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Maxwell on Molecules February 22, 2010

Posted by isotopeeffect in Chemistry, Physics.
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This post continues the art theme, at least momentarily.

While refreshing my memory about the works of the Pre-Raphaelite Brotherhood I stumbled upon the rather excellent Victorian Web, a web site gathering links on areas of interest from that era ranging from fine art through science, from religion through economics, a veritable liberal arts goldmine.

There is plenty of material on the site to recommend itself under the heading of science, but I was particularly struck by an essay by James Clerk Maxwell entitled “Molecules”, first published in Nature in 1873.

Maxwell opens with a reference to the ancient Greeks, by distinguishing the philosophy of the atomists, represented by Democritus, from that of Socrates’ teacher Anaxagoras (a somewhat less well-known – because “wrong” – philosophy called Homoiomereia). Neither philosophical stance counts as what we would now call science; both are essentially entirely speculative in nature. Anaxagoras’ perspective was that substances are homogeneous and thus can be subdivided indefinitely without losing their character, while that of the atomists was that a limit would be reached where matter can no longer be so subdivided, but has been reduced to its fundamental units, atoms, objects which (literally) cannot be cut.

Of course, for all the ingenuity of the ancient Greeks they had no concept of molecules (their idea of “elements” more closely resembling our modern idea of “states of matter”), so jumping to a more modern concept, Maxwell quickly defines the molecule as the simplest unit of a particular substance that retains the composition of the whole before moving on to his main arguments.

Maxwell’s overall goal in this essay is to introduce the public to recent research in the field of molecular science, areas of study that we would nowadays call statistical thermodynamics and the kinetic-molecular theory of gases.

The first part of Maxwell’s argument is the modern counterpart to the opinion of Lucretius that bodies are in ceaseless motion (on what we would now call the atomic scale), even when they appear to be at rest. Tracing a historical thread that runs through the “usual suspects” Boyle and Charles to Bernoulli, Clausius, Joule, and Boltzmann, not to mention Maxwell himself, Maxwell shows how once gas pressure is recognized as a product of atomic motion and collisions a theory can be built that allows us to calculate the speeds at which molecules move (“about seventeen miles per minute”), the average distance between molecules (“about the tenth part of the length of a wave of light”), and the number of collisions undergone per second (“hundreds of millions”).

The next section of the essay contrasts the “historical method”, in which the history of an individual is followed through time, with the “statistical method”, in which general results relating to groups or populations are obtained. This section is remarkable enough to quote at length:

“The equations of dynamics completely express the laws of the historical method as applied to matter, but the application of these equations implies a perfect knowledge of all the data. But the smallest portion of matter which we can subject to experiment consists of millions of molecules, not one of which ever becomes individually sensible to us. We cannot, therefore, ascertain the actual motion of any one of these molecules, so that we are obliged to abandon the strict historical method, and to adopt the statistical method of dealing with large groups of molecules.

“The data of the statistical method as applied to molecular science are the sums of large numbers of molecular quantities. In studying the relations between quantities of this kind, we meet with a new kind of regularity, the regularity of averages, which we can depend upon quite sufficiently for all practical purposes, but which can make no claim to that character of absolute precision which belongs to the laws of abstract dynamics.

“Thus molecular science teaches us that our experiments can never give us anything more than statistical information, and that no law deduced from them can pretend to absolute precision. But when we pass from the contemplation of our experiments to that of the molecules themselves, we leave the world of chance and change, and enter a region where everything is certain and immutable.

“The molecules are conformed to a constant type with a precision which is not to be found in the sensible properties of the bodies which they constitute. In the first place the mass of each individual molecule, and all its other properties, are absolutely unalterable. In the second place the properties of all molecules of the same kind are absolutely identical.”

Maxwell here points out the astonishing fact that molecules resemble one another in a manner very different from that in which, for example, apples or golf balls do; while the latter may be similar but for a nick or blemish, the former are (up to the point of being in different quantum states) completely identical in every respect.

The indistinguishability of atomic or molecular states has ramifications even Maxwell did not think of. In the world of quantum mechanics, the wavefunction of a system of particles must show a distinct symmetry with respect to the pairwise interchange of particle labels. For particles like electrons, for example, the requirement that the wavefunction be antisymmetric is known as the Pauli principle, which every chemistry student knows as the principle determining that an orbital can contain only two electrons. A slightly more elusive implication of the antisymmetry principle, however, is that if a helium atom is in the 1s2s state, we cannot say which of the two electrons is the 1s electron. In statistical thermodynamics, symmetry requirements impact which combinations of states can exist and lead to phenomena such as Bose-Einstein condensation (see here for NIST’s BEC site) and degeneracy pressure (the phenomenon that keeps white dwarf stars from collapsing).

Maxwell goes on to show that the identical nature of all molecules (or atoms) of a particular type is precisely that which allows us to use the methods of spectroscopy to make discoveries not only about samples in front of us in the laboratory but about any object in the universe from which light can be detected. Here’s Maxwell again:

“But in the heavens we discover by their light, and by their light alone, stars so distant from each other that no material thing can ever have passed from one to another, and yet this light, which is to us the sole evidence of the existence of these distant worlds, tells us also that each of them is built up of molecules of the same kinds as those which we find on earth. A molecule of hydrogen, for example, whether in Sirius or in Arcturus, executes its vibrations in precisely the same time.”

In this latter point, Maxwell is not quite right. Light from distant objects beyond our galaxy exhibits a redshift due to the velocity at which these objects are receding from us. (In their own reference frame, as Maxwell states, they emit radiation of the same frequency as molecules on Earth. ) There is an approximately linear correlation between recession velocity and distance which led Georges Lemaître to his theory that the universe that we now know evolved from a “primeval atom” (his term) in which all the mass-energy now present was compressed into a tiny volume, via a cosmic explosion now universally known as the Big Bang.

Quoting Maxwell again:

“Natural causes, as we know, are at work, which tend to modify, if they do not at length destroy, all the arrangements and dimensions of the earth and the whole solar system. But though in the course of ages catastrophes have occurred and may yet occur in the heavens, though ancient systems may be dissolved and new systems evolved out of their ruins, the molecules out of which these systems are built — the foundation stones of the material universe — remain unbroken and unworn.”

To revise Maxwell, our current understanding is that atoms (specifically, hydrogen atoms) did not exist until about 400,000 years after the Big Bang. Before that, there was too much energy present in the compact early universe for atoms to be stable, so any atoms formed would instantly ionize. Traveling further back in time, a few minutes after the Big Bang energy was so concentrated that protons and neutrons themselves could not exist; the tiny universe was an extremely hot “quark soup”. In the first microsecond after the Big Bang quarks themselves were not stable. An enhanced understanding of what happened in those first few instants after the Big Bang is a research goal of the Large Hadron Collider.

It is an interesting irony that in this essay Maxwell seems to so strongly espouse a “particulate” picture of the universe. Most physics students first meet Maxwell through his famous equations describing the behavior of electromagnetic fields. The modern descendents of Maxwell’s work are the various forms of quantum field theory that describe the universe in a manner in which “particles” are viewed as excited states of a quantum field, an entity that spreads throughout space. Perhaps the followers of Anaxagoras were not entirely wrong.