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Popular Science Monthly/Volume 62/December 1902/The Motive Power of Heat

THE MOTIVE POWER OF HEAT.
By C. K. EDMUNDS,

JOHNS HOPKINS UNIVERSITY.

MR. ALFRED RUSSEL WALLACE, in his 'Wonderful Century,' describes those great material and intellectual achievements which especially distinguish the nineteenth century from any and all of its predecessors, and shows how fundamental is the change they have effected in our life and civilization. From a comparative estimate of the number and importance of these achievements, he concludes that not only was the century just passed superior to any one that had gone before, but that it must in its results be compared with the whole preceding historical period. It, therefore, marks the beginning of a new era of human progress.

There appears, however, upon looking back through the long, dark vista of human history, one step in material progress that seems to be really comparable with several steps of modern times. It was when fire was first utilized and became the servant and friend rather than the master and enemy of man. From that day to this, fire, in various forms and in ever-widening spheres of action, has been the greatest, the essential factor in that increase of man's power over nature, which has in turn been a chief means of developing what we term civilization. As Mr. Fernald, in his 'Gulf of Fire,'[1] points out, all men, however widely separated by millenniums of time and by utmost range of space, by mountains, deserts and oceans, by color, language and occupation, by custom and religion, all agree in this they make fire a servant and a friend. Mr. Fernald shows how the firebrand draws an impassable line between man and the brute creation, and graphically depicts the part played by the ancient element fire in aiding man along that upward path which, having entered, he had only to follow on to make the universe his own.

Steam engines in their infancy were known as 'fire' (i. e., heat) engines; and, in point of fact, the older term is the more correct, because the water or steam is used only as a convenient medium through which the form of energy which we call heat is made to perform the required mechanical operations. The claims of the steam engine (as locomotive, marine and stationary) to the greatest share in the marvelous material progress of the nineteenth century are too well known and acknowledged to need recounting here. We wish merely to call to mind one upon whose theoretical deductions all the advances in the science of thermodynamics since his day have been based: Sadi Carnot, a young French military engineer.

Appreciation of genius is a mark of the civilization and culture of a people, and the world can ill afford to neglect the memory not only of the men who by practical application have helped to make that civilization possible, but also of those who by theoretical deduction have pointed out the way of progress. The world must needs remember those by whose brain power, as Mr. Fernald says, we have been

PSM V62 D124 Nicholas Leonard Sadi Carnot.png

Nicholas Leonard Sadi Carnot (1796-1832), Father of the Science of Thermodynamics.

enabled to cross the gulf of fire to the paradise of invention and achievement that lies beyond, of which none can see the farther bound. Every schoolboy knows of James Watt, of the Stephensons and their 'Rocket,' but who, save the special student, has ever heard of Sadi Carnot?

Watt made his great improvements in the steam engine (really almost invented it) during the last quarter of the eighteenth century. But for the next fifty years, including those covered by the life of Carnot, the development was seemingly the result of chance. The recognition of this fact led Carnot, when but twenty-eight years old, to undertake to lay the foundation on which all future progress was to rest, by publishing in 1824 a small memoir, 'Reflexions sur la puissance motrice du feu' ('Reflections on the Motive Power of Heat'). George Stephenson made his wonderfully simple, but exceedingly effective, improvements in the locomotive from 1814 to 1829, at a time when the world was still ignorant of the value of Carnot's ideas. To-day, however, when physicists fully realize the importance of his work, the world at large is still ignorant of Carnot's biography; and so it is the purpose of this article to recall the life and character of him whose name must always be intimately associated with the development of the modern 'heat engine' and its influence upon civilization.

Carnot's father, Lazare Nicholas Marguerite Carnot, soldier and statesman, was a member of one of the oldest and most distinguished families in France. Educated as a military engineer, he gained a second lieutenancy at eighteen and in later life won renown as a mathematical writer. Minister of War under Napoleon, he took a prominent part in the French Revolution and was regarded by his countrymen as the genius and organizer of victory, exhibiting the talents later illustrated by the German, Von Moltke. He voted against the extension of the consulate and against the Empire, and was forced into private life in the early days of the latter and died, proscribed, at Magdeburg in 1823. Carnot's brother, Lazare Hippolyte Carnot, was twice a member of the Chamber of Deputies and also Minister of Public Instruction. He died as recently as 1888. A fact which brings Carnot's life still closer to us of to-day is that the late president of France, Sadi Carnot, who was assassinated in 1890, was a grand-nephew of the founder of thermodynamics, the subject of this sketch.

Nicholas-Leonard-Sadi Carnot was born June 1, 1796, in the smaller Luxembourg palace, a part of which was occupied by his father as a member of the Directory. Christened 'Sadi' after the celebrated Persian poet and moralist, he merited the name in that his nature proved to be highly artistic as well as philosophic. Hardly a year after Carnot's birth, in consequence of his father's proscription and enforced exile, his mother took refuge at her homestead in Saint Omer. The boy's delicate constitution was so affected by the vicissitudes of his mother's life that he regained his bodily powers later on only by judicious exercise. He was of medium stature, gifted with extreme sensibility and at the same time with extreme energy, generally reserved, sometimes timid, but singularly quick upon occasion. Whenever he believed that he was encountering injustice, nothing served to restrain him. An incident, which his brother has described, exhibits him in this light even as a child.

The Directory giving place to the Consulate, Carnot senior, after two years of exile, reentered France, being called to the Ministry of War by Bonaparte, who, remembering Carnot 's services to him at the beginning of his career, wished to continue the intimate relations that had existed between them during the Directory. Often when the minister came to work with the consul he brought his son, now about four years of age, and left him in the charge of Madame Bonaparte, who was very fond of him. On one such occasion, Madame Bonaparte and some of her ladies, mounted on a little raft, were paddling about PSM V62 D126 Lazare Nicholas Marguerite Carnot.pngLazare Nicholas Marguerite Carnot,
Father of Nicholas Leonard Sadi Carnot.
upon a pond in the palace court. Napoleon, happening along, began to amuse himself by throwing stones at the raft so as to splash the water over the clean dresses of the would-be sailors. The latter feared to manifest their displeasure, but the little boy, after watching the procedure for a while, suddenly faced the conqueror of Marengo and, shaking a stick at him, cried: 'Animal of a First Consul, are you not ashamed to torment these ladies!' Sadi showed such interest in machinery and applications of physics that his father early directed his studies toward science, and he was just sixteen when he entered the Ecole Polytechnique in 1812. He made rapid progress, graduating the next year with first rank in the artillery. But he was thought too young for the military school at Metz, and was allowed to continue his studies at Paris for another year. Having fought with his gallant fellow-students at Vincennes in March, 1814, he returned when peace was established to his studies at the Polytechnique, but left in October with the rank of sixth-cadet of engineers and repaired to Metz as a sublieutenant in the school of practical fortifications.

The events of 1815 brought Carnot senior again upon the political field during the 'Hundred Days.' This was the occasion for Sadi to make a test of men, of which he never spoke afterwards without disgust. His little quarters of sublieutenant were visited by certain superior officers, who did not hesitate to mount three flights of stairs in order to greet the son of the new minister. Waterloo put an end to all this. The Bourbons reestablished upon the throne, General Carnot was proscribed, and Sadi was sent successively to several forts as engineer, to count bricks, to repair walls and to draw up plans destined to lie buried in the official archives. However, he worked conscientiously, although his name, which but now had won for him so many official pleasantries, served to retard his due advancement for some time. In 1819, desiring greater leisure for private study, Carnot presented himself as a candidate for a new staff-corps then forming, and received an appointment as lieutenant on the general staff. His duties brought him to Paris and the surrounding country, and he led a studious life, interrupted but once by several happy months spent with his brother and exiled father at Magdeburg.

He followed original lines in all his work, and was a constant enemy to the traditional and conventional. Upon his table were found only Pascal, Molière or La Fontaine, and he knew these favorites by heart. With him music was a passion inherited from his mother; not content in attaining a superb execution on the violin, he must needs plunge into the study of theory. His insatiable intelligence led him to follow assiduously courses at the Collège de France, at the Sorbonne and at the École des Mines. He visited manufacturing plants and familiarized himself with the different processes. Mathematical sciences, natural history, industrial arts, political economy, all these were cultivated with ardor. Not only did he indulge in gymnastic exercises, but he investigated the theory of fencing, swimming, dancing and skating.

Toward the end of 1826 he requested and obtained his return to the corps of engineers, receiving by reason of seniority the rank of captain. However, military service was onerous to him, jealous as he was of his liberty, and in 1828 he resigned in order to devote himself more fully to science.

His manuscript notes show that he had perceived the relation which is believed to exist between heat and mechanical work; and after having established the principle which now bears his name, he began researches which would have established with surety the principle of equivalence of mechanical energy and heat had they not been suddenly interrupted by his enthusiastic participation in the revolution of 1830.

An anecdote which shows his impetuous nature as a man, even as we have seen it exhibited as a child, is also given us by his brother. On the day of the funeral of General Lamarque, Carnot was strolling for curiosity's sake in the neighborhood of the insurrection. A cavalier who headed a charge and who appeared intoxicated passed down the street at a gallop, flourishing his saber and striking the pedestrians. Carnot rushed forward nimbly avoiding the soldier's sword, seized him by the leg, threw him to the earth, tumbled him into the gutter and went on his way amid the shouts of the crowd, who were amazed at this exhibition of dexterity and strength.

The hopes of the Democracy, however, appeared to be short-lived, and Carnot returning to his scientific studies applied to them his pent-up political ardor. He undertook important researches upon the physical properties of gases and vapors, especially upon their elastic tensions. Unfortunately his tables were not completed. His excessive application was followed by an attack of scarlet fever in June, 1832, and while convalescing from this attack he was seized on the twenty-fourth of August with the epidemic of cholera and died in a few hours. As if by a sinister presentiment he had been watching the advance of the epidemic very closely, when without previous warning he was carried away upon its tide in the very prime of life, being but thirty-six years of age.

Although the one work that he published is sufficient to keep his name from being forgotten among scientists, yet it is from portions of his note-book that we learn of the activity of his spirit, the variety of his knowledge, his love for humanity and his clear ideas of justice and liberty. In these notes we find rules of practical conduct; observations later embodied in his memoir; some thought that happened especially to strike him, sad or gay; sometimes also, though seldom, an outburst of ill feeling against men and society; thoughts on general political economy or on taxation in particular; and on morals and religion. Some of the ideas contained in these notes remind one not a little of 'Poor Richard's Almanac,' and are so quaintly set that it will doubtless be of interest to quote a few.

The promptness with which a resolution comes to one generally accords with the justice of it.

Never feign a character that you do not possess, and never assume a personality that you will not be able to sustain.

Speak little of that which you know; not at all of that which you do not know. Why not the more often say: 'I do not know'?

Hope is the greatest of blessings; it is necessary, therefore, in order to be happy, to sacrifice the present to the future.

I do not know why one always confounds the two expressions: 'Good sense' and 'common sense.' Nothing is less common than good sense.

People speak of the laws of war, as if war were not the destruction of all law.

Men attribute to chance that of which they do not know the cause. If they come to divine the cause, the chance disappears. To say that a thing happens by chance is to say that we have not been able to foresee it. What is chance for an ignorant man, may not be chance for a man more instructed.

Carnot possessed a repugnance toward publicity, so that, except in conversation with a small number of intimate friends, those among whom he lived were entirely ignorant of the fund of knowledge he had accumulated. His brother, who was called upon to read the manuscript of his memoir on the motive power of heat in order to see that it was clear enough to be understood by others than scientists, says that he never did understand why Carnot made this one exception. It seems that his solitary life in small garrisons, in the office and in the laboratory served to increase his natural reserve. Yet he was not in the least reticent in a small company; he took part willingly in the gayest joys and abandoned himself to the liveliest conversation. His language was then full of witticism, biting but not malignant, original but not eccentric, sometimes paradoxical, but never with any other pretension than that of an active mind.

It was in 1824 while still an officer on the general staff that Carnot published his 'Reflections on the Motive Power of Heat.' Struck with the fact that chance alone seemed to direct the construction of steam engines, he undertook to raise to the rank of a science the art that was still so imperfect in spite of its importance. He investigated the phenomena of the production of motion by heat from the most general point of view, independent of any particular mechanism and of any particular agent. It was only some years after his death that the value of his work was revealed to his fellow countrymen by an echo from England. However, it did merit the attention of a few French scientists, notably the celebrated engineer, Clapeyron, who in 1834 published in the Journal École Polytechnique a paper which was a comment upon and an extension of the ideas of Carnot, in which he called attention to Carnot 's reasoning, represented Carnot 's processes in an analytical form and arrived at some new results, usefully applying, and for the first time, the principle of Watt's indicator diagram to the geometrical exhibition of the different quantities involved in the cycle of operations by which work is derived from heat by the temporary changes it produces in the volume and molecular state of bodies. It was through this work of Clapeyron that Carnot 's ideas became known to Lord Kelvin, who presented them to the world in 1848, pointing out that they enabled us to give for the first time an absolute definition of temperature, i. e., a thermodynamic scale of temperature which is independent of the properties of any particular substance. On this scale the absolute values of two temperatures are defined to be in the same ratio as the amounts of heat-energy taken in and rejected by a perfect (i. e., reversible) thermodynamic engine, working with its source and its refrigerator at the higher and lower of these temperatures respectively. Lord Kelvin showed that the ratio between these quantities of heat-energy depends only on these two working temperatures and is independent of the substance used in the engine, and so the scale of temperature thus defined may be termed absolute. Moreover, such a scale leads at once to the idea of an absolute zero, for no engine could be supposed to convert more heat-energy into work than it received; and, choosing the temperature of the refrigerator (calling it T), so that no heat-energy is rejected by the engine, but all the heat-energy taken from the source is turned into work, it is impossible for T to be negative, else the engine would have an efficiency greater than 100 per cent. Therefore zero is the smallest algebraic value the temperature of the refrigerator can have, and temperatures reckoned from this zero are called absolute. The size of the degrees on this scale is arbitrary, and has been conveniently chosen so that there are one hundred degrees between the temperatures of boiling and freezing water. If now a reversible engine be worked between these temperatures and the quantities of heat-energy received and rejected be measured, the temperature of boiling water on the absolute scale may be found. It is not necessary actually to try the experiment, for the work done by the expansion of a substance which obeys the ordinary laws of gases may be calculated by the methods of the infinitesimal calculus. In this way Kelvin's thermodynamic scale has been shown to be practically identical with that of a perfect gas thermometer, which shows the absolute zero of the thermodynamic scale to lie about 273 degrees below the zero of the Centigrade scale.

Professor Tait has said:"Without this work of Carnot, the modern theory of energy, and especially that branch of it which is at present by far the most important in practice, the dynamical theory of heat, could never have attained in so few years its now enormous development. Carnot 's claims to recognition are of an exceedingly high order, because they depend not merely upon his method, which is one of startling novelty and originality and is not confined to the subject of heat; but upon the fundamental principle upon which he based his mode of comparing the heat employed with the work procured from it. Every reasoner who has applied himself to the subject of heat since Carnot has gone right so far as he attended to Carnot 's principle; but has inevitably gone wrong when he forgot or did not attend to it."

The two things which Carnot introduced, which were entirely original with him and which left his hands in an almost perfect form, were the idea of a 'Cycle of Operations' and the further idea of a 'Reversible Cycle,' giving also the notion of a 'Reversible and Perfect Engine,' showing that the efficiency of such depends only on temperature.

The peculiar merit of Carnot's reasoning consists in the idea of bringing the body back to its initial state as to temperature, density and molecular condition, after a cycle of operations, before making any assertion as to the amount of heat-energy gained or lost. This he accomplished by causing the working substance (a gas in Carnot's case) to pass through two isothermal changes, the first at a higher and the second at a lower temperature, alternated with two adiabatic[2] changes by which the temperature of the working substance is allowed to fall and then raised again. Each separate step was itself reversible and so the whole cycle was reversible. The great virtue in this is that at the close of the cycle of operations the intrinsic energy of the body is exactly the same that it was at the beginning; and so we make no mistake in saying that the difference between the quantity of heat-energy given out by the body during the isothermal change at the higher temperature and that absorbed by it during the isothermal change at the lower temperature is exactly equal to the amount of external work done by the body in the course of the cycle.

By applying this principle Carnot showed that the production of motive power is possible wherever there is a difference of temperature, the motive power being due to a transfer of heat-energy from the hotter to the colder body, its quantity being independent of the agents employed to develop it, but depending solely upon the temperatures of the bodies between which the transfer occurs, provided the process is reversible.

The most striking fact concerning this memoir is that Carnot used hardly any mathematics at all, but arrived at his conclusions by sheer logical exercise of his mind, expressing the different processes entirely in words and using only such terms as would be clear to one not a scientist. Some of his conclusions are incorrect on account of the erroneous assumption of the materiality of heat, but sometimes he is led to conclusions correct in form, although the deduction is erroneous. Instinct seems to have led him in the right direction.

It may be of interest to go through Carnot's memoir and pick out the various important statements as he himself italicized them. They are in part as follows:

The production of motive power in the steam-engine is not due to a real consumption of the caloric, but to its transfer from a hotter to a colder body.

Wherever there is a difference in temperature the production of motive power is possible, and conversely.

The maximum motive power resulting from the use of steam is also the maximum motive power which can be obtained by any other means.

The motive power of heat is independent of the agents employed to develop it; its quantity is determined solely by the temperature of the bodies between which in the final result (i. e., upon the completion of the cycle) the transfer of caloric occurs.

When a gas passes without change of temperature from one definite volume and pressure to another, the quantity of caloric absorbed or emitted is always the same, irrespective of the nature of the gas chosen for the experiment.

The difference between the specific heat under constant pressure and the specific heat under constant volume is the same for all gases.

When the volume of a gas increases in geometrical progression its specific heat increases in arithmetical progress.

Of course these last two statements are now known to be incorrect, it being established that the difference between Cp and Cv is a constant for any one gas, but not for all gases; and also that the specific heat of permanent gases is independent of pressure and temperature. These conclusions were obtained by Carnot on account of the erroneous assumption of the materiality of heat. Moreover, the assumption of the change of specific heat with volume led him to incorrect conclusions in other cases.

The deductions from Carnot 's work made by Clapeyron are correct by reason of the fact that he used differential equations in the extension of Carnot 's ideas. For, although Carnot in considering the energy changes of a body subjected to a Carnot 's cycle made the mistake of equating the amount of heat-energy (H) given out by the body during the isothermal change of volume and pressure at the higher temperature to the heat-energy (h) absorbed by the body during the isothermal change at the lower temperature, Clapeyron was correct in his equations because they dealt only with infinitesimal changes in temperature, and hence the difference H — h, which is the area included between the two adjacent adiabatics and the two isothermals, is an infinitesimal of the second order as compared with the length of the adiabatic included between the two adjacent isothermals, which was taken itself as an infinitesimal of the first order.

It is fortunate that Clapeyron was mathematician enough to use differential equations in expressing these processes analytically. Indeed, in contrast to Carnot he used such a method wherever he could throughout all his memoirs, and always to good advantage.

Carnot used the materialistic theory of heat; but it must not be supposed that he was throughout a believer in the same. For even in his memoir as published in 1824 he gives more than a suspicion of its falsity, and in the extracts from his laboratory note-book,[3] published by his brother after his death, we have direct evidence that he not only foresaw the dynamical theory of heat, but even went so far as to calculate the mechanical equivalent and to plan the very experiments since carried out by later workers. To give emphasis to this statement, we have but to consider the following translation of his own words:

Heat is nothing else than motive power, or rather, motion which has changed its form. It is a movement of the particles of a body. Wherever there is a destruction of motive power, there is at the same time the production of a quantity of heat precisely proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is the production of motive power. We can lay down the general proposition that motive power is a quantity

PSM V62 D133 Carnot notes on the transformation of heat to motive power.png

Facsimile of a Page of Carnot's Note-book Relative to the Transformation of Heat into Motive Power, containing a valuation (the first known to be made) of the so called "mechanical equivalent of heat." This estimate gives for the mechanical equivalent of heat 370 kilogram-meters, which is nearer the correct value (420, Rowland) than Mayer's later determination (365).

invariable in nature, that it is never, properly speaking, either produced or destroyed. In truth, it simply changes form—sometimes producing one kind of motion, sometimes another; but it is never annihilated.

This is a clear and positive statement of the now well-known 'Principle of the Conservation of Energy'; and yet, by reason of the fact that these notes were not published by their author and did not come to light for half a century after his death, the world awaited the enunciation of this universal principle till the day of Mayer, Helmholtz and Joule. Shall all honor be denied Carnot simply because his work remained undiscovered so long? While we ascribe great and merited praise to those philosophers who were fortunate enough first to present the doctrine of energy to the world, we must not forget him who by reason of the much earlier day in which he lived, made a far greater stride in arriving at the same conclusion.

We complete our quotations by giving some of the passages in which Carnot outlines experiments for determining the mechanical equivalent of heat:

Stir vigorously a mass of water in a small barrel or in the cylinder of a double-action pump, the piston of which is pierced with small holes. Experiments of the same kind on the agitation of mercury, of alcohol, of air, and of other gases. Measure the motive power consumed and the heat produced. . . . Allow air to enter a vacuum or a space occupied by air more or less rarefied; the same for other gases or vapors; examine the rise in temperature. Estimate the error of the thermometer by noting the time taken for the temperature of the air to vary a given number of degrees. These experiments will serve to measure the changes of temperature produced in a gas by changes in volume; they will furnish, among other things, the means of comparing these changes with the quantities of motive power produced or consumed. . . . Allow a quantity of air compressed in a larger reservoir to escape therefrom, and check its velocity by having it escape through a large tube containing a number of solid bodies; measure the temperature when it has become uniform. See if it is the same as that in the reservoir. Same experiments with other gases and with vapor formed under various pressures.

How effectually such experiments did accomplish what Carnot expected is fully attested by the subsequent researches of Joule, Kelvin, Hirn, Regnault and others.

Carnot 's work was followed up by the epoch-making papers of Sir William Thomson (now Lord Kelvin) in England, and of Rudolph Clausius in Germany.

The science of thermodynamics, founded on the labors of these three illustrious men, has led to the most important development in all departments of physical science. It has pointed out relations among the properties of bodies which could scarcely have been anticipated in any other way; it has laid the foundation for the science of chemical physics; and, taken in connection with the kinetic theory of gases, as developed by Maxwell and Boltzmann, it has furnished a general view of the operations of the universe which is far in advance of any that could have been reached by purely dynamical reasoning.
  1. Harper's Monthly Magazine, July, 1902.
  2. By isothermal changes are meant changes in volume and pressure involving changes in the heat-energy of the working substance, but unaccompanied by any changes in temperature.

    By adiabatic changes are meant changes in volume and pressure involving changes in temperature, but unaccompanied by any gain or loss of heat-energy of the working substance.

  3. These MS. notes, and also the MS. copy of the 'Reflexions' are on file in the 'Academie des Sciences,' to which body they were presented in 1878 by Carnot's brother, H. Carnot. The notes were entirely unknown to the public until that late date, forty-six years after their author's death.