Born in Paris, Sadi Carnot was the first son of the eminent military leader and geometer, Lazare Nicholas Marguerite Carnot, elder brother of Hippolyte Carnot, and uncle of Marie François Sadi Carnot (President of the French Republic (1887-1894), son of Hippolyte Carnot). His father named him for the Persian poet Sadi of Shiraz, and he was always known by this third given name. From age 16 (1812), he lived in Paris and attended the École polytechnique where he and his contemporaries, Claude-Louis Navier and Gaspard-Gustave Coriolis, were taught by professors such as Joseph Louis Gay-Lussac, Siméon Denis Poisson and André-Marie Ampère. After graduation, he became an officer in the French army before committing himself to scientific research, becoming the most celebrated of Fourier's contemporaries who were interested in the theory of heat. Since 1814, he served in the military. Following the final defeat of Napoleon in 1815, his father went into exile. He later obtained permanent leave of absence from the French army. Subsequently, he spent time to write his book.

The historical context in which Carnot worked was that there had been almost no scientific study of the steam engine, and yet the engine was actually pretty far along in its development. It had risen to a widely recognized economic and industrial importance. Newcomen had invented the first piston-operated steam engine over a century before, in 1712. Some 50 years after that, Watt made his celebrated improvements which greatly increased the efficiency and practicality of the engine. Compound engines (engines with more than one stage of expansion) had already been invented. There was even a crude form of internal-combustion engine, with which Carnot was familiar and which he described in some detail in his book. Amazing progress on the practical side had been made, so at least some intuitive understanding of the engine's workings existed. The scientific basis of its operation, however, was almost nonexistent even after all this time. In 1824 the principle of conservation of energy was still immature and controversial, and an exact formulation of the first law of thermodynamics was still more than a decade away. The mechanical equivalent of heat was not identified for another two decades. The prevalent theory of heat was the caloric theory, which regarded heat as a sort of weightless, invisible fluid that flowed when out of equilibrium.

Engineers in Carnot's time had tried various mechanical means, such as high pressure steam, or the use of some fluid other than steam, to improve the efficiency of their engines. In these early stages of engine development, the efficiency of a typical engine -- the useful work it was able to perform when a given quantity of fuel such as a lump of coal was burnt -- was a mere 3%. Carnot sought to answer two questions about the operation of heat engines: "Is the work available from a heat source potentially unbounded?" and "Can heat engines in principle be improved by replacing the steam with some other working fluid or gas?" He attempted to answer these in a memoir, published as a popular work in 1824 when he was only 28 years old. It was entitled Réflexions sur la puissance motrice du feu ("Reflections on the Motive Power of Fire"). The book was plainly intended to cover a rather wide range of topics about heat engines in a rather popular fashion. Equations were kept to a minimum and called for little more than simple algebra and arithmetic, except occasionally in the footnotes, where he indulged in a few arguments involving a little calculus. He discussed the relative merits of air and steam as working fluids, the merits of various aspects of steam engine design, and even threw in some ideas of his own on possible practical improvements. But the most important part of the book was devoted to a quite abstract presentation of an idealized engine that could be used to understand and clarify the fundamental principles that are of general applicability to all heat engines, independent of the particular design choices that might be made.

Perhaps the most important contribution Carnot made to thermodynamics was his abstraction of the essential features of the steam engine as it was known in his day into a more general, idealized heat engine. This resulted in a model thermodynamic system upon which exact calculations could be made, and avoided the complications introduced by many of the crude features of the contemporary steam engine. By idealizing the engine, he could arrive at clear, indisputable answers to his original two questions. He showed that the efficiency of this idealized engine is a function only of the two temperatures of the reservoirs between which it operates. He did not, however, give the exact form of the function, which was later shown to be ^{(T₁−T₂)}⁄_T1, where T₁ is the absolute temperature of the hotter reservoir. (Note: This equation probably came from Kelvin.) No thermal engine operating any other cycle can be more efficient, given the same operating temperatures. He saw very clearly, intuitively, that he could give very definite answers to the two questions set before the reader. The Carnot cycle is the most efficient possible engine, not only because of the (trivial) absence of friction and other incidental wasteful processes; the main reason is that it assumes no conduction of heat between parts of the engine at different temperatures. He knew that conduction of heat between bodies at different temperatures is a wasteful, irreversible process and must be eliminated if the heat engine is to have the maximum efficiency. Regarding the second point, he also was quite certain that the maximum efficiency attainable did not depend upon the exact nature of the working fluid. He stated this for emphasis as a general proposition: "The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which the transfer of caloric takes place." For his "motive power of heat", we would today say "the efficiency of a reversible heat engine," and rather than "transfer of caloric" we would say "the reversible transfer of heat." He knew intuitively that his engine would have the maximum efficiency, but was unable to state what that efficiency would be. In his ideal model, the heat of caloric converted into work could be reinstated by reversing the motion of the cycle, a concept subsequently known as thermodynamic reversibility. Carnot however further postulated that some caloric is lost, not being converted to mechanical work. Hence no real heat engine could realise the Carnot cycle's reversibility and was condemned to be less efficient.

Though formulated in terms of caloric, rather than entropy, this was an early insight into the second law of thermodynamics. Carnot’s book apparently received very little attention from his contemporaries at first. The only citation within a few years after his publication was a review of it in a periodical “Revue Encyclopédique,“ which was a journal that covered a wide range of topics in literature. The work only began to have a real impact when modernised by Émile Clapeyron, in 1834 and then further elaborated upon by Clausius and Kelvin, who together derived from it the notion of entropy and the second law of thermodynamics.

Carnot died in a cholera epidemic when he was only 36 in 1832. Because of the concern of cholera, many of his belongings and writings were buried together with him after his death. Thus only a handful of his scientific writings survived besides his book.

After the publication of his book in 1824, it quickly went out of print and for some time was very difficult to obtain. For example, Kelvin had great difficulty in getting a copy of Carnot's book. Nowadays, his book in French or English can be downloaded electronically. An English translation of it by R. H. Thurston in 1890 has been reprinted in recent decades by Dover and by Peter Smith, most recently by Dover in 2005. Some of his posthumous manuscripts have also been translated into English and included.

Carnot published his book in the days of steam engines. His theory explained why steam engines using superheated steam were better because of the higher temperature of the hot reservoir involved. Carnot's theory did not help to improve the efficiency of steam engines in the beginning; his theory only helped to explain why one existing practice was better. It was only towards the end of the nineteenth century that Carnot's idea -- that a heat engine can be made more efficient if the temperature of its hot reservoir is increased -- was put into practice by, for example, Rudolf Diesel (1858-1913), who was fascinated by Carnot's theory and designed an engine (diesel engine) in which the temperature of the hot reservoir is much higher than that of a steam engine, resulting in an engine which is more efficient than a steam engine. Thus, though it took time, Carnot's book eventually had a real impact on the design of practical engines.