Sadi Carnot

How to use this course

Sadi Carnot published a single short book, Réflexions sur la puissance motrice du feu (Reflections on the Motive Power of Fire), in 1824. Almost nobody read it for twenty years. Yet it contains the reversible cycle, the principle that no engine can beat a reversible one, and the conclusion that the work obtainable from heat depends only on the temperatures between which it flows. These are the core of the second law of thermodynamics. Carnot reached them while believing that heat was an indestructible fluid, caloric. Much of this course is about how a correct law came out of a wrong theory, and how Rudolf Clausius and William Thomson later separated the two.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Part II follows the order of the Réflexions: the question, the cycle, the principle, the temperature dependence, gases, vapours, and finally the notes in which Carnot abandoned caloric. Modern notation is used throughout, with Carnot’s own formulation given where it differs. For his father’s mechanics, which shaped his reasoning, see the course on Lazare Carnot.

Timeline

The Luxembourg and the lycée, 1796–1812

Nicolas Léonard Sadi Carnot was born on 1 June 1796 in the Petit Luxembourg palace in Paris. His father, Lazare Carnot, was then one of the five Directors who governed France, three years after he had organized the armies of the Republic. An elder brother, also called Sadi, had died in infancy. Lazare named both boys after Saadi, the thirteenth-century Persian poet of Shiraz, whose moral verse he admired. A younger brother, Hippolyte, was born in 1801.

The family’s fortunes followed Lazare’s politics. He fell from the Directory in the coup of 1797 and fled abroad, returned under the Consulate, served briefly as minister of war, and then opposed Napoleon’s accession to the Empire. From 1807 he lived in retirement, and he used the time to teach his sons himself: above all mathematics and physics. Lazare was the author of the first general theory of machines, and his central idea, that a machine works best when its motions are continuous and could be reversed, would reappear in his son’s book.

Sadi then prepared for the entrance examination of the École polytechnique at the Lycée Charlemagne in Paris.

Polytechnique, Metz and the engineer corps, 1812–1818

Carnot entered the École polytechnique in 1812, at 16, the minimum age, ranked 24th of 179 candidates. The school, which his father had helped to found, was the centre of French science. Among his teachers were Siméon Denis Poisson, André-Marie Ampère and François Arago, and the geometer Michel Chasles was a classmate. Carnot rose to 20th at the end of the first year and graduated 10th in 1814.

The Empire was collapsing. In March 1814, as the allied armies reached Paris, the students of the École polytechnique were mobilized, and Carnot fought with them in the defence of Vincennes, east of the city. It was his only battle. Later that year he entered the École d’application at Metz, the school for military engineers, as a student sub-lieutenant, for the two-year course.

During the Hundred Days of 1815 Lazare served Napoleon as minister of the interior. After Waterloo the restored monarchy proscribed him, and he went into exile, first to Warsaw and then to Magdeburg in Prussia, where he spent the rest of his life. The name Carnot now counted against a young officer. Sadi was posted from fortress to fortress, inspecting fortifications and writing reports that his superiors seem to have ignored. In September 1818 he took six months’ leave to prepare for the examination of the new General Staff Corps in Paris.

Paris on half pay, 1819–1824

Carnot passed the examination and joined the General Staff Corps in January 1819 as a lieutenant. Almost at once he went on leave at half pay, which allowed him to live in Paris, in his father’s former apartment, while remaining available for duty. He used the next eight years to educate himself in the sciences and arts of industry. He attended lectures at the Sorbonne and the Collège de France, and at the Conservatoire des arts et métiers he followed the chemistry course of Nicolas Clément and the political economy of Jean-Baptiste Say. He visited factories and workshops, and studied the economics of French and British industry.

Clément and his father-in-law Charles Desormes became his closest scientific friends. They were industrial chemists who had measured the ratio of the specific heats of air in 1819 and had written on the theory of steam engines. From them Carnot took precise data on steam, and probably the conviction that the work of an engine is governed by temperature differences rather than by pressure.

The question that held him was practical and national. Britain’s wealth rested on steam engines, transformed since the 1770s by the separate condenser and expansive working of James Watt; France had coal but used it poorly. Engines were improved by trial and error, and nobody could say whether steam was the best working substance, whether higher pressures helped, or whether there was any limit to the work a given amount of fuel could yield. His father died in exile at Magdeburg on 2 August 1823. Within a year his son had published the answer.

The problem of the heat engine

For the general reader. By 1824 steam engines pumped mines, drove mills and moved ships, but nobody understood why they worked as well as they did, or how much better they could be made. Carnot set aside the details of boilers and valves and asked a general question. Given some heat, how much work can it produce, in any engine, with any working substance? His key observation was that heat produces work only when it passes from a hot body to a colder one.

The question

Carnot measured work, which he called motive power, as a weight raised through a height, in units of a cubic metre of water raised one metre. A heat engine is any device that takes heat from a hot source, such as a furnace, and produces work. He noticed that every engine also rejects heat to a cold body, the condenser or the atmosphere. In the caloric theory of heat, taught by Antoine Lavoisier and Pierre-Simon Laplace, heat was a weightless fluid that could be neither created nor destroyed. In that picture the engine borrows caloric from the furnace and returns all of it to the condenser. What it produces is a change of level, from high temperature to low.

The waterfall

Carnot compared heat falling from a high temperature to water falling from a height. The power of a waterfall depends on the quantity of water and the height of the fall; he proposed that the power of heat depends on the quantity of caloric and the fall of temperature. He was careful to say that he did not know whether the work was proportional to the temperature difference, or whether a fall from 100° to 50° gave the same work as a fall from 50° to 0°. That question is the subject of Lesson 07.

In modern terms the analogy is half right. Energy conservation (the first law) says that the heat rejected \(Q_c\) is less than the heat taken in \(Q_h\), and the difference is the work: \(W=Q_h-Q_c\). Heat is not conserved. But the quantity \(Q/T\), the entropy, does play the role of Carnot’s caloric in a reversible engine (Lesson 10).

Efficiency

The thermal efficiency of an engine is \(\eta=W/Q_h\). In 1824 good engines converted only a few percent of the heat of their fuel into work. Carnot’s book was an attempt to show what limited this figure and how it could be raised.

The Carnot cycle

For the general reader. To compare all possible engines, Carnot imagined an ideal one. Its working gas exchanges heat only with bodies at its own temperature, and changes temperature only by expanding or being compressed, never by contact with something hotter or colder. Such an engine wastes nothing, and it can run backwards. Every textbook on thermodynamics still begins with it.

Carnot’s description

Carnot considered a gas in a cylinder under a piston, and two large bodies: \(A\), kept at a high temperature, and \(B\), kept at a low one. In modern order:

1. Isothermal expansion. In contact with \(A\), the gas expands slowly at the temperature \(T_h\) of \(A\), absorbing heat \(Q_h\).
2. Adiabatic expansion. Removed from \(A\), the gas expands further without exchange of heat, cooling until it reaches the temperature \(T_c\) of \(B\).
3. Isothermal compression. In contact with \(B\), the gas is compressed at \(T_c\), giving up heat \(Q_c\).
4. Adiabatic compression. Removed from \(B\), the gas is compressed without exchange of heat until it returns to \(T_h\) and its initial state.

Expansion happens at higher pressure than compression, so the gas does more work on the piston than the piston does on the gas. Carnot did not draw a diagram; the pressure–volume picture, in which the cycle is bounded by two isotherms and two adiabats and the work is the enclosed area, was added by Émile Clapeyron in 1834.

Derivation

On an isotherm the internal energy of an ideal gas is constant, so the heat absorbed equals the work done: \(Q=\int p\,dV=nRT\ln(V_\text{final}/V_\text{initial})\). On an adiabat \(TV^{\gamma-1}\) is constant. Let the four corners be \(V_1,V_2,V_3,V_4\). The two adiabats give

Hence \(Q_c=nRT_c\ln(V_3/V_4)=nRT_c\ln(V_2/V_1)\). The work on the two adiabats cancels, since each equals \(\pm nC_V(T_h-T_c)\). The ratio \(Q_c/Q_h=T_c/T_h\) follows.

Reversibility

Every step can be run in reverse by an infinitesimal change in the pressure on the piston. Run backwards, the cycle consumes work \(W\), takes \(Q_c\) from \(B\) and delivers \(Q_h\) to \(A\). The condition that makes this possible is that heat never passes between bodies at different temperatures. Carnot stated it as a design rule: any contact between bodies of different temperature, without a change of volume, is a pure loss of motive power. This is the thermal analogue of his father’s rule that a machine should transmit motion without shock.

Carnot’s principle

For the general reader. Carnot proved that no engine working between two given temperatures can do better than his reversible one, and that all reversible engines between those temperatures do equally well, whatever substance they use: air, steam, alcohol. The proof is a thought experiment. If a better engine existed, it could be coupled to a reversed Carnot engine to make work from nothing.

Carnot’s proof

Suppose an engine \(E\) obtains more work than the reversible engine \(R\) from the same fall of a given quantity of caloric from \(A\) to \(B\). Run \(E\) forwards, and use part of its work to drive \(R\) backwards, which restores all the caloric from \(B\) to \(A\). The reservoirs are back where they started, and work is left over. Repeating the cycle gives unlimited motive power without consuming anything. Carnot rejected this as contrary to the accepted principles of mechanics and of sound physics. Part (b) follows by applying (a) in both directions to two reversible engines.

The modern proof

Carnot’s proof relies on caloric being conserved, which is false. Clausius and Thomson kept the structure and changed the premise. The second law, in the form Thomson gave it in 1851, states:

Now suppose \(\eta_E>\eta_R\). Adjust \(E\) so that it takes the same heat \(Q_h\) from \(A\) as \(R\) delivers to \(A\) when reversed. Then \(E\) produces \(W_E=\eta_EQ_h\), and \(R\) reversed consumes \(W_R=\eta_RQ_h<W_E\). The hot reservoir is unchanged. By energy conservation the net work \(W_E-W_R>0\) has been drawn as heat from \(B\) alone, contradicting Kelvin’s statement. So \(\eta_E\le\eta_R\).

Consequence

Since all reversible engines between temperatures \(t_1\) and \(t_2\) share one efficiency, there is a universal function \(\eta(t_1,t_2)\), independent of any material. Carnot saw this as the central result: the motive power of heat is fixed solely by the temperatures of the bodies between which the caloric passes. For an ideal gas Lesson 05 computed \(\eta=1-T_c/T_h\), so this is the formula for every reversible engine. Carnot himself could not write it down, because he lacked both the first law and an absolute temperature scale.

Carnot’s function and absolute temperature

For the general reader. Carnot asked whether a degree of temperature fall is worth the same amount of work at every temperature. From the scattered data of his day he computed the work obtainable from a one-degree fall of heat with air, steam and alcohol, and found that it decreased slowly as the temperature rose. His numbers are close to the modern answer, which says the work per degree is inversely proportional to the absolute temperature. William Thomson used this idea to define a temperature scale independent of any thermometer.

Carnot’s function

Consider a reversible engine working between \(t\) and \(t-dt\), on any temperature scale. By Carnot’s principle its work is proportional to the heat taken in and to \(dt\), with a coefficient depending only on \(t\):

The function \(\mu(t)\), which Thomson later called Carnot’s function, is universal. Carnot proposed that the fall of caloric produces more motive power at lower temperatures than at higher ones, so \(\mu\) decreases with \(t\).

Proof. For a reversible engine between \(T\) and \(T-dT\), Lesson 05 gives \(\eta=1-(T-dT)/T=dT/T\), so \(dW=Q\,dT/T\).

Carnot’s numbers

Carnot estimated the work per unit of heat for a one-degree fall in several cases. J. Güémez, C. Fiolhais and M. Fiolhais (2002) recomputed his efficiencies as pure numbers. The table compares them with the modern value \(1/T\).

At equal temperatures alcohol and water agree to within a few percent, as Carnot’s principle requires, and the values fall as the temperature rises. The air figure is less good, because Carnot’s calculation for air used assumptions about specific heats that later proved wrong (Lesson 08).

Thomson’s absolute scales

In 1848, still accepting caloric, Thomson proposed a scale on which a unit of heat falling through one degree gives the same work at every temperature, so that \(\mu\) is constant. After accepting energy conservation, Thomson and James Prescott Joule defined in 1854 the scale now used, on which \(Q_c/Q_h=T_c/T_h\) for reversible engines. The second scale is the kelvin scale.

Gases, adiabatic heating and air engines

For the general reader. Compress air quickly and it gets hot; let it expand and it cools. Carnot used these facts, then poorly measured, to draw general conclusions about gases from his principle. Some of his conclusions are correct and now standard. One rested on caloric and turned out wrong. He ended with practical advice: use the widest possible temperature range, and consider engines that burn fuel inside compressed air.

Two theorems on gases

Modern versions: for an ideal gas, (1) is \(Q=pV\ln(V_2/V_1)\), the same for any gas, logarithmic in the ratio. (2) is \(C_p-C_V=nR=pV/T\), which Julius Robert Mayer used in 1842 to estimate the mechanical equivalent of heat. Carnot derived both from his principle by imagining small cycles with two different gases.

The error

Carnot also concluded, using caloric, that the specific heat of a gas per unit mass increases in arithmetic progression as its volume increases in geometric progression. He tabulated it for air, from 1.84 times the atmospheric value at 1/1024 atmosphere down to 0.16 at 1024 atmospheres, and doubted the extremes himself. For an ideal gas the specific heat per unit mass does not depend on pressure at all. Henri Victor Regnault’s later measurements confirmed this. The error came from treating heat as a conserved quantity of the gas.

Adiabatic heating

For a reversible adiabatic change of an ideal gas, \(TV^{\gamma-1}\) is constant, so for a small change

Carnot took from the theory of sound, as corrected by Laplace, the datum that air compressed by 1/116 of its volume warms by about one degree.

Practical conclusions

From the principle Carnot drew engineering advice. The work depends on the temperature fall, so an engine should take in heat as hot as possible and reject it as cold as possible. Steam at high temperature requires dangerous pressures; air does not, because its pressure and temperature are not tied together. He proposed heating air by burning fuel within it, after compressing it with a pump. Rudolf Diesel cited Carnot when he designed the compression-ignition engine in the 1890s, with the aim of approaching the Carnot cycle.

Vapours and the Clapeyron equation

For the general reader. Water boils at a lower temperature on a mountain because the air pressure is lower. How fast does the boiling point fall? Carnot’s method answers this. Run his cycle with water and steam instead of a gas: evaporate a little water at one temperature, condense it at a slightly lower one. Carnot used this cycle to compute his function from steam data. Clapeyron turned it into an equation in 1834, and Clausius corrected it in 1850.

The small cycle

Take unit mass of water in equilibrium with its vapour at temperature \(T\) and vapour pressure \(p(T)\). Evaporate it entirely at \(T\) and \(p\): the volume grows from \(v_l\) to \(v_g\) and heat \(L\) (the latent heat) is absorbed. Lower the temperature by \(dT\) adiabatically, so the pressure drops by \(dp\); condense at \(T-dT\); return. In the \(p\)–\(V\) plane the cycle is a thin parallelogram of width \(v_g-v_l\) and height \(dp\).

Clapeyron wrote the first form with Carnot’s unknown function. Clausius identified \(\mu=1/T\) (with \(J\)) and so obtained the second. Read in reverse, the equation gives \(\mu\) from measured \(L\), \(v_g\) and \(dp/dT\), which is essentially how Carnot found his steam values in Lesson 07.

The integrated form

Neglect \(v_l\), treat the vapour as an ideal gas \(v_g=RT/(Mp)\), and take \(L\) constant:

From caloric to energy and entropy

For the general reader. After 1824 Carnot kept thinking. His surviving notes show that he came to doubt caloric, decided that heat is a form of motion, and estimated how much work a unit of heat is worth, years before the experiments of Joule and Mayer. He published none of this. Twenty-five years later Clausius showed that Carnot’s principle survives the death of caloric if one adds energy conservation, and that the quantity Carnot thought was conserved was entropy.

The notes

Hippolyte Carnot published extracts from his brother’s notes in 1878. In them Sadi wrote that heat is simply motive power, or motion that has changed form, and that motive power is never created or destroyed but only changes form. He proposed experiments to test this: measuring the heat produced by friction in metals and by stirring water, and the heat produced by compressing gases. He gave a value for the motive power equivalent to a unit of heat, about 370 kilogram-force metres per kilocalorie.

Clausius’s synthesis (1850)

Clausius accepted Joule’s equivalence of heat and work, \(W=Q_h-Q_c\), and kept Carnot’s principle as a separate law: heat cannot by itself pass from a colder to a hotter body. Carnot’s proof then went through with the new premise (Lesson 06), and his efficiency became \(\eta=1-T_c/T_h\). In 1854 and 1865 Clausius extended the analysis to arbitrary cycles.

Caloric as entropy

In a reversible Carnot engine \(Q_h/T_h=Q_c/T_c\): the entropy taken from the hot body equals the entropy given to the cold body. So the quantity that Carnot’s argument needs to be conserved is entropy. Hugh Longbourne Callendar pointed this out in 1911. Reread with this substitution, much of the Réflexions is correct for reversible processes: the work of an engine is the entropy it carries times the fall in temperature, \(W=S\,(T_h-T_c)\), as in the waterfall analogy, with \(S\) for the quantity of water and \(T\) for the height. In irreversible processes entropy is produced, which has no caloric analogue.

Silence, illness and death, 1824–1832

The Réflexions appeared on 12 June 1824, published by Bachelier at Carnot’s own expense in 600 copies, a book of 118 pages with five figures and almost no mathematics. Hippolyte had read it in draft and urged his brother to make it accessible to a wide public. On 26 July the engineer Pierre-Simon Girard gave a favourable report on it to the Academy of Sciences, later printed in the Revue encyclopédique. Clément mentioned it in a lecture in 1825. There the response stopped. Engineers found it too abstract, physicists found it about engines, and its author did not promote it.

He also left an unpublished manuscript seeking a formula for the motive power of steam. In 1827 he was recalled to active service with the rank of captain, and after about fifteen months of duty, at Lyon and Auxonne, he resigned in 1828 to return to his studies in Paris. A directory of former polytechnicians from that year lists him as a builder of steam engines, though no patent or machine of his is known.

He was a liberal in politics, as his father had been, and welcomed the July Revolution of 1830. He played the violin and read Pascal, Molière and La Fontaine. He never married. His notes from these years, the source of Lesson 10, show him moving beyond caloric.

In June 1832 he fell seriously ill, apparently with scarlet fever, followed by a mental collapse. On 3 August he was admitted to a private asylum at Ivry, outside Paris, directed by the alienist Jean-Étienne Esquirol. Cholera was then sweeping Paris. The asylum register records that he was cured of his mania and died of cholera, on 24 August 1832. He was 36. Most of his belongings were burned, as was the practice after cholera, and few of his papers survive. He was buried in the old cemetery of Ivry.

Rediscovery and legacy

In 1834 Émile Clapeyron, an engineer and fellow polytechnicien, published a memoir in the Journal de l’École polytechnique that restated Carnot’s theory in analytic form, with the pressure–volume diagram and the equation of Lesson 09. Translations appeared in English in 1837 and German in 1843. In Paris in 1845 William Thomson read Clapeyron, but could not find Carnot’s book; he obtained a copy only in 1848. That year he proposed his first absolute temperature scale, and in 1849 he published An Account of Carnot’s Theory. His brother James Thomson used Carnot’s reasoning to predict the lowering of the freezing point of water by pressure.

Thomson saw the conflict between Carnot and Joule but could not resolve it. Clausius, who knew Carnot through Clapeyron, did so in 1850, and Thomson reached the same synthesis in 1851. William Rankine worked in parallel. Out of their work came the two laws of thermodynamics and, in 1865, entropy. The Réflexions was reprinted in the Annales scientifiques de l’École normale supérieure in the early 1870s, and in 1878 Hippolyte reissued it with a memoir of his brother and extracts from the notes. Robert Henry Thurston’s English translation followed in 1890.

Legacy

• Physics: the reversible cycle, the efficiency limit, the universality of the efficiency function and the idea of an absolute temperature are the core of classical thermodynamics. Clausius, Kelvin, Josiah Willard Gibbs and Ludwig Boltzmann all built on his principle.
• Engineering: the Carnot efficiency remains the benchmark for power stations, engines, refrigerators and heat pumps; the advice to raise the upper temperature guided the development of high-pressure steam, gas turbines and the Diesel engine.
• Method: reasoning about all possible processes from the impossibility of perpetual motion, rather than from a model of matter, became the style of thermodynamics.
• Honours: the lunar crater Carnot (named 1970) and the minor planet 12289 Carnot carry his name. His nephew Marie François Sadi Carnot, Hippolyte’s son, was President of France from 1887 to 1894.

Connections

People linked to Sadi Carnot across this collection. A name without a link has no page here yet. See Sadi Carnot in the network of all 337 lives →

Teachers and advisors

• André-Marie Ampère 1812-1814
• François Arago 1812-1814
• Siméon Denis Poisson 1812-1814

Influenced by

• Antoine Lavoisier
• James Watt
• Pierre-Simon Laplace

Influenced

• Rudolf Clausius 1850-1865
• William Thomson, Lord Kelvin 1848-1851
• James Thomson 1849
• Rudolf Diesel
• Émile Clapeyron 1834

Friends

• Michel Chasles
• Charles Desormes 1819-1832
• Nicolas Clément 1819-1832

Family

• Lazare Carnot
• Hippolyte Carnot

Also mentioned in this life

• Ludwig Boltzmann
• Enrico Fermi
• Josiah Willard Gibbs
• James Prescott Joule
• Julius Robert Mayer
• Blaise Pascal
• William P. Thurston

Fields

• thermodynamics 12 lives
• military engineering 3 lives

On CoursesHub

• Thermodynamics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Nicolas Léonard Sadi Carnot, MacTutor History of Mathematics.
2. Nicolas Léonard Sadi Carnot, Wikipedia; Sadi Carnot (physicien), Wikipédia (French).
3. H. Carnot, Notice biographique sur Sadi Carnot, in S. Carnot, Réflexions sur la puissance motrice du feu, Gauthier-Villars, 1878.
4. C. C. Gillispie and R. Pisano, Lazare and Sadi Carnot: A Scientific and Filial Relationship, Springer, 2014.
5. R. Fox, introduction and critical edition of the Réflexions, Vrin, 1978; English edition, Manchester University Press, 1986.

Physics

1. S. Carnot, Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance, Bachelier, Paris, 1824.
2. S. Carnot, Reflections on the Motive Power of Heat, trans. R. H. Thurston, 1890 (Project Gutenberg).
3. Reflections on the Motive Power of Fire, Wikipedia.
4. É. Clapeyron, Mémoire sur la puissance motrice de la chaleur, Journal de l’École polytechnique 14 (1834).
5. W. Thomson, An account of Carnot’s theory of the motive power of heat, Trans. Roy. Soc. Edinburgh 16 (1849).
6. R. Clausius, Über die bewegende Kraft der Wärme, Annalen der Physik 79 (1850).
7. J. Güémez, C. Fiolhais and M. Fiolhais, Sadi Carnot on Carnot’s theorem, American Journal of Physics 70 (2002).
8. H. L. Callendar, The caloric theory of heat and Carnot’s principle, Proc. Phys. Soc. London 23 (1911).
9. D. S. L. Cardwell, From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age, Heinemann, 1971.
10. E. Fermi, Thermodynamics, Dover, 1956, chapters 3–4.