Carnot cycle theorem

It is an immediate consequence of Carnot s theorem that the ratio of the quantities of heat absorbed and rejected by a perfectly reversible engine working in a complete cycle, depends only on the temperatures of the bodies which serve as source and refrigerator. 

A hypothetical cycle for achieving reversible work, typically consisting of a sequence of operations (1) isothermal expansion of an ideal gas at a temperature T2 (2) adiabatic expansion from T2 to Ti (3) isothermal compression at temperature Ti and (4) adiabatic compression from Ti to T2. This cycle represents the action of an ideal heat engine, one exhibiting maximum thermal efficiency. Inferences drawn from thermodynamic consideration of Carnot cycles have advanced our understanding about the thermodynamics of chemical systems. See Carnot s Theorem Efficiency Thermodynamics... 

Consider a cycle consisting of a reversible extension of the surface by unit area at a temperature T+dT and a contraction at T. At T- -dT the quantity of heat put into the system is q8- -dq8, and the surroundings do work on it y dy. At T the system does work on its surroundings y, and gives up the heat q8. The net work done by the system on its surroundings is —dy, and a quantity of heat q8 has fallen from T+dT to T. Therefore, by Carnot s theorem, ... 

Nicolas Leonard Sadi Carnot, the French engineer and physicist, was bom in Paris in 1796. His father, Lazare Nicolas Marguerite Carnot, was in the French military service. Sadi Camot is considered as the founder of modem thermodynamics. Famous for his invaluable contributions to science and thermodynamics, Sadi Camot was honored with the title Father of Thermodynamics. Some of his noteworthy contributions to thermodynamics are the concepts of Camot heat engine, Camot cycle, Carnot s theorem, Camot efficiency, and reversible cycle. 

Sadi Carnot s most valuable contribution to thermodynamics is Carnot s ideal heat engine operating with Carnot cycle. His works on ideal heat engine provided the foundation for quantitative mathematical formulation of Carnot efficiency based on Carnot s theorem. However, Carnot s research findings were not well known until another scientist Benoit Pierre Emile Clapeyron followed in his footsteps and experimented with the change in pressure and volume of the processes of a cycle and its effect on work done. Clapeyron developed Carnot s idea of the efficiency of... 

Carnot s theorem (see later) states that tj ax is the maximum of a set of efficiencies from all those heat cycles with those (extreme) working temperatures. Following from (24) in a reversible Carnot cycle it is valid that... 

Therefore in the case of a reversible Carnot cycle, Clausius integral, the algebraic sum of all the by-temperature-reduced heats [both delivered (directly) to the cycle and drained off (directly) from the cycle], equals 0. According to Thomson-Planck s formulation there must be both heat AQ,=w delivered into the cycle O and heat AQ,=o drained off from the cycle. As a consequence of the existence of these heats (the input AQw and the "residual" AQo). and in accordance with the 1. Principle of Thermod5mamics, it must be valid that rjmax < 1-Another formulation of the It. Principle of Thermodynamics is Carnot s theorem, the fitst part of which states The efficiencies of all reversible Carnot cycles with the working temperatures Tyj and To are equal. 

The second part of Carnot s theorem states The efficiency of any irreversible heat cycle with (extreme) working temperatures Ty/ and Tq, Tw>Tq > 0, is less than the efficiency of a reversible Carnot cycle with those same working temperatures. 

Following the first part of Carnot s theorem the efficiencies of all reversible Carnot cycles with the working temperatures Tw and Tq, Ty Tq > 0, are ecjual we must conclude that in the opposite case we would be able to couple two Carnot machines with different efficiencies j/i, 72/ 7l 7 72 in such a way that the resulting machine would be the perpetuum mobile cf the II. order (machine changing cyclically, permanently the whole input heat AQw in the output work AA) AQw = AA). So it must be valid that rj = tj2. 

The equality in the relation (40) is valid for all reversible Carnot cycles (with temperatures and To) viewed informationally, and can be considered to be an information formulation of the first part of the Carnot s theorem. 

If the temperature remains fixed, it follows from (3.3.3) that for a reversible flow of heat Q, the change in entropy is QjT. In terms of entropy, Carnot s theorem (3.2.3) becomes the statement that the sum of the entropy changes in a reversible cycle is zero ... 

The results of these thermodynamic investigations of heat engine processing have resulted in Carnot s theorem (1850) the thermodynamic efficiency e of the ideal engine working on the Carnot cycle depends only on the heater and cooler T2 temperatures and does not depend on the engine construction and on the kind of working body used. 

Theorem.—A process yields the maximum amount of available energy when it is conducted reversibly.—Proof. If the change is isothermal, this is a consequence of Moutier s theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectlyJ reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot s cycles (Fig 11). 

Sadi Carnot s principle. Generalization of this principle by Clausius.— In 1824 di Carnot published a short work on the mechanical effects of heat depending on the one hand upon the impossibility of perpetual motion, on the other hand upon the principle, then accepted without question, that aroimd a closed cycle a i stem undergoes losses and absorptions of heat which exactly compensate each other, he demonstrated a theorem of the greatest importance both for the theory of heat and for the applications of this science to heat-engines. 

In the form that we have stated it, this principle applies to a closed cycle only this is a troublesome condition for its application. The principle of the equivalence of work and heat, in the first form stated, possessed the same inconvenience (Chap. II, Art 21) we transformed it in such a way as to remove this inconvenience, and it is this transformation which introduced into our reasonings the quantity called internal energy. We shall transform the theorem of Carnot and dautius in an analogous manner, and this trans-... 

In order to prove the Carnot theorem, it will be assumed that there exist two reversible heat engines I and II, working between the same two temperatures, but having different efficiencies. Suppose that in each cycle the machine I takes in heat Q2 from the source at converts an amount W into work, and gives up the remainder Q2 — W — Qi to the sink at Ti, The machine II, on the other hand, is supposed to convert a smaller amount of the heat Q2 taken in at into work, returning a quantity Q2 — TF = Ql, which is greater than Qi, to the sink at Let the machines be coupled... 

Carnot s cycle has been thus far discussed for an ideal gas. Similar reversible cycles can be performed on other materials, including solids and liquids, and the efficiency of these cycles determined. The importance of the reversible cycle for the ideal gas is that, as has just been seen, it gives us an extremely simple expression for the efficiency, namely (T, Tc)/Th. Similarly, a theorem of Carnot shows that the efficiency of all reversible cyclj S,.Qperating.betwe.en, t % is the same, namely... 

---