Carnot’s Cycle

Sadi Carnot had the same ambitions as his father. He hoped to abstract, from
the detailed complexities of real machinery, general principles that dictated the
best possible performance. Lazare’s analysis had centered on ideal mechanical
operation; Sadi aimed for the mechanical ideal, and also for ideal thermal
operation.
He could see, first of all, that when heat was dropped from a high temperature
to a low temperature in a heat engine it could accomplish something. His conceptual
model was based on an analogy between heat engines and water engines.
He concluded that for maximum efficiency a steam engine had to be designed so
it operated with no direct fall of heat from hot to cold, just as the ideal water
engine could not have part of the water stream spilling over and falling directly
rather than driving the waterwheel. This meant that in the perfect heat engine,
hot and cold parts in contact could differ only slightly in temperature. One can
say, to elaborate somewhat, that the thermal driving forces (that is, temperature
differences) in Carnot’s ideal heat engine have to be made very small. This designhad more than an accidental resemblance to Lazare Carnot’s principle of continuity
in the transmission of mechanical power.
To make it more specific, Carnot imagined that his ideal heat engine used a
gaseous working substance put through cyclic changes—something like the
steam in the pistons of the Woolf steam engine. Carnot’s cycles consisted of four
stages:
1. An isothermal (constant-temperature) expansion in which the gas absorbed
heat from a heat “reservoir” kept at a high temperature t2.
2. An adiabatic (insulated) expansion that lowered the temperature of the gas
from t2 to t1.
3. An isothermal compression in which the gas discarded heat to a reservoir
kept at the low temperature t1.
4. An adiabatic compression that brought the gas backto the original high temperature
t2.
Stages 1 and 3 accomplish the heat fall by absorbing heat at a high temperature
and discarding it at a low temperature. More workis done by the gas in the
expansion of stage 1 than on the gas in the compression of stage 3; and amounts
of workdone on and by the gas in stages 2 and 4 nearly cancel each other. Thus,
for each turn of the cycle, heat is dropped from a high temperature to a low
temperature, and there is net workoutput.

Heat Engines, Then and Now

The heat engines of interest to Sadi Carnot were steam engines applied to such
tasks as driving machinery, ships, and conveyors. The steam engine invented by
a Cornishman, Arthur Woolf, was particularly admired in France in the 1810s
and 1820s. Operation of the Woolf engine is diagrammed in figure 3.1. Heat Q2

was supplied at a high temperature t2 by burning a fuel, and this heat generated
steam at a high pressure in a boiler. The steam drove two pistons and they provided
the workoutput W1. (In this chapter and elsewhere in this part of the book,
keep in mind that the symbol t represents temperature and not time, as in chapters
1 and 2.) The steam leaves the pistons at a decreased pressure and temperature.
Heat Q1 was then extracted in a condenser where the steam was further
cooled to a still lower temperature t1 and condensed to liquid water. Finally, the
liquid water passed through a pump, which restored the high pressure by expending
work W2, and low-temperature, pressurized water was returned to the
boiler. This is a cycle of operations, and its net effect is the dropping of heat
from the high temperature t2 to the low temperature t1, with workoutput W1
from the pistons and a much smaller workinput W2 to the pump.
The Woolf steam engine and its variations have evolved into a vast modern
technology. Most contemporary power plants operate similarly. The scale is
much larger in the modern plants, the operating steam pressures and temperatures
are higher, and the working device is a turbine rather than pistons. But the
concept of heat falling between a high and a low temperature with net work
output again applies.

Lazare Carnot

Although he always worked on the fringes of the scientific world of his time,
Sadi Carnot did not otherwise live in obscurity. His father, Lazare, was one of
the most powerful men in France during the late eighteenth and early nineteenth
centuries. Sadi was born in 1796 in the Paris Luxembourg Palace when Lazare
was a member of the five-man executive Directory. Lazare Carnot served in highlevel
positions for only about four years, but his political accomplishments and
longevity were extraordinary for those turbulent times. Before joining the government of the Directory, he was an influential member of the all-powerful Committee
of Public Safety led by Maximilien de Robespierre. In that capacity, Lazare
was responsible for the revolutionary war efforts. His brilliant handling of logistics
and strategy salvaged what might otherwise have been a military disaster; in
French history textbooks he is known as “the great Carnot” and “the organizer
of victory.” He was the only member of the Committee of Public Safety to survive
the fall of Robespierre in 1794 and to join the Directory. A leftist coup in 1797
forced him into exile, but he returned as Napoleon’s war minister. (He had given
Napoleon command of the Italian army in 1797.) Napoleon’s dictatorial ways
soon became evident, however, and Lazare, unshakable in his republican beliefs,
resigned after a few months. But he returned once more in 1814, near the end of
the Napoleonic regime, first as the governor of Antwerp and then as Napoleon’s
last minister of the interior.
Lazare Carnot’s status in history may be unique. Not only was he renowned
for his practice of politics and warfare; he also made important discoveries in
science and engineering. A memoir published in 1783 was, according to Lazare’s
biographer, Charles Gillispie, the first attempt to deal in a theoretical way with
the subject of engineering mechanics. Lazare’s goal in this and in later workin
engineering science was to abstract general operating principles from the mechanical
workings of complicated machinery. His aim, writes Gillispie, “was to
specify in a completely general way the optimal conditions for the operation of
machines of every sort.” Instead of probing the many detailed elements of machinery
design, as was customary at the time, he searched for theoretical methods
whose principles had no need for the details.
Lazare Carnot’s main conclusion, which Gillispie calls the “principle of continuity
of power,” asserts that accelerations and shocks in the moving parts of
machinery are to be avoided because they lead to losses of the “moment of activity”
or workoutput. The ideal machine is one in which power is transmitted
continuously, in very small steps. Applied to water machines (for instance, waterwheels),
Lazare’s theorem prescribes that for maximum efficiency there must
be no turbulent or percussive impact between the water and the machine, and
the water leaving the machine should not have appreciable velocity.
Lazare’s several memoirs are not recognized today as major contributions to
engineering science, but in an important sense his worksurvives. His approach
gave his son Sadi a clear indication of where to begin his own attackon the
theory of heat engines. Lazare’s views on the design of water engines seem to
have been particularly influential. Waterwheels and other kinds of hydraulic machinery
are driven by falling water, and the greater the fall, the greater the machine’s
workoutput per unit of water input. Sadi Carnot’s thinking was guided
by an analogy between falling water in water engines and falling heat in heat
engines: he reasoned that a heat engine could not operate unless its design included
a high-temperature body and a low-temperature body between which heat
dropped while it drove the working parts of the machine.

A Tale of Two Revolutions Sadi Carnot

Reflections
The story of thermodynamics begins in 1824 in Paris. France had been rocked to
its foundations by thirty-five years of war, revolution, and dictatorship. A king
had been executed, constitutions had been written, Napoleon had come and gone
twice, and the monarchy had been restored twice. Napoleon had successfully
marched his armies through the countries of Europe and then disastrously into
Russia. France had been invaded and occupied and had paid a large war
indemnity.
In 1824, a technical memoir was published by a young military engineer who
had been born into this world of social, military, and political turmoil. The engineer’s
name was Sadi Carnot, and his bookhad the title Reflections on the
Motive Power of Fire. By “motive power” he meant work, or the rate of doing
work, and “fire” was his term for heat. His goal was to solve a problem that had
hardly even been imagined by his predecessors. He hoped to discover the general
operating principles of steam engines and other heat engine devices that supply
workoutput from heat input. He did not quite realize his purpose, and his work
was largely ignored at the time it was published, but after Carnot’s workwas
rediscovered more than twenty years later it became the main inspiration for
subsequent workin thermodynamics.

Thermodynamics Historical Synopsis

“flow” of heat from a high
temperature to a low temperature. Eighteenth-century engineers
knew that with cleverly designed machinery, this heat flow could be
used in a “heat engine” to produce useful workoutput.
The basic premise of the caloric theory was that heat was
“conserved,” meaning that it was indestructible and uncreatable;
that assumption served well the pioneers in heat theory, including
Sadi Carnot, whose heat engine studies begin our story of
thermodynamics. But the doctrine of heat conservation was attacked
in the 1840s by Robert Mayer, James Joule, Hermann Helmholtz, and
others. Their criticism doomed the caloric theory, but offered little
guidance for construction of a new theory.
The taskof building the rudiments of the new heat science,
eventually called thermodynamics, fell to William Thomson and
Rudolf Clausius in the 1850s. One of the basic ingredients of their
theory was the concept that any system has an intrinsic property
Thomson called “energy,” which he believed was somehow
connected with the random motion of the system’s molecules. He
could not refine this molecular interpretation because in the mid–
nineteenth century the structure and behavior—and even the
existence—of molecules were controversial. But he could see that
the energy of a system—not the heat—was conserved, and he
expressed this conclusion in a simple differential equation.
In modern thermodynamics, energy has an equal partner called
“entropy.” Clausius introduced the entropy concept, and supplied
the name, but he was ambivalent about recognizing its fundamental
importance. He showed in a second simple differential equation
how entropy is connected with heat and temperature, and stated
formally the law now known as the second law of thermodynamics:
that in an isolated system, entropy increases to a maximum value.
But he hesitated to go further. The dubious status of the molecular
hypothesis was again a concern.Thermodynamics had its Newton: Willard Gibbs. Where Clausius
hesitated, Gibbs did not. Gibbs recognized the energy-entropy
partnership, and added to it a concept of great utility in the study of
chemical change, the “chemical potential.” Without much guidance
from experimental results—few were available—Gibbs applied his
scheme to a long list of disparate phenomena. Gibbs’s masterpiece
was a lengthy, but compactly written, treatise on thermodynamics,
published in the 1870s.
Gibbs’s treatise opened theoretical vistas far beyond the theory of
heat sought by Clausius and Thomson. Once Gibbs’s manifold
messages were understood (or rediscovered), the new territory was
explored. One of the explorers was Walther Nernst, who was in
search of a theory of chemical affinity, the force that drives chemical
reactions. He found his theory by taking a detour into the realm of
low-temperature physics and chemistry.

Nearer the Gods

Biographers and other commentators have never given us a consensus view of
Newton’s character. His contemporaries either saw him as all but divine or all
but monstrous, and opinions depended a lot on whether the author was friend
or foe. By the nineteenth century, hagiography had set in, and Newton as paragon
emerged. In our time, the monster model seems to be returning.
On one assessment there should be no doubt: Newton was the greatest creative
genius physics has ever seen. None of the other candidates for the superlative
(Einstein, Maxwell, Boltzmann, Gibbs, and Feynman) has matched Newton’s
combined achievements as theoretician, experimentalist, and mathematician.
Newton was no exception to the rule that creative geniuses lead self-centered,
eccentric lives. He was secretive, introverted, lacking a sense of humor, and prudish.
He could not tolerate criticism, and could be mean and devious in the treatment
of his critics. Throughout his life he was neurotic, and at least once
succumbed to breakdown.
But he was no monster. He could be generous to colleagues, both junior and
senior, and to destitute relatives. In disputes, he usually gave no worse than he
received. He never married, but he was not a misogynist, as his fondness for
Catherine Barton attests. He was reclusive in Cambridge, where he had little
admiration for his fellow academics, but entertained well in the more stimulating
intellectual environment of London.
If you were to become a time traveler and meet Newton on a trip back to the
seventeenth century, you might find him something like the performer who first
exasperates everyone in sight and then goes on stage and sings like an angel. The
singing is extravagantly admired and the obnoxious behavior forgiven. Halley,
who was as familiar as anyone with Newton’s behavior, wrote in an ode to Newton prefacing the Principia that “nearer the gods no mortal can approach.” Albert
Einstein, no doubt equal in stature to Newton as a theoretician (and no paragon),
left this appreciation of Newton in a foreword to an edition of the Opticks:
Fortunate Newton, happy childhood of science! He who has time and tranquility
can by reading this book live again the wonderful events which the great
Newton experienced in his young days. Nature to him was an open book, whose
letters he could read without effort. The conceptions which he used to reduce
the material of experience to order seemed to flow spontaneously from experience
itself, from the beautiful experiments which he ranged in order like playthings
and describes with an affectionate wealth of details. In one person he
combined the experimenter, the theorist, the mechanic and, not least, the artist
in exposition. He stands before us strong, certain, and alone: his joy in creation
and his minute precision are evident in every word and in every figure.

More Disputes

Newton was contentious, and his most persistent opponent was the equally contentious
Robert Hooke. The Newton story is not complete without two more accounts
of Newton in rancorous dispute. The first of these was a battle over astronomical
data. John Flamsteed, the first Astronomer Royal, had a series of
observations of the Moon, which Newton believed he needed to verify and refine
his lunar perturbation theory. Flamsteed reluctantly supplied the requested observations,
but Newton found the data inaccurate, and Flamsteed took offense at
his critical remarks.
About ten years later, Newton was still not satisfied with his lunar theory and
still in need of Flamsteed’s Moon data. He was now president of the Royal Society,
and with his usual impatience, took advantage of his position and attempted
to force Flamsteed to publish a catalogue of the astronomical data. Flamsteed
resisted. Newton obtained the backing of Prince George, Queen Anne’s
husband, and Flamsteed grudgingly went ahead with the catalogue.
The scope of the project was not defined. Flamsteed wanted to include with
his own catalogue those of previous astronomers from Ptolemy to Hevelius, but
Newton wanted just the data needed for his own calculations. Flamsteed stalled
for several years, Prince George died, and as president of the Royal Society, Newton
assumed dictatorial control over the Astronomer Royal’s observations. Some
of the data were published as Historia coelestis (History of the Heavens) in 1712,
with Halley as the editor. Neither the publication nor its editor was acceptable
to Flamsteed.
Newton had won a battle but not the war. Flamsteed’s political fortunes rose,
and Newton’s declined, with the deaths of Queen Anne in 1714 and Montague
in 1715. Flamsteed acquired the remaining copies of Historia coelestis, separated
Halley’s contributions, and “made a sacrifice of them to Heavenly Truth” (meaning
that he burned them). He then returned to the project he had planned before
Newton’s interference, and had nearly finished it when he died in 1719. The task
was completed by two former assistants and published as Historia coelestis britannica
in 1725. As for Newton, he never did get all the data he wanted, and
was finally defeated by the sheer difficulty of precise lunar calculations.
Another man who crossed Newton’s path and found himself in an epic dispute
was Gottfried Leibniz. This time the controversy concerned one of the most precious
of a scientist’s intellectual possessions: priority. Newton and Leibniz both
claimed to be the inventors of calculus.
There would have been no dispute if Newton had published a treatise composed
in 1666 on his fluxion method. He did not publish that, or indeed any
other mathematical work, for another forty years. After 1676, however, Leibniz
was at least partially aware of Newton’s work in mathematics. In that year, Newton wrote two letters to Leibniz, outlining his recent research in algebra and on
fluxions. Leibniz developed the basic concepts of his calculus in 1675, and published
a sketchy account restricted to differentiation in 1684 without mentioning
Newton. For Newton, that publication and that omission were, as Westfall puts
it, Leibniz’s “original sin, which not even divine grace could justify.”
During the 1680s and 1690s, Leibniz developed his calculus further to include
integration, Newton composed (but did not publish) his De quadratura (quadrature
was an early term for integration), and John Wallis published a brief account
of fluxions in volume 2 of his Algebra. In 1699, a former Newton prote´ge´,
Nicholas Fatio de Duillier, published a technical treatise, Lineae brevissimi (Line
of Quickest Descent), in which he claimed that Newton was the first inventor,
and Leibniz the second inventor, of calculus. A year later, in a review of Fatio’s
Lineae, Leibniz countered that his 1684 book was evidence of priority.
The dispute was now ignited. It was fueled by another Newton disciple, John
Keill, who, in effect, accused Leibniz of plagiarism. Leibniz complained to the
secretary of the Royal Society, Hans Sloane, about Keill’s “impertinent accusations.”
This gave Newton the opportunity as president of the society to appoint
a committee to review the Keill and Leibniz claims. Not surprisingly, the committee
found in Newton’s favor, and the dispute escalated. Several attempts to
bring Newton and Leibniz together did not succeed. Leibniz died in 1716; that
cooled the debate, but did not extinguish it. Newtonians and Leibnizians confronted
each other for at least five more years.