where s is distance, u is initial velocity, and a is acceleration. I suppose that should have been easy for me, but I am not a formula monkey. I may have applied it in the intervening years, but only when turning 60, and starting to learn and teach History of Mathematics, did I come across our Oxford friends and their mean speed theorem. They were the first to draw anything we would recognize as a graph, would almost certainly have worked it out using geometrical reasoning, based on a construction like this following (without the algebraic notation, though).

Doing university chemistry, I encountered thermodynamics (a beautiful subject, let me tell yez). But I could not get hold of partial molar quantities. I had some idea of what they were supposed to mean. Take this diagram of the ethanol-water mixture from Atkins’ Physical Chemistry by Peter Atkins and Julio de Paula (Oxford University Press).

Leave the ethanol on one side for the moment (boozers especially) and look at the blue curve for water. On the left hand side we see the partial molar volume of water when it is pure is just over 18cm³, which makes sense because a mole of water has a mass of (just over) 18g and its specific gravity is a little bit less than 1. But is there any physical meaning to its partial molar volume of about 14 cm³ when there’s next to none there in the absolute ethanol? I think there is. We all know that ice floats on water. This is because the molecules in solid ice are linked together in an open hydrogen-bonded three-dimensional lattice, leaving lots of space between molecules. Much of this hydrogen-bonded structure is locally preserved when ice melts, although the molecules are constantly changing partners like in a barn dance. This means that the water molecules are still not as closely packed as space allows. But when they’re all alone in a sea of ethanol, they’ve got no water partners to form this open structure with, and so they take up, you might say, only their own volume of space.

Very nice then. "So whatcher worryin’ about?" I hear Oscar the Grouch say. Go to any thermodynamics textbook and you will get a chunk of mathematics (particularly partial differentials) and a chunk of verbiage, but I had never been able to relate one to the other. But then William Heytesbury (ca. 1313 – 1372/3), comes along with a verbal description of what we would regard as a differential: "In nonuniform motion . . . the velocity at any given instant will be measured by the path which would be described by the . . . point if, in a period of time, it were moved uniformly at the same degree of velocity which it is moved in that give instant, whatever [instant] be assigned."

With this still in my mind, I come across Atkins’ textbook (actually an old 2nd Student edition). And there is the method of intercepts, for extracting partial molar quantities from a graph.

For those that haven’t met moles, they’re quite simple. The molecular weight of ethanol is 46.07, so 1 mole of ethanol is 46.07 grams of the stuff. Its density is 0.789 g/cm³, so 1 mole occupies 58.39 cm³ of space (as in the Atkins graph). Now look at our hypothetical mixture of two liquids, A-one and B-ane. At A and B we simply read off the volume of 1 mole of either liquid. Now take the tangent at point M, our mixture. Thinking à la Heytesbury, imagine we are extracting a a few molecules of B-ane and replacing them with an equal number of molecules of A-one. There will be a tiny tiny change in volume. Now imagine that if we were to carry that process on, with the same volume change, until the liquid was all A-one, we would achieve a volume of A'.

Of course that wouldn’t happen in practice, because the molecular environment for both types of molecules would be changing as the composition changed. But with Heytesbury to help me, a concept that had bugged me for 40 years is now ‘detoxicated’.

Before leaving this topic altogether, look at the shape of the partial molar volume curves for both water and ethanol for low concentrations of the latter. Weird indeed! Is this some indication of why things go the way they do in the brain of a drinker?

References:

A History of Mechanics (Dover Classics of Science&Mathematics) (Paperback) by Rene Dugas ISBN-10: 0486656322 http://www.amazon.co.uk/History-Mechanics-Classics-Science-Mathematics/dp/0486656322/

This was the book that really put me on to our medieval friends and their mechanics. Dugas also has an excellent chapter on Galileo.

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The graphical method of intercepts is, alas, rather rough-and-ready. Here’s a more mathematical way of doing the same thing, from one of my favourite journals:

Regression Methods To Extract Partial Molar Volume Values in the Method of Intercepts by Leon F. Loucks