Background

When Carnot began working on his book, steam engines had achieved widely recognized economic and industrial importance, but there had been no real scientific study of them. Newcomen had invented the first piston-operated steam engine over a century before, in 1712; some 50 years after that, James Watt made his celebrated improvements, which were responsible for greatly increasing the efficiency and practicality of steam engines. Compound engines (engines with more than one stage of expansion) had already been invented, and 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. Although there existed some intuitive understanding of the workings of engines, scientific theory for their operation was almost nonexistent. In 1824 the principle of conservation of energy was still poorly developed and controversial, and an exact formulation of the first law of thermodynamics was still more than a decade away; the mechanical equivalence of heat would not be formulated for another two decades. The prevalent theory of heat was the caloric theory, which regarded heat as a sort of weightless and invisible fluid that flowed when out of equilibrium.

Engineers in Carnot's time had tried, by means such as highly pressurized steam and the use of fluids, to improve the efficiency of engines. In these early stages of engine development, the efficiency of a typical engine—the useful work it was able to do when a given quantity of fuel was burned—was only 3%.

Carnot cycle

Carnot wanted 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 27 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 Carnot indulged in a few arguments involving some calculus. He discussed the relative merits of air and steam as working fluids, the merits of various aspects of steam-engine design, and even included some ideas of his own regarding possible practical improvements. The most important part of the book concentrated on an abstract presentation of an idealized engine that could be used to understand and clarify the fundamental principles that are generally applied to all heat engines, independent of their design.

Perhaps the most important contribution Carnot made to thermodynamics was his abstraction of the essential features of the steam engine, as they were known in his day, into a more general and 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 and 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.

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. Carnot knew that the conduction of heat between bodies at different temperatures is a wasteful and irreversible process, which must be eliminated if the heat engine is to achieve 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 is effected, finally, the transfer of caloric.

— Carnot 1890, p. 68

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 entropy ∆S" or "the reversible transfer of heat at a given temperature Q/T". Carnot knew intuitively that his engine would have the maximum efficiency, but was unable to state what that efficiency would be.

He concluded:

The production of motive power is therefore due in steam engines not to actual consumption of caloric but to its transportation from a warm body to a cold body.^[5]

— Carnot 1890, p. 46

and

In the fall of caloric, motive power undoubtedly increases with the difference of temperature between the warm and cold bodies, but we do not know whether it is proportional to this difference.^[6]

— Carnot 1890, p. 61

In an idealized model, the caloric transported from a hot to a cold body by a frictionless heat engine that lacks conductive heat flow, driven by a difference of temperature, yielding work, could also be used to transport the caloric back to the hot body by reversing the motion of the engine consuming the same amount of work, a concept subsequently known as thermodynamic reversibility. Carnot further postulated that no caloric is lost during the operation of his idealized engine. The process being completely reversible executed by this kind of heat engine is the most efficient possible process. The assumption that heat conduction not driven by a temperature difference can exist, so that no caloric is lost by the engine, guided him to design the Carnot-cycle to be operated by his idealized engine. The cycle is consequently composed of adiabatic processes where no heat/caloric ∆S = 0 flows and isothermal processes where heat is transferred ∆S > 0 but no temperature difference ∆T = 0 exists. The proof of the existence of a maximum efficiency for heat engines is as follows:

As the cycle named after Carnot doesn't waste caloric, the reversible engine has to use this cycle. Imagine now two large bodies, a hot and a cold one. He postulates now the existence of a heat machine with a greater efficiency. We couple now two idealized machines —but of different efficiencies— and connect them to the same hot and the same cold body. The first and less efficient one lets a constant amount of entropy ∆S = Q/T flow from hot to cold during each cycle, yielding an amount of work denoted W. If we use now this work to power the other —more efficient— machine, it would, using the amount of work W gained during each cycle by the first machine, make an amount of entropy ∆S' > ∆S flow from the cold to the hot body. The net effect is a flow of ∆S' − ∆S ≠ 0 of entropy from the cold to the hot body, while no net work is done. Consequently, the cold body is cooled down and the hot body rises in temperature. As the difference of temperature rises, now the yielding of work by the first is greater in the successive cycles and due to the second engine difference in temperature of the two bodies stretches by each cycle even more. In the end this set of machines would be a perpetuum mobile that cannot exist. This proves that the assumption of the existence of a more efficient engine was wrong, so that a heat engine that operates the Carnot cycle must be the most efficient one. This means that a frictionless heat engine that lacks conductive heat flow driven by a difference of temperature shows maximum possible efficiency.

Carnot concludes further that the choice of the working fluid, its density or the volume occupied by it cannot change this maximum efficiency. Using the equivalence of any working gas used in heat engines, he deduced that the difference in the specific heat of a gas measured at constant pressure and that measured at constant volume must be constant for all gases. By comparing the operation of his hypothetical heat-engines for two different volumes occupied by the same amount of working gas he correctly deduces the relation between entropy and volume for an isothermal process:

${\displaystyle \Delta S\propto \ln {\frac {V}{V_{0}}}.}$