Carnot's scientific work attracted little attention during his lifetime, but in 1834 it became the object of a detailed commentary and explanation by another French engineer, Émile Clapeyron.

Some of the difficulties that Sadi faced in his own career might have been connected to the persecution of his family by the restored Bourbon monarchy after the fall of Napoleon in 1815.

Sadi Carnot was born in Paris on the 1st of June 1796, at the Petit Luxembourg palace, where his father Lazare resided as one of the five members of the Directory, the highest governing body of the French First Republic in the immediate aftermath of the Thermidorian Reaction.

"Sadi" is the only given name that appears in the second-born's civil birth certificate, dated 14 prairial, year IV in the French Republican calendar.

In 1811, at the age of 16 (the minimum allowed) Sadi Carnot became a cadet of the École polytechnique, where his classmates included the future mathematician Michel Chasles.

Carnot graduated in 1814 and was admitted at the École d'application de l'artillerie et du génie ("School of Applied Artillery and Military Engineering") in Metz, where he completed a two-year course.

Sadi's position in the army, under the restored Bourbon monarchy of King Louis XVIII, became increasingly difficult.

[7] He remained on call for military duty, but from then on he dedicated most of his attention to private intellectual pursuits and received only two-thirds pay.

[8] In Paris, Carnot befriended Nicolas Clément and Charles-Bernard Desormes and attended lectures on physics and chemistry at the Sorbonne and the Collège de France.

[9] Carnot became interested in understanding the limits to improving the performance of steam engines, which led him to the investigations that became his Reflections on the Motive Power of Fire, published at his own expense in June 1824.

Carnot was finally promoted to his former rank of captain in September of 1827, but the following April he quit the army, having completed only fifteen months of active service and without right to a pension.

[10] In a directory of alumni of the École polytechnique published by Ambroise Fourcy in 1828, Carnot is listed as "maker of steam engines".

This and some other indications suggest that Carnot may have been involved in a practical scheme for the improvement of steam engines, but no patents or other concrete evidences of that work have emerged.

[14] According to his brother Hippolyte, there was some discussion among leaders of the new regime of incorporating Sadi to the Chamber of Peers, as he could be considered to have inherited the Imperial title of "Count Carnot" that Napoleon had bestowed on his father Lazare in 1815.

"[17] Hippolyte also described his brother as a talented violin player, interested principally in the music of Jean-Baptiste Lully and Giovanni Battista Viotti, who also cultivated gymnastics, fencing, swimming, dancing, and skating.

[18] According to historian of science James F. Challey, "although sensitive and perceptive", Carnot "appeared extremely introverted, even aloof, to all but a few close friends.

[22] Sadi Carnot's contribution to the development of thermodynamics is contained in his only published work, a short book titled Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance ("Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power") published in Paris in June of 1824 by Bachelier, with Carnot himself paying for the printing of the 600 copies.

This resulted in an idealized thermodynamic system upon which exact calculations could be made, and avoided the complications introduced by many of the crude features of the contemporary steam engines.

Carnot considered an idealized process in which heat from a thermal reservoir at a high temperature flows very slowly (and thus reversibly) into the gas contained in a cylinder enclosed by a movable piston.

Otherwise, the more efficient engine could run Carnot's cycle in reverse as a refrigerator, thus returning all of the "caloric" from the colder back to the hotter reservoir, with some positive amount of work left over to perform a further useful task.

Carnot assumed that such a process, in which no net "caloric" was consumed while positive work could be done forever, would be a perpetual motion and therefore forbidden by the laws of physics.

This argument led Carnot to conclude that 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 is effected, finally, the transfer of caloric.

[29]Later in his book, Carnot considered a heat engine operating very close to the boiling point of water, alcohol, or some other working fluid.

By requiring that the volume change associated with such a transition not be available to construct what he characterized as a perpetual motion device, Carnot arrived at what would later be formalized mathematically as the "Clausius–Clapeyron relation".

James Thomson's prediction was later confirmed experimentally by his brother (the future Lord Kelvin), who found that the data agreed fully with Carnot's analysis.

In the opinion of historian of science Arthur Birembaut, the "smokescreen" that Hippolyte drew over his brother's life makes it impossible now to reconstruct the details of Sadi's career, his relationship with other physicists and engineers, and the circumstances of his death.

[37] Among the private notes published by Hippolyte in 1878 there is material indicating that Sadi Carnot had, by the spring of 1832, rejected the caloric theory and accepted the equivalence of heat and work.