The entropy of any crystalline solid would therefore be zero, except for the atoms’ vibrations about their crystalline positions. The solid will still have some temperature, no matter how low, so the atoms in the crystal will be shaking back and forth. We will truly have no uncertainty, and the entropy will be exactly zero, only when all vibrations of every atom in the solid cease. The fact that the entropy is zero only when the temperature is also zero is termed the Third Law of Thermodynamics. In the zero-temperature state, none of the atoms have any kinetic energy at all. At this point we say that the solid has a temperature of absolute zero degrees. We use the prefix “absolute” because no matter what type of thermometer you use, it will read zero average kinetic energy at this point. Note that not even outer space is this cold. There is background light and stray cosmic rays even in the vacuum of space, and they carry energy. In fact, the radio-wave background radiation that is a remnant of the Big Bang origin of the universe has an energy characterized by an average temperature of 3 degrees above absolute zero. So, even outer space has a temperature, and hence an entropy. The only way to beat the Second Law of Thermodynamics is to use systems with zero entropy, but this can only be realized at absolute zero. But if everything in your machine is at zero degrees, how would there be any heat flow to power our engine? The three laws of thermodynamics almost conspire to prevent us from constructing perfectly efficient machines, and just like supervillains in a comic book, we must reconcile ourselves to inevitable loss.
Our discussion of entropy has relied so heavily on the fluctuations of the constituent atoms that it is striking that the Second Law was formulated long before most scientists were convinced that matter was indeed composed of atoms. From the mid 1800s onward, some scientists had been taking the atomic theory of matter more and more seriously, while others remained unconvinced of the reality of atoms. These critics felt that the idea of matter as composed of atoms was useful in simplifying many of the calculations about the properties of fluids and gases, but that it was nonetheless meaningless to ascribe physical reality to entities that were too small to ever be seen. Many of the elder statesmen of physics in the late nineteenth century, notably Ernst Mach (for whom the speed of sound in air, the Mach number, is named) held this view.
Nevertheless, the atomic hypothesis eventually won out, via the same time-tested strategy by which all revolutionary ideas succeed. As Max Planck, himself a young Turk of the quantum revolution (about whom we will have much more to say in Section 3), once remarked, “A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it.”
A key development that convinced these younger scientists that atoms were real, regardless of what the older establishment claimed, was the explanation for the jitter that small objects underwent due to their random bombardment from still-smaller atoms and molecules striking them from all sides. This phenomenon is termed “Brownian motion” after Robert Brown, the botanist who observed the excursions of a pollen grain in a drop of water using a new scientific instrument, the microscope. While Brownian motion had been known since 1828, it was not until 1905 that a satisfactory theoretical description was provided by another young upstart, Albert Einstein. Einstein was able to quantitatively calculate the excursions of a pollen particle due to the collisions with the water in which it was suspended, and also relate the magnitude of the fluctuations to the temperature of the surrounding medium. The close agreement between Einstein’s calculations and experimental observations convinced many physicists that the atomic hypothesis was indeed correct. This work (a related aspect of which formed Einstein’s Ph.D. thesis) may not be as revolutionary as his subsequent discovery of relativity published the same year. Nevertheless, his elucidation of the statistical nature underlying Brownian motion would ensure that Einstein would be well known to physicists even if he never got mixed up with relativity and quantum mechanics.
As the Atom shrinks down so that he is roughly the size of a pollen grain—say, a hundredth to a tenth of a millimeter, less than the diameter of a human hair—he will begin to experience the back-and-forth motion that Brown first observed. This size is critical: When he is larger, the average bombardment is negligible; when much smaller, he fits between the atoms in the air,45 so that as long as he can avoid being struck, he will be fine. To go back to the example of the Atom floating on air currents, at one particular instant there may be more molecules striking him from below so that he will suddenly be pushed upward, while the following moment may see a downward thrust, perhaps not as severe as the last upward swing, or possibly even more extreme. This is a very slow way to get anywhere, and the Atom will need some Dra mamine before too long.
One doesn’t have to be as small as the Atom to directly experience Brownian motion. The random collisions of the air on our eardrums produce deflections that are just at the limit of our hearing. Sit in a soundproof room for about thirty minutes and your hearing will improve (just as your eyes’ sensitivity to stray light increases when you’ve acclimated to a darkened room) enough for you to be able to detect the deflection of your eardrums by the motion of the air atoms. It is possible in a very quiet room to hear the background noise emanating from the entropy of the air—in essence, to hear the temperature of the room. Super-h earing—it’s not just for Kryptonians anymore!
14
MUTANT METEOROLOGY— CONDUCTION AND CONVECTION
STAN LEE, head writer and editor of nearly all Marvel comics in the Silver Age, was fond of using radiation as a source of his heroes’ and villains’ superpowers. The bite of a radioactive spider gave Peter Parker the proportionate strength and abilities of a spider; the Fantastic Four were bombarded by cosmic rays (high-energy protons from as near as the sun and as far as other galaxies); Bruce Banner turned into the Hulk when he was belted by gamma rays (more radiation); and Matt Murdock was struck in the eyes with a radioactive isotope that literally fell off a truck, blinding him but endowing him with a “radar sense” and enhancing his other senses so that he could fight crime as Daredevil. After all of this radioactivity, Lee was tired of trying to come up with origins for bizarre superpowers. Hence, in 1963, when he co-created with Jack Kirby a new team of superpowered teens, the X-Men, he essentially threw in the towel, claiming that they were mutants,46 and hence simply born with their strange abilities and attributes.
One of the original X-Men, first appearing in X-Men # 1,47 was Bobby Drake, code- named Iceman. Drake’s mutant power was the ability to lower the temperature of his body and his immediate surroundings to less than 32 degrees Fahrenheit. His body would thereby acquire a protective coating of frozen water. As explained in X Men # 47, Bobby does not generate the ice that covers his body or projects from his hands when he employs his mutant power. Instead, by lowering the temperature around him, he condenses the water vapor that is always present in the air.
By now, I don’t need to belabor the point that any heat—that is, any kinetic energy—that Bobby is able to subtract from his surroundings must be compensated for by an amount of heat delivered somewhere else and, given the Second Law of Thermodynamics, the heat added is most likely greater than the heat removed. Refrigerators remove heat from an enclosed space, but this heat must then be deposited elsewhere. In addition, the motors in the compressors of the refrigerator require energy, and some of this electrical energy is not converted into useful work but takes the form of “waste heat.” The waste heat of a refrigerator is ejected from its rear; usually placed against a wall. (If you ever want to heat your kitchen, just leave the refrigerator door wide open. As the appliance struggles to lower the temperature of the room, it will deposit more heat into the kitchen than it is able to remove, thanks to the Second Law of Thermodynamics.) Where Iceman deposits the excess heat generated when he lowers the temperature of his surroundings remains a mystery.
Iceman’s body covering initially took the form of fluff
y snow, and when he had gained further control over his power in X-Men # 8, he acquired an icy, crystalline appearance. The difference between snow and ice is in the arrangement of the water molecules that results when the water freezes into the solid state. Snowflakes are constructed from aggregations of water within clouds. When water molecules condense from the vapor phase, they release energy, warming the surrounding air. The lower-density, warmed air keeps the cloud aloft, as in a hot-air balloon. When too many water molecules come together, frequently coalescing around a dust grain in the cloud, they form a droplet. When the temperature in the cloud is above 32 degrees Fahrenheit, the droplet can fall from the cloud in the form of rain, converting its potential energy into kinetic energy. A frozen rain droplet is called sleet. The construction of a snowflake is a more delicate affair. A snowflake is created when water vapor slowly freezes around a dust particle.
The chemical properties of atoms dictate how they are arranged in a solid. The chemical forces between metal atoms makes them stack in a solid, like oranges in a grocery store display. Water molecules have a more V-shaped geometry, with an oxygen atom at the vertex bonded with two hydrogen atoms, protruding like a rabbit-ear antenna. The shape of the water molecule determines the geometry of its packing, which turns out to be a hexagon.
Chemical stacking explains the sixfold symmetry of snowflakes, but how do they form their characteristic lacy structure? In the snowflake-generating clouds that have a relatively low humidity, the water molecules must diffuse to the growing flake before they can be incorporated into its structure. The water molecules in the cloud are not being pushed in any given direction, but are undergoing Brownian motion as they fluctuate one way and another. This type of “random walk” is a very slow way to get anywhere, since you are just as likely to take a step away from your goal as toward it. When you hold your hand over a hot object, the warmth you feel is carried by air molecules that collided with air diffusing away from the high-temperature region. This method of conveying energy from one location to another is termed “conduction,” and is fairly inefficient. In general, unless the object in question is glowing white hot, you typically have to place your hand very close before a significant energy transfer is detected.
Einstein’s equation for how far a fluctuating atom moves in a given period of time, derived in his 1905 paper on Brownian motion, indicates that it will take a hundred times longer for the water in a growing snowflake to diffuse a distance of one centimeter than it will take for it to travel one millimeter. Consequently, those regions of the growing snowflake that extend farther out from the body of the flake will accrete water faster, because they shorten the distance the molecules must random-walk to reach the flake. The six pointed regions at the corners of the hexagon will therefore grow first by adding water molecules, and as they extend farther, they continue to grow more rapidly than neighboring regions. The exact details of how the flake develops—how the dendritic branches grow secondary offshoots, the role that the energy deposited by the diffusing water molecules plays in locally melting and then refreezing the growing flake—will all depend sensitively on the humidity and temperature within the cloud. The final structure of the flake will also depend on the unique details of how the flake forms around a dust particle, such that no two snowflakes will ever be exactly alike, though it is possible to find flakes that are strikingly similar. But at its heart, the beautiful symmetry and near-perfect order of a snowflake arises from the disordered, statistical fluctuations underlying Brownian motion.
As his mastery of his mutant ability progressed, Iceman was able to project “freeze rays” from his hands, icing up another person or object, or even creating a large mountain of ice beneath his feet. To get around while fighting such evil mutants as Magneto, the Blob, or the sinister, robotic Sentinels, Bobby would frequently generate an “ice slide” underneath his feet on which he would glide, as illustrated in fig. 22. Bobby would, in principle, create a large ice mountain underneath himself and then generate a ramp on which he would slide to his intended destination. Well, this in itself would not violate any physics principles, assuming, of course, that one could indeed control local temperatures in this manner and that there was sufficient humidity in the air to create all this ice. What is troubling, however, is the apparent stability of Bobby’s ice slides, no matter how long they became. At some point, Bobby would get out in front of the center of mass of his ice slide—a point at which one would expect bad things to happen.
The “center of mass,” also referred to as the “center of gravity,” is the point at which an object behaves as if all of its mass, no matter how much or how unevenly distributed, were concentrated at this one location. A yardstick would have its center of mass located exactly at its center. You can balance the yardstick across your index finger, held so that it lies parallel to the ground, but only if your finger is underneath this midpoint. Placing your supporting finger too close to either end will cause the stick to rotate and fall. The center of mass depends on the distribution of matter in the object. A baseball bat, thicker and heavier at one end, will have its center of mass closer to the wider end than to the narrower handle.
To see why Bobby’s ice slides cannot extend too far without crumbling, place a book on a table. The weight of the book is directed toward the ground and is balanced by a force from the table. The center of mass is in the middle of the book, and as long as it is on top of the table, the book is stable. But now slide the book near the edge of the table. At first a small fraction of the book can hang over the edge of the table with no problem. The portion of the book that’s over the edge creates a twisting force, a torque. This unsupported weight of the book tries to rotate the book, but there is more weight over the table, trying to rotate the book in the other direction, so the book remains stationary. But as you slide the book farther, so that its center of mass is no longer above the table but over the side, the book will rotate and flip onto the floor.
Fig. 22. Scene from Amazing Spider-Man # 92, where the mutant X-Man Iceman battles Spider-Man due to a mistaken impression (such confusion that would lead two heroes to fight until they realized that they were on the same side occurred no more often in Marvel comics than once or twice—a month). It’s no surprise that Spidey has never seen anything like Iceman’s ice ramps, as they display questionable stability when the mutant hero ventures far beyond the ramps’ centers of gravity.
Similarly, when Iceman’s slides become too extended, he moves too far from the center of mass of the ice mountain he’d originally generated. Rather than flipping his ice ramp and mountain, it is more likely that the torque that results from him gliding along its edge becomes greater than the strength of the ice slide, and his slide should break off. In order to satisfy basic mechanics, Bobby Drake should continually reinforce the underside of his ice slides with ice pillars, in order to avoid getting too far in front of their center of mass. Sometimes the implausible mechanical stability of Iceman’s constructs would be acknowledged, as indicated in fig. 23 from a backup feature “I, the Iceman” in X-Men # 47. If any “wiseguy physics majors” wonder what keeps Iceman’s ladders and ramps aloft, the answer, which no one can refute, is “a hulkin’ helping of imagination!”
Fig. 23. A scene from X-Men # 47, where Bobby Drake (Iceman) talks directly to the reader about the mechanisms underlying his mutant freezing powers and gives a knowing wink that he recognizes the physical implausibility of some (OK, all) of his uncanny feats.
The X-Men comic in its first incarnation in the 1960s was not a major sales success, and original stories ceased publication in 1970. Five years later, new management at Marvel Comics decided to try a revival of the X-Men, and the All-New, All-Different X-Men (written by Len Wein and drawn by Dave Cockrum), as their debut giant-size issue was titled, was a financial success from the very beginning. Despite the promise implied by the title, some of the X-Men characters featured in this issue were from the original team, but many of the new characters that debuted, such as Or
oro Munroe (Storm); Logan (Wolverine); Peter Rasputin (Colossus); and Kurt Wagner (Nightcrawler) would go on to become fan favorites.
Not everyone was impressed, however. Stan Lee has been quoted as complaining about the implausibility of Storm’s mutant ability to control the weather. He did not seem bothered by Colossus’s ability to transform his skin into “organic steel” (I have absolutely no idea what this phrase means, or how this power could possibly work in the physical world) or Nightcrawler’s teleportation power (well, if you give me enough “miracle exceptions,” I suppose I could try to make this work), but “weather powers” seemed too implausible for the man who brought the world the Incredible Hulk and the Silver Surfer (if he travels through outer space, what exactly is he surfing on?).48 But Stan should not be too quick to throw stones at superhero houses. The same mutant power manifested by one of Stan’s own creations, Iceman of the original X-Men—the ability to generate and control thermal gradients—also enables Storm to influence meteorological phenomena.
At its core, the weather is simply a matter of the atmosphere absorbing energy in the form of sunlight, and is in fact so simple that it is nearly impossible to accurately predict. When one thinks of weather, terms such as “wind,” “rain,” and “snow” come to mind (particularly if, like me, you live in Minnesota). All are governed by spatial differences in temperature, driven by variations in the amount of sunlight energy absorbed by the atmosphere.
The Physics of Superheroes: Spectacular Second Edition Page 20