A Brief Guide to the Great Equations (17 page)

Lazare Carnot (1753–1823), a military engineer whose talent is uncovering and eliminating administrative and mechanical inefficiency, publishes a treatise on water-powered machines,
General Principles of Equilibrium and Motion
. Follow the water, he writes: the maximum power depends on how great a distance it falls. Track down and eradicate sources of waste, he also counsels, to make your machine work better. But Lazare can’t pursue these insights. He’s forced back to military duties; later he seduces a woman betrothed to another and ends up in jail. He’s released as the French Revolution begins and joins the revolutionaries, who nickname him ‘Organizer of Victory’ for the innovative way he mobilizes, trains, and supplies troops. He has two sons, whom he homeschools and who will carry on his legacy: Sadi, a military engineer (named after a Persian poet), and Hippolyte, a journalist and politician.

Count Rumford (1753–1814), soldier of fortune and amateur scientist, is in Munich, momentarily between courtships of wealthy widows. Keen to reveal the mysteries of heat, he puts a 6-pound brass cannon barrel in a vat of water, inserts a drill bit driven by a winch, hitches up a horse to the winch, and finds that this generates enough heat through the drilling to boil the water in 2½ hours. The caloric theory formulated by Lavoisier (the former husband of one of Rumford’s mistresses) is wrong, Rumford proclaims, for the seemingly inexhaustible amount of heat generated in the process is not coming
from either the brass or the water, but is clearly a form of motion coming from the friction between the bit and the cannon. He counts the candles it takes to boil the same amount of water, to compare the amount of heat and mechanical force. Reporting to the Royal Society, he implicitly likens himself to Newton, saying the laws of heat are as important as those of gravity. But Rumford is no Newton. His arguments are not entirely convincing and he has no overall theory of heat, just a very suggestive set of observations. Yet his idea that one can quantitatively compare various kinds of work that create the same amount of heat (candles, horses), and the work that heat does in different forms, helps to set up the looming conflict between conservation and conversion.

Sadi Carnot (1796–1832), a quiet engineer, returns from his father Lazare’s deathbed to the apartment he has just inherited. Determined to carry on his father’s work, Sadi sets to work composing a treatise,
Reflections on the Motive Power of Heat
, on ways to make steam engines more practical and efficient. Fearing that his prose is too convoluted to appeal to the general audience that he covets, he has his brother Hippolyte edit his manuscript and steady his prose. Steam engines, he begins, ‘seem destined to produce a great revolution in the civilized world.’ Nevertheless, he continues, ‘their theory is very little understood.’ Such a theory must begin by considering the general question of what the most efficient way is to use steam. One key thing to consider in a machine, Carnot realizes, is its maximum duty, or maximum output; for instance, how high a given amount of coal in the machine can raise a given amount of water. Follow the heat, he writes. Caloric in a heat engine, like water in a
water engine, is conserved as it flows from the hot to the cold places, and the maximum power depends on the magnitude of the temperature drop. The most efficient machine is modeled by an ideal cycle of expansion and compression in which the engine works reversibly, the caloric being conserved in going back and forth between the two temperature endpoints with no heat diverted (wasted) to friction or dissipation. This is a key insight, but
Reflections
is almost totally ignored. Carnot publishes nothing more, catches scarlet fever, brain fever, cholera, and dies, aged thirty-six, in a madhouse.

James Prescott Joule (1818–1889), who as a youth built a home lab in his parents’ brewery, manages to get highly accurate measurements of various conversions of heat and electrical, mechanical, and chemical energy into each other; for instance, the temperature increase that rotating paddles, stirring up water, produce in the water thanks to friction. He determines the mechanical equivalent of heat: 772 foot-pounds of work make a 1-degree F rise in 29 cubic inches of water.

The conflict between conservation (Carnot’s approach) and conversion (Joule’s) begins to come to a head. Young William Thomson (later Lord Kelvin, 1824–1907), the polymathic, trilingual, and farsighted son of a mathematics professor, travels to Paris, where he reads the only published comment on Sadi Carnot’s work and is so impressed he tries in vain to find a copy of the original. Then he attends a conference in Oxford, where he hears Joule. Joule is treated badly by the conference organizers, who instruct him to be brief. But Joule’s words jolt Thomson. How can heat be converted to something else when Carnot’s spectacular work relies on the fact that the amount of caloric in an engine is constant? Joule’s work must have ‘great flaws’, Thomson decides, and he resolves to find them.

Thomson, still convinced that Carnot’s conservation theory is right and that something must be wrong with Joule’s work, gets another jolt. He reads a paper by German physicist Rudolf Clausius (1822–1888), who has also noticed the conflict between the approaches of Carnot and Joule. Clausius has been examining the kinetic theory according to which heat and gases consist of tiny particles in constant motion. And Clausius says that the conflict between Carnot and Joule is only apparent, and is not a conflict in reality because
two
principles are in play. One involves the
conservation
of something (not heat, and soon called energy) in exchanges of heat and mechanical work; the other the
conversion
of heat into energy, and the property that heat cannot flow spontaneously from colder to warmer bodies. Thomson, inspired, begins to leapfrog works with Clausius on the new heat-mechanics. In 1854 Thomson names it
thermodynamics
, after the Greek for heat and force. Some heat in every engine, Thomson writes, ‘is irrevocably lost to man, and therefore ‘wasted’ although not
annihilated
’ – his version of the second of Clausius’s two principles. Clausius embarks on a series of papers that culminate in 1865, when he names the tendency of the energy transfer process to occur spontaneously (disorder, we now say) ‘entropy’, after the Greek for ‘transformation’; he referred to entropy as
S
, a function of the state of a system, and uses the formula ∫
dQ
/
T
≤ 0. In 1867, Thomson and his collaborator Tait compose thermodynamics’
Principia
, the
Treatise on Natural Philosophy
. In 1872, Clausius formulates what becomes known as the two laws of thermodynamics this way: ‘The energy of the world is constant; the entropy of the world strives toward a maximum.’

Priority battles erupt. In 1847, German physician Robert Mayer (1814–1878) reads a paper by Joule on the conversion of heat into mechanical energy and says he discovered it first. Seven years previously, as a doctor on a Dutch ship in the East Indies, Mayer had realized that the unusual redness of the blood of the crew – meaning it was oxygen-rich – was due to the fact that human metabolism is slower in the tropics. This had inspired him to write a paper on the interchangeability of mechanical work and heat to
Annalen der Physik und Chemie
, the leading German science journal, but the poorly written paper had been treated as a crackpot letter by the editor and Mayer didn’t receive a reply, though he later revised it and published it elsewhere. Depressed when Joule disputes his priority, Mayer flings himself out a third-floor window, and is committed to an asylum in a straitjacket. Meanwhile, another German physicist, Hermann von Helmholtz (1821–1894), is also a contender for discovering the first law of thermodynamics thanks to an 1847 paper on ‘the conservation of force.’ Tait and Clausius battle over who discovered various principles of thermodynamics, slinging mud at each other in journals and books.

Another battle breaks out, this time over which of the two laws of thermodynamics is more important. For they seem to conflict. The first law (conservation of heat/energy) implies that processes are reversible – that the ‘before’ and ‘after’ states of a physical process cannot be distinguished, for each can turn into the other. The second law (heat cannot be completely turned back into work) implies irreversibility, or what is later known as the ‘arrow of time’, that change tends
to go in one direction only. The problem comes to a head in Clausius’s specialty, the kinetic theory of gases. A gas is a ‘big thing’ governed by irreversible processes and the second law, but is composed of ‘little things’ – atoms and molecules – each of which obeys reversible Newtonian principles governed by the first law. In 1859, James Clerk Maxwell (1831–1879) comes across Clausius’s paper on the kinetic theory of gases, and decides that the statistical methods he had just used for studying Saturn’s rings as a collection of small bodies might also apply to gases. Maxwell realizes that a gas’s jostling molecules do not end up, or reach equilibrium, with all of them having exactly the same speed; rather, they have a range of speeds clustered about one value. Imagine a dense crowd of people randomly milling about in a train station: the people are not all moving at exactly the same speed, but most are moving at about the same speed, with only a handful dead-still or going very fast. To understand the behaviour of the gas, furthermore – or of a crowd – it is not necessary to track the positions and velocities of each and every individual in it, but suffices to know the distribution of positions and velocities. Using only statistical methods and assumptions of Newtonian mechanics, Maxwell comes up with an equation to describe the spectrum of velocities of the gas molecules. The plot describes a bell-shaped curve: few lie at the extremes, moving almost not at all or very fast, but most cluster about an average velocity with the numbers tapering off at higher and lower velocities. But Clausius’s 1865 paper, and Maxwell’s own experimental work, force him to modify the theory, and he publishes a revision in 1867. Maxwell concludes that the second law is merely statistical, true only when vast quantities of particles are involved and not true of individual motions. The second law, he writes, is true for the same reason as is the statement that ‘if you throw a tumblerful of water into the sea, you cannot get the same tumblerful out again (i.e., exactly the same molecules as before).’
¹
At the atomic level, reversibility is possible and the second law does not hold, he thought. But why can’t reversibility be possible for large bodies in principle; why cannot heat flow sometimes from a cold to a hot body? In 1867, writing
to Tait, Maxwell demonstrates this imaginatively and theatrically by a thought experiment involving a little ‘being’ that can detect faster-moving molecules in a gas, and by opening and shutting a door at the right time gets the faster ones on one side of a barrier, thus causing heat to flow to one side of the box. In this way the creature seems to refute Thomson’s idea of dissipation by getting heat to flow from a colder to a hotter place. Maxwell publishes this idea in 1871, in a short section entitled ‘Limitations of the Second Law of Thermodynamics’ in his
Theory of Heat
. That seems to end the matter; it is all a question of statistics.

Ludwig Boltzmann (1844–1906) extends Maxwell’s work. In 1868, a year after Maxwell’s paper, he produces an expression for the distribution of energy among molecules of a gas valid for gases of any kind. To derive it, he relies on a key assumption known as the equipartition theorem, according to which a molecule stores energy by spreading it equally among all the avenues (‘degrees of freedom’) available to it. This work also includes a now-famous term, Boltzmann’s constant, now referred to as
k
, 1.38 × 10
⁻²³
joules/Kelvin. The result is a thoroughly statistical interpretation of thermodynamics. In 1872, Boltzmann generalizes this work still further, in a revolutionary paper with a banal title, ‘Further Researches on the Thermal Equilibrium of Gas Molecules.’ In it, he derives a function related to entropy, now called the H-function, whose value almost always increases with time until it reaches a maximum – an entirely novel and innovative approach to proving the second law, and in a way that explicitly demonstrates irreversibility, or how it increases with time. But this work is subject to friendly fire: from Thomson, in an 1874 paper that refers to Maxwell’s little creature as a ‘demon’ and, in 1876, from Boltzmann’s former mentor Josef Loschmidt, for not having eliminated certain puzzles involving the relation between the second law and the first. Even complex many-body systems, such as the deployment of the planets around the sun, are cyclical, with
the same patterns eventually recurring, so why doesn’t this happen in thermodynamic systems? Also, if two gases mixed, following the H-curve, entropy increases – but if you then reverse the velocities of all the gas molecules, wouldn’t it then make the H-curve, the ‘arrow of time’, go the other way, violating his theorem? Boltzmann replies (1877) that when a big state corresponds to many equally probable little states, its probability is related to the number of little states. This all but forces big states to evolve in the direction of their more probable states. Boltzmann’s approach is an explicit probabilistic interpretation of entropy, introduces probability into electromagnetism, and proves the centrality of irreversibility to thermodynamics. Newton’s laws + objects made of myriads of pieces + laws of probability = the arrow of time. The forbidden becomes the highly unlikely. On large scales, you play dice, and statistics rule. In 1879, this work is extended by Boltzmann’s former teacher Stefan into the Stefan-Boltzmann law, a law relating the dependence of radiation on temperature in black bodies. But Boltzmann becomes vulnerable to depression late in life, from both personal and professional setbacks, and in 1906, on vacation near Trieste, he hangs himself while his wife and daughter are out swimming. On his tombstone is engraved his equation, not in the form he wrote it but as Max Planck would:
S
=
k
log
W
.