Units and stoichiometry

As part of my group’s current work for HHMI (Project NEXUS) to build a physics course for biology majors and pre-meds, physicists and biologists at the University of Maryland are holding extensive discussions. One of the things we like to talk about is the way we each look at the world – what feels like “real physics” or “real biology” to us and what just looks fake. (“Fake biology in physics” is just using biological organisms as the physical objects in a physics example, but not learning anything useful for biology – like using a spherical cow as a projectile.)  In that context, my colleague Wolfgang Losert and I had an interesting discussion about the nature of units and dimensional analysis.

In thinking about how we use math to model the world I like to use a little diagram that helps me make explicit some of the features we often take for granted.[1]  Here is it.

We start on the lower left with some physical system we want to model mathematically. We have some property in mind that we want to describe. We choose an abstract mathematical structure to map onto, creating a mathematical model of the system. From that math, we inherit lots of generative and processing tools that allow us to do things that we can’t easily do with purely conceptual thinking. Once we’re done processing, we have to interpret our results back in the world and then evaluate whether it works – or whether we have to modify our model. [Note this this is meant neither as a description of the cognitive processes of how we actually use math in science nor as a normative way of teaching how to model with math in science. We actually do things in a much more blended and integrated fashion. The modeling model is more of a philosophical analysis rather than a cognitive one.]

Applying this model to units and dimensional analysis is interesting. When we decide how to quantify something in the physical world via measurement, we are modeling. For example, if I decide that a length or distance can be assigned a number by counting the number of times a standard unit fits into that length, I have decided to model length and distance using positive real numbers. With this, I inherit addition, subtraction, multiplication, and division. Addition and subtraction lets me create coordinate systems and vectors and interpret negative distances. Division lets me create fractions of my standard and imagine – at least for a bit – that distance is “exactly” described by the line of real numbers. (We often forget that this is a model.) The fact that I inherit multiplication lets me generate numbers that I can assign to areas and volumes. We use the resulting model both to describe objects (their linear dimensions, areas, volumes) and positioning in space.

But we often forget the limitations of the model. First, lengths of objects are not perfectly modeled by real numbers when you look closely. No object has a perfectly sharp edge. Most are “fuzzy” – if you measured to high resolution you would get slightly different numbers by measuring at different places. At one level this comes from the process to construct the object (How carefully were the boards the table is made of sanded?), at another from the fundamental physics of the structure (Atoms and molecules are discrete – and quantum fuzzy.). If we use distances in space, inheriting the math of the real number line means that we are assuming we are measuring in a Euclidian line, plane, or volume. If that turns out not to be the case (e.g., 2-D maps on the surface of the earth), we need to choose a different mathematical model.

Each time we choose a different concept to measure (and typically an operational definition to go with it), we have a new set of numbers to decide about. If, for example, we are thinking about mass, we map to the positive real numbers since (so far) we don’t have an interpretation for what negative mass would be. (To fit into our current mathematical structures consistently, it would have to have anti-gravity when interacting with positive masses.) We inherit addition and subtraction, but multiplication doesn’t do us much good (except when calculating a gravitational force). We don’t have any physical quantity that we map onto “square mass” so we have to treat the unit of mass differently than we treat a unit of length. (See Solomon Golomb’s cute little problem, “Proving a penny equals a dollar”.) Also, when we put two masses together, they don’t quite add. Their gravitational potential energy reduces their joint mass somewhat (using E = mc² on the negative gravitational potential energy) yielding a “gravitational binding energy” that can be significant for astronomical sized objects. It's usually small and we can ignore it in almost all of physics, but it speaks to the limitations of the “real number” model of mass.

I find it particularly interesting that although each of these two models maps properties in the real world into real numbers, they behave differently in the way the real numbers can be used correctly (squares are meaningful in one and not in the other, negative values are meaningful in one and not in the other). This is one of the points students often don’t get. They think math in science is just about assigning numbers and they prefer to drop the units until the end. They miss that the way you can use those numbers changes depending on which physical dimension you are talking about. And we almost never tell them.

This is why we physicists think that dimensional analysis and units belong to us and why learning to think with units is so much to the essence of “thinking like a physicist;” because with the units, you are supposed to bring along your knowledge of the physical nature of the measurement being considered and constrain your use of the number assignments appropriately.

But the comment that started me thinking about this again and how it fits with my model for analyzing modeling, was Wolfgang’s comment that “Stoichiometry is chemistry’s dimensional analysis.” I thought that was a really exciting insight.

Now I think many of my readers are probably physicists and, if you are like me, you might not have studied much chemistry. (I only studied chemistry in high school when there were a lot fewer elements in the periodic table!) I had to ask him to remind me what stoichiometry was. What he meant was what Wikipedia calls “reaction stoichiometry” – the way the ratio of the numbers of atoms in a chemical reaction need to be adjusted so they react together properly. Basically it’s the statement that in a chemical reaction the number of each kind of atom doesn’t change – they just rearrange. If I’m building water molecules out of hydrogen and oxygen molecules, since hydrogen and oxygen atoms come in pairs (usually, as H₂ and O₂), and since water is H₂O, you need to combine two hydrogen molecules with one oxygen molecule and you wind up with two water molecules:

2H₂ + O₂ ⟷ 2H₂O.

From the point of view of my model of modeling, the chemists are mapping each kind of atom onto the set of positive integers. Wolfgang's insight is that this is like introducing a “unit check” for every distinct atom – using integers rather than the real numbers. We inherit addition and subtraction (you can move things from one side of the reaction arrow to the other) but multiplication, division, and fractions don’t associate with new physical quantities.

It’s interesting to evaluate this integer model and think about why it works. It’s not that the atoms remain inert when they react. The electronic structure changes as atoms undergo chemical reactions. In some sense, the reason you get away with stoichiometry in chemistry is that chemical reactions don’t affect atomic nuclei at all. So what you are counting are the different nuclei and saying you have the same number of each kind throughout a chemical reaction. Despite the fact that we talk as if the atoms were staying fixed in the chemical reaction, it’s really the nuclei that remain conserved. The electron states of the “atom” are shifted around, so although it’s a reasonable approximation to treat the atoms as fixed in a chemical reaction, it’s not quite correct. The stoichiometric equations and the “ball-and-stick” models chemists use are highly symbolic and rely on applying the mathematical counting model described here. The chemists’ “space filling models” are a little more realistic, showing the atoms overlapping and deforming a little. But for some circumstances it must be necessary to keep in mind that the electrons are shared and no longer may belong to a particular atom. We might have to think of some of the electrons are really shared over an entire molecule, producing a kind of “band structure” like we have in conducting crystals.

So even some of the simplest mathematical ideas in physics and chemistry – unit checks and counting of atoms – reveal themselves to be (very) useful models, but to only tell a part of the story.

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[1] Problem Solving and the Use of Math in Physics Courses, E. F. Redish, in Proceedings of the Conference, World View on Physics Education in 2005: Focusing on Change, Delhi, August 21-26, 2005