Einstein and the Quantum (7 page)

This simple picture, that gas pressure arises from the collisions of enormous numbers of molecules with the walls of the container, along with simple ideas of classical mechanics, allows Maxwell to derive Boyle's law, that the pressure of the gas is proportional to its density. It also allows him to understand the observation that the ratio of volumes of any two gases depends only on the ratio of temperatures of the gases. The relation of temperature to volume of a gas is critical: in this view absolute temperature (what we now call the kelvin scale) is related to molecular motion and is proportional to the average of the square of the molecular velocity in a gas. Since the energy of motion for any mass, called kinetic energy, is just one-half its mass times the square of the velocity, this also means that for a gas its energy is just proportional to temperature. As Einstein had noted in his letter to Maric, in the Maxwell-Boltzmann theory, the entire energy of a gas is the kinetic energy of moving molecules. This principle of the Maxwell-Boltzmann theory, that the energy of each molecule is proportional to the temperature, applies even in the solid state, in which the molecules vibrate back and forth around fixed positions instead of moving freely throughout the substance. This property of the theory would perplex Einstein later, when he was trying to make sense of Planck's radiation law.

“
The most important consequence
which flows from [our theory],” Maxwell continues, “is that a cubic centimetre of every gas at standard temperature and pressure contains the same number of molecules.” This fact about gases was conjectured by the Italian scientist Amadeo Avogadro in 1811. In 1865 Josef Loschmidt, a professor in Vienna and later a colleague of Boltzmann, had estimated this actual number, which is very large: 2.6 × 10
¹⁹
, or roughly five billion
squared
. (This “Loschmidt number” is closely related to Avogadro's number, which is the number of molecules in a mole of any gas—both Einstein and Planck were very interested in accurately determining these numbers). With all this information about gas properties, it was possible for Maxwell to determine the average velocity of a molecule in air. He found it to be roughly one thousand miles per hour. He described the implications most picturesquely:

If all these molecules were flying
in the same direction, they would constitute a wind blowing at the rate of seventeen miles a minute, and the only wind which approaches this velocity is that which proceeds from the mouth of a cannon. How, then, are you and I able to stand here? Only because the molecules happen to be flying in different directions, so that those which strike against our backs enable us to support the storm which is beating against our faces. Indeed, if this molecular bombardment were to cease, even for an instant, our veins would swell, our breath would leave us, and we should, literally, expire…. If we wish to form a mental representation of what is going on among the molecules in calm air, we cannot do better than observe a swarm of bees, when every individual bee is flying furiously, first in one direction, and then in another, while the swarm as a whole … remains at rest.

Maxwell goes on to describe how his own experiments and others have determined that the molecules in a gas are continually colliding with one another, moving only about ten times their diameter before changing direction again through a collision, leading to a kind of random motion called diffusion. Because of this constant changing of direction, the actual distance moved from the starting point during a given time is much less than if the molecule were moving in a straight line. This explained why, when Maxwell took the lid off a vial of ammonia in the lecture, its characteristic odor was not immediately detected in the far reaches of the lecture hall. The same kind of diffusion occurs in liquids such as water, but much more slowly. Maxwell then throws off a poetic but profound comment: “
Lucretius … tells us
to look at a sunbeam shining through a darkened room … and to observe the motes which chase each other in all directions…. This motion of the visible motes … is but a result of the far more complicated motion of the invisible atoms which knock the motes about.” Exactly this process occurs to small particles suspended in a liquid but visible under a microscope, so-called Brownian motion. In one of his four masterpieces of 1905 Einstein would actually take the suggestion of Lucretius and Maxwell seriously and, by careful analysis, turn this into a precise method for determining Avogadro's number! Experiments by the
French physicist Jean Perrin would confirm Einstein's predictions and determine that number very precisely; as a result Perrin received the Nobel Prize for Physics in 1926, long after his work had permanently put to rest doubts about the existence of atoms.

The kind of complex, essentially random motion characteristic of gas molecules gave rise to a new way of doing physics, described by Maxwell in the same lecture. “
The modern atomists have
therefore adopted a method which I believe new in the department of mathematical physics, though it has long been in use in the section of statistics.” Thus was born the discipline of
statistical mechanics
. Maxwell could only assume that the invisible molecules obeyed Newtonian mechanics; he had no reason to doubt this. But in describing what would happen in a gas, he realized that one must inevitably encounter the weak point in Laplace's grandiose dictum. Laplace had imagined an intellect that “at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed.” Maxwell realized that getting all the necessary information and using it to predict the future was an absurd proposition. “
The equations of dynamics
completely express the laws of the historical [Laplacian] 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 becomes individually sensible to us … so that we are obliged to abandon the historical method and to adopt the statistical method of dealing with a large group of molecules.” Maxwell's point is that for all practical purposes one doesn't want to know what each molecule is doing anyway; for example, to find the pressure exerted by a gas one needs only to know the
average
number of molecules hitting the wall of a container per second, and how much momentum (mass times velocity) they transfer to the wall.

This was the key insight of Maxwell and Boltzmann: to predict the physical properties of a large aggregation of molecules, one needed only to find their average behavior, assuming they were behaving as randomly as allowed by the laws of physics. Calculating these properties was relatively easy for a gas, where most of the time the molecules are not in close
contact; for liquids and solids it was much harder and in certain cases still challenges the physicists of the twenty-first century. Tied up with this insight was a new understanding of the laws of thermodynamics. The First Law says that heat is a form of energy, and that the total energy (heat plus mechanical) always stays the same (is “conserved”) even when one form is being changed into the other. For example, when a car is moving at 60 miles per hour, it has a lot of mechanical energy, specifically kinetic energy, ½
m
v
²
, where
m
is the mass of the car and v is its speed (60 mph in this case). When you slam on the brakes, that kinetic energy doesn't disappear; it is turned into heat in your brakes and tires, due to friction. From the point of view of statistical mechanics, that heat is just mechanical energy transmitted to the molecules of the road and tires, distributed in some complicated and apparently random manner among them. So heat is just random, microscopic mechanical energy, stored in various forms in the atoms and molecules of gases, liquids, and solids.

This view sheds light on the Second Law, which states that disorder always increases and is measured by a quantity called entropy. This law now can be interpreted as saying that in any process where something changes (e.g., the car coming to a stop), you can never perfectly “reorganize” all the energy that goes into the random motion of molecules. It is always too hard to retrieve all of it in a useful form. Before the car stopped, all its molecules (in addition to some random motion due to its non-zero temperature) were moving together in the same direction at 60 mph, providing a kinetic energy that could be used to do useful work, such as dragging a heavy object against friction. As the car stops, that energy is transformed into the less usable form of heat. It is not that we can't turn heat back into usable energy (e.g., use it to get the car moving again); it is just that we can't do it perfectly. We could run some water over the hot brake discs of our stopped car, which could generate steam, which could turn a turbine, and, presto, we would get back some useful mechanical energy. This of course is not the best-designed heat engine one could imagine. But the Second Law says that no matter how carefully or cleverly you design an engine to turn heat into useful mechanical energy, you will always find that you have to put more heat energy in than you get back.

To make this all precise and tractable in a mathematical theory, the German physicist Rudolph Clausius, while a professor at our familiar Zurich Poly in 1865, introduced the notion of entropy, which is a measure of how much the microscopic disorder increases in every process involving heat exchange. The word
entropy
was chosen from the Greek word for “transformation,” and indeed Clausius was guided by just the picture we have been painting: heat is the internal energy of atoms or molecules, which can be partially but never fully transformed to usable energy. Now, with their new statistical mechanics, Maxwell and Boltzmann were trying to make this idea of the internal energy of a trillion trillion rocking and rolling molecules precise, and in so doing come to understand entropy and the laws of thermodynamics on the basis of atomic theory. This program was so controversial that even by the end of the century, thirty years later, Planck, the thermodynamicist par excellence, was reluctant to adopt it. It was only his quantum conundrum that forced him to overcome his scruples, as we will see.

The key point is that the statistical mechanics of Maxwell and Boltzmann was still
Newtonian mechanics
, just applied to a system so complicated that one imagines it behaving like a massive game of chance, in which each molecular collision with a wall or with another molecule is like a coin being tossed (heads you go to the right, tails you go to the left). The
worldview
is the same as that of Newton and Laplace; only the method is different. Maxwell, had he lived another two decades, might have begun to recognize the leaks springing in this optimistic vessel, since the basic inconsistency in this view appeared at the intersection of his two great inventions, the theory of electromagnetic radiation and the statistical theory of matter. However, that was not to be; he would pass away a mere five years after his spectacular lecture on molecular science, having spent those final years occupied by his administrative duties. At the end of that same lecture, having anticipated the next twenty-five years of physical theory, the devout Maxwell makes one of the great historical appeals for intelligent design:

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 … the molecules out of which these systems are built … remain unbroken and unworn.

They continue this day as they were created, perfect in number and measure and weight, and from the ineffaceable characters impressed on them we may learn that those aspirations after accuracy in measurement, truth in statement, and justice in action, which we reckon among our noblest attributes as men, are ours because they are essential constituents of the image of Him Who in the beginning created, not only the heaven and the earth, but the materials of which heaven and earth consist.

The next century would demonstrate in many ways, culminating in the awesome demonstration of August 1945, that atoms are not as indestructible as Maxwell had supposed. And Einstein would be the first to understand, through his most famous equation,
E
=
mc
²
, just how much energy would be released when the perfect instruments of the Creator were disassembled.

CHAPTER 6

MORE HEAT THAN LIGHT

“
I have again made the acquaintance
of a sorry example of that species—one of the leading physicists of Germany. To two pertinent objections which I raised about one of his theories and which demonstrate a direct defect in his conclusions, he responds by pointing out that another (infallible) colleague of his shares his opinion. I will shortly give that man a kick up the backside with a hefty publication. Authority befuddled is the greatest enemy of truth.”

Such was the feisty mood of Einstein as he wrote in July of 1901 to an old friend, Jost Winteler. The object of his ire was Paul Drude, theorist and chief editor of
Annalen der Physik
, the most prestigious physics journal in the world at that time. Drude himself was the author of a well-respected text on optics and Maxwell's equations (in fact it was Drude who introduced the universal symbol
c
for the speed of light in vacuum). The “infallible” colleague mentioned by Einstein was none other than Ludwig Boltzmann. Einstein, characteristically, seemed oblivious to the potential consequences of offending such prominent scientists, one of whom was editor of the journal to which he would submit all his original research papers for the next six years.

Einstein wrote those lines from the small city of Winterthur, about twenty miles from Zurich, where he had a two-month position teaching physics and mathematics at the Technical College while the regular instructor was performing his military service. The teaching load was quite heavy, thirty hours a week, but, undeterred, he reassured Mileva that “
the Valiant Swabian is not afraid
.” In fact he found that he enjoyed the teaching much more than he had expected, and despite the busy schedule he managed time to study research questions, such as
Drude's new “electron theory of metals.” It was only four years earlier, in 1897, that the English physicist J. J. Thomson had confirmed the existence of electrons, negatively charged particles much lighter than the hydrogen atom itself, and he hypothesized that electrons were constituents of atoms. By 1899 Thomson had shown that electrons could be pulled off atoms (a process we now call “ionization”) and hence that the atom was in this sense divisible. This represented the first crack to appear in the indestructible atoms of Maxwell.