Proceedings of the 16th International Heat Transfer Conference, IHTC-16
August 10-15, 2018, Beijing, China

IHTC16-FL01
HISTORICAL DEVELOPMENT
IN THE THOUGHT OF THERMAL SCIENCE—HEAT AND ENTROPYPREFACE AND CHAPTERS 18‒20(250 YEARS AFTER JAMES WATT AND
200 YEARS AFTER SADI CARNOT)
Yoshitaka Yamamoto1
English Translation by Hideo Yoshida2*
1

Surugadai Gakuen, Chiyoda-ku, Tokyo, 101-0062, Japan
Kyoto University, Nishikyo-ku, Kyoto, 615-8540, Japan

2

ABSTRACTThis paper is an English excerpt of the Japanese book “Historical Development in the Thought of Thermal
Science—Heat and Entropy” by Yoshitaka Yamamoto (first edition, Gendai-Sugakusha, Kyoto, 1986; revised
edition, Chikumashobo, Tokyo, 2009). The subtitle “250 Years after James Watt and 200 Years after Sadi Carnot”
was added to this excerpt by the translator, because the separate condenser for the steam engine was invented(patented) by Watt in 1769 and the historical paper “Réflexions sur la Puissance Motrice du Feu (Reflections on
the Motive Power of Fire)” was published by Carnot in 1824. In the Fourier Lecture at IHTC-16, the translator
presented one of the most important and interesting parts of Yamamotoʼs masterpiece, in which he thoroughly
studied the development of thermal science in the seventeenth‒nineteenth centuries and shed light on its true
nature; this excerpt comprises Chapters 18‒20, about one-tenth of the original book. In Chapter 18, Carnotʼsoriginal problem setting of the motive power of heat is discussed in conjunction with Wattʼs pioneering work on
the steam engine. In Chapters 19 and 20, the essence of Carnotʼs theory is described, and its profound significancein physics is analyzed in detail.

KEY WORDS: History of thermal science, James Watt, Sadi Carnot, Heat, Caloric theory, Steam engine, Motive
power, Carnot cycle, Thermodynamics
PREFACE TO THE FIRST EDITION
Through the fevered Age of Great Voyages in Western Europe, human activity extended to the entire globe
and discovered an entirely renewed world. On the basis of this world view, modern Europe advocated anddeveloped heliocentrism and modern physics.
Modern physics was launched from the mechanisms proposed by Galileo and Descartes. It succeeded inproviding a quantitative understanding of nature by regarding natural objects as homogeneous inactive bodies
having only a geometrical shape and motion; this was a giant leap from Aristotelian physics, which focusedprincipally on qualities. Simultaneously, it achieved a breakthrough from Aristotleʼs dualistic world view (i.e.,a terrestrial world and celestial world), which was deduced from daily experiences that motion on the groundinevitably decelerates and ceases unlike the eternal and regular motion of heavenly bodies. Contrary to
Aristotleʼs view, the modern mechanism treated celestial and terrestrial bodies on an equal footing, and suchan approach was initially realized by neglecting the friction and drag associated with motion on the ground.

*Corresponding Author: sakura@hideoyoshida.com
1

IHTC16-FL01
It was Newton who overcame the limitation of the primitive mechanism. He recognized that the decelerationof motion by friction and drag is inevitable in nature, and furthermore introduced the concept of a force acting
between bodies; this concept was necessary to compensate for deceleration. However, Newton also regarded
ordinary matter as being passive and inactive, so forces were considered to be exerted by means of ʻaetherʼ as
an active agent. This was one of the means of treating activity in the natural world, which newly introduced adualistic materialism into the monistic one previously homogenized by the primitive mechanism.
The ʻaetherʼ of Newton as an active agent soon became the ʻairʼ of Hales, ʻmateria ignisʼ of Boerhaave,ʻelectric matterʼ of Franklin, ʻphlogistonʼ of Stahl, and finally the material of heat or ʻcaloriqueʼ of Cleghorn,
Lavoisier, and others. Therefore, the caloric theory in the late eighteenth century was not a simple alternative
to the dynamical theory of heat but connoted the pan-thermal world view, which regarded heat as the ultimate
source of natural activity. It was an alternative, or at least a complement, to the mechanism in the seventeenth
century. Furthermore, the material theory of heat, namely the considerations of heat as a special material, after
the establishment of fundamental calorimetry by Black, paved the way for quantification in thermal sciencethrough its conservation and equilibrium. Although caloric theory itself was proved to be incorrect in thenineteenth century, we should note that its impact on thermal science and its legacy were profound.
Affected by the pan-thermal world view as well as caloric theory, and stimulated by the development of steam
engines since Watt, in 1824, Carnot first proposed and solved the problem of whether the motive power (work)
produced from heat has a limitation governed by principles. According to Carnot, to produce motive powerfrom heat, a temperature difference between the heat source and environment is necessary, and its maximum
efficiency is determined by their temperatures.
On the other hand, Mayer and Joule insisted that heat and work are equivalently interchangeable and their total
is conserved. Such an approach advanced the unification of heat and work into energy, and apparently
contradicted Carnotʼs principle that the conversion from heat to work is realized under a special condition, i.e.,
the existence of a temperature difference. Although the conservation of heat on which Carnotʼs work was
based was later rejected, it is impossible to extract work by simply cooling a uniform-temperature body, eventhough such extraction of work does not violate the law of energy conservation.
By resolving the contradiction between Carnot and Mayer/Joule, Clausius and Thomson established almost
perfect thermodynamics. Its principle is based on the argument that in nature there exist irreversible
processes—entropy production in addition to energy conservation. Previously, to understand terrestrial
phenomena in the same manner as planets and to construct a unified view of the cosmos, the mechanism
neglected friction, drag, and diffusion/dissipation due to conduction/mixing. Clausiusʼs and Thomsonʼsthermodynamics, however, affirmed that such phenomena are essential natural principles.
The creation of thermodynamics was a giant step toward understanding terrestrial phenomena. Looking backover the history of thermal science, from the very beginning, it developed with the aim of understanding thephenomena on earth, which is a unique environment for humanity and also a huge thermal engine. Thermalscience has retained its profound significance to this day, even though mainstream physics has shifted frommacroscopic to microscopic subjects. The existence of humanity depends on the entropy balance of the globe.
The purpose of this book is to clarify the scope of thermal science and its importance in modern times byhistorically and demonstratively reviewing the development of thought on thermal science from the birth ofmodern physics to the early twentieth century. This book is basically a history of ideas that was also writtenwith the aim of its use a textbook on thermal science. I sincerely hope that this book will be read from various
viewpoints.
November 1986
Yoshitaka Yamamoto2

IHTC16-FL01
CONTENTS
Preface to the First Edition
November, 1986
Remarks to the Revised Edition
August, 2008
Part 1 Materialism and Dynamical Reductionism
Chapter 1
Mechanical View of Nature and Heat—Galilei
Chapter 2
ʻCorpuscular Philosophyʼ and Proposal of Dynamical Theory of Heat—Boyle
Chapter 3
ʻBoyleʼs Lawʼ—Boyle, Hooke, and Newton
Chapter 4
Formation of Attraction-Repulsion Paradigm—Newton and Hales
Chapter 5
End of Monistic Materialism—Desaguliers
Chapter 6
ʻAetherʼ as Active Agent—Another Newton

Part 2 Formation of Caloric Theory
Chapter 7
Impoundable Fluid and Its Conservation Law—Boerhaave and Franklin
Chapter 8
Formation of the Scotland School—Maclaurin, Hume, and Cullen
Chapter 9
Establishment of Heat Capacity and ʻQuantity of Heatʼ Concept—Cullen and Black (1)Chapter 10
Latent-heat Concept and Conservation of Heat—Cullen and Black (2)Chapter 11
Formation and Branching of Material Theory of Heat—Black, Cleghorn, and Irvine
Chapter 12
Caloric Theory and Phlogiston Theory—Early LavoisierPart 3 Calorimetry and Conservation of Heat
Chapter 13
Proposal of the Principle of Calorimetry—Laplace and Lavoisier
Chapter 14
Thermal Expansion of Gas and Criticism on Concept of Temperature—Laplace, Gay-Lussac, andDaltonChapter 15
Adiabatic Process and Specific Heat of Gas
Chapter 16
Completion of Analytical Calorimetry—Laplace and Poisson
Chapter 17
Why ʻDynamical Theory of Heatʼ was Incompetent—Rumfordʼs Myth3

IHTC16-FL01
Part 4 Motive Power of Heat—Carnot and Joule
Chapter 18
Setting of a New Problem—ʻMotive Powerʼ of Heat—Carnot and Watt
Chapter 19
Theory of Ideal Heat Engine—Carnotʼs Theorem
Chapter 20
Structure of Carnotʼs Theory and Its Extension—The Start of Thermodynamics
Chapter 21
Intermezzo—Formation of Wave Theory of Heat and Its Intrinsic Limit—Young, Whewell, and Carnot
Chapter 22
Conservation of ʻKraftʼ and Mechanical Equivalent of Heat—Mayer
Chapter 23
Proving of Universal Interchangeability between Heat and Work—Joule
Chapter 24
Peculiarity of Heat and Universality of Energy Conversion—Thomsonʼs DilemmaPart 5 Proposal of the Principles of Thermodynamics
Chapter 25
Principle of Universality of Heat—Establishment of the First Law of Thermodynamics—Clausiusʼs
Paper in 1850 (1)Chapter 26
Principle of Peculiarity of Heat—Proposal of the Second Law of Thermodynamics—Clausiusʼs Paper
in 1850 (2)Chapter 27
Carnot Function and Absolute Temperature—Thomsonʼs Awareness
Chapter 28
Joule‒Thomson Effect and Definition of Absolute Temperature—Thomson in 1852‒54
Chapter 29
Mathematical Expression of the Second Law of Thermodynamics—Thomson and Clausius in 1854

Part 6 Energy and Entropy
Chapter 30
From the Second Law of Thermodynamics to Entropy—Clausiusʼs New Horizon
Chapter 31
Toward Systematization of Thermodynamics—Available Energy and Equilibrium Condition
Chapter 32
Free Energy and System of Thermodynamics—Gibbs
Chapter 33
Nernstʼs Theorem and the Third Law of Thermodynamics—Nernst and Planck
Chapter 34
Thermal Science and Thermal World View—Aims Pursued by Thermal Science

Postscript to the First Edition
November 9, 1986
Postscript to the Revised Edition
January 1, 2009

4

IHTC16-FL01
IMPORTANT PERSONS IN THIS BOOK
Jean-Baptiste Joseph Fourier (1768‒1830)
Thomas Young (1773‒1829)
Charles Bernard Desormes (1777‒1838)
Louis Joseph Gay-Lussac (1778‒1850)
Nicolas Clément (1779–1841)
Siméon Denis Poisson (1781‒1840)
Michael Faraday (1791‒1867)
William Whewell (1794‒1866)
Nicolas Léonard Sadi Carnot (1796‒1832)
Émile Clapeyron (1799‒1864)
Born in the nineteenth century
Henri Regnault (1810–1878)
Carl Holtzmann (1811‒1865)
Julius Robert von Mayer (1814‒1878)
James Prescott Joule (1818‒1889)
Hermann Ludwig Ferdinand von Helmholtz (1821‒1894)
Rudolf Julius Emanuel Clausius (1822‒1888)
William Thomson, 1st Baron Kelvin (1824‒1907)
Julius Thomsen (1826‒1909)
Pierre Eugène Marcellin Berthelot (1827‒1907)
James Clerk Maxwell (1831‒1879)
Josiah Willard Gibbs (1839‒1903)
August Horstmann (1842‒1929)
Ludwig Eduard Boltzmann (1844‒1906)
Jacobus Henricus vanʼt Hoff (1852‒1911)
Max Karl Ernst Ludwig Planck (1858‒1947)
Walther Hermann Nernst (1864‒1941)

Born in the sixteenth century
Galileo Galilei (1564‒1642)
Pierre Gassendi (1592‒1655)
René Descartes (1596‒1650)
Born in the seventeenth century
Robert Boyle (1627‒1691)
Robert Hooke (1635‒1703)
Isaac Newton (1642‒1727)
Thomas Savery (ca.1650‒1715)
Thomas Newcomen (1664‒1729)
Herman Boerhaave (1668‒1738)
Stephen Hales (1677‒1761)
John Theophilus Desaguliers (1683‒1744)
Colin Maclaurin (1698‒1746)
Born in the eighteenth century
Benjamin Franklin (1706‒1790)
William Cullen (1710‒1790)
David Hume (1711‒1776)
Joseph Black (1728‒1799)
James Watt (1736‒1819)
William Irvine (1743‒1787)
Antoine Laurent Lavoisier (1743‒1794)
Pierre-Simon Laplace (1749‒1827)
Benjamin Thompson, Count von Rumford (1753‒1814)
Lazare Carnot (1753–1823)
William Cleghorn (1754‒1783)
John Dalton (1766‒1844)

Fig. 1 Dates of important persons in this book. Note that this year, 2018, marks the 250th anniversary of
Fourierʼs birth and the 200th anniversary of Jouleʼs birth. (This section and Fig. 1 have been supplemented by
Yoshida.)5

IHTC16-FL01
CHAPTER 18 SETTING OF A NEW PROBLEM—“MOTIVE POWER” OF HEAT—CARNOT AND WATT
CONTENTS
I.
Neglect of Carnotʼs paper during his lifetime......................................................................................6
II. Carnotʼs premise and problem setting ..................................................................................................8
III. Caloric theory and cosmology based on heat .................................................................................... 11
IV. Carnotʼs view of nature and of society based on heat .................................................................... 12V.
Development of steam engine and its defects before Watt............................................................ 14
VI. Wattʼs improvement—separate condenser ....................................................................................... 17VII. Wattʼs improvement—expansive principle ....................................................................................... 19
VIII. P‒V diagram and expression for work ............................................................................................... 21
IX. High-pressure engine and its development in France .................................................................... 23
References in chapter 18 ........................................................................................................................... 24
I. Neglect of Carnotʼs paper during his lifetime
It was on August 24, 1832 that Sadi Carnot (Fig. 18.1) died of cholera at his age of 36. Although there is noconnection, only three months prior to his death, the mathematician Évariste Galois was also killed before his 21stbirthday. France experienced the successive early deaths of two great geniuses in physics and mathematics.
Without doubt, the driving force of their work was the French intellectual atmosphere of that age, as if “the wholeof Europe was illuminated by the light of science which emanated from Paris during the first third of thiscentury”[1].
It was many years later, however, that the significance of their pioneering studies was understood and recognizedby people; at that time, people, science, and society were all immature.
Sadi Carnotʼs short life is only known from a biographical sketch written in 1878 by Hippolyte Carnot (1801–
1888), who was Sadiʼs younger brother and a French statesman [2]. Sadiʼs and Hippolyteʼs father was Lazare
Carnot (1753–1823), who was a French politician known as “the Organizer of Victory” in the French
Revolutionary Wars. Since he acted as a republican even in the anti-revolution period, he was purged and forced
into exile; although he was a famous statesman, he was also an eminent mathematician and an engineer withexpertise in applied mechanics. Sadi, who was born in 1796, graduated from the École Polytechnique in Paris,
1814. After becoming an army officer, he started research on physics and engineering perhaps being influenced
by his father. Although he was isolated from the circle of the scientific elite such as Laplace (1749‒1827), GayLussac (1778‒1850), and Poisson (1781‒1840) as well as the French scientific establishment, he was acquainted
with Clément (1779–1841), one of the first chairs in chemistry at the Conservatoire des Arts et Metiers in Paris.
According to Hippolyte Carnot, “He had such a repugnance to bringing himself forward that, in his intimate
conversations with a few friends, he kept them ignorant of the treasures of science which he had accumulated”[2].
Despite his reticence, however, he was a quarter of a century ahead of the physicists at that time.
He left us with only three manuscripts; “Réflexions sur la Puissance Motrice du Feu (Reflections on the Motive
Power of Fire) (1824), hereafter referred to as the Réflexions”[3]1, which was published during his lifetime (Fig.
18.2), “Notes sur les Mathématiques, la Physique et autres sujets (1878),” which was published nearly half acentury after his death, and “a draft of the Réflexions,” which was discovered in 1964. His other remaining
manuscripts left behind were burned to prevent cholera infection.

1

Since reference [3] in this chapter is frequently referred to throughout this paper, it is expressed as [C, page
number in Foxʼs English version (1986)], for example, [C, pp. 66-67].

6

IHTC16-FL01

Fig. 18.1 Sadi Carnot (1796‒1832) at the age of 17.

Fig. 18.2 Cover page of “Réflexions sur la
Puissance Motrice du Feu” (1824).

In physics, if a problem has been properly defined, its solution is generally near at hand. In this sense, it is no
exaggeration to say that the birth and consolidation of thermodynamics and its foundations were almost entirely
due to a single paper: the Réflexions—a booklet of 118 pages, a small number of copies of which were publishedin 1824.
The paper, however, was neglected by physicists and engineers of the age, and completely forgotten after Carnotʼsdeath. “Possibly the most astonishing fact about Sadi Carnotʼs work is that it was received in complete silence by
the world of French science”[4]. “A few days after its publication, Carnotʼs book was formally presented to the
Académie des Sciences; this was at the meeting of Monday, June 14, 1824. It was presented by Girard, a prominentengineer. (omit) Among the Academicians present were Arago, Fourier, Laplace, Ampère, Gay-Lussac, Poinsot,
Fresnel, Legendre, Poisson, Cauchy, Dulong, as well as Navier and Prony. Later this review was printed in the
Revue Encyclopédique (23, 411, 1824). (omit) To summarise, the Réflexions received adequate publicity before
the most eminent physicists and engineers of the Académie and the same enthusiastic report was published in awell-known journal. In spite of this auspicious beginning, one may look in vain for the slightest mention of
Carnotʼs work in any of the journals”[4]. “In fact, Carnotʼs book made hardly a ripple on the surface of the mainstream of science”[5].
That is, as Herivel [6] wrote, “It is too fanciful to suppose that many of the members may have allowed theirattention to wander away from the work of the son to the life of the father? Some would have remembered with asudden start the friends and relations on whose death warrants had figured the name of Lazare Carnot.”The exception is “in a manuscript book of notes taken by a student who attend the course of lectures given by
Clément on Applied Chemistry at the Conservatoire des Arts et Métiers in 1824‒25 and later years. The lectures
were apparently open to the public”[7]. As Kuhn [8] pointed out, the mainstream of the Paris Académie had
traditionally been interested in pure mathematical science since DʼAlembert (1717‒1783), Laplace, and Lagrange(1736–1813), while Lazare and Sadi Carnot belonged to the relatively minor stream based on practical engineering.
In 1834, ten years after Carnotʼs death, the engineer Émile Clapeyron (1799‒1864) took up the Réflexions andgave it graphical and mathematical expressions in Mémoire sur la Puissance Motrice de la Chaleur (Cf. Reference[2] in Chapter 20). The Réflexions, however, still continued to be neglected. Kerker [9] wrote “F. Arago, as
sécrétaire perpetuelle of the Academy of Sciences, read a 97-page éloge of Lazare Carnot in 1837, Comptes
Rendus. 1837, 5; 294. There is no mention of Sadi in this biography which Arago submitted for publication without

7

IHTC16-FL01
change in 1850, Mémoires de lʼAcadémie des Sciences. 1850 (2), 22; 1. Aragoʼs neglect of Sadi is all the more
remarkable because of his interest in the history of the steam engine. First published in 1829 in LʼAnnuaire duBureau de Longitudes, this “Notice Historique sur les Machines à Vapeur” was reissued in 1830, 1837 and 1860.
(The 4th edition forms volume V of Aragoʼs Œuvres Complètes.) The purpose of this 117-page work is to negate
the assertion by many Englishmen that the steam engine was entirely an English invention. Arago emphasizes,
especially, the role of Papin, but the contributions of Carnot are completely ignored, both in this work and in the
biography of Lazare.” Also, de Pambourʼs Théorie de la Machine à Vapeur (1939), which was well known toengineers and translated into English, does not refer to the Réflexions or reflect its influence. Furthermore, “Mostsurprisingly, the lithographic edition of course on steam engines given by Clapeyron at the Ecole des Ponts et
Chaussées in 1844‒45 makes no mention of the Réflexions nor uses its ideas”[10]. It was in 1844 that James
Prescott Joule (1818‒1889) disputed Carnotʼs theory on the basis of anti-caloric theory.
In the end, it was William Thomson (later Lord Kelvin) (1824‒1907) who, after Clapeyron, rehabilitated the
Réflexions, making it widely known among physicists. The following episode, however, clearly demonstrated theignorance of the Réflexions by the scientific community at that time:
In 1844 or 1845 Thomson, who was born in the year that the Réflexions was published, learned about it for the
first time from Clapeyronʼs paper. After he was elected a fellow of Peterhouse, Cambridge, he moved to Paris.
“The perusal of this mémoire incited Thomson to refer to the original tract of Carnot. In vain did he inquire for itin the Library of the Collège de France. No one could tell him even where a copy might be seen. But a copy hemust have, even if he searched all Paris to find it. (omit) I (= Thomson) went to every book-shop I could think of,asking for the Puissance motrice du feu, by Carnot. “Caino ? Je ne connais pas cet auteur.” With much difficulty
I managed to explain that it was “r” not “i” I meant. “Ah! Ca-rrr-not! Oui, voici son ouvrage,” producing a
volume on some social question by Hippolyte Carnot; but the Puissance motrice du feu was quite unknown. (omit)
Not until the end of the year 1848 did he see the book, a copy of it being sent by Professor Lewis Gordon”[11]. In
this way, the Réflexions finally reached a reader with a stature comparable to its author, in other words, a reader
who could understand its significance, 24 years after its publication.
Compered with the understanding in France, other countries were far behind. In 1850, the German Rudolf Clausius(1822‒1888), who developed the second law of thermodynamics almost simultaneously with Thomson, testified
the following: “I have not been able to procure a copy of this work; I know it solely through the writings ofClapeyron and Thomson, from which latter are taken the passages hereafter cited”[12].2
In 1872, Carnotʼs paper was republished in Annales Scientifiques–École Normale Supérieure (pp. 393–457). On
the other hand, “Notes sur les Mathematiques, la Physique et autres sujets” was first published in 1878 as anappendix of the Réflexions.

II. Carnotʼs premise and problem setting
Various reasons why Carnotʼs paper was neglected by physicists at that time have been conjectured by researchers,
for example, the method of publication was not effective, or Carnot intentionally did not use analytical expressions
to develop his concept. Although these are, of course, contributing factors, the critical reason is its contents, thatis, his problem setting was essentially different from those of the physicists of that time.
The year of 1824 when Carnot published the Réflexions was one year after the completion of analytical calorimetryby Laplace and Poisson during the heyday of caloric theory. Caloric theory, especially the theory of specific heatand latent heat, succeeded in forming a paradigm. Carnot himself made the following comment on the ʻamount ofcaloric.ʼ
2

In this connection, there was a perversion that Carnot conceptually distinguished ʻcaloriqueʼ and ʻchaleurʼ;
he regarded the former as the quantity conserved in a reversible cycle, and thus equivalent to entropy. For the
background of this incorrect interpretation, Clausiusʼs testimony was misapplied as “Clausius did not knowthat Carnot had defined ʻcaloriqueʼ in a manner equivalent in principle to his own definition of entropy, aconcept of Clausius introduced in 1865”[13].

8IHTC16-FL01“We think it unnecessary to explain here what is meant by the terms ʻamount of caloric (calorique)ʼ or ʻamount ofheat (chaleur)ʼ (we use the two terms interchangeably) or to describe how to measure these quantities with acalorimeter. Nor shall we explain the meaning of latent heat, temperature, specific heat. etc. The reader should befamiliar with these terms from elementary treatises on physics or chemistry”[C, pp. 66-67].
As demonstrated above, at least in the Réflexions, he accepted the paradigm of caloric theory with specific heatand latent heat, and developed his theory on the basis of its premises.
Note that the analytical calorimetry developed by Laplace is a dynamical theory which understands thermal
phenomena in terms of ʻmolecules of caloricʼ and the force acting on them; attempts to break away from such
dynamical reductionism, however, were also made. In Fourier (1768‒1830)ʼs “The Analytical Theory of Heat,”which was published in 1822 (two years prior to the Réflexions), the following description can be found:“From what precedes it is evident that a very extensive class of phenomena exists, not produced by mechanicalforces, but resulting simply from the presence and accumulation of heat. This part of natural philosophy cannot be
connected with dynamical theories, it has principles peculiar to itself, and is founded on a method similar to thatof other exact sciences”[14].
Without doubt, Carnot studied the above-mentioned “principles peculiar to itself.” Actually, the caloric theory of
Carnot was one based on the pure ʻconcept of functions.ʼ Briefly, the purpose of the basic outline of his caloric
theory was simply to approve the conservation of heat. The following description by Carnot clearly demonstrates
how pure his caloric theory was:“In our proof, we make the implicit assumption that when a body has undergone its various changes and after
passing through a number of stages, has returned precisely to its original state (its state here being defined in termsof its density, temperature, and mode of aggregation), it contains the same quantity of heat as it did at the start.
This is tantamount to assuming that the quantities of heat lost and gained in the different processes cancel one
another out. This assumption has never been called into question; it was originally just taken for granted, but it has
since been amply verified by numerous calorimetric experiments. To reject the assumption would be to overturnthe entire theory of heat, which rests upon it.”[C, p. 76].
This description on the conservation of heat directly expresses  dq  0 (13.3)". This is the best expression alongwith that of Poissonʼs (16-III).3 This is completely different from the description based on the substantialism
frequently repeated since Cleghorn (1754‒1783) and Lavoisier (1743‒1794), in which a caloric is neither anarising nor a ceasing peculiar elastic fluid having repulsion with itself but bonds with ordinary substances.
From only this feature, however, Carnotʼs paper can also be regarded as an extension of the analytical calorimetry
purified by Poisson, which cannot have been so unfamiliar to the pivotal members of French academia. The
critically novel factor which prevented them from understanding Carnotʼs paper was the unique style of Carnotʼs
problem setting itself. That is, what is the theoretical limitation on the operational ability of heat? Carnot addressed
this problem as follows:First, “The question whether the motive power of heat is limited or whether it is boundless has been frequently
discussed. Can we set a limit to the improvement of the heat engine, a limit which, by the very nature of things,
cannot in any way be surpassed? Or, conversely, is it possible for the process of improvement to go on
indefinitely?”[C, p. 63].
Second, “is the motive power of heat fixed in quantity, or does it vary with the working substance that is used?
Does it vary with the intermediary material that we subject to the action of heat?”[C, p. 66].

3

However, as described later, Carnot finally rejected the conservation of heat and claimed the first law ofthermodynamics.

9

IHTC16-FL01
Here, ʻmotive power (puissance motrice)ʼ means ʻworkʼ in terms of modern physics, and its magnitude is the
product of the force and the distance in the direction of the force. Actually, Carnot clearly defined it as follows:“The expression ʻmotive powerʼ is used here to mean the useful effect that an engine is capable of producing. The
effect can always be expressed in terms of a weight being raised to a certain height, it is measured, of course, by
the product of the weight and the height to which the weight is considered to have been raised.”[C, p. 63].
At that time, however, those who recognized the importance of motive power (work) and had grasped its concept
clearly were researchers of applied mechanics and engineers rather than those of mathematical physics. In France,
this concept was developed and established in mechanics by Lazare Carnot, Hachette (1769–1834), Navier (1785–
1836), and Poncelet (1788–1867) from the eighteenth to early nineteenth century, while, in England, the sameprocess was under way with the improvement and commercialization of the steam engine by James Watt (1736‒1819) in the late eighteenth century [15]; Watt and his business partner Matthew Boulton (1728–1809) definedhorsepower as a unit of power, while a unit of work [kgwꞏm] was first introduced by Poncelet and Navier.
The idea of not only studying the theoretical limitation of the operational ability of heat but also regarding heat as
the source of motive power was motivated by the requirements of engineering and technology stimulated by thedevelopment of the fire (steam) engine. This was completely different from previous thermal science, i.e., caloric
theory based on substantialism starting from natural philosophy (especially chemistry). This is the crucial point
where Carnotʼs theory is distinguished from the previous caloric theory.
Thermal energy has been utilized since the time when humans first learned to make and control fire. The usage,
however, was limited to direct use such as for heating, cooking, pottery manufacturing, and metallurgy, with the
exception of guns. On the other hand, hydraulic power and animal power were used for motive power. The
invention of the fire engine provided a means of converting heat to motive power. Prior to that time, humans usedfire but also used various tools and machines independently of fire; the invention of fire engine integrated themboth. Consequently, new issues regarding the relationship between heat and motive power were raised forengineers.
On the other hand, the central concern of the thermal science developed by Black (1728‒1799), Lavoisier, Dalton(1766‒1844), Gay-Lussac, and Laplace was always clarification of the change in substances with heat. That is, its
main theme and purpose were to explain the expansion/contraction of matter, phase transitions, andexothermic/endothermic chemical reactions (especially combustion) as the union or separation of caloric withmatter. It promoted the measurement of specific heat and contributed to the theory of the speed of sound andadiabatic changes.
Regardless of the sophistication of this approach, however, it did not go beyond ʻcalorimetryʼ; the relationshipbetween heat and ʻmotive powerʼ was of little concern. For example, at that time, regarding adiabatic changes,
which typically indicate the relationship between heat and work from our modern viewpoint, the direct cause ofgeneration/absorption was considered to be the change in volume itself. That is, people of that time did not consider
that in the process of compression, mechanical work is necessary for heat generation. Also, they did not realize
that, in the expansion process, they should distinguish the cases with and without work for an external system. As
already stated in Chapter 15-I, Dalton wrote “Now a condensation of volume is a certain mark of diminution of
capacity and increase of temperature, whether the condensation be the effect of chemical agency, as in the mixtureof sulphuric acid and water, or the effect of mechanical pressure, as with elastic fluids”(New System of Chemical
Philosophy, pt. l, p. 50, 1808, https://archive.org/details/newsystemofchemi01daltuoft [accessed May 17, 2018]),and stressed that there is no relationship between temperature changes and mechanical work.
Carnotʼs concern about the quantitative relationship between heat and motive power was overlooked by not only
Dalton but also Lavoisier, Laplace, and Gay-Lussac. Carnotʼs proposition was, so to speak, a Copernican
revolution, even if it was based on caloric theory. This is the real reason why Carnotʼs paper was neglected byacademic physicists of the day.

10

IHTC16-FL01
III. Caloric theory and cosmology based on heat
The origin of caloric theory could be ascribed to the ʻpneumaʼ of Anaximenes and the ʻfireʼ of Heraclitus proposedin ancient times, but it can more directly be ascribed to the ʻaetherʼ of Newton. Although there is no need to dwell
on Newton, let us briefly review his concept.
Classical mechanics was inaugurated when Newton introduced the concept of action at a distance between material
bodies. The key concept of the cosmic system considered by Newton was the gravitational force among planets.
This concept of force was far from the previous mechanism, in which materials were considered to be passive. In
Descartesʼ mechanism, there was no origin of motion other than that given by God, and there was no action
between materials other than caused by direct contact.
On the other hand, Newton did not base his work simply on his mechanism. For him, observed motion and changes
could not be understood from the viewpoint of passive materials. That is, “they (= these particles) are moved bycertain active Principles, such as is that of Gravity, and that which causes Fermentation, and the Cohesion ofBodies” (Optics, p. 376, https://archive.org/details/opticksortreatis1730newt [accessed May 17, 2018]), “And if itwere not for these Principles, the Bodies of the Earth, Planets, Comets, Sun, and all things in them, would growcold and freeze, and become inactive Masses” (ibid., p. 375). It was the aether and spirits that Newton regarded assuch active agents.
As a result, the aether was considered not only to be an intermediate force but also to be responsible for the
expansion of materials, the reflection and refraction of light, electromagnetic phenomena, and even the functionsof muscles and nerves of animals. In the final analysis, all actions in nature were ascribed to the aether. Its
background was a ʻpan-aetherialʼ view of nature, in which nature is considered as a process of circulationconsisting of the evaporation/condensation and bonding/separation of aether, the essential driving force andultimate material (Chapter 6-V).
Newtonʼs philosophy was based on Cambridge neoplatonism, which regarded ordinary materials as passive andinactive, while active materials belonged to spirits (God); for Newton, nature was ascribed to the providence of
God as an infinite spirit which is omnipresent in space. Consequently, aether and spirits as active agents wererather understood as a spiritual reality which embodies and/or expresses Godʼs activity.
In the development of materialism during the Age of Enlightenment after Newton, the ʻspiritual realityʼ for aether
given by neoplatonism gradually disappeared, while dualism, consisting of ordinary passive materials and active
principles, was retained. Thus, in the eighteenth century, through the ʻairʼ of Hales (1677‒1761), the ʻmateria
ignisʼ of Boerhaave (1668‒1738), and the ʻelectric matterʼ of Franklin (1706‒1790), materials of heat such as
the ʻfireʼ of Cleghorn and the ʻcaloriqueʼ of Lavoisier were born. In all these materialisms, ʻair,ʼ ʻfire,ʼ andʻcaloriqueʼ were regarded as active substances distinguished from ordinary materials, and because of suchspecial features, their conservation law held true.
Consequently, on the basis of this historical background, caloric theory does not simply mean that heat is a material.
It was based on, from the very beginning, the “pan-thermal” view of nature in which heat is regarded as the originof all the activity in nature, namely the motive force for material circulation and vital phenomena.
On the other hand, owing to the progress in hydrology, meteorology, and oceanography in the seventeenth andeighteenth centuries, water circulation (evaporation of ocean water, formation of clouds, rainfall and snowfall) andatmospheric circulation (trade winds and westerlies) were understood to be caused by a thermal driving force.
Thus, a cosmology based on heat, which regards both the earth and the solar system as a heat engine, was formed.
The most typical statement of such cosmology based on heat can be found in the paper “On Fire” by Cleghorn,
one of the pioneers of caloric theory. Let us cite its rather lengthy introduction:“If we consider how important a part fire plays in Nature and how widespread it is, nothing will seem moredeserving of the attention of philosophers. Inert matter acted on by fire puts on new forms. From it plants havedrawn their life and living creatures their breath. It has given wings to the winds and movement to the waters, and11

IHTC16-FL01
it has raised the mountains that tower above the earth. What beautiful, what splendid things are thus displayed andbrought into being! The earth continually changes its appearance. Now the fields are hard with frost and stiff withsnow: now they smile and are bedecked with flowers, grass and leaves. Now the rivers are frozen into ice; now
they flow down through the lands. Nations are terrified by clouds that blot out the day by the violent whirlwind,
and the hearts of men are filled with terror.
How many and how beautiful are the things that fire produces in the sea! It produces vapours, which, driven aloft,form the rainbow by refracting the rays of light and those meteors that strike wondering men dumb withamazement. From the vapours raised into the air are formed the clouds which adorn the heavens and which,dispersed in showers, maintain the springs and rivers.
Heat is indispensable to the life of animals, to whom Nature has given the wonderful power of preserving theirown temperature unchanged in the changing temperatures of the air. The same power exists in some measure alsoin plants: for experiments show that their juices resist the strength of cold below the freezing point; and no one
doubts that they diminish the heat in hot countries. But if the external heat is increased or decreased to such an
extent that the temperature of animals or plants is greatly changed, they soon perish.
Nor does fire seem to be confined within the bounds of this earth. It is probable that it flows through the wholesolar system and extends to Saturn, the most distant of the planets. By analogy we conclude also that the fixedstars, which shine without number in the heavens, are bodies similar to the Sun and pour forth fire and light toother planets wheeling around them.
On this ordinance of God all thing, animate and inanimate, depend. If fire ceased to be, air would lose its spring,water its fluidity, animals their life, and all Nature would become a desolate and inert mass; if it increased greatly,
the whole earth would dissolve, all things would disperse into vapour, and Nature would sink into chaos”[16].
This was written in 1779. No additional explanation is necessary. According to Cardwell [17], writing about theseventeenth and eighteenth centuries, “Indeed, a kind of cosmic machine of a second type came to be recognized.
In asserting this we are extending the seventeenth century notion of a cosmic machine of the sort that was central
to the thoughts of Galileo, Descartes and Newton: a strictly mechanical piece of clockwork. Fundamental to the
new and complementary cosmic machine was heat.”IV. Carnotʼs view of nature and of society based on heat
In Carnotʼs paper, published about half a century after Cleghornʼs paper, the pan-thermal view of nature was also
enthusiastically claimed. Both Carnotʼs viewpoint and the accented point, however, were essentially different fromthose of Cleghorn. To compare them, the first part of Carnotʼs paper is cited below:“It is generally known that heat can be the cause of motion and that it possesses great motive power. The steam
engines in widespread use today are visible proof of this.
We must attribute to heat the great movements that we observe all about us on the Earth. Heat is the cause ofcurrents in the atmosphere, of the rising motion of clouds, of the falling of rain and of other atmosphericphenomena; likewise, of the waters which furrow the surface of the globe and which mankind has succeeded in
exploiting in some small measure. Heat is the cause too of earthquakes and volcanic eruptions.
From this vast reservoir we can draw the moving force that is indispensable to us. By providing us everywherewith fuel, nature has given us the means to produce at any time and in any place both heat and the motive power
to which heat gives rise. It is the purpose of heat engines to develop this power and to harness it for our use.
The study of these engines is of the utmost interest. Their importance is immense, and their use is increasing daily.
They seem destined to bring about a great revolution in the civilized world. The heat engine is already at work in
the exploitation of our mines, for driving our ships, digging out our ports and rivers, forging steel, fashioning wood,
milling grain, spinning and weaving cloth, transporting the heaviest loads, etc. It seems that one day it must becomea universal source of power and in this respect supplant animals, water and wind. Over the first of these sourcesof power the heat engine has the advantage of economy; over the other two, the invaluable advantage that it canbe employed and remain in uninterrupted use irrespective of either time or place.
If one day the heat engine is so perfected that it becomes cheap to erect and economical to run, it will combine all
the qualities we could wish, and advance industrial techniques to an extent that can scarcely be predicted”[C, p.
61].

12

IHTC16-FL01
Carnotʼs view was the same as the previous cosmology based on heat as far as he also regarded heat as the motive
force for the circulation of materials in nature. The crucial difference, however, is that Carnot extended the viewto that of society.
While Cleghorn just (passively) praised the universality of the activity of heat in nature, Carnot attached great
importance to the motive power of the heat engine, and consistently contemplated how to control and use thepower for human society. It can also be said that, regarding heat, Carnot left the natural philosophical viewpoint
and introduced political and economic viewpoints. The contrast between Cleghorn and Carnot demonstrates thata half-century including the French revolution and industrial revolution caused a change in the views of nature andof society, namely, the recognition of the superiority of humans to nature.
As previously stated, after the revolution, the French government promoted science and technology. Victor Hugo
(1802–1885) wrote about the dream which fascinated the young after the revolution in “Quatrevingt-Treize”(Ninety-Three), in which a young Republican who had formerly been an aristocrat speaks as follows:“Utilize nature, that gigantic auxiliary; enlist every breeze, every waterfall, every magnetic current, in your service.
This globe has a subterranean network of veins, through which flows a marvellous circulation of water, oil, andfire; pierce this vein of the globe, and let the water feed your fountains, the oil your lamps, and the fire your hearths.
Consider the action of the waves,—the ebb and flow of the tides. What is the ocean? A prodigious force wasted.
How stupid is the earth, to make no use of the ocean!”[18].
At the time of the French revolution, young intellectuals must have been fascinated by the dream of enhancing
productivity by controlling and exploiting nature, as well as (or much more than) by the idea of freedom, equality,and fraternity. It is reasonable to claim that, after the revolution, Saint-Simon (1760–1825)ʼs industrialism, atechnocratʼs dream, had been realized. Furthermore, it was correctly thought to be the heat engine (steam engine)that had achieved the dream. In 1839, Marc Seguin (1786–1875), who was later regarded as one of the pioneersof the first law of thermodynamics, wrote the following:“The old world has shaken off the yoke of its old habits. It is refreshing and remaking itself. So look, everythingis changed around us—the towns, the face of the countryside, the course of the rivers, the work of the peoples, theproduction of the soil and industry, the distribution of property—everything has taken a new face. And just where
the direct force of the material power of men has shown itself insufficient to accomplish its work and to persevere
in progress; where his will seems to be broken against insurmountable obstacles, just there a drop of water turned
into steam acts to supplement his weakness, to create for him a power of which we cannot now, nor yet for a longtime to come, measure the extent.
From now on with the help of this agent these prodigies have been accomplished, and the wonders which ourfathers would not have thought realizable with the united efforts of all their magicians have become the ordinary
run of things”[19].
Also, as can be understood from the above citation, the influence of the progress in the heat engine on Carnot
would have been profound and direct. In “The Industrial Revolution in the Eighteenth Century: An Outline of the
Beginnings of the Modern Factory System in England,” Paul Mantoux wrote, “With the steam engine, science firstmade its appearance, and the empirical period of the industrial revolution was followed by the scientific one”[20].
Thermodynamics was established following the development of the steam engine.
Actually, it goes without saying that Carnotʼs problem, “The question whether the motive power of heat is limited,”
was stimulated by the earlier development of the heat engine, and conversely it was aimed at further improving
the heat engine.
Furthermore, to solve the problem, Carnot used the improvement of the steam engine by James Watt as a clue,and directly attempted to grasp its process theoretically. This is because Wattʼs improvement of the steam engine
was based on his purposive consideration, which clarified the essential features of the heat engine, unlike those
based on ad hoc experience and intuition by craftsmen of the day. As a result, in the early nineteenth century, theheat engine had already been conceptualized, making it suitable for theoretical consideration. That is, since Watt

13

IHTC16-FL01
practically pursued an ʻideal engine,ʼ he could provide an object which was indispensable and most suited to thework of Carnot, who also theoretically pursued an ʻideal engine.ʼ

V. Development of steam engine and its defects before Watt
Here, let us briefly look back on the development of the steam engine.
Huygens (1629–1695) was the first person to invent a prototype of the internal combustion engine using
gunpowder as its fuel. Controlling the reaction of gunpowder, however, was so difficult that Papin (1647–1713)
used steam as an alternative working fluid. In his steam-driven water-lifting machine, a cylinder filled with water
was heated, and a piston lifted by steam was then cooled by water. The subsequent downward motion of the piston
was employed to lift water through the use of a lever. Although their engines can be demonstrated as desk machines,
they could not be applied to practical use.
At the time, an effective means of removing and lifting spring water from mines was an important problem. Thatis because of the increased demand for metals owing to the modernization of weapons and the development of amoney economy, which made water drainage in European mines on urgent issue. Many lifting pumps weredepicted in “De re Metallica” by Agricola (1494–1555) in German and “The Various and Ingenious Machines”
of Ramelli (1531–ca.1610) in Italy. Also, it is said that of “the fifty-five patents for inventions granted during thereign of Elizabeth, 1561-99, one in seven is for the raising of water”[21]. Since these hydro-, wind-, animal-, andhuman-power machines, had many limitations and were inefficient and uneconomical, an alternative was strongly
required. This was a particularly critical problem in England in the eighteenth century.
The first commercially used steam powered device was an engine to raise water invented by Thomas Savery(ca.1650–1715) in England, as shown in Fig. 18.3. First, by opening cock O1, steam generated in boiler L is
introduced to container P1. When P1 is occupied by the steam, the lower valve R3 is shut and the upper valve R1
is opened. Next, cock O1 is shut, and the outside of P1 is cooled by water. Then, the steam inside P1 is condensed,
the space inside P1 becomes nearly vacuum, valve R1 is closed, valve R3 is opened, and water is lifted fromunderground through pipe T. Next, cock O1 is opened, and the water inside P1 is exhausted from pipe S by theforce of the steam, and the system returns to the initial state. In P2, a similar process to that in P1 follows with ahalf-cycle delay. (This device originally only had P1 then was improved to continuously lift water by using thedual system as shown in Fig. 18.3.) That is, this operation consists of lifting water (from T to P1 and P2) byatmospheric pressure and exhausting water (from P1 and P2 to S) by steam pressure, where the former process
involves water suction due to vacuum.

Fig. 18.3 Thomas Saveryʼs steam engine(from Farey, J., A Treatise on the Steam Engine: Historical, Practical, and Descriptive, (1827)
https://archive.org/details/treatiseonsteame01fareuoft [accessed May 17, 2018]).

14

IHTC16-FL01
This device had a defect that it could not work in principle more than 10m column of water, in addition to thepractical limitations owing to the immature high-pressure technology at the time and the low efficiency associatedwith cooling from the outside of the container. Therefore, this device could not practically contribute to liftingwater in mines.
On the basis of Papinʼs idea, Thomas Newcomen (1664–1729) separated the cylinder and boiler. Newcomenʼspractically useful engine was completed at the beginning of the eighteenth century. In 1712 the valve operations
were automatized, which is considered to have revitalized the coal mines in Newcastle.
In Newcomenʼs engine (Figs. 18.4 and 18.5), a piston rod and a pump rod are connected to both ends of a beam
oscillating about a central pivot. When a cylinder is filled with steam of nearly atmospheric pressure, the beam isFig. 18.4 Mechanism of Thomas Newcomenʼs steam engine. (I have added an exhaust valve to the figures inLilley, S., Men, Machines and History: A Short History of Tools and Machines in Relation to Social Progress,
London: Cobbett Press, pp. 76-77, (1948) and Bernal, J.D., The Extension of Man: A History of Physics before theQuantum, Cambridge: The M.I.T. Press, p. 261, (1972).)

Fig. 18.5 Newcomenʼs steam engine designed by John Smeaton. Cooling water is stored in the upper left tank.
Water exhausted from the cylinder is returned to the boiler (from Farey, J., ibid., (1827)).

15

IHTC16-FL01
counterweighted at the other end by the weight of the pump rod so that it tends to pull the piston up, drawing steam
after it into the cylinder. When the cylinder is full, the steam is turned off and cold water is sprayed into it so thata vacuum is formed; the weight (pressure) of the atmosphere drives the piston to the bottom of the cylinder, andwater is lifted on the opposite side of the beam. Since the maximum steam pressure was roughly atmosphericpressure, this machine could be built using the primitive technology at that time.
At the time, the steam engine was called an “Invention for raising Water by Fire”[22]. Indeed, Hales also called it“the engine to raise water by fire”[22].
What distinguishes Newcomenʼs engine is that it was operated by atmospheric pressure so can be called ʻanatmospheric engine.ʼ Furthermore, at the time, it was understood that the vacuum inside the cylinder generated bysteam condensation jacks up the piston, rather than that atmospheric pressure pushes down the piston; steam, i.e.,
heat, was considered to merely have the peripheral role of creating the vacuum. In other words, steam wasconsidered to perform work not by expansion but by contraction. From this understanding, it was doubtful that the
crucial fact that ʻheat produces workʼ was correctly understood.
Regarding Newcomenʼs engine, John Smeaton (1724‒1792) later tried to improve its efficiency by changing theshape and size of the boiler and cylinder, although no essential improvements were made until those by Watt a
half-century later.
In 1763, James Watt became interested in Newcomenʼs engine. Since 1757, Watt had been making a mathematical
instruments in a workshop at the University of Glasgow, where he became friendly with Joseph Black and John
Robison (1739‒1805), who later edited Blackʼs lectures. Later, Watt said, “You (= Robison) must have been muchmisinformed concerning my attending Dr. Blacks Lectures. I never did so, and certainly would have possessedmore solid knowledge of chemistry than I do, if I had been wise enough to have done so. Every thing I learnt fromhim was in conversation and by doing small mechanical jobs for him. These Conversations and those I had withyou served to give me true notions in Science and to develope the powers of my Mind, such as they were”[23].
On the other hand, Black and Robison also described Watt as their colleague. Robert Dick Junior, who wasProfessor of Natural Philosophy at Glasgow University from 1751 to 1757, supported Watt, and also Black lent
research funds for putting his steam engine into practical use.
The fact that in the university a craftsman and scientists could have discussions and cooperate was due to not only
the intellectual atmosphere in Glasgow but also Wattʼs own temperament. Actually, his way of thinking andworking was more of a scientist than that of a workman. “Robison, who met him for the first time in 1758 (he wasthen twenty-two), was struck by his extensive knowledge and by intellectual grasp: ʻI saw a workman and expected
no more: I found a philosopherʼ ”[24]. He was surprised at the depth of Wattʼs learning and the width of his vision.
Later, Robison acknowledged Watt when he edited Blackʼs lecture notes he edited. A century later, Balfour
Stewart, a physicist of Manchester, wrote, “In 1763 James Watt, who was philosophical instrumentmaker to the
University of Glasgow”[25].
Watt encountered Newcomenʼs engine by chance when he was asked to repair a model of the engine in the
University of Glasgow. During the repair, he detected an essential shortcoming of the engine. His memoirs clearlypointed out the defects:“As I had not at that time paid much attention to the subject and had other avocations, it was then pursued nofarther; but in the Winter of 1763, having occasion to repair a Model of Newcomenʼs Engine which belonged to
the Natural Philosophy class of the University of Glasgow; when I had got it in working order, I found that the
Boiler, though large in proportion to the Cylinder could not supply it with Steam to work at a proper rate unlessthe fire was violently urged with bellows.
I observed also, that the Cylinder which was about 2 Inches diameter, could only give power enough to work apump of the same diameter and about 2 feet high, which effect was much less in proportion to the Cylinder than
was generally performed by larger \Steam/ Engines.
On considering the causes of these defects it appeared, 1st that as Steam is condensed by the contact of bodies
colder than itself and in proportion to the quantities of such cold bodies with which it comes into contact, and the16

IHTC16-FL01
internal surface of any cylinder being many times greater in proportion to its contents than that of the large Engines
employed in Collieries, much more Steam must be condensed before the Cylinder could be filled so as to permit
the piston to rise; 2dly that as water was supposed to boil in vacuo at much lower heats than it did under thepressure of the atmosphere, a vacuum could not be produced in the cylinder unless that vessel were cooled by the
injection below the point at which water would boil in an exhausted Vessel”[26].
That is, Watt considered one of the reasons why the miniature model required more heat than the real one to be as
follows. The net amount of steam required for the operation (A) is proportional to the volume of the cylinder (V),while an amount of steam (A) proportional to the inner surface area of the cylinder (S) is cooled by contact withthe cylinder wall and is condensed without being used for work. As a result, the ratio of wasted steam is given by
ΔA aS a
 ,AVlwhere l is the characteristic length of the model and a is a constant with the dimension of length, which means thatthe ratio increases with decreasing size of the model. By observing the miniature steam engine model, the essential
defect of Newcomenʼs engine that a large amount of steam is lost per cycle was brought to the fore.

VI. Wattsʼ improvement—separate condenser
The above conclusion was so logical that Shin-ichiro Tomonaga (1906‒1979; Nobel Prize in Physics 1965)
commented that “This fact clearly demonstrates that Watt was not merely a merchant but a scientist”[27].
At that time, Watt wrote, “But before anything could be known certainly about the best manner of Constructing
Fire engines there were several facts necessary to be determined”[28]. Also, his later approach to the problem wasakin to the style of a scientist rather than that of a craftsman.
He first found by measurement that the volume of water expands by a factor of 1600 when it evaporates, and hecalculated the actual volume of steam supplied to the cylinder per unit time, i.e., volume of water consumed perunit time  1600 . Watt compared this amount with the required net amount of steam, i.e., volume of the cylinder cycles per unit time. As a result, he found that the former is about five times the latter, that is, most of the steam
was consumed only to heat the cylinder wall.
To reduce the wasteful use of steam and therefore fuel, it is clear that the cylinder must be kept at as hightemperature as possible.
Simultaneously, Watt found that not only does steam contains air but also that water cooling nearly reaches theboiling temperature ( 212F ) after it is injected into steam because a large amount of heat is released when steam
condenses. Regarding this point, although it may be argued that Watt learnt about latent heat from Black rather
than discovering it himself, it was not the case [29].4
As noted above, under atmospheric pressure, water boils at a temperature below its usual boiling point.
Consequently, the high-temperature water produces so-called backpressure and, together with the air contained in
the steam, destroys the vacuum and decreases the output power. Hence, to increase the output power using thismechanism, when the piston moves down, it is not sufficient to simply reduce the temperature inside the cylinder
to lower than the boiling temperature; the temperature must be kept as low as possible.
To reduce fuel consumption, the cylinder and its inside should be maintained at a high temperature, while, for highmechanical performance, effective cooling is necessary in each cycle. This produces a dilemma if we adhere to
the idea based on the above mechanism. Watt, however, found a method to simultaneously satisfy both
Later, in 1781, Watt measured the latent heat of evaporation of water by himself. The results were 945.5 ,922.5 , 935.5 , 963.5 , 942.5 , 960.5 , 940 , 937 , and 943.375 F, an average of 524 C. namely 524
cal/gr. This value is much more accurate than that obtained by Black 450 C (the actual value is 540 cal/gr). We
can see that Watt was a remarkably skilled experimental researcher [30].
417IHTC16-FL01
requirements by spatially and substantially separating the expansion process and contraction process. That is, theessence of Wattʼs improvement is that, by separating the condenser (cooler) from the cylinder, it is possible forthe condenser to be operated with a sufficiently low temperature and for the cylinder to be steadily maintained at
a high temperature. Although this is very simple from the present perspective, it was “the greatest single
improvement ever made in the engine”[31].
Furthermore, to prevent the cylinder from cooling by coming in contact with the surrounding air, Watt introduced
a new mechanism so that the piston could be pushed by the steam itself, resulting in a real steam engine. This
improvement is said to have reduced the fuel consumption by “almost 75 per cent”[31]. Since steam contains air,an exhaust pump was added downstream of the condenser.
The details of the cycle and mechanism of Wattʼs engine are as follows. In Fig. 18.6, when the piston, moved
upward by counterweighted beam, reaches the top of the cylinder, the gate valve (which separates the upper andlower parts of the cylinder) is closed and the outlet valve (which is connected to the condenser) is opened and thelower part of the cylinder is evacuated. Simultaneously, by opening the inlet valve (which is connected to theboiler), steam is introduced into the upper part of the cylinder. As a result, the piston is driven downward. Whenthe piston reaches the bottom, the inlet and outlet valves are closed and the gate valve is opened, then steam spreads
into both sides of the piston while the piston is driven up again by the counterweighted beam. In the first half ofthe process, when the vacuum pump is in operation, the working substance is pure water vapor in the upper partof the cylinder, while in the second half of the process, the piston is driven by gravity. This mechanism patented
in 1769 is called single-acting engine because steam is the driving force during only half of the cycle (Fig. 18.7).

Fig. 18.6 Single-acting steam engine
developed by JamesWatt.

Fig. 18.7 Single-acting steam engine
developed by James Watt.
A condenser and pump for the exhaust are
equipped on the left side below the cylinder(from Farey, J., ibid., (1827)).

Furthermore, in 1782, Watt invented a double acting engine, which was operated by steam in both the upward anddownward directions. In other words, a 100% steam engine or heat engine, whose operation was independent of
atmospheric pressure and gravity, had been developed. It not only doubled the output power but also made rotating
action possible instead of reciprocating action; as a result, its applicability was tremendously widened.
Watt and Boulton were highly successful in their commercialization of the steam engine. Boulton wrote thefollowing in a letter to Erasmus Darwin (1731‒1802; grandfather of Charles Darwin):

18

IHTC16-FL01“Steam Engines of B & W construction differ from Newcomins in 6 principles
st
1. The acting power is Steam on ye piston. & Not the Atmosphere as in ye Comn Engne
2. The Steam is condensed in a separate Vessell, & Not in the Cylinder as in D°
3. The Cylinder is kept as hot as Steam, & Not alternately heated & coold as in D°
4. The air is extracted by an air pump at each Stroke & not blown out at Snifting Clak
5. The packing of the piston is preserved air & steam tight by oyls ... & not by Water
6. The Vacuum is alternately made above & below the piston & thereby obtain double power [32]This description clearly shows, in a deeply satisfying way, how well Watt and Boulton understood theirachievement.

VII. Wattʼs improvement—expansive principle
In the patent of 1782, one more improvement, which is very important in terms of thermodynamics, was based on
Wattʼs expansive principle. Its idea was discussed in a letter written in 1769 [33].
Watt detected that when high-pressure and high-temperature steam freely expands from a cylinder to a lowtemperature cooler, power potentially contained in the steam is wasted. Hence, Watt considered that by stopping
the supply of steam from the boiler in the middle of the stroke, followed by adiabatic expansion of the steam, thesteam will be gradually cooled while simultaneously carrying out work with the energy contained in the steam.
Since this point is very interesting, let us cite his rather lengthy description in the patent specification (1782):“MY FIRST NEW IMPROVEMENT in steam or fire engines consists in admitting steam into the cylinders or
steam vessels of the engine only during some certain part or portion of the descent or ascent of the piston of thesaid cylinder, and using the elastic forces, wherewith the said steam expands itself in proceeding to occupy larger
spaces, as the acting powers on the piston through the other parts or portions of the length of the stroke of the said
piston; and in applying combinations of levers, or other contrivances, to cause the unequal powers wherewith thesteam acts upon the piston, to produce uniform effects in working the pumps or other machinery required to be
wrought by the said engine: whereby certain large proportions of the steam hitherto found necessary to do thesame work are saved. To explain which principle or improvement, I have delineated a section of a hollow cylinder
at figure 1st (Fig. 18.8 in this paper) in the annexed drawing. (omit) These things being thus situated, and the piston
placed as near as may be to the top of the cylinder, let the space of the cylinder under the piston be supposed to be
exhausted or emptied of air, steam, and other fluids; and let there be a free passage above the piston for the entryof steam from the boiler, and suppose that steam to be of the same density or elastic force as the atmosphere, orable to support a column of mercury of thirty inches high in the barometer. Then I say that the pressure or elastic
power of the said steam on every square inch of the area or upper side of the piston, will be nearly fourteen pounds
avoirdupois weight, and that if the said power were employed to act upon the piston through the whole length ofits stroke, and to work a pump or pumps, either immediately by the piston rod prolonged, or through the medium
of a working beam or great lever, as is usual in steam engines, it would raise through the whole length of its stroke
a column or columns of water, whose weight should be equal to ten pounds for each square inch of the area of thepiston, besides overcoming all the frictions and vis inertia, of the water and the parts of the machine or engine. But
supposing the whole distance from the under side of the piston to the bottom of the cylinder to be eight feet, andthe passage which admitted the steam from the boiler to be perfectly shut when the piston has descended to thepoint K two feet, or one‑fourth of the length of the stroke or motion of the said piston, I say that when the piston
had descended four feet, or one‑half of the length of the stroke, the elastic power of the steam would then be equal
to seven pounds on each square inch of the area of the piston, or one‑half of the original power; and that when thepiston had arrived at the point P, the power of the steam would be one‑third of the original power, or four pounds
and twothirds of a pound on each square inch of the pistonʼs area; and that when the piston had arrived at thebottom or end of its stroke, that the elastic power of the steam would be one-fourth of its original power, or threepounds and one‑half pound on each square inch of the said area. And I further say, that the elastic power of thesteam at the other divisions marked in the lengths of the said cylinder, are represented by the lengths of thehorizontal lines or ordinates of the curve KL, also marked or delineated in the said cylinder, and are expressed in

19

IHTC16-FL01
decimal fractions of the whole original power by the numbers written opposite to the said ordinates or horizontal
lines. And I also say, that the sum of all these powers is greater than fifty‑seven hundred parts of the original power
multiplied by the length of the cylinder; whereby it appears that only one‑fourth of the steam necessary to fill thewhole cylinder is employed, and that the effect produced is equal to more than one‑half of the effect which would
have been produced by one whole cylinder full of steam, if it had been admitted to enter freely above the piston
during the whole length of its descent”[34].

Fig. 18.8 Wattʼs diagram showing pressures of steam expanding in a cylinder (from his patent specification,(1782)).
In the Réflexions, Carnot highly appreciated the significance of Wattʼs expansive principle as follows: “Watt, who
was responsible for nearly all the main improvements of the steam engine and who so perfected the engine thatfurther advances would now be difficult, was also the first to use steam at progressively diminishing pressures”[C,p. 105]. That is, “The mark of a good steam engine, therefore, must be not only that it uses steam at a high pressure
but also that it uses it at pressures which are not constant but which vary substantially from one moment to thenext and progressively decrease.”[C, p. 104]. In other words, if high-pressure and high-temperature steam
gradually performs work as its pressure and temperature decrease, then under a fully low-pressure and lowtemperature condition it is guided to a condenser, the waste of energy is minimized. As discussed later, this leads
to Carnotʼs theory that a quasi-static reversible change is ideal for maximizing work.
Although the aim of Wattʼs expansive principle was to improve fuel efficiency, from the theoretical viewpoint, it
clarified that the role of steam in the steam engine is not ʻto create a vacuum by condensationʼ but ʻto work by
expansion.ʼ Also, by flexibly controlling and stopping the supply of steam into the cylinder, and thus by promoting
the independence of the working substance, Wattʼs engine perfected the separation of the boiler and cylinder, theidea originating from Papin and Newcomen.
Combined with the invention of a separate condenser (cooler), Watt clarified the following three points. First, aheat engine consists of three elements, a furnace, cooler, and working substance; the working substance can freely
be separated from and forced to come in contact with the furnace and cooler. Second, to perform work, not only ahigh temperature but also a low temperature is necessary. That is, heat is carried by the working substance froma high-temperature region (furnace) to a low-temperature region (cooler). Third, work is performed by the volume
change of the working substance. That is, the existence of a high temperature and a low temperature is a necessary
but nor sufficient condition for performing work; to perform work, the potential power of the high-temperatureworking substance should be extracted by expanding its volume.

20

IHTC16-FL01
As described in the next chapter, these three points gave Carnot important hints when deriving his theory and areincorporated in his theory.

VIII. P‒V diagram and expression for work
It should be noted that in Wattʼs description of expansive principle, he erroneously assumed that Boyleʼs law canbe applied to adiabatic changes. Leaving this aside until later, what is important and interesting is that Watt actually
recognized the importance of the P‒V diagram and the concept of work.
Since the scale depicted on the side of the cylinder in Fig. 18.8 is the distance moved by the piston, it can also be
regarded as the volume of the steam because the cross-sectional area of the cylinder is constant. On the other hand,
the horizontal distance along the curve MKPL gives the pressure. Consequently, this figure is basically a P‒V
diagram, as widely used in later thermodynamics.
The P‒V diagram was first introduced into thermal science in 1834 by Clapeyron, and it is often considered tooriginate from the indicator diagram of John Southern in 1796: “The practical solution of this problem is very
simple and was found by John Southern in 1796. As one of Boulton and Wattʼs business acquaintances wrote: ʻI
am like a man parchʼd with thirst in the Expectation of relief or a woman dying to hear (or tell) a Secret—to knowSouthernʼs mode of determining Power ...ʼ It was simple enough. A sheet of paper is pinned to a board, whichmoves to and fro with the reciprocating motion of the piston; at the same time a spring-loaded pressure gauge
causes a pencil, held in a lever arm, to move at right angles to the motion of the board. As a result the pencil
automatically traces out the curve, relating the pressure of steam in the cylinder to the change of volume: that is tosay it traces out the indicator diagram; and Gilbertʼs analysis showed that the area under the curve must be
proportional to the work done or the power being developed by the engine”[35]. I, however, cannot understandwhy scientific historians did not attribute the origin of the P‒V diagram to Wattʼs diagram in his patent
specification of 1782.
In his specification, Watt wrote “that the sum of all these powers is greater than fifty‑seven hundred parts of theoriginal power multiplied by the length of the cylinder; whereby it appears that only one‑fourth of the steam
necessary to fill the whole cylinder (V2) is employed”[35]. Here, Watt evaluated the work by expressing one tickof the cylinder length as the unit volume ( ΔV  1 volume),20

 P ΔV  11.54 (atmꞏvolume),ii 1where Pi is one of the figures shown on the right side of Fig. 18.8. That is, Watt considered that the work done by
the steam is expressed as  PdV or the area enclosed by the curve MKPL and both axes, which was calculated bynumerical integration. Then, he compared it with the work under constant atmospheric pressure,20 P V  1 20 (atmꞏvolume),
(0)

i

i 1

and obtained the result of 11.54 20  57.7 %. Thus, by using a quarter of the original amount of fuel, more thanhalf of the work was available.
For more precise evaluation, since steam of pressure P0 and volume V1 is first supplied and then adiabatically
expands to volume V2 , the total work is (erroneously assuming Boyleʼs law)
V2



V2

 V2 

 PdV  P V   PdV  P V 1  ln V  .
0 1

0

0 1

V1

1

Comparing this with the work when steam of constant pressure P0 and volume V2 is supplied,V2

 PdV  P V ,
0

0

21

2

(18.1)

IHTC16-FL01
for V2  4V1 , the ratio of V2 
V1 
1  ln   0.59  59%
V2  V1 
is obtained. Wattʼs numerical calculation result (57.7%) is in good agreement with the analytical value.
Clément first pointed out Wattʼs misconception about the application of Boyleʼs law (isothermal change of idealgas) to the adiabatic process in 1819 [36]. However, Clément was also incorrect. He and Desormes (1777–1838)
experimentally found the law for the adiabatic change of steam, which had been called Wattʼs law. Wattʼs law
states that the absolute amount of heat (the sum of sensible heat and latent heat) contained in steam at saturatedpressure is a constant independent of temperature, although it is uncertain whether or not Watt first discovered thelaw. For example, Dalton (1766‒1844) wrote the following in Chemical Philosophy:“I owe to Mr. Ewart the first hint of the idea respecting elastic fluids, which I have endeavoured to expand in thepresent section; he suggested to me some time ago, that it was probable steam of any low temperature, as 32°, of
maximum density, contained the same quantity of absolute heat as the like weight of steam of 212° of maximum
density; and that consequently if it could be gradually compressed without losing any heat, that is, if the containg
vessel kept pace with it in increase of temperature, there would never be any condensation of steam into water, but
it would constantly retain its elasticity”[37]. (Here, the unit of temperature is Fahrenheit, and ʻsteam of maximum
densityʼ corresponds to the state at the saturated pressure.) Dalton also wrote, “Since writing the above, Mr. Ewart
informs me that the idea respecting steam, which I had from him, is originally Mr. Watt”[37].
In 1845, after graduating from Glasgow and Cambridge, William Thomson spent some time studying experiments
on gases in Henri Regnault (1810–1878)ʼs laboratory. He wrote, “When, eighty-one years later, a student under
Regnault in his laboratory in the college of France, I used to hear him speaking of “la loi de Watt,” and telling usthat it was the nearest approach to the truth which he found among the results of previous experiments, I felt somepride in thinking that the experiments on which it was founded had been made in Glasgow College”[38].
According to scientists who studied gas characteristics at that time, Wattʼs law was considered to be approximately
correct. As Dalton wrote, on the basis of Wattʼs law, it was considered that saturated steam remains in the saturated
state after an adiabatic change, and this fact was thought to provide the scientific rationale for treating saturated
steam in the same manner as an ideal gas. Clément and Desormes were of the same opinion. In the Réflexions
Carnot wrote,“According to a law discovered by MM. Clément and Desormes—a law which they established by direct
experiment—a given weight of steam always contains the same amount of heat at whatever pressure it is created.
This is the same as saying that saturated steam will remain saturated whenever it is compressed or expandedmechanically in such a way that there is no loss of heat. Steam in these conditions may therefore be treated as apermanent gas, subject to all the normal gas laws (italicized by Yamamoto). [C, p. 90]
We shall see in Chapter 26-IV and V that the decision of whether the law of Clément and Desormes, i.e., Wattʼslaw, is correct or not serves as a litmus test to ascertain whether the conservation of heat is correct or not.
Since the first law of thermodynamics is far ahead of the present subject, let us return to the concept of work.
“Watt and Boulton decided that their engine should be paid for by a royalty, or as they termed it a ʻpremiumʼ,
based very appropriately on the saving in fuel effected by the engine as compared with the consumption of acommon engine doing the same work. Boulton and Watt stipulated that they should receive one-third of the valueof the fuel saved”[31] for 25 years, meaning that both the establishment of a unit of work and the accurate
evaluation of work were crucially important. According to Cardwell [39], in Wattʼs litigation with Jonathan
Hornblower (1753‒1815) in 1792, Davies Gilbert (the sheriff for Hornblower) first obtained a mathematicalexpression for the work  PdV . In particular, the analytical expression for the expansive principle, eq. (18.1), was

22

IHTC16-FL01
first referred to by Robison in the 1797 edition of Encyclopaedia Britannica. We should, however, realize that
Watt understood the quantitative concept and its expression for the work before them. Incidentally, it was Wattand Boulton who defined one horsepower as 33,000 (originally 32,572) ft lbf min .
In any case, Wattʼs patent specification of 1782 has much greater historical importance.

IX. High-pressure engine and its development in France
Since Watt only used low-pressure (atmospheric pressure) engines, he could not apply the expansive principle toactual devices. The advantage of the expansive principle only emerges when engines are operated at a highpressure; the practical realization of high-pressure engines started in the nineteenth century.
Wattʼs invention of the separate condenser was in 1765 and his expansive principle was patented in 1782. During
the patent term, innovation leading to engines superior to Wattʼs engine was inhibited. After its expiration in 1800,
however, the Boulton‒Watt monopoly was broken, and in 1802, Richard Trevithick (1771‒1833), an engineer
from Cornwall, invented a high-pressure engine. In 1814, the Cornish engineer Arthur Woolf (1776‒1837)
produced a high-pressure engine based on the expansive principle.
In the Cornish mines, far from any coal mining areas, since the efficiency (ratio of output power to fuelconsumption) of the steam engine was crucially important, Woolfʼs engine, whose efficiency was twice that of
Wattʼs engine, had a significant impact. This success of the high-pressure engine demonstrated that there wasplenty of scope for improving steam engines, although nobody could understand why the high-pressure engine
could attain high efficiency [40].
The fact that the year in which Woolfʼs engine was perfected was1814 had special significance to Carnot, who
graduated from École Polytechnique in the same year and was pondering over his future.
This is because in 1815 the Napoleonic Wars were over and diplomatic relations between England and France wasrestored. Much to the surprise of the French, the steam engine in England had achieved amazing progress.
Attention was particularly directed to the high efficiency of the high-pressure engine [41], resulting in Woolfʼsengine plant being constructed as early as in 1815 in France. “on the Continent far more emphasis was placed on
fuel economy than in Britain. From the very start the Woolf compound engine ... which made use of high pressure
to operate two cylinders alternately and offered a fuel economy over the Watt engine of about 50 per cent foundits greatest market in France”[42].
Consequently, after 1815, steam engines, particularly high-pressure engines, spread rapidly through France, and it
is considered that by 1824, the year Carnot published his paper, about 300 steam engines were being operated in
France.
Since the high-pressure engine could be readily downsized, it was suitable for use in transportation. In England,
Trevithick produced the first steam locomotive in 1804. George Stephenson (1781–1848)ʼs steam locomotive was
first practically applied to a traveling engine designed for hauling coal in 1814, and he also opened the public
railway between Stockton and Darlington (21 km) in 1825.
Regarding steamboats, in 1807 the American engineer Robert Fulton (1765–1815) surprised the world by sailinga steamboat on the Hudson river from New York City to Albany, a distance of 240 km, in 32 hours. In France,
although railways were introduced later, steamboats were already in operation by the 1810s. Victor Hugo wrotein “Les Misérables” that steamboats were operating on the Seine in 1817. In “Une vie” by Guy de Maupassant(1850–1893) a passage is contained in which the heroine went to Corsica on her honeymoon in 1819 on asteamboat [43].
The strong impression that steam engines had on French people was passionately expressed by Saint-Simonians
who dreamed of the construction of an industrial society. Actually, “A substantial proportion of the first French

23

IHTC16-FL01
railway-builders were fervent disciples of Saint-Simon”[44]. One of them, Michel Chevalier (1806‒1879)
described his dream as follows:“By the aid of mechanical contrivances, this poor weak creature (= human), reaching out his hands over theimmensity of nature, takes possession of the rivers, of the winds of heaven, of the tides o the ocean. By them, hedrags forth from the secret bowels of the earth their hidden stores of fuel and of metals, and masters thesubterranean waters, which there dispute his dominion. By them, he turns each drop of water into a reservoir ofsteam, that is, into a magazine of power, and thus he changes the globe, in comparison with which he seems anatom, into a labourious, untiring submissive slave, performing the heaviest tasks under the eye of its master. Is
there any thing which gives a higher idea of the power of man, than the steam-engine under the form in which itis applied to produce motion on railroads? It is more than a machine, it is almost a living being; it moves, it runs
like a courser at the top of his speed”[45].
Chevalier wrote this in 1836, and the following year another disciple of Saint-Simon, the financier, Émile Pereire(1800‒1875), completed the first passenger railway between Paris and St. Germain in France.
In western Europe, which rushed headlong toward a highly industrialized society in the nineteenth century, thepredominance of human beings over nature was first recognized; this presumptuous philosophy was based on thedevelopment of steam engines.
With this background, Carnot published the Réflexions. “If you were now to deprive England of her steam engines,
you would deprive her of both coal and iron; you would cut off the sources of all her wealth, totally destroy hermeans of prosperity, and reduce this nation of huge power to insignificance”[C, p. 62]. This specific comment
reveals Carnotʼs accurate recognition of the state of the art and the influence of the steam engine on his thinking.
In particular, the words “both coal and iron” are interesting. In the eighteenth century, the three generations of
Abraham Darby (I (1678–1717), II (1711–1763), III (1750–1789)) in England produced marketable pig iron in acoke-fired furnace instead of one using charcoal. Their technology was completed almost at the same time as the
improvement of the steam engine by Watt. Through this achievement, despite its limited supply of lumber, themass production of iron became possible in England. For the new ironmaking process, a large supply of coal was
required as well as high-performance blowers to supply air to large furnaces; both these requirements were met by
Wattʼs engines. Thus, in England, the era of large-scale manufacturing started, and, in particular, the iron industry
owed its existence to Wattʼs engines, as Carnot pointed out.
One of the motivations of Carnotʼs study was to clarify the reason why high-pressure engines were superior to
low-pressure engines. This had been a major issue not only for Carnot but also for all the researchers and engineersof steam engines in France since 1815.
Only Carnot, however, could thoroughly investigate this issue from the principles, and a single paper of Carnot
provided the basis for thermodynamics. Carnotʼs work was originally stimulated by a technical issue that was notof interest to physicists. Simultaneously, its approach was so fundamental and theoretical that it was not understoodby engineers, and was therefore neglected.
In the next chapter, I will refer to the fact that in Carnotʼs investigation of heat engines, he depended on the analogy
between heat engines and hydro-pressure engines.

References in chapter 18[1] Merz, J.T., A History of European Scientific Thought in the Nineteenth Century, Edinburgh and London: William Blackwood andSons, Vol. 1, p. 61, Cf. p. 155, (1904), https://archive.org/details/ahistoryeuropea06merzgoog [accessed May 17, 2018].
[2] Carnot, H., Notice biographique sur Sadi Carnot, The Life of N.-L.-Sadi Carnot, tr. by Thurston, R.H., New York: John Wiley &Sons, London: Chapman & Hall, p. 29 (1897), https://www3.nd.edu/~powers/ame.20231/carnot1897.pdf [accessed May 17, 2018].
[3] Carnot, S., Réflexions sur la Puissance Motrice du Feu, (1824), References fromhttps://fr.wikipedia.org/wiki/Sadi_Carnot_(physicien) or
https://en.wikipedia.org/wiki/Reflections_on_the_Motive_Power_of_Fire [accessed May 17, 2018]24

IHTC16-FL01
Carnot, S., Reflexions on the Motive Power of Fire, A Critical Edition with the Surviving Scientific Manuscripts, translated andedited by Fox, R., Manchester: Manchester University Press, (1986).
[4] Mendoza, E., “Contributions to the Study of Sadi Carnot and his Work,” Archives internationales dʼhistoire des sciences, vol. 12,
pp. 389ff, (1959);
Mendoza, E., “The Writing of the Réflexions and their Reception,” In Sadi Carnot et lʼEssor de la Thermodynamique, Paris:
Centre National de la Recherche Scientifique, p. 177f, (1976).
[5] Cardwell, D.S.L., From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Ithaca: Cornell University
Press, p. 213, (1971).
[6] Herivel, J., “Carnot and the French Scientific «milieu» around 1824,” In Sadi Carnot et lʼEssor de la Thermodynamique, Paris:
Centre National de la Recherche Scientifique, pp. 83-85, (1976).
[7] Mendoza, E., “Contributions to the Study of Sadi Carnot and his Work,” Archives internationales dʼhistoire des sciences, vol. 12,
p. 394, (1959);
Mendoza, E., “The Writing of the Réflexions and their Reception,” In Sadi Carnot et lʼEssor de la Thermodynamique, Paris:
Centre National de la Recherche Scientifique, p. 177, (1976).
[8] Kuhn, T.S., “Engineering Precedent for the Work of Sadi Carnot,” Archives internationales dʼhistoire des sciences, vol. 13, pp.
251-255, (1960);
Gillispie, C.C., “The Scientific Work of Lazare Carnot, and its Influence on that of his Son,” In Sadi Carnot et lʼEssor de laThermodynamique, Paris: Centre National de la Recherche Scientifique, pp. 23-34. (1976);
Merz, J.T., A History of European Scientific Thought in the Nineteenth Century, Edinburgh and London: William Blackwood andSons, Vol. 2, p. 100, n. 2 (1907), https://archive.org/details/historyofeuropea00merz [accessed May 17, 2018].
[9] Kerker, M., “Sadi Carnot and the Steam Engine Engineers, Isis A Journal of the History of Science Society, vol. 51, no. 3, p. 259
footnote, (1960).
[10] Mckeon, R., “Sadi Carnot and the French Engineering Community of the 1820s: the Mechanical Theory of Heat Applied to HeatEngines as Developed by Navier,” In Sadi Carnot et lʼEssor de la Thermodynamique, Paris: Centre National de la Recherche
Scientifique, p. 98, (1976).
[11] Thompson, S.P., The Life of Lord Kelvin, 2 vols. (1910), reprinted, New York: Chelsea Publishing Company, p. 133, (1976); Cf.
Scott, W.L., The Conflict between Atomism and Conservation Theory 1644‒1860, London: Macdonald, New York: Elsevier, p.
288, n. 24 (1970).
[12] Clausius, R., “Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten
lassen,” Annalen der Physik und Chemie, Bd. 79, p. 368 (1850), References fromhttps://de.wikipedia.org/wiki/Rudolf_Clausius or https://en.wikipedia.org/wiki/Rudolf_Clausius [accessed May 17, 2018];
Clausius, R., The Mechanical Theory of Heat with its Applications to the Steam-Engine and to the Physical Properties of Bodies,edited by Archer Hirst, T., London: J. Van Voorst, p. 14, (1867), References fromhttps://de.wikipedia.org/wiki/Rudolf_Clausius or https://en.wikipedia.org/wiki/Rudolf_Clausius [accessed May 17, 2018];
Scott, W.L., ibid., p. 289, n. 24, (1970)[13] La Mer, V., “Some Current Misinterpretation of N. L. Sadi Carnotʼs Memoir and Cycle. II,” American Journal of Physics, vol.
23, p. 99, (1955); Cf.
Remarks related to Reference [13] in Chapter 20.
[14] Fourier, J., The Analytical Theory of Heat, (1822), translated by Freeman A., (1878), reprinted, New York: Dover Publications,p. 23, (1955).
[15] Cardwell, D.S.L., ibid., pp. 79ff, pp. 164ff, (1971);
Kuhn, T.S., “Energy Conservation as an Example of Simultaneous Discovery,” In Critical Problems in The History of Science,
Proceedings of the Institute for the History of Science, pp. 321-356, (1959);
Takayama, S., “Relationship between Physics and Engineering in the 19th century first half—From the viewpoint of ʻTheory ofPowerʼ and ʻTheory of Machineʼ,” In the Study of the 19th Century History of Physics (in Japanese), no.1, pp. 2-13, (1986);
Merz, J.T., A History of European Scientific Thought in the Nineteenth Century, Edinburgh and London: William Blackwood andSons, vol. 2, p. 100, n. 2 (1907), https://archive.org/details/historyofeuropea00merz [accessed May 17, 2018].
[16] Cleghorn, W., “Wiliam Cleghornʼs De Igne,” (1779) by McKie, D. and Heathcote, N.H.V., Annals of Science, vol. 14, p. 12f.,(1958).
[17] Cardwell, D.S.L., ibid., pp. 186-189, (1971).
[18] Hugo, V., Quatrevingt-Treize, Ninety-Three, part. 3, book V, (1874), http://www.gutenberg.org/ebooks/49372 [accessed May 17,
2018].
[19] Bernal, J.D., Science and Industry in the Nineteenth Century, London: Routledge & Kegan Paul, p. 53f, (1953).
[20] Mantoux, P., La Révolution industrielle au XVIIIe siècle: Essai sur les commencements de la grande industrie moderne enAngleterre, Paris: Société Nouvelle de Libha1hie et Dʼédition (1906);
Mantoux, P.,The Industrial Revolution in the Eighteenth Century: An Outline of the Beginnings of the Modern Factory System inEngland, translated by Vernon, M., London: Jonathan Cape, p. 318, (1959).
[21] Dickinson, H.W., A Short History of the Steam Engine, 2nd. ed., London: Frank Cass & Co. Ltd., p.16, (1938/1963).
[22] Dickinson, H.W., ibid., p.47, (1938/1963);
Hales, S., Vegetable Staticks, (1727), (reprinted, The Scientific Book Guild, 1961),
http://www.biodiversitylibrary.org/item/ 188038#page/14/mode/1up [accessed May 17, 2018], p.65.
[23] Robinson, E. and McKie, D. ed., Partners in Science: Letters of James Watt to Joseph Black, London: Constable, p. 416, (1970).
[24] Mantoux, P., ibid., p. 326, (1906/1959).
[25] Stewart, B., Lessons in Elementary Physics, London: Macmillan and Co., p. 222, (1886).
[26] Robinson, E. and McKie, D. ed., ibid., p. 418, (1970).
[27] Tomonaga, S., Butsurigaku toha Nandarouka (in Japanese) (What is Physics?), Part 1, Tokyo: Iwanami, p. 146, (1979).
[28] Robinson, E. and McKie, D. ed., ibid., p. 436, (1970).

25

IHTC16-FL01[29] It was frequently pointed out that Watt could improve steam engine, because he learned the theoretical background for the separate
condenser from Black; for example,
Dickinson, H.W. & Jenkins, R., James Watt and the Steam Engine, London: Oxford Univ. Press, p. 22, (1927),
https://ia800801.us.archive.org/21/items/jameswattsteamen01dick/jameswattsteamen01dick.pdf [accessed May 17, 2018];
Dickinson, H.W., James Watt: Craftsmans & Engineer, Cambridge University Press, p.35, (1935);
McKie, D., and Heathcote, N.H. de, Discovery of Specific and Latent Heat, p. 122, (1935) (reprinted, Arno Press, 1975);
Actually, however, Watt found the effect of latent heat by himself, and thus it .is not always true; at least, for the invention of theseparate condenser, the theory by Black was not necessary; about this point, see,
Smiles, S., Lives of Boulton and Watt, London: John Murray, Albemarle Street., p. 122, (1865);
https://ia802300.us.archive.org/26/items/livesofboultonwa00smilrich/livesofboultonwa00smilrich.pdf [accessed May 17, 2018];
Fleming, D., “Latent Heat and the Invention of the Watt Engine,” ISIS, vol. 43, pp. 3-5 (1952);
Cardwell, D.S.L., ibid., p.50f., (1971);
Cardwell, D.S.L., Technology Science and History, A Short Study of the Major Developments in The History of Western
Mechanical Technology and Their Relationships with Science and Other Forms of Knowledge, London: Heinemann Educational,(1972).
Cardwell, D.S.L., edited by Porter, R., Man Masters Nature: Twenty-five Centuries of Science, New York: G. Braziller, (1987).
Partington, J.R., A History of Chemistry, vol. III, New York: Martino Publishing, p. 153, (1962).
[30] Robinson, E. and McKie, D. ed., ibid., p. 105, (1970).
[31] Dickinson, H.W., A Short History of the Steam Engine, 2nd. ed., London: Frank Cass & Co. Ltd., p.66, (1938/1963).
[32] Tann, J. ed. Selected Papers of Boulton & Watt, Boulton to Erasmus Darwin (Jan. 4, 1790), Cambridge: M.I.T. Press, vol. 1, p.
72, (1981);
The detailed description about improved Watt engine is recorded in An Act for vesting in James Watt, Engineer...the sole Use andProperty of certain Steam Engines...of his Invention...throughout his Majesty's Dominions for a limited time,https://www.parliament.uk/about/living-heritage/transformingsociety/tradeindustry/industrycommunity/collections/steamengines-act/ [accessed May 17, 2018];
Robinson, E. & Musson, A.E. ed., James Watt and the Steam Revolution, London: Adams & Dart, pp. 76-80, (1969),
http://himedo.net/TheHopkinThomasProject/TimeLine/Wales/Steam/JamesWatt/RobinsonMusson/JamesWattPatents.htm[accessed May 17, 2018][33] Muirhead, J.P. ed., The Origin and Progress of the Mechanical Inventions of James Watt, 3 vols., London: John Mueeat, (Mr.
Watt to Dr. Small, Glasgow, May 28, 1769) pp. 60ff., (1854),
https://books.google.co.jp/books/about/The_Origin_and_Progress_of_the_Mechanica.html?id=jFzVAAAAMAAJ&redir_esc=y[accessed May 17, 2018][34] Robinson, E. and Musson, A.E. ed., idem, p. 98, (1969),
http://himedo.net/TheHopkinThomasProject/TimeLine/Wales/Steam/JamesWatt/RobinsonMusson/JamesWattPatents.htm[accessed May 17, 2018][35] Cardwell, D.S.L., From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Ithaca: Cornell University
Press, p. 80, (1971).
[36] Fox, R., “Wattʼs Expansive Principle in the Work of Sadi Carnot and Nicolas Clément,” Note and Records of the Royal Society,vol. 24, pp. 235-238, (1970);
Fox, R., Introduction of “Carnot, S.: Reflexions on the Motive Power of Fire, A Critical Edition with the Surviving Scientific
Manuscripts,” Manchester: Manchester University Press, p. 10, (1986).
Fox, R., “The Challenge of a New Technology; Theorists and the High-pressure Steam Engine before 1824,” Sadi Carnot etlʼEssor de la Thermodynamique, Édition du Centre National de la Recherche Scientifique, p. 161, (1976);
Lervig, P. “Sadi Carnot and Nicolas Clément,” In Human Implications of Scientific Advance, edited by E.G. Forbes, Edinburgh:
Edinburgh University Press, p. 294, (1977).
[37] Dalton, J., New System of Chemical Philosophy, Pt. l, Pt. 2 (reprinted, William Dawson & Sons Ltd.), p. 131 footnote, (1808,
1810)., http://digbib.ubka.uni-karlsruhe.de/volltexte/digital/1/277.pdf [accessed May 17, 2018], p. 131.
[38] Thomson, W. (Lord Kelvin), “James Watt: An Oration,” (1901), Mathematical and Physical Paper of William Thomson,
Cambridge: Cambridge University Press, vol. 6, p. 357 (1911/2011), https://archive.org/details/mathematicalphys06kelvuoft,[accessed May 17, 2018][39] Cardwell, D.S.L., ibid., (1971);
Fox, R., ibid., p. 161, (1976).
[40] Cardwell, D.S.L., ibid., pp. 155ff, (1971).
[41] Fox, R. ibid., p. 151, (1976).
[42] Landes,. D.S., “Technological Change and Development in Western Europe, 1750‒1914,” in The Cambridge Economic Historyof Europe, edited by Habakkuk, H.J., and Postan, M., VI, The Industrial Revolutions and After: Incomes, Population andTechnological Change (I), Part I, Cambridge: Cambridge University Press, pp. 410ff., (1965); Cf.
Cardwell, D.S.L., ibid., p. 157, (1971).
[43] Hugo, V., Les Misérables, (1862), http://www.gutenberg.org/ebooks/135 [accessed May 17, 2018];
Maupassant, G., Une vie, (1883), http://www.gutenberg.org/ebooks/7114 [accessed May 17, 2018].
[44] Bernal, J.D., Science and Industry in the nineteenth Century, London: Routledge & Kegan Paul Ltd. p. 51, (1953); Cf.
Charléty. S., Histoire du Saint-Simonisme (1825‒1864), Paris: Hachette et cie., (1896) Paris, Paul Hartmann, (1931).
[45] Chevalier, M., Society, Manners and Politics in the United States: Being a Series of Letters on North America, Boston: Weeks,
Jordan and Company, pp. 134-135, (1839), https://archive.org/details/societymannersa00chevgoog [accessed May 17, 2018].

26

IHTC16-FL01
CHAPTER 19 THEORY OF IDEAL HEAT ENGINE—CARNOTʼS THEOREMCONTENTS
I.
Purpose of Carnotʼs paper ..................................................................................................................... 27
II. Carnotʼs preliminary theorem and its background (the first and second premises) .................... 28
III. Carnot cycle ................................................................................................................................................ 32
IV. Carnotʼs theorem (the third premise) .................................................................................................... 35
References in chapter 19 ........................................................................................................................... 36
I. Purpose of Carnotʼs paper
Carnot raised the problem of motive power not from the viewpoint of thermal science as a field of physics but
from that of the technological development of steam engines. “Carnot owed far more to his contact with the worldof power engineering than he did to the physics of his day”[1].
Nevertheless, it should be noted that, when both setting and solving the problem, he stuck to fundamentals andtheory. As a result, his study was not limited to the technical theory of steam engines, and by clarifying theuniversal limitation in the relationship between heat and work, he developed a new field of physics.
Carnotʼs finding was that in steam engines, “The steam serves simply as a means of transporting the caloric” [2][C, p. 65]. This can be also confirmed in the Réflexions, in which he wrote, “We draw a distinction here between
the steam engine and the heat engine in general. The latter may employ any working substance, steam or anythingelse, in order to develop the motive power of heat”[C, p. 64], and he clearly distinguished ʻheat engines (machineà feu)ʼ and ʻsteam engines (machine à vapeur).ʼ
Although Cardwell wrote, “The steam-engine, on the other hand, manifests in dramatic and clear form the capacityof heat to perform work”[3], such a comment is based on Carnotʼs recognition that the steam engine is a heatengine—the true origin of power is not steam but heat. Although this may sound obvious, it was not always so.
For example, in “A Course of Lectures on Natural Philosophy and the Mechanical Arts” by Thomas Young (1773–
1829) in 1807 [4], which also deals with technical issues as indicated by its title, heat is discussed in Part 3(“Physics”), while steam engines are discussed as an example of a pneumatic machine in Part 2 (“Hydrodynamics”); pneumatic machines include an air pump and “are principally dependent, in their operation, upon theproperties of elastic fluids.” That is, the power source of steam engines was considered to be the steam itself. Since
Young, as described later, believed in the vibration (wave) theory of heat, he did not accept caloric, which might
have made it difficult for him to ascribe the power source to heat.
Thus, it is considered that the problem of the motive power of heat was raised not by a proponent of the dynamical
theory of heat but by Carnot, who believed that caloric theory was not paradoxical but natural.
In any case, solely by regarding steam simply as a means of transporting heat and by also regarding steam engines
essentially as heat engines, Carnotʼs universal approach allowed the nature of heat to be studied as a source ofpower independent of the peculiarities of working substances and of the individual structures of devices.
By grasping and analyzing the status quo as follows: “The phenomenon of the production of motion by heat (laproduction du mouvement par la chaleur) has not been treated from a sufficiently general point of view”[C, p. 63],
Carnot set out what needed to be achieved as follows:“In order to grasp in a completely general way the principle governing the production of motion by heat, it isnecessary to consider the problem independently of any mechanism or any particular working substance.

27

IHTC16-FL01
Arguments have to be established that apply not only to steam engines but also to any conceivable heat engine,
whatever working substance is used and whatever operations this working substance is made to perform”[C, p.
64].
Such a surprisingly radical assertion may be the privilege of a young man in his 20s. Here, Carnot intended toestablish “a complete theory (théorie complète)” based on a comprehensive universal principle at once. As
guidance, the existing mechanical theory was referred to. Carnot illustrated a complete theory using mechanicaltheory.
“A mechanical theory permits a very detailed study of those machines that do not derive their motion from heatbut are driven by men or animals, a fall of water, wind, etc. Every situation can be foreseen, and every conceivable
movement is subject to some well-founded and universally applicable general principle. It is this whichcharacterizes a complete theory”[C, p. 64].
Therefore, it was necessary to extend the theory of the heat engine to the mechanical theory. Actually, themechanical theory in France had been almost completed by Lazare Carnot. In contrast, how random (au hasard)
the improvement of the heat engines (particularly high-pressure engines after Watt) was!—The young Carnot wasacutely aware of the lack of fundamental theory.
“but such a theory is plainly lacking in the case of the heat engine. It will only be achieved when the laws ofphysics are sufficiently extensive and general for us to be able to predict all the effects that are produced whenheat acts in a particular way on any given substance”[C, p. 64].
Carnot achieved his objectives; he had taken the first step toward establishing a “universally applicable generalprinciple” in thermal science. On the basis of these objectives, his paper, which “is perhaps the most original inphysical science”(J. Larmor, On the Nature of Heat, P.T.R.S. (1916‒18), p. 326), was published.
Although Carnotʼs paper was motivated by technical requirements, it was extremely fundamental and universal,
and was not written for engineers nearly interested in short-sighted practicability. Consequently, it did not have
any impact on them. Truesdell wrote that “It is easy to say, and it has often been said, that Carnot wrote forengineers, but I can find no evidence that any engineer ever read and applied his results”[5].

II. Carnotʼs preliminary theorem and its backgroundAs described earlier, Carnot tackled the problems of “whether the motive power of heat is limited or whether it isboundless,” and “is the motive power of heat fixed in quantity, or does it vary with the working substance that isused?” He sought clues to these problems by the analysis of actual steam engines. That is, “The production of
motion in the steam engine always occurs in circumstances which it is necessary to recognize, namely when theequilibrium of caloric is restored, or (to express this differently) when caloric passes from a body at one
temperature to another body at a lower temperature”[C, p. 64]. On the basis of his recognition of this simple but
essential fact, he made the following premises:
The first premise is that “Wherever there is a difference in temperature, motive power can be produced”[C, p.
67]; this can be summarized to mean that a temperature difference is a necessary condition to produce motive
power. The detailed description is as follows:“So the production of motive power in a steam engine is due not to an actual consumption of caloric but to its
passage from a hot body to a cold one. It is due, in other words, to a restoration of the equilibrium of caloric afterthat equilibrium has somehow been disturbed, for example by a chemical reaction such as combustion, or by someother means. We shall soon see that the same principle can be applied to any engine whose motion is due to heat”[C,p. 65].
The discovery that not only a high temperature but also a low temperature is necessary was Carnotʼs breakthrough.

28

IHTC16-FL01
The second premise is that “Obviously heat can only be a source of motion in so far as it causes substances toundergo changes in volume or shape”[C, p. 66]; this can be summarized to mean that a change in the volume ofsubstances is an indispensable condition for producing motive power. That is, upon the direct contact between
high- and low-temperature substances, only heat transport occurs and no work is done. Work is done not merely
by heat transport but also through the cooling of the working substance accompanied by its expansion.5 That is,“For there is nothing in nature that does not undergo changes in volume, contracting and then expanding as it
experiences cold or heat; there is nothing which, in doing so, cannot act against a resisting force and there bydevelop motive power”[C, p. 66]. This is precisely the theoretical specification of Wattʼs expansive principle; theonly difference is that Watt assumed the expansion process on the basis of Boyleʼs law, while Carnot correctly
understood the temperature drop associated with the adiabatic process.
As a result of the above two premises, the maximum possible work with respect to a specified difference intemperature is obtained “if no changes in temperature occur unless they are caused by changes in volume; in otherwords, bodies at sensibly different temperatures should not come into contact with one another”[C, p. 77].
“Since any process in which the equilibrium of caloric is restored can be made to yield motive power, a process inwhich the equilibrium is restored without producing power must be regarded as representing a real loss (perte).
Now, a momentʼs reflexion will show that any change in temperature that is not due to a change in the volume ofa body is necessarily one in which the equilibrium of caloric is restored profitlessly. Hence the necessary condition
for the achievement of maximum effect is that the bodies used to produce motive power should undergo no change
in temperature that is not due to a change in volume. Conversely, whenever this condition is fulfilled, the effectproduced will be a maximum”[C, p. 70].
Here, I will daringly refer to this as Carnotʼs preliminary theorem, although this name has not been used
elsewhere.
Furthermore, as a corollary to this theorem, we can derive the following: an ideal engine which achieves the
maximum work is simultaneously a reversible one because it does not include thermal conduction between
regions with a finite temperature difference. This played a very important role in deriving Carnotʼs theorem.
Here, ʻreversibleʼ is used with a narrow meaning that the process can be reversely operated by adding the sameamount of work to the previously obtained work.
Carnotʼs idea for his preliminary theorem might have been directly stimulated by the development of heatengines, particularly the improvement of the steam engine by Watt. The first premise, i.e., to obtain work byusing steam, not only a high temperature but also a low temperature is necessary, is exactly the same principle
as used in the theoretical work understanding Wattʼs invention of the separate condenser—purification of theheat engine by separating the furnace and cooler. The second premise followed Wattʼs expansive principle,
in which steam separated from a boiler expands, reducing its pressure and temperature, and enters a cooler
after fully converting its potential activity to work.
Simultaneously, however, in the background of Carnotʼs argument, there was accumulated technology andtheory on hydraulic machines, which were developed in parallel with the steam engine during the industrial
revolution.
In particular, there was remarkable progress in the column-of-water engine (machine à colonne dʼeau), which
was invented in the eighteenth century by modeling Newcomenʼs engine. As shown in Fig. 19.1, a piston is
operated by using hydro pressure instead of steam pressure, and hence it was also known as a water-pressureengine. This engine was popular in European countries with few coal resources but abundant flowing water5

In 1821, the German physicist Seebeck (1770–1831) discovered the phenomenon that two junctions of a loop
consisting of two dissimilar metals produces an electric current when the two junctions are kept at different
temperatures. If a motor is operated by using this current, the temperature difference will produce work without
a volume change. Carnot, however, did not know of this phenomenon. Also, we do not consider thisphenomenon here.

29

IHTC16-FL01
with high drops. “Between 1750 and 1825, the column-of-water engine represented an ingenious and scientificallyadvanced method of harnessing water power”[6].
Carnotʼs recognition that both high and low temperatures are necessary for heat to produce work was considered
to be based on the analogy with the following column-of-water engines, which require a fall of water to produce
work. This is confirmed from the following statement in the Réflexions: “From the ideas that have been establishedso far, we are sufficiently justified in comparing the motive power of heat with that of a fall of water. ...... The
motive power of a fall of water depends on its height (chute dʼeau) and on the amount of liquid. The motive powerof heat likewise depends on the amount of caloric that is used and on what might be termed—in fact on what weshall call—the height of its fall (sa chute); it depends, in other words, on the difference in temperature of the bodies
between which the passage of caloric occurs”[C, p. 72].

Fig. 19.1 Early column-of-water engine produced by Gensanne. (from Cardwell, D.S.L., From Watt to Clausius:
The Rise of Thermodynamics in the Early Industrial Age, Ithaca: Cornell University Press, Plate VI, (1971))
After Carnot, this analogy became more popular. A textbook by Balfour Stewart (1866) contains the following
description: “Carnot, a French philosopher, who was the first to study this subject, very ingeniously likened themechanical capability of heat to that of water, remarking that just as water on the same level can produce nomechanical effect, so neither can bodies at the same temperature; and just as we require a fall of water from ahigher to a lower level in order to obtain mechanical effect, so likewise we must have a fall of heat from a body ofhigher to one of lower temperature”[7].
Regarding the analogy with a current of water, there were other elements described below. One more basis of
Carnotʼs preliminary theorem was the studies by the French military engineer Jean-Charles Borda (1733–1799)and his father Lazare Carnot.
By the middle of the eighteenth century, it was known that an overshot water wheel based on gravitation is much
more efficient than an undershot water wheel. The English engineer John Smeaton (1724–1792) found that theefficiency of an overshot water wheel is three times that of an undershot water wheel. Furthermore, he found thatwhen the blades of an undershot water wheel receive an impact by a current of water, a major loss arises. Also, foran overshot water wheel, it is desirable that buckets filled with water are driven not by impact but by gravitation,
as slowly as possible [8].

30IHTC16-FL01
Similarly, Borda, who studied the output power of the water wheel, found that when fluid subjected to a suddenchange becomes turbulent and loses its vis viva (kinetic energy), “Then the water, instead of suddenly impacting
the blade, would enter without impact, as prescribed by Borda, and would slowly give up its momentum as it
ascended the curve of the blade. The water would gradually slow down, stop, and then descend the blade, picking
up its initial velocity, but in a direction opposite that of the wheel. If the wheelʼs velocity and the length of its
blades were proportioned correctly, the water as it left the wheel would have a velocity equal to that of the wheel,
but in an opposite direction, so that their relative velocity would be zero. Thus Bordaʼs requirement that the water
leave the wheel without velocity could be met as well”[9].
The principle of energy loss associated with inelastic collision was generalized by Lazare Carnot. In his “Essay
upon Machines in General” (1782), he wrote,“We may conclude from this, for example, that the method of producing the greatest possible effect in a hydraulicmachine moved by a current of water, is not to adapt a wheel to it, the wings of which receive the shock of thefluid. In fact, two good reasons prevent us from producing in this way the greatest effects: the first is, as we havealready said, because it is essential to avoid every kind of percussion whatever; the second is, because after theshock of the fluid there is still a velocity which remains to it as a pure loss, since we should be able to employ thisremainder in still producing a new effect to be added to the first. In order to make the most perfect hydraulicmachine, i.e. capable of producing the greatest possible effect, the true difficulty lies, 1st, In managing so as thatthe fluid may lose absolutely all its movement by its action upon the machine, or at least that there should only
remain precisely the quantity necessary for escaping after its action; 2d, Another difficulty occurs in so far as it
loses all this movement by insensible degrees, and without there being any percussion, either on the part of thefluid, or on the part of the solid parts among themselves: the form of the machine would be of little consequence;
for a hydraulic machine which will fulfil these two conditions will always produce the greatest possible effect: but
this problem is very difficult to resolve in general, not to say impossible; it may even happen that in the physicalstate of things, and in respect of their simplicity, there can be nothing better than wheels moved by shocks: and in
this case as it is impossible to fulfil at once the two conditions most desirable, the more we wish to make, the fluid
lose of its movement in order to attain the first condition, the stronger will be the shock; the more, on the contrary,
we wish to moderate the shock in order to approach the second, the less will the fluid lose of its movement. We
perceive that there is a medium, by means of which we shall determine, if not in an absolute manner, at least,
having regard to the nature of the machine, that method which will be capable of the greatest effects”[10; Carnot,
L.N.M., vol. 31, p. 300].
Sadi Carnotʼs statement6 about the maximum work for a given heat is almost a counterpart of Lazare Carnotʼsstatement about the maximum work for a water wheel. In the former, the contact between substances having atemperature difference reduces the power of the heat, and in the latter, the impact between the blades and the fluidwith a velocity difference reduces the power of the water current. We can see the remarkable analogy between an
ideal hydraulic machine which changes its state as slowly as possible and an ideal heat engine which operates
through a quasi-static process.
When Sadi Carnot wrote, “A mechanical theory permits a very detailed study of those machines that do not derive
their motion from heat”[C, p. 64], he must have recalled his fatherʼs study.

6

From only the following statement, the reason why a high-pressure engine is advantageous can be explained.
“Since coal is able to produce temperatures above 1000° when it is burned, and since the water most commonly
available to us in our climate is at roughly 10°, it is easy to obtain a fall of caloric of 1000°, though of this fall
only 120° arc used in a steam engine. Even then, the 120° are not utilized to the full. There are always
substantial losses caused by a failure to restore the equilibrium of caloric in a useful way. The reasons for thesuperiority of what we call high-pressure engines over engines operating at a lower pressure are now evident.
The superiority lies essentially in their ability to utilize a greater fall of caloric. Since steam that is produced
at a higher pressure is also at a higher temperature, and since the temperature of condensation always remains
much the same, the fall of caloric is obviously greater”[C, p. 104].

31

IHTC16-FL01
III. Carnot cycle
On the basis of the preliminary theorem, Carnot proposed the so-called Carnot cycle, a thermal cycle for gas withthe maximum efficiency which operates between a high-temperature furnace and a low-temperature cooler. He
was wise enough to consider not only a single process but also an entire cycle to clarify the work done by heat; atthat time, all other scientists and engineers considered only the expansion process directly producing work.
The most important feature of the cycle is as follows: When the working substance has returned to the initial state,
the heat added at the furnace—and the heat exhausted to the cooler in a Carnot cycle—is considered to becompletely used for work to the outside. If the state has not returned to the initial state, however, part of the heat
may be used to induce a physical internal change of the working substance, or conversely, the physical internalchange of the working substance may be used to produce work.
William Thomson in 1849 and Rudolf Clausius in 1850 highly appreciated Carnotʼs insightful view [11], while
Ernst Mach (1838–1916) simply wrote, “We may remark that the choice of a cyclic process for the derivation ofthis theorem was a particular happy one (eine besonders glückliche)”[12]. Since Carnot intentionally considered
the cycle, Machʼs comment was inappropriate. On the basis of the preliminary theorem, in the operation of an
ideal engine, the temperature of the working substance must be equal to that of the constant (high)-temperature
furnace during heat absorption, and to that of the constant (low)-temperature cooler during heat discharge, in order
not to include thermal conduction due to the finite temperature difference between the working substance and thedevice. Thus, the ideal engine must consist of isothermal and adiabatic processes. This cycle forms the skeletal
structure of the heat cycle proposed by Carnot.
Strictly, of course, the temperature of the working substance must be lower than that of the furnace and higher
than that of the cooler. “But the most minute difference in temperature is enough to bring about condensation, and,for the purpose of my argument, that is all that is necessary. It is just as in differential calculus, where we achieve
rigorously demonstrated proofs by ignoring quantities that arc infinitesimally small relative to the ones that we
retain in our equations”[C, p. 68].
In such a case, the working substance and the furnace or the cooler are effectively in thermal equilibrium, and heatis transported infinitely slowly, i.e., “jiwa jiwa” in Japanese (which means bit by bit) [13], from the furnace to theworking substance and from the working substance to the cooler.
Similarly, when the working substance (for example, air) expands, its pressure must be higher than the externalpressure, and when it contracts, its pressure must be lower than the external pressure. To prevent the loss of power
during these processes, and more specifically to avoid the acceleration of the piston and the induced flow of theworking substance (air), these processes must proceed as slowly as possible. To this end, the difference between
the internal and external pressures must be infinitely small.
Here, we define the quasi-static change as the change under the conditions of an infinitesimally small difference
from the external pressure as well as an infinitesimally small temperature difference from the heat source. In thiscase, inside the working substances, the state variables such as pressure and temperature—which were originallydefined only for the equilibrium state—are also defined at every moment of the change. Furthermore, since thereverse process is also quasi-static, this process is reversible.
As the working substance, Carnot considered air (or gas) contained in a cylinder with a movable piston. The
bottom of the cylinder is made of a material with perfect heat conduction, while the side walls and the piston are
made of a material with perfect thermal insulation (Fig. 19.2).
To clarify the point to be discussed, let us consider the explanation by Clapeyron. First, the cylinder is set on the
furnace, and air with the same temperature as the furnace (   ) is expanded by adding heat ( q ) (isothermal
expansion: A → B). Second, the cylinder is separated from the furnace, its bottom is closed by an insulator, andthe air is cooled to   (<   ) by expansion (adiabatic expansion: B → C). Third, the cylinder is set on a cooler,

32

IHTC16-FL01
and air with the same temperature as the cooler (   ) is made to contract by extracting heat ( q ) (isothermal
compression: C → D). Finally, the cylinder is separated from the cooler, the bottom is closed by an insulator, andthe air is heated to   by compression (adiabatic compression: D → A).
In this operation, each process is quasi-static. Therefore, for the air, the pressure P, temperature  and volume V
are defined at every moment; thus, the equation of state P  P , V  holds and, furthermore, on the basis of
Carnotʼs premise, the heat contained in the air is a state variable ( Q  Q ,V  ). From these two relationships, if
Q and  return to their initial values, P and V will also return to their initial values. Consequently, the air returns
to the initial state and a perfect cycle is achieved.

Fig. 19.2 Carnot cycle.

According to this explanation, it might be considered that the existence of the heat function Q  , V  , i.e., theconservation of heat, is necessary to close the cycle; however, this is not always the case. Note that the above
explanation is based on that of Clapeyron, which was different from that of Carnot.
Carnot started his cycle from a gas of volume V0 with temperature   ( A 0 in Fig. 19.3). The gas is isothermallyexpanded to B upon coming in contact with the furnace with the same temperature   . From B, the gas isadiabatically expanded to C with temperature   . From C, the gas is isothermally compressed to D with the initial
volume V0 by coming in contact with the cooler of temperature   . Furthermore, from D, the gas is heated totemperature   by adiabatic compression; of course, the volume at A is less than V0. From A, the air can be
isothermally expanded to V0 by coming in contact with the furnace of temperature   , and the cycle is closed at
A0 .
This is known as the Carnot cycle.

33

IHTC16-FL01

Fig. 19.3 P‒V diagram of Carnot cycle.

Consequently, to close the Carnot cycle, the conservation of heat is not required. Let define the sum of the heatabsorbed from the furnace during processes A → A 0 → B as q and the heat discharged to the cooler in process
C → D as q . Even if q   q  , this cycle is closed, which enabled the Carnot cycle to play a key role in thedevelopment of thermal science; the approach by Clapeyron detracts from the significance of Carnotʼs reasoning[14].7
Upon reading Carnotʼs original paper after studying thermodynamics, it may be considered that his description of
the above five processes is somewhat convoluted as well as strange; however, we should recognize that such a
description was necessary.

Fig. 19.4 Carnot cycle depicted in Clapeyronʼs paper.

7

If we know the P-V relationship for an adiabatic process—Poissonʼs formula P1    const . orPV   const . (16.15)—, we can determine point D as the point which can adiabatically return to point A
without using the relation q  q , as employed by Clapeyron. Thus Clausius succeeded in removing the
restriction of the conservation of heat. However, if we start from point D or B, the discussion is much simpler.
Clapeyronʼs paper first expressed the Carnot cycle as an indicator diagram in the P‒V plane as shown in Fig. 19.4.
Clapeyron also pointed out that the area enclosed by the cycle lines is equal to the work.

34

IHTC16-FL01
Actually, in the Réflexions, Carnotʼs discussion was based on the conservation of heat, while he was much more
careful about the necessary condition than Clapeyron by adding the following words: “We believe, nevertheless,that if the fundamental law that we set out to confirm is to be placed beyond all doubt, new tests are called for.
The law is based on the theory of heat as we understand it today, and it must be admitted that we do not consider
this as an entirely solid foundation”[C, p. 100].8“In these different processes, the air in the cylinder exerts pressure on the piston. The pressure varies, since theelastic force of the air is affected by changes both in its volume and in its temperature. But it will be seen that for
any given volume, that is, when the piston is in a particular position, the temperature is higher while the air is
expanding than while it is being compressed. During expansion, therefore, the elastic force of the air is greater,and hence the amount of motive power produced is also greater than the amount that is expended in order to bringabout compression. In this way, there will be a net production of motive power which can be put to some use. The
air has thus been made to act as a heat engine; moreover, it has been used in the most advantageous way possible,
since there has been no useless restoration of the equilibrium of caloric”[C, p. 75].

IV. Carnotʼs theorem
In this cycle (hereafter referred to as C), according to Carnot, heat q flows from the furnace (   ) to the cooler(   ), work W is produced, and there are no other changes. Furthermore, the cycle is reversible from the corollaryof the preliminary theorem, that is, “The whole sequence of operations just described can be reversed, and carriedout in the opposite order”[C, p. 75]. In other words, if work W is provided from the outside, heat q is pumped
up from the cooler to the furnace. This point is crucial in the following discussion.
That is, the specific structure and the gas properties of the working fluid of a Carnot cycle are not important but
the fact that the cycle is ideal and thus reversible is essential. (A Carnot cycle is reversible in a narrow sense
because all four processes are reversible. In the following discussion, however, the cycle is required to be reversible
only in the broad sense that it can be reversed irrespective of the above-mentioned four processes.)
Here, we go back to the original problems of “whether the motive power of heat is limited or whether it isboundless,” and “is the motive power of heat fixed in quantity, or does it vary with the working substance that isused?”To this end, we assume that there is a cycle (referred to as C') between the temperatures   and   whose
efficiency is superior to the Carnot cycle C. That is, there is a cycle C' which can produce work W  (> W) for thesame q . In this case, as shown in Fig. 19.5, by transferring the work W (  W  ) generated by C' to the reversed
cycle C, heat q can be pumped from the cooler to the furnace. Overall, without any change in the furnace andcooler or the working substance, net work W   W  0 can be produced.

8

As demonstrated in a later chapter, in the recently discovered notes of Carnotʼs manuscript, he reached thefirst law of thermodynamics—he rejected the conservation of heat and he insisted that to generate the motive
power, heat must be consumed. Medoza, who carefully studied his notes, wrote, “The remarkable change of
opinion, from believing that the theory of heat was «beyond doubt» to feeling that «its basis is not unshakablesolidity», again indicates that many of the doubts expressed in the Manuscript Notes must have begun to assail
Carnot even while he was writing book”[15]. If this is the case, Carnot might have been so strongly prepossessedby the analogy between the heat engine and the water-column machine that he assumed that heat is conserved
from the high-temperature furnace to the lower-temperature cooler even though he doubted caloric theory.

35

IHTC16-FL01

Fig. 19.5 Carnotʼs proof of his theorem.

Carnot said, “We should then have a case not only of perpetual motion (mouvement perpétuel) but of motive
power being created in unlimited quantities without the consumption of caloric or of any other agent.9 Creation ofthis kind completely contradicts prevailing ideas, the laws of mechanics (mécanique), and sound physics; it isinadmissible”[C, p. 69]. That is, if we regard the principle that a perpetual motion machine of the first kind is
impossible as the third premise (for the first and second premises, see pp. 28-29 of this paper), we can conclude
that the Carnot cycle attains the maximum efficiency independent of the properties of the working substance andthe mechanism of the machine.
“We have chosen air as the substance with which to develop the motive power of heat, but it is clear that theargument would be the same for any other gas. Indeed, it would be the same for any body whose temperature canbe changed through successive decreases or increases in volume, that is for any substance in nature or at least forany substance with which motive power can be developed. From this, the following general proposition may bestated:The motive power of heat is independent of the working substances that are used to develop it. The quantity isdetermined exclusively by the temperatures of the bodies between which, at the end of the process, the passage ofcaloric has taken place. (La puissance motrice de la chaleur est indépendante des agens mis en œuvre pour laréaliser; sa quantité est fixée uniquement par les températures des corps entre lesquels se fait en dernier résultat
le transport du calorique.)”[C, p. 76].
This is Carnotʼs theorem. It is the greatest discovery in the history of thermal science, from which
thermodynamics began.
Note that the following three items are also important discoveries of Carnot in addition to his theorem: first, theconcept of quasi-static and reversible changes; second, the consideration of his cycle; and finally, the proof bycontradiction by considering a combination of the normal and reversed cycles.

References in chapter 19[1] Fox, R., The Caloric Theory of Gases from Lavoisier to Regnault, Oxford: Clarendon Press, p. 177, (1971); Cf.
9“perpetual motion ...is inadmissible” is the conclusion strongly asserted Lazare Carnot. (“The Academy of
Paris, in 1775, passed a resolution not to consider any plans intended for perpetual motion; at the same time
appeared a paper from Carnot (Principes Fondamentaux de lʼéquilibre et du Mouvement),” Par., 1803, Sec.
281)”[16].)36

IHTC16-FL01
Fox, R., “Wattʼs Expansive Principle in the Work of Sadi Carnot and Nicolas Clément,” Note and Records of the Royal Society,vol. 24, p. 234, (1969b);
Cardwell, D.S.L., “Power Technologies and the Advance of Science, 1700-1825,” Technology and Culture, vol. 6, pp. 188-207,(1965);
Kuhn, T.S., “Engineering Precedent for the Work of Sadi Carnot,” Archives internationales dʼhistoire des sciences. vol. 13, pp.
251-255, (1960);
Kuhn, T.S., “Sadi Carnot and the Cagnard Engine,” ISIS, Vol. 52, pp. 567-574, (1961).
[2] Carnot, S., Réflexions sur la Puissance Motrice du Feu, (1824), References fromhttps://fr.wikipedia.org/wiki/Sadi_Carnot_(physicien) or
https://en.wikipedia.org/wiki/Reflections_on_the_Motive_Power_of_Fire [accessed May 17, 2018]
Carnot, S., Reflections on the Motive Power of Fire, A Critical Edition with the Surviving Scientific Manuscripts, translated andedited by Fox, R., Manchester: Manchester University Press, (1986).
[3] Cardwell, D.S.L., From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Ithaca: Cornell University
Press, p. 189, (1971).
[4] Young, T., A Course of Lectures on Natural Philosophy and the Mechanical Arts, Vol. I, Lecture XXIX (1807),
https://archive.org/details/lecturescourseof02younrich [accessed May 17, 2018][5] Truesdell, C., The Tragicomical History of Thermodynamics 1822-1854, New York, Heidelberg, Berlin: Springer-Verlag, p. 80,(1980).
[6] Cardwell, D.S.L. “Power Technologies and the Advance of Science, 1700-1825,” Technology and Culture, vol. 6, pp. 188-207,
p. 200, p. 192, (1965);
Cardwell, D.S.L., From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Ithaca: Cornell University
Press, pp. 85-88, (1971).
[7] Stewart, B., Lessons in Elementary Physics, London: Macmillan and Co., p. 219, (1886),
https://archive.org/details/lessonsinelemen02stewgoog [accessed May 17, 2018][8] Reynolds, T.S., Stronger than a hundred men: a history of the vertical water wheel, Baltimore: Johns Hopkins University Press,
pp. 241-244, (1983);
Wilson, P.N., “Water-Driven Prime Mover,” In Engineering Heritage: Highlights from the History of Mechanical Engineering,
vol. 1, Heinemann Educational, p. 29f, (1964);
Stowers, A., “Watermills, 1500-1850,” In A History of Technology, vol. IV: The Industrial Revolution, c.1750 to c.1850, editedby Singer, C., Holmyard, E.J., Hall, A.R. and Williams, T.I., New York: Oxford University Press, pp. 199-213, (1958).
[9] Reynolds, T.S., ibid., p. 261f, (1983).
[10] Carnot, L.N.M., “Essay upon Machines in General”, Edinburgh and Dublin Philosophical Magazine (1807-09), vol. 30, pp. 8-15,
pp. 154-158, pp. 207-221, pp. 310-320, vol. 31, pp. 28-36, pp. 136-144, pp. 220-228, pp. 295-305, vol. 32, pp. 124-130, (1782),
http://www.biodiversitylibrary.org/item/15594#page/9/mode/1up [accessed May 17, 2018],
http://www.biodiversitylibrary.org/item/52931#page/7/mode/1up [accessed May 17, 2018],
http://www.biodiversitylibrary.org/item/123271#page/9/mode/1up [accessed May 17, 2018];
Reynolds, T. S., ibid., p. 263f, (1983);
Gillispie, C.C., The edge of objectivity: an essay in the history of scientific ideas, Princeton: Princeton University Press, p. 362(in 1990 version), (1960);
Scott, W.L., The Conflict between Atomism and Conservation Theory 1644‒1860, London: Macdonald, New York: Elsevier, p.
99f., (1970).
[11] Thomson, W. (Lord Kelvin), “An Account of Carnotʼs Theory of the Motive Power of Heat: with Numerical Results deduced
from Regnaultʼs Experiments on Steam,” (1849), Mathematical and Physical Paper of William Thomson, Cambridge: Cambridge
University Press, vol. 1, p. 115 (1882/2011), https://archive.org/details/mathematicaland01kelvgoog [accessed May 17, 2018];
Clausius, R., “Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten
lassen,” Annalen der Physik und Chemie, Bd. 79, p. 379, (1850), References fromhttps://de.wikipedia.org/wiki/Rudolf_Clausius or https://en.wikipedia.org/wiki/Rudolf_Clausius [accessed May 17, 2018];
Clausius, R., edited by Archer Hirst, T., The Mechanical Theory of Heat, with Its Applications to the Steam-engine and to the
Physical Properties of Bodies, London: John van Voorst, p. 23, (1867), References fromhttps://de.wikipedia.org/wiki/Rudolf_Clausius or https://en.wikipedia.org/wiki/Rudolf_Clausius [accessed May 17, 2018].
[12] Mach, E., Die Principien der Wärmelehre, Historisch-kritisch entwickelt, Leipzig: Verlag von Johan Ambrosius Barth, (1896);
https://archive.org/details/dieprincipiender00mach [accessed May 17, 2018]
Mach, E., Principles of the Theory of Heat: Historically and Critically Elucidate, Springer Nature, (1986).
[13] Tomonaga, S., Butsurigaku toha Nandarouka (in Japanese) (What is Physics?), Part 1, Tokyo: Iwanami, p. 162, (1979);
the detailed description about ʻquasi-static changeʼ is included in pp. 162-164.
[14] Klein, M.J., “Closing the Carnot Cycle,” Sadi Carnot et lʼÉssor de la Thermodynamique (Édition du Centre National de laRecherche Scientifique), pp. 213-219, (1976).
[15] Mendoza, E., “Contributions to the Study of Sadi Carnot and his Work,” Archives internationales dʼhistoire des sciences, vol. 12,
pp. 380-389. p. 385 (1959); Cf.
Mendoza, E., “The Writing of the Réflexions and their Reception,” Sadi Carnot et lʼÉssor de la Thermodynamique (Édition duCentre National de la Recherche Scientifique), p. 170f., (1976).
[16] Dircks, H., Perpetuum Mobile; or, A History of the Search for Self-Motive Power from the 13th to the 19th Century, p. 210, (1861,70), https://archive.org/details/perpetuummobile01dircgoog [accessed May 17, 2018];
Scott, W.L., ibid., p. 100, (1970).

37

IHTC16-FL01
CHAPTER 20 STRUCTURE OF CARNOTʼS THEORY AND ITS EXTENSION—THE START OF THERMODYNAMICSCONTENTS
I.
Premises of Carnotʼs theory (premises A, B, and C) ........................................................................ 38
II. Analytical expression for Carnotʼs theorem (expressions I and II) .............................................. 40
III. Experimental determination of Carnot function ............................................................................. 44
IV. Extension of Carnotʼs theory and gas theorem (gas theorems 1, 2, and 3) ................................ 48
V.
Further discussion and significance of thermodynamics .............................................................. 51
References in chapter 20 ..................................................................................................................................... 54
I. Premises of Carnotʼs theory
Carnotʼs theory states that there is an upper limit to the efficiency of a heat engine which is determined only by
the temperature, irrespective of the working substance and the structure of the engine. In the previous chapter, aproof of the theorem was given on the basis of the Carnot cycle, and the emphasis was laid on the historicaldevelopment, particularly in connection with the improvement of the steam engine by Watt. In this chapter, we
focus on the internal logical structure of Carnotʼs theorem and its extension.
The basis of Carnotʼs heat theory in the Réflexions is the same as that in the work of Laplace and Poisson in
Chapter 16 except that the taking temperature  and volume V are considered as the variables; they are summarized
by the following three premises:
Premise A: paradigm of specific heat and latent heat

For a unit mass of a material, the added heat q is related to the changes in temperature and volume asdq  CV d  ΛD dV ,(13.5)rwhere CV is the specific heat for a constant volume and ΛD is the latent heat of expansion.
Carnot stated that “When a gaseous fluid is rapidly compressed, its temperature rises; and when, on the other hand,
it is rapidly expanded, there is a fall in temperature. This is one of best demonstrated of all pieces of experimental
evidence, and it will be taken as a starting-point for my proof”[C, p. 73]. and “if the gas is expanded, its temperature
can be prevented from falling if we supply to it an appropriate quantity of caloric. We shall call the caloric that isused in these circumstances, where the temperature remains constant, the caloric due to the change in volume(calorique dû au changement de volume)”[C, p. 74].
The former statement means that in the case of a rapid change, i.e., dq  0 , then d  0 for dV  0 and d  0
for dV  0 , hence ΛD CV  0 . The latter statement means that in isothermal heating, i.e., d  0 , then dV  0
for dq  0 , hence ΛD  0 . Therefore,ΛD  0 , CV  0 .
This restriction is postulated as Premise A.
Premise B: existence of the equation of state

For a gas, the existence of an equation of state, i.e., a relationship among P, , and V, is assumed. That is,38(20.1)

IHTC16-FL01
P  P ,V  .

(20.2)

In particular, for most ordinary gases, the Boyle‒Gay-Lussac law, namely
PR 1  V(20.2)'

holds, where  1  267C , R   PV  0C, P 1atm . As shown later in Fig. 20.3, this equation was used by Carnot
himself. Carnot regarded this equation as being applicable to all gases; hereafter, the gas expressed by eq. (20.2)'
is called an ʻideal gas.ʼ10
Premise C: conservation of heat

This means that a heat function exists as a state variable Q ,V  . In other words, dq in eq. (13.5) is a total
derivative, and the single-valued function

Q ,V    ,V  0 ,V0 dq(13.4)r

can be defined; in this case, dQ  dq holds.
In most of the books about the history of thermal science, Premise A (paradigm of specific heat and latent heat)
and Premise C (conservation of heat) are inseparably linked to each other. Although this is historically true, it is
not logically true. Premise A is not always associated with Premise C. Even if the conservation of heat does nothold (i.e., dq is not the total derivative), ex. (13.5) for dq is allowed; q   dq is obtained if the integration path isspecified for a quasi-static change. Or, one can understand that when the state changes along the curvef  (t ), V (t )  on the  ,V  plane, dq  CV dt  ΛDVdt can be determined [1].
Here, let us consider the condition required to realize a Carnot cycle.
In a Carnot cycle, the working substance isothermally expands when it comes in contact with the furnace oftemperature   , then it is cooled to   by adiabatic expansion; it is isothermally compressed when it comes incontact with the cooler of temperature   , then it is heated to   by adiabatic compression. Its P‒V diagram isshown in Fig. 20.1 (Fig. 19.4).
Therefore, dP   dP 

 
 0
 dV Q  dV (20.3)

is necessary. (The subscripts indicate the variables which are kept constant; Q means adiabatic and  means
isothermal.)10

In the following, the apostrophe is used to distinguish an equation for an ordinary gas (eq. (20.2)) and one
for an ideal gas (eq. (20.2)').

39

IHTC16-FL01

Fig. 20.1 Carnot cycle for ideal gas.

For an ideal gas,P
 dP 
  ,
V
 dV 

P R  d  dP     ,V V  dV  Q
 dV  Q
and from eqs. (13.5) and (20.1), when Q is constant dQ  dq  0 , d dV Q   ΛD CV  0 , hence eq. (20.3) isproved, which enables the realization of a Carnot cycle.11

II. Analytical expression for Carnotʼs theorem
In a Carnot cycle (Fig. 20.1), according to Carnot, the working substance receives heat q in the isothermal process
A → B, and the same amount of heat is discharged to the cooler. Associated with the decrease in heat, the volumeof the working substance changes and work W   PdV is generated. In general, the work for this case isexpressed as
W  W q ,  ,  ,VA ,VB  .
11

It seems that Clapeyron, who first illustrated the Carnot cycle on the P‒V plane, did not know Poissonʼsformula for an adiabatic process, eq. (16.15) PV   const. Actually, he wrote, that regarding curve BC, “itspressure decreases more rapidly according to an unknown law”, and regarding the curve DA, “the unknown law,of how the pressure varies when the volume of the gas is reduced”[2]. He, however, proved eq. (20.3), andtherefore could depict a qualitatively correct diagram.

40

IHTC16-FL01
VB

Strictly speaking, from eq. (13.5), for the isothermal process A → B, q   ΛD dV ; as a result, VB is determined
VA

by VA and q , and hence is not independent.
Here, Carnotʼs theorem demonstrates that the work generated by a Carnot cycle W is maximum when heat q
decreases from   to   irrespective of the properties of the working substance and the mode of operation.
Then, this theorem indicates that the work is independent of VA and VB .
Or, returning to Carnotʼs proof (Chapter 19-IV), we can state the following. First, we assume that for a Carnot
cycle in a different volume range (VA ‒ VB ) , more work is generated. That is, for the same q , W VA   W VA  .
In this case, first we operate a normal cycle starting from VA , and second by using part of the generated work
W VA  , we can operate the reverse cycle for VA . As a result, we can obtain work W VA   W VA  without any
change; however, this violates the principle that a perpetual engine is impossible, i.e., the third premise.
Consequently, W is independent of the volume range of the working substance.
Furthermore, it can be deduced that in a Carnot cycle the work produced by using the amount of heat nq is ntimes as great as W produced by using q . This is because Carnotʼs theorem includes the statement that a cyclewith heat nq , and n cycles, each with q , are equivalent. That is, W nq ,  ,    nW q ,  ,   . Hence,W q ,  ,    q .
From the above discussion, Carnotʼs theorem is expressed as follows. The maximum efficiency of a heat engine
operated between   and   is determined asW Ψ   ,   ,q

(20.4)

where Ψ is a universal function independent of the working substance; this is known as Carnotʼs theorem:
expression I. Here,   ,   ,Ψ   ,     0 ,Ψ   ,     0 .

Furthermore, Carnot considered the case of     ,      Δ , where Δ  is very small ( 0  Δ    ) as
shown in Fig. 20.2.
In this case, we let the produced work be ΔW and introduce the function1Ψ  ,   
.
lim 

      Θ    

(20.5)

Then, Carnotʼs theorem is expressed using a universal function Θ  asΔWΔ.
 Ψ  ,  Δ  qΘ  

(20.6)

This is known as Carnotʼs theorem: expression II. Here, we call this universal function Θ  the Carnot
function. The existence of this universal function is the key point of Carnotʼs theorem, and it is the ultimate basisof thermodynamics.

41

IHTC16-FL01

Fig. 20.2 Carnot cycle with infinitesimally small temperature difference.

It is often said that Carnot did not use mathematical expressions, but that is a misconception; he wrote the above
equation in a footnote (Fig. 20.3). As shown in the figure, Carnot expressed 1 Θ  as Ft and RΘ  as T.
Although the Carnot function is usually denoted by C   after Clapeyron or 1    after Thomson, I did not use
C   because it could be confused with specific heat. Actually, I wanted to use T   similarly to Carnot; however,to avoid the possible misunderstanding that T   is the absolute temperature, Θ  is used following
Takabayashiʼs book “History of Thermal Science” in Japanese (first edition, Nihonkagakusya, Kyoto, 1948;
second edition, Kaimeisya, Tokyo, 1999).

42

IHTC16-FL01

Fig. 20.3 Part of a footnote of Carnotʼs paper; N denotes R, r denotes W, 1 Ft  T N denotes Θ , e denotes q .

43

IHTC16-FL01
III. Experimental determination of Carnot function
The core of Carnotʼs theory is that there is the principle restriction on the work produced from heat. Theoretically,
it is reduced to the fact that the Carnot function Θ  is a universal function independent of the working substance.
Therefore, to examine whether Carnotʼs theory is correct, it is sufficient to directly determine Θ  for various
working substances and compare them. Thus, Carnot wrote, “In order to test our basic proposition, to test whether
the nature of the working substance that is used for developing motive power really has no bearing on the amountof power produced, we shall consider a number of substances in turn. These substances are air, steam, and alcohol
vapour”[C, p. 94].
As the first example, let us consider a cycle for a unit amount of air with an infinitesimally small temperature
difference ( Δ ), A → B → B' → A' → A (Fig. 20.2). In this case, the work done in the adiabatic processes (B→ B', A' → A)12 can be neglected because it is an infinitesimal higher-order quantity, and hence the net work
done by the air isP    P   Δ dV  V  P  ΔdV  P  Δ VB  VA  .
V

ΔW  

VB

VB

A

A

   V   V(20.7)

Here, as shown in Fig. 20.4, consider the intersection M of an adiabatic line through points A and A' and an isobaricline through point B (temperature   Δ 0 ). If we consider the curve AB where the relationships Δ  Δ 0 andΔ 0   1  267 C are satisfied, by using

Fig. 20.4 Explanation of Carnotʼs calculation.

VB  VM
1  V      V     VΔ


P
0
,


1
V
V
VA



M
      V    Q
VΔ 0


12

The difference between the work done to the outside in B → B' and that from the outside in A' → A.

44

(20.8)

IHTC16-FL01
the volume change is expressed as

VB  VA         VΔ 0 .
(20.9)(Here, since a small factor Δ 0 is included in the R.H.S., V in the R.H.S. can be VA , VB , and VM .)On the other hand, if we assume the conservation of heat (the existence of a heat function) along the adiabatic
process corresponding to MA, QA   QM  ; since MB is an isobaric process, the heat absorbed by the air inprocess AB is expressed by using the specific heat at constant pressure as follows:q  QB  QA   QB  QM   C P Δ 0 .

(20.10)

Actually, even if we do not assume the conservation of heat, in the process M → A → B → M,  dq is the second
order of a minutely small quantity; since we now only consider the first order of a minutely small quantity, thisresult is obviously correct.13
Therefore, the Carnot function is obtained asΘ   qCP Δ 0   Δ
Δ
CP
.


ΔW VB  VA P  V Δ       V P  VHere, if we assume that air is an ideal gas, P  V  R V   PV 0 V and      1  
we obtainΘ 0C  CP   0 PV 0.

 , then   0C ,1(20.11)

  0C , P  1 atm  10.4  10 3 kgw m 2 , V 0   0.77 m 3 for 1 kg of air, the measured value of Delaroche andBérard for the specific heat of gases of C P  0.267 C water  0.267 kcal C (Table 15.2),   1 267 C of GayLussac, and  0  1 116 C of Poisson ((eq.15.6))14, and obtainedΘ0C  

1  kcalC  .
1.39  kgw  m 

For later discussion, we rewrite the above result as follows:Θ0C  

 calC 
1 10 3  calC  .
  307 
1.39 9.8  J 
 4.185 J (A factor of 4.185 was introduced to account for knowledge obtained later. If we introduce the precise values ofV 0   0.773 m 3 , 1 atm  10.3  10 3 kgw m 2 , C P  0.240 kcal C ,   1 273C , and  0  1 109 C , weobtain13

I consider this fact to be self-explanatory. However, there is a paper stating that the result expressed in eq.
(20.10) is “surprenant” and needlessly studying the reason why it is correct [3].

Strictly speaking, although  0  1 116 C obtained by Poisson is the value at 7.5C , Carnot regarded it asconstant.
1445

IHTC16-FL01
Θ0C  

 calC 
1  kcalC  .
  273 
1.56  kgw  m 
 4.185 J 

Readers can anticipate what Θ  means.)
As the second example, Carnot considered a cycle in which 1kg of water at   100C is evaporated and thenliquefied by compressing it at   99C . In this cycle, both the temperature and pressure are constant in theevaporation and liquefaction processes, as shown as Fig. 20.5, which illustrates a Carnot cycle consisting of two
isothermal and two adiabatic processes. For m  1 kg , the variations in the volume and pressure are expressed as

ΔV  Vvapor  Vwater  Vvapor  1.7 m 3 ,ΔP  P 100C   P 99C   0.36  10 3 kgw m 2 ,q  ΛE m  550 kcal ,where ΛE denotes the latent heat of evaporation, the subscript E means evaporation, andΘ100C   q calC ΔΔ
1  kcalC  .
  384  ΛE mΔW
ΔV ΔP 1.112  kgw  m 
 4.185 J 

Fig. 20.5 Carnot cycle in the case of using steam.

As the third example, for 1kg of ethyl alcohol at its boiling point   78.8C , he obtainedΘ 78.8C   calC 
1  kcalC  .
  347 
1.230  kgw  m 
 4.185 J 

Simply by comparing values at different temperatures, however, it is impossible to clarify that Θ  is a universal
function for air, water, and alcohol. Carnot himself recognized that “It would have been desirable to be able to
make other similar comparisons, to calculate, for instance, the motive power developed by the action of heat on
solids and liquids, by the freezing of water, etc. But, at present, physics does not provide the information we needfor this purpose”[C, p. 100].

46

IHTC16-FL01
Clapeyron, who popularized Carnotʼs theory 10 years after it was published, also wrote, “The function C (Carnot
function Θ  ) is, as we have seen, of great importance: it is the common link between the phenomena caused byheat in solid bodies, liquids and. gases”[4], and focused on its importance. Following Carnot, Clapeyron tried todetermine the values of the function.
Clapeyron expressed the work for the cycle shown in Fig. 20.5 as
ΔW  V gas  Vliquid ΔP  V gas  Vliquid 

dP
Δ .
d

where P is the saturated vapor pressure (a function of only ). On the basis of the above equation, using q  ΛE m ,and applying Carnotʼs theorem eq. (20.6), Claperon obtained
V gas  Vliquid dP
ΔW
1
,



Θ  q Δ
mΛE
d

i.e.,
mΛE
ΛE
dP
1
1
,




d V gas  Vliquid Θ   v gas  vliquid Θ (20.12)where v gas  V gas m . This equation is called Clapeyronʼs foumula, which “was the first physico-chemical
application of what we now call the second law of thermodynamics”[5].
Using this formula, Clapeyron evaluated Θ  for water, ethyl alcohol, diethyl ether (sulfuric ether: boiling point
35 .5C ), and oil of turpentine (boiling point 156 .8C ). In Fig. 20.6, Clapeyronʼs results are plotted by open
triangles15, while Carnotʼs are plotted by open circles.

Fig. 20.6 Determination of Carnot function.
(open circles: Carnot, open triangles: Clapeyron, a straight line: Thomson)
15

Actually, Clapeyron obtained two kinds of values. Cf. Table 25.1 including the values evaluated by
Helmholtz (1821–1894).

47

IHTC16-FL01
From these limited results, it is obviously impossible to confirm the universality of the Carnot function. Clapeyron
himself stated, “Unfortunately there are no experiments which allow us to determine the values of this function at
all values of the temperature”[6].
Another person who focused on the importance of the Carnot function was William Thomson in 1849. “Thecomplete theoretical investigation of the motive, power of heat is thus reduced to the experimental determinationof the coefficient (  1 Θ  ); and may be considered as perfect, when, by any series of experimental researches
whatever, we can find a value of  for every temperature within practical limits. The special character of theexperimental researches, whether with reference to gases, or with reference to vapours, necessary and sufficientfor this object, is defined and restricted in the most precise manner, by the expressions (6) for , given above”[7].
For the sake of comparison, in Fig. 20.6, I included Thomsonʼs results as a straight line, which he evaluated Θ 
on the basis of Regnaultʼs measurements and Clapeyronʼs formula for   0 ‒ 230C . In view of the lower
accuracy of Carnotʼs and Clapeyronʼs evaluations, we can safely conclude that the agreement with the results ofThomson is reasonable [8].16

IV. Extension of Carnotʼs theory and gas theorem
To confirm the correctness of a basic theory such as Carnotʼs theorem, however, a direct proof is not necessarily
required. On the contrary, its extension, i.e., what is theoretically deduced from it, is more important.
Carnot wrote his paper with the aim of solving technological issues on thermal efficiency. Simultaneously,
however, he was well aware of the main theme of research, which at that time was dominated by the central
academy of France. He successfully demonstrated the effectiveness of his theory by applying it to the gas theory,
i.e., a central issue of the day, which attracted interest from French top academics.
In fact, he succeeded in deriving several theorems which could not be explained by the simple calorimetry of theday, although some of them were incorrect. In other words, for the first time, he demonstrated the power ofthermodynamics.
Later, Clapeyron wrote, “S. Carnot, avoiding the use of mathematical analysis, arrives by a chain of difficult andelusive arguments at results which can be deduced easily from a more general law which I shall attempt toprove”[9]. On the strength of this comment, it was later considered that Carnot did not use mathematics.
Furthermore, Truesdell, who liked mathematical formulation, made the strong criticism, “Among physicists of thefirst rank, Carnot is the first who was not in at least equal measure a mathematician”[10]. It is indeed true that
Carnotʼs paper is hard to read, and cannot avoid criticism for lacking in mathematics.
Actually, however, Carnot presented a mathematical discussion. Certainly, in the main text, he developed “a chainof difficult and elusive arguments” without mathematics, and only in the footnotes did he add very briefmathematical explanations; because of his poor description, he suffered an unreasonable amount of criticism.
Figure 20.3 shows part of one of his footnotes.
Here, to provide a clear understanding, let us consider Carnotʼs gas theory on the basis of his footnote.
The work done when a gas expands by dV is given by dW  PdV . In particular, for an ideal gas, it is expressed
as dW  R  1   dV V . As a result, the work done by an ideal gas in an isothermal process A → B is given by





16

The unit used by Thomson is converted as follows: 1  BTUC kcal  C
cal  C    lbw  ft  kgw  ft  334.8 J . Note that Thomson expressed lbw simply as lb.
 48

IHTC16-FL01



W     R  1  
VB

VA

V 
 dV
R   ln  .
V
V
1

B

 A

(20.13)

In a Carnot cycle with an infinitesimal temperature difference ( Δ ), the net work done isV 
ΔW  W    W   Δ   RΔ ln B  .
 VA 

(20.14)

The amount of heat q which the gas absorbed in the process A → B is given by this equation and Carnotʼstheorem: expression II (eq. (20.6)), and is expressed by using the Carnot function as follows:q  Θ V ΔW
 RΘ  ln B  ,Δ VA 

(20.15)where Θ  is a universal function, and hence q is identical irrespective of the kind of gas if we consider ideal
gases. Consequently, the following theorem is deduced. (The “gas” Carnot referred to was an ideal one.)Gas theorem 1“When a gas passes from a particular volume and pressure to another specified volume and pressure, without
undergoing a change in temperature, the amount of caloric absorbed or given out is always the same, whatever
the gas on which the observation is made”[C, p. 78].
Gas theorem 2“When the volume of a gas changes at a constant temperature, the amounts of heat absorbed or released by thegas will follow an arithmetical progression when the increases or decreases in volume follow a geometricalnprogression”[C, p. 83]. (This means that if q q 0 for VB VA   , then q  nq 0 for VB VA   .)Of course, this theorem can be deduced from the first law of thermodynamics and the fact that the internal energyof an ideal gas is only a function of temperature. At that time, however, these facts were not known. It is surprisingthat such a general relationship for q as eq. (20.15) can be obtained only from the constraint on the work generated
by heat (Carnotʼs theorem). It was four years later that Dulong (1785‒1838) experimentally found gas theorem
1. When Mayer (1814‒1878) proposed the equivalent conversion between heat and work, this fact was one of theimportant bases.
Furthermore, Carnot considered the case of the isobaric process M → B and the constant-volume process M → N
in Fig. 20.4. Assuming that heat is a state variable, he set
M → B: C P Δ 0  QB  QM  ,M → N: CV Δ 0  QN   QM  ,and thus obtained

C P  CV Δ 0  QB  QN  .

(20.16)

However, since the right hand side is the amount of heat added in the isothermal process N → B, if the gas isregarded as ideal, it is concluded from gas theorem 1 that the left hand side is independent of the type of gas. As
a result, the following theorem is obtained:
Gas theorem 3“The difference between the specific heat at constant pressure and the specific heat at constant volume is the samefor all gases”[C, p. 80], [11].

49

IHTC16-FL01
Although the proof was based on the conservation law of heat, as described earlier, it is correct because thediscussion was made for an infinitesimally small cycle.
Note that if we combine eqs. (20.15) and (20.16), because VM  VN , we obtain

C P  CV Δ 0  R ln VB  ,
 VM 

Substituting the first relationship of eq. (20.8), i.e.,
VB  1    Δ 0 VM
into the above equation gives   

1  V 1.
  1V    P   

Since   Δ 0  Δ 0 Δ 0 267C  1 , the right-hand side of the above equation becomesR  ln1    Δ 0   R   Δ 0 ,and finally we obtainC P  CV  RΘ     RΘ  .
 1  (20.17)This is the mathematical expression for gas theorem 3, which was first derived by Clapeyron [12].
Although Carnot derived various theorems, some of them were incorrect because he based them on erroneous
experimental data; thus, I will omit them here.
Clapeyron presented full mathematical expressions corresponding to Carnotʼs discussion on the basis of theexistence of the heat function as described below.
Let us consider a Carnot cycle with an infinitesimally small temperature difference and volume. In Fig. 20.2,
imagine that state B is infinitesimally close to state A. By introducing VB  VA  ΔV in eq. (20.7), the net work is
expressed as P 
ΔW  ΔV ΔP   Δ  ΔV .
  V(20.18)

In particular, for an ideal gas, for which P  V  R V ,ΔW  R V Δ  ΔV .

(20.18)'At this point we assume the existence of the heat function Q  Q  , V  . Although Clapeyron expressed it as P,V  ,let us write it as  , V  for convenience in the following discussion. In this case, the amount of heat absorbed in
the isothermal process A → B is q   ΔQ   Q V  ΔV . Therefore, substituting it into Carnot theorem:
expression II, we obtain

50

IHTC16-FL01
ΔW P  V Δ ΔVΔ,Q V  ΔV Θ qwhich, for an ideal gas, leads toR P 
 Q   Θ   .
  Θ  V   V V  
By integrating this equation, the heat function for an ideal gas is obtained as follows:(20.19)Q ,V   f    RΘ lnV ,where f   is an unknown function of temperature. As a result, the amount of heat absorbed in the isothermal
process A → B is given byq  Q ,VB   Q ,VA   RΘ  lnVB VA  .

(20.20)This is simply the eq. (20.15). Furthermore, if we consider the difference between the following two equations:df  dΘ  R
 V 
 Q 
ln V  Θ  
CP  
R
 ,
 
d
d
V
   P   P
df  dΘ   Q CV  
R
ln V , dd   V
RV V 
, we again arrive at eq. (20.17). Thus, gas theorems 1, 2, and 3 have been proved
using    1   P P   anew.

V. Further discussion and significance of thermodynamics
As shown above, Clapeyron proved the gas theorems firmly on the basis of the existence of the heat function.
Consequently, after the conservation of heat was later rejected, these theorems were often regarded as having
been obtained as a result of multiple errors cancelling each other out. On the other hand, a contrived
interpretation in which the “caloric” referred to by Carnot should be regarded as entropy was historically
proposed [13].
As stated earlier, however, Premise A: paradigm of specific heat and latent heat and Premise C:
conservation of heat are different; most of Carnotʼs theory was obtained from Premise A, and thus Premise
C is not necessary.
Even if the conservation of heat does not hold, provided we consider an infinitesimally small cycle,

 dq  infinitesimal quantity of the second order  0 ,thermodynamically correct results are often derived in practice.
Most of Carnotʼs mistakes are based on erroneously measured data relating to the pressure dependence ofspecific heat. Thus, using only Premise A, let us reconstruct Carnotʼs theory.
Consider a Carnot cycle with an infinitesimally small temperature difference and volume change. From
Premise A, i.e., eq. (13.5),

51

IHTC16-FL01q   Δq   ΛD ΔV .
As described earlier, even if dq in eq. (13.5) ( dq  CV d  ΛD dV ) is not total a derivative, for a quasi-static change,
the integration along a path can be expressed as  Δq path . On the other hand, since ΔW is given by eqs. (20.18)
and (20.18)', from Carnotʼs theorem: expression II
ΔW P  V ΔΔV
Δ


,
q
ΛD ΔV
Θ 

we obtain P 
ΛD  Θ   .
   V(20.21)

In particular, for an ideal gas, the following important relation is derived:ΛD  Θ  R.
V

(20.21)'

According to Truesdell, eqs. (20.21) and (20.21)' deserve to be called the general Carnot‒Clapeyron
theorem [14].
From this, the amount of heat absorbed during an isothermal change is given byVBV q   ΛDdV  RΘ ln B  ,VA
 VA 

(20.22)

which is identical to eq. (20.15).
If the heat function exists, these two equations are, of course, the same as eqs. (20.19) and (20.20), respectively.
Yet even if the conservation of heat does not hold and eqs. (20.19) and (20.20) are not available, eqs. (20.21)
and (20.22) are correct because they were proved by using an infinitesimally small cycle. (This point is
explained in detail in Chapter 27-IV.)
Carnot wrote, “What is the cause of the difference between the specific heats at constant volume and at constantpressure? It is the caloric required to bring about expansion in the latter case”[C, p. 86]. That is, from eq. (13.5),
 V 
C P Δ   Δq P  CV Δ   ΛD 
 Δ ,
   P V 
C P  CV  ΛD  ,   P(20.23)

In particular, for ideal gases,
C P  CV  ΛD RV.
 ΛD 1P (20.23)'This is obtained only from Premise A: paradigm of specific heat and latent heat. If Clapeyronʼs theorem (eq.
(20.21)) is substituted into eq. (20.23), we obtain

52

IHTC16-FL01
 P   V 
C P  CV  Θ   .
    V    P(20.24)For ideal gases, it goes without saying that this is identical to eq. (20.17).
Additionally, if we apply the definition ΛD   Δq ΔV  , to the evaporation of water ( is constant during
evaporation), we obtain
mΛE
.
ΛD 
Vgas  Vliquid
On the other hand, in the generalized form of the Carnot‒Clapeyron theorem (eq. (20.21)), the saturation vapor
pressure is only a function of temperature, and we obtainΛD  Θ  

dP
.
d

From these two equations, we also arrive at Clapeyronʼs formula (eq. (20.12)); this was pointed out by
Clapeyron in 1864 [15].
Further, if we denote  0    in eq. (20.11), we obtainC P      PV 0 Θ0       R Θ0 .
On the other hand, from eq. (20.17),

C P  CV  R Θ0 .

In addition to these two equations, if we take data measured by Gay-Lussac and the calculated values ofPoisson, the ratio of the heat capacities becomes 

C P    267  116
1
,



CV

267
0.70

(20.25)which was shown by Carnot in the Réflexions. (This result is readily obtained if we set   0 and k    1 ineq. (15.4),
1  dV 
1  d 1

.
(15.4)r)      V  d  Q   d  Q k 1  
The above comprises all the results and extensions of Carnotʼs theory for the present moment.
Note that Carnotʼs theory is crucially different from the previous caloric theory. The significance of Carnotʼstheory—thermodynamics—is as follows.
For example, if we compare the equation of state for a gas eq. (20.2.2)', i.e., the Boyle‒Gay-Lussac law, with
the Carnot‒Clapeyron theorem (eq. (20.21)) and Clapeyronʼs formula (eq. (20.12)), we find an essential
difference: even if the equation of state PV  R  1    seems to be valid for various gases, it is only
approximately valid, and, strictly speaking, the functions and constants vary depending on the gas. That is, thestatement is true only within the measurement accuracy.
On the other hand, Carnotʼs statement expressed by eqs. (20.12), (20.21), and (20.24) means that although thelatent heat Λ and specific heat C may depend on the gas, the relationship based on the universal function Θ 

53

IHTC16-FL01
is absolutely correct for all gases; this is the statement of Carnot, namely of thermodynamics. Because of thisfeature, thermodynamics is called ʻthe physics based on the principles.ʼ Truesdell wrote, “Carnot was the first to
see that a theory connecting heat and work imposes restrictions upon the constitutive function of bodies (, C,
P, etc.). The discovery of such restrictions has been the essence of thermodynamics from his day to ours”[16].
That is, only eqs. (20.12), (20.21), and (20.24) are called thermodynamic equations.

As shown above, Carnot can be credited with the birth of thermodynamics. His theorem (eqs. (20.4) and (20.6))
gave the expressions clarifying the quantitative relationship between heat and work performed by heat.
Twenty years later, Mayer and Joule found another quantitative relationship between heat and work (the first lawof thermodynamics). Then, by unifying these two relationships which at first glance seemed to be incompatible,
the basis of thermodynamics was established.
In the next chapter, prior to describing the work of Mayer and Joule, we will survey the present state of the kinetic
theory of heat.

References in chapter 20[1] Truesdell, C., The Tragicomical History of Thermodynamics 1822-1854, New York, Heidelberg, Berlin: Springer-Verlag, 2C,(1980).
[2] Clapeyron, É., “Memoir on the Motive Power of Heat,” (1834), in Reflections on the Motive Power of Fire by SADI CARNOT
and other Papers on the Second Law of Thermodynamics by É. Clapeyron and R. Clausius, translated and edited by Mendoza, E.,
New York: Dover Publications, pp. 76-77, (1960).
[3] Hoyer, U., “La Théorie de Carnot. Première et seconde approximations de la thermodynamique,” Sadi Carnot et lʼEssor de laThermodynamique, Paris: Centre National de la Recherche Scientifique, p. 224, (1976).
[4] Clapeyron, É., ibid., p. 101, (1834/1960).
[5] Lewis, G.N. & Randall, M., Thermodynamics and the Free Energy of Chemical Substances, New York and London: McGrawHill Book Company, p. 134, (1923).
[6] Clapeyron, É., ibid., p. 97, (1834/1960).
[7] Thomson, W. (Lord Kelvin), “An Account of Carnotʼs Theory of the Motive Power of Heat: with Numerical Results deduced
from Regnaultʼs Experiments on Steam,” (1849), Mathematical and Physical Paper of William Thomson, Cambridge: Cambridge
University Press, vol. 1, p. 135, (1882/2011), https://archive.org/details/mathematicaland01kelvgoog [accessed May 17, 2018].
[8] Thomson, W. (Lord Kelvin), ibid., (1849), p. 139, (1882/2011).
[9] Clapeyron, É., ibid., p. 74, (1834/1960).
[10] Truesdell, C., ibid., p. 136, (1980); Cf.
Truesdell, C., ibid., p. 80, (1980).
[11] Note that “gas theorems 1, 2, and 3” are renamed by Yamamoto. Carnot explained them in the order of 1, 3, and 2.
[12] Clapeyron, É., ibid., p. 88, (1834/1960).
[13] La Mer, V., “Some Current Misinterpretation of N. L. Sadi Carnotʼs Memoir and Cycle,” American Journal of Physics, vol. 22,
pp. 20-27, (1954);
La Mer, V., “Some Current Misinterpretation of N. L. Sadi Carnotʼs Memoir and Cycle. II,” American Journal of Physics, vol.
23, pp. 95-102, (1955);
Hirshfeld, M.A. “On “Some Current Misinterpretations of Carnotʼs Memoir”,” American Journal of Physics, vol. 23, pp. 103-105,(1955);
Bernal, J.D., Science and Industry in the Nineteenth Century, London: Routledge & Kegan Paul Ltd. p. 47, (1953);
Bernal, J.D., The Extension of Man: A History of Physics before the Quantum, Cambridge: The M.I.T. Press, p. 306, (1972);
Criticisms on the above references are as follows:
Kuhn, T.S., “Carnotʼs Version of “Carnotʼs Cycle”,” American Journal of Physics, vol. 23, pp. 91-95, (1955a);
Kuhn, T.S., “La Merʼs Version of Carnotʼs Cycle,” American Journal of Physics, vol. 23, pp.387-389, (1955b);
Elkana, Y., The Discovery of the Conservation of Energy, London: Hutchinson Educational, p. 180f., (1974);
Truesdell, C., ibid., p. 81, n. 5, p. 129, n. 3, (1980);
The brief description on the development of this controversy is given in
Kerker, M., “Sadi Carnot,” Scientific Monthly, vol. 85, p. 148f., (1957).
[14] Truesdell, C., ibid., p. 111, (1980);
Truesdell, C. and Bharatha, S., The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Rigorously
Constructed upon the Foundation Laid by S. Carnot and F. Reech, New York, Heidelberg, Berlin: Springer-Verlag, p. 73, (1977).
[15] Clausius, R., edited by Archer Hirst, T., The Mechanical Theory of Heat, with Its Applications to the Steam-engine and to the
Physical Properties of Bodies, London: John van Voorst, p. 48, footnote, (1867),
Publications from https://en.wikipedia.org/wiki/Rudolf_Clausius [accessed May 17, 2018].
[16] Truesdell, C., ibid., p. 113, (1980); Cf.
Truesdell, C., ibid., p. 134f., (1980).

54