, once things start resonating, and you add something to it, then the results can altogether start resonating in their own ways, and it can multiply exponentially to end up with a world that is totally flipping out. A cell is full of such resonating wonders. If resonance is taken literally, a cell is not just one thing but rather it is resonating amongst countlessly many ways of being real.

But fortunately, there are two other solutions to Kekule’s problem.

bond order

Once technology progressed to the point that we could measure the distance between the carbon atoms, then we could discover that a double bond was shorter than a single bond. So, now with benzene, we could look to see, with its alternating single and double bonds, which one came first. Was the first double bond between carbons 1 & 2 or between carbons 2 & 3? But it turns out it is neither. The length of the bonds between all of the carbons is the same and with a length in-between that of the single and double bonds. If we call the bond length of a single bond a “1,” and with a double bond it is a “2,” then the bonds in benzene are a “1.5.”

Reality is “in-between” the two ways we can describe it mathematically or in drawings on paper. More exactly, carbon is observed to have a valence of four—it can attach to four things—which logically seems to require either single or double bonds as the two possible choices. Yet benzene is made of neither single bonds nor double bonds but of something in-between.

Even so, it can still react with other molecules to turn into things which might have single or double bonds (although how it does so remains an interesting question).

So, how can we picture all of that? Starting from scratch—if we had no sense of what the world is doing—how could we begin to visualize what is happening?

What we can do is describe the two “bookends” and say that the world is in-between. Either it flips back and forth between the bookends, or it exists in-between them. But we start out with a notion that is wrong (the bookends) because that is what we can describe, and then we go from there. With the benzene molecule, the goal is to be able to predict what the outcome of a reaction will be, which we can sort of start to do using these bookends.

The thing in the middle is sometimes called a “hybrid” of the bookends. So hybridization is the second explanation, besides Kekule’s resonance structures.

quantum mechanics

Nowadays, the answer to understanding benzene is in terms of quantum mechanics. We think of the molecule as sharing electrons equally among all the carbons, and as such it becomes “that which can take on various forms” as it is “measured” differently via the act of joining different things to it. This sharing of the electrons can change into single or double bonds when in the presence of other molecules it is reacting with. In that way, we can explain where “Y” is likely to end up, given the presence of “X.” (If that reminds us of a quantum “wave collapse” upon a measurement, it should. The vague sharing of electrons among all six carbons becomes, upon being near another molecule, a specific arrangement of the electrons into single or double bonds).

Stated more technically according to quantum mechanics, the carbon electrons are “delocalized,” but they can become “localized” into single or double bonds upon being measured in the sense of having another molecule react and join with it. And we now know that many molecules exhibit this behavior, not just benzene.

But there is a catch. We can still only predict that some results will be “more likely” than others, and we can give the percentages of each result that can happen. But quantum mechanics doesn’t give automatic results. It is another case of that old saw in chemistry, “I cannot tell you what just one molecule will do, but I can tell you the proportions of what 30 million will do.”

A certain percentage will end up one way, and another percentage will end up another way. And this changing from delocalized to localized is instead of it flipping back and forth or instead of it existing in multiple universes to keep things rigidly literal.

Still, the percentages will depend on the circumstances. So that is what chemists do; they adjust the circumstances to get mostly what we want. The molecules themselves are acting randomly, as per a wave collapse, but within certain setups some outcomes will be (because of those setups) more probable. It is like how flipping a weighted coin will (because of how it is organized) yield one outcome more often than another, although we still cannot predict deterministically what any individual outcome will be. 

Or, as my old organic teacher once put it (I have quoted him here before), “There is no such thing as A causes B. There is only A causes B under Circumstances C.” (I will discuss this role of arrangement in turning random actions into Newtonian actions more fully in a later post).

But that is the ultimate answer to describing benzene. What it even “is” is “none of the above” —it is delocalized—until we factor in the circumstances. Benzene is that which, when measured under different conditions, appears to be a different thing. But it literally isn’t any of those different things until it is actually measured under those conditions.

And yet that seems to be a foreign concept to many philosophers upon hearing about quantum wave collapse. They think that there is only one way for things to be, if they are “real.” But benzene is a practical example showing how it works.

What benzene even “is” depends on how we measure it. It is that which does not fit any of the macro-scale choices until it is measured in some macro-scale way. But then it can seem to fit into our macro-scale way of thinking.

The problem is with assuming that the world has to fit with our ways of describing it.

more complex

And of course the great fun is when we realize that there can be more than just two “bookends” to our understanding of a problem. A phenomenon might best be described as in-between eight ideals that we can use to picture it, or twenty, or a hundred. That might well be how we come to have a sense of the meaning of words; it is by experiencing countless examples of them, each one making sense in the context of that situation, but none of them being “perfect” pure essences of what the thing is always about. Reality doesn’t live in the single perfect description.

That is what it means to have “a world in-between.”

A further non-quantum mechanics example of that is the ideal gas laws. No gas acts in an ideal manner, but by pretending that it does, we can formulate equations that let us start to get a handle on what is happening.

Yet I want to make clear that I am not talking about the math itself. I am talking about how we take the measure of the world. And I am the first to admit that this business of saying that “the world lies in-between how we can measure it” can be overdone. But still, when we are talking about structures, and about what it means to have a physical structure more generally, I am wondering if the lesson here with benzene is this: Physically, a structure need not be stuck in one perfect way forever.

It can conform to how we measure it. It is malleable. It is dynamic.

Also, it can be a means of intuiting quantum wave collapse by visualizing an example. A measurement can transform an indeterminate delocalized structure (a vague sharing of electrons within a ring) into a definite localized outcome (having organized single or double bonds), albeit randomly but still subject to the overall circumstances.

And that brings me full circle back to Kekule. When it comes to trying to actually synthesize new molecules out of benzene, it is often easiest to think of benzene as having alternating single and double bonds, for purposes of predicting how things will end up, even though we know full well that that isn’t accurate.

For acquiring our results, it doesn’t bother us that the “real” molecule exists changeably—it is itself different when alongside different neighbors—it being “really” somewhere in-between the rigid ways that we might describe it.

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