The Speed of Atoms and Kinetic Theory

James Clerk Maxwell

James Clerk Maxwell (1831–1879) was born in Edinburgh, Scotland, in 1831. His family moved to a small country estate in Middlebie, Galloway (southwestern Scotland) that his father, John Clerk inherited (the addition of the name “Maxwell” was required to satisfy legalities of this inheritance). When he was eight, James’ mother died, of abdominal cancer; she was forty-eight. John Clerk Maxwell was an attentive and perhaps overly protective father. Unfortunately, he made the mistake of entrusting James early education to a tutor who employed beatings as a teaching tactic. Fortunately, a visit from his maternal aunt, Jane Cay, discontinued this abusive treatment, as she was able to convince Maxwell’s father to allow him to continue his education at Edinburgh Academy.

His initial experiences at the academy were not so pleasant either, as he endured bullying (his strange dress and strong Galloway accent made him an easy target), and acquired the nickname, “Dafty”, similar to “weirdo” as used today. Nonetheless, Maxwell endured and even made lasting friendships.

As an enthusiast of science and mathematics himself, Maxwell’s father was very encouraging of his son’s passion for science, and together they often attended meetings of the Edinburgh Society of Arts and the Edinburgh Royal Society. At the age of fourteen, Maxwell wrote a paper on a new method for constructing ovals. Although Descartes had already described a majority of this work, some of it was original. Maxwell’s father brought the work to the attention of James Forbes, a professor of natural philosophy at the University of Edinburgh and this propelled Maxwell’s career.

In 1847 at the age of sixteen, Maxwell began attending the University of Edinburgh where he studied with Forbes and William Hamilton. As sworn enemies, Forbes and Hamilton pretty much disagreed on everything. However, agreeing he was gifted, they gave Maxwell the special attention they felt he deserved.

A skilled experimentalist, Forbes gave Maxwell free reign of his laboratory, while Hamilton, a philosopher, honed Maxwell’s conceptualization skills by stressing the use of idealized models to provide insight into real phenomenon. Although after his first term Maxwell had the opportunity to attend Cambridge (a favored choice at the time for those pursuing mathematics), he stayed and completed his undergraduate education at Edinburgh. Maxwell didn’t find the classes very challenging, and found time to pursue many topics on his own, such as his experiments with polarized light.

In 1850, he attended Peterhouse (a constituent college of the University of Cambridge, England) but after his first term transferred to Trinity College (Cambridge). It was during this time that Maxwell began to fit in (despite his quirky sense of humor and eccentricities), developing a number of friendships. In 1854, Maxwell graduated from Trinity with a degree in mathematics. He remained on the staff at Trinity for two more years, receiving a fellowship during his last year.

In 1856, a professorship opportunity became available at Marischal College, Aberdeen, Scotland. Among other things, this would provide a good opportunity to be near his father, whose health was now declining. His father assisted him in preparing the needed references for the position, dying just prior of Maxwell learning he had the job.

Maxwell was now twenty-five and a decade and a half younger than any other professor at Marischal. Although his written works (papers, formal lectures, and books) exemplified clarity, Maxwell was unsuccessful as a teacher, unable to convey lecture material to his students. In 1860, reorganization left Maxwell without a job, and he accepted an appointment at King’s College in London. His five years here were perhaps the most creative of his life, and his work gave him much satisfaction. Of Maxwell’s many contributions, what interests us here is his work on the motion of atoms.

Maxwell and Kinetic Theory

Maxwell’s initial efforts in kinetic theory began while still a professor at Marischal College, Aberdeen. Maxwell wrote to, fellow physicist and mathematician, George Gabriel Stokes (1819–1903) on May 30, 1859 describing the work as an “exercise in mechanics”. Apparently, Maxwell had serious doubts for a kinetic theory that correctly described the properties of gases. Nonetheless, in 1860 he published a revolutionary paper, where he proved what Clausius had only speculated about: the speeds of gas atoms come from a specific distribution when the system is at equilibrium.

The condition that the system is at equilibrium is an important one. In this case, it means that the temperature, the number of gas atoms, and the volume of the container holding them, remain unchanged; they’re all constant. Maxwell realized that, for all practical purposes, the collisions occurring between the atoms in a gas result in an overall random motion. Therefore, much to the frustration of his contemporaries, he completely ignores the explicit collisions occurring between the gas atoms (and the walls of the container) and instead appeals to probability theory. The laws of probability govern such things as: the odds of winning the lottery being very small; the next turn of a card giving you twenty-one (or not) in a game of blackjack; a flipped coin giving heads (or tails) fifty percent of the time (on the average), and the alike. Maxwell showed that these same ideas apply to understanding the properties of a very large number of colliding gas atoms.

Maxwell considered the velocity of the atoms of a gas in equilibrium. Now, for the velocity (since it’s a vector) it’s necessary to consider both the amount and the direction of the motion. By assuming that each atom of a gas has the same likelihood, or probability, to move in any of the available directions, he showed that the atoms of a gas do indeed have a range of velocities available to them. Therefore, they don’t all travel at the same speed, but rather a very small number travel very slow or very fast, while the majority travel at more intermediate speeds. In other words, according to the Maxwell distribution, when the system is at equilibrium there’s a higher probability of finding an atom in a gas traveling at an intermediate speed than either at a very slow or very fast speed. Now, in practice we find that the Maxwell distribution applies to a gas that behaves as an ideal gas, where the atoms of the gas don’t experience the attractive (“pull”) and repulsive (“push”) interactions that a real gas does.

In 1860, Maxwell needed merely a single page to derive this amazing result, which also allowed him to calculate other important properties of a gas that matched with experimental observation. To be sure, a system that is not in equilibrium will also take on a distribution for the velocities of the gas atoms. However, there’s a catch: it won’t be a Maxwell distribution.

Consider, our system of gas atoms once again, where the number of atoms and the volume of the container are unchanged, but the temperature is changing in time. Let’s assume our nonequilibrium system is evolving towards equilibrium, and therefore the temperature will eventually be constant. As the system evolves so does its velocity distribution, and until the system finally reaches equilibrium, the distribution will continue to change over time. However, once the system has reached equilibrium, the velocities will acquire a Maxwell distribution. This is regardless of the initial state of the system or how it was initially prepared. Finally, the tendency of our nonequilibrium system to evolve towards equilibrium is a spontaneous process, which means the entropy is increasing irreversibly along the way. The evolution of our system to a Maxwell distribution at equilibrium plays a vital role in understanding certain physical processes.

Probability and Physical Processes

Consider a hot bowl of soup. Blowing on a spoonful prior to placing it in your mouth does cool it down. This is because the soup is made up of atoms comprised of a range of speeds. The atoms with the higher speeds – the hotter ones – are on top, hovering above the atoms with the lower speeds – the cooler ones. So, the end result of blowing on the spoonful of soup is that the hotter atoms are blown away, while the cooler ones are left behind. If the atoms in the soup were all traveling at the same speed there would be no hot or cold atoms, they would all be the same. In this case, you could blow until you’re blue in the face, but the temperature of the soup would never change.

Before Maxwell, probability and statistics had been used for data analysis (in the social sciences and physics). However, Maxwell’s approach distinguished itself because it used these methods to correctly describe the actual physical process itself. Maxwell’s work in thermodynamics was done over a period of years and yet he never wrote any major papers on the topic. We know of his efforts from his correspondence with Thomson and Tait, and from Theory of Heat, first appearing in 1870 and running for eleven editions thereafter. In addition to his own contributions, Maxwell was instrumental in clarifying the ideas of Gibbs (1839–1903), Boltzmann, and Clausius, and in reconciling disparate views among Clausius, Tait, and Thomson.

Maxwell’s Later Life

Aside from his eccentric character, Maxwell was generous, unselfish, and had a deep sense of duty. When his wife, Katherine was sick (she was often in frail health), Maxwell sat by her bed for three weeks, and attended to his responsibilities at his laboratory, all the while his own health was failing. When reviewing another’s paper for publication, Maxwell’s comments often offered more insight into the topic than the paper itself. His comments on a paper by William Crookes (1832–1919) offered advice that, had it been pursued, may have led to the discovery of the electron. Perhaps surprising was Maxwell’s sense of the metaphysical. In a letter to a friend he writes:

“… that the relation of parts to wholes pervades the invisible no less than the visible world, and that beneath the individuality which accompanies our personal life there lies hidden a deeper community of being as well as of feeling and action.”

On November 5, 1879 Maxwell died, at the age of forty-eight, in Cambridge of abdominal cancer (the same cancer that killed his mother at the same age). Maxwell’s statistical approach to the dynamics of gases opened the door for Ludwig Boltzmann to arrive at a microscopic interpretation of entropy, thus going far beyond Clausius’ thermodynamic definition of entropy of “heat over temperature”.


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