JH van ‘t Hoff and the Gaseous Theory of Solutions; also, Pricing Games

Do you ever think about why stuff dissolves? Like, why a spoon of sugar in a glass of water should seem to disappear instead of turning into a slight change in the water’s clarity? Well, sure, in those moods when you look at the world as a child does, not accepting that life is just like that and instead can imagine it being otherwise. Take that sort of question and put it to adult inquiry and you get great science.

Peter Mander of the Carnot Cycle blog this month writes a tale about Jacobus Henricus van ‘t Hoff, the first winner of a Nobel Prize for Chemistry. In 1883, on hearing of an interesting experiment with semipermeable membranes, van ‘t Hoff had a brilliant insight about why things go into solution, and how. The insight had only one little problem. It makes for fine reading about the history of chemistry and of its mathematical study.

In other, television-related news, the United States edition of The Price Is Right included a mention of “square root day” yesterday, 4/4/16. It was in the game “Cover-Up”, in which the contestant tries making successively better guesses at the price of a car. This they do by covering up wrong digits with new guesses. For the start of the game, before the contestant’s made any guesses, they need something irrelevant to the game to be on the board. So, they put up mock calendar pages for 1/1/2001, 2/2/2004, 3/3/2009, 4/4/2016, and finally a card reading $\sqrt{DAY}$. The game show also had a round devoted to Pi Day a few weeks back. So I suppose they’re trying to reach out to people into pop mathematics. It’s cute.

Knot.

It’s a common joke that mathematicians shun things that have anything to do with the real world. You can see where the impression comes from, though. Even common mathematical constructs, such as “functions”, are otherworldly abstractions once a mathematician is done defining them precisely. It can look like mathematicians find real stuff to be too dull to study.

Knot theory goes against the stereotype. A mathematician’s knot is just about what you would imagine: threads of something that get folded and twisted back around themselves. Every now and then a knot theorist will get a bit of human-interest news going for the department by announcing a new way to tie a tie, or to tie a shoelace, or maybe something about why the Christmas tree lights get so tangled up. These are really parts of the field, and applications that almost leap off the page as one studies. It’s a bit silly, admittedly. The only way anybody needs to tie a tie is go see my father and have him do it for you, and then just loosen and tighten the knot for the two or three times you’ll need it. And there’s at most two ways of tying a shoelace anybody needs. Christmas tree lights are a bigger problem but nobody can really help with getting them untangled. But studying the field encourages a lot of sketches of knots, and they almost cry out to be done out of some real material.

One amazing thing about knots is that they can be described as mathematical expressions. There are multiple ways to encode a description for how a knot looks as a polynomial. An expression like $t + t^3 - t^4$ contains enough information to draw one knot as opposed to all the others that might exist. (In this case it’s a very simple knot, one known as the right-hand trefoil knot. A trefoil knot is a knot with a trefoil-like pattern.) Indeed, it’s possible to describe knots with polynomials that let you distinguish between a knot and its mirror-image reflection.

Biology, life, is knots. The DNA molecules that carry and transmit genes tangle up on themselves, creating knots. The molecules that DNA encodes, proteins and enzymes and all the other basic tools of cells, can be represented as knots. Since at this level the field is about how molecules interact you probably would expect that much of chemistry can be seen as the ways knots interact. Statistical mechanics, the study of unspeakably large number of particles, do as well. A field you can be introduced to by studying your sneaker runs through the most useful arteries of science.

That said, mathematicians do make their knots of unreal stuff. The mathematical knot is, normally, a one-dimensional thread rather than a cylinder of stuff like a string or rope or shoelace. No matter; just imagine you’ve got a very thin string. And we assume that it’s frictionless; the knot doesn’t get stuck on itself. As a result a mathematician just learning knot theory would snootily point out that however tightly wound up your extension cord is, it’s not actually knotted. You could in principle push one of the ends of the cord all the way through the knot and so loosen it into an untangled string, if you could push the cord from one end and if the cord didn’t get stuck on itself. So, yes, real-world knots are mathematically not knots. After all, something that just falls apart with a little push hardly seems worth the name “knot”.

My point is that mathematically a knot has to be a closed loop. And it’s got to wrap around itself in some sufficiently complicated way. A simple circle of string is not a knot. If “not a knot” sounds a bit childish you might use instead the Lewis Carrollian term “unknot”.

We can fix that, though, using a surprisingly common mathematical trick. Take the shoelace or rope or extension cord you want to study. And extend it: draw lines from either end of the cord out to the edge of your paper. (This is a great field for doodlers.) And then pretend that the lines go out and loop around, touching each other somewhere off the sheet of paper, as simply as possible. What had been an unknot is now not an unknot. Study wisely.

The Liquefaction of Gases – Part II

The CarnotCycle blog has a continuation of last month’s The Liquefaction of Gases, as you might expect, named The Liquefaction of Gases, Part II, and it’s another intriguing piece. The story here is about how the theory of cooling, and of phase changes — under what conditions gases will turn into liquids — was developed. There’s a fair bit of mathematics involved, although most of the important work is in in polynomials. If you remember in algebra (or in pre-algebra) drawing curves for functions that had x3 in them, and in finding how they sometimes had one and sometimes had three real roots, then you’re well on your way to understanding the work which earned Johannes van der Waals the 1910 Nobel Prize in Physics.

Future Nobel Prize winners both. Kamerlingh Onnes and Johannes van der Waals in 1908.

On Friday 10 July 1908, at Leiden in the Netherlands, Kamerlingh Onnes succeeded in liquefying the one remaining gas previously thought to be non-condensable – helium – using a sequential Joule-Thomson cooling technique to drive the temperature down to just 4 degrees above absolute zero. The event brought to a conclusion the race to liquefy the so-called permanent gases, following the revelation that all gases have a critical temperature below which they must be cooled before liquefaction is possible.

This crucial fact was established by Dr. Thomas Andrews, professor of chemistry at Queen’s College Belfast, in his groundbreaking study of the liquefaction of carbon dioxide, “On the Continuity of the Gaseous and Liquid States of Matter”, published in the Philosophical Transactions of the Royal Society of London in 1869.

As described in Part I of…

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