Carnot theorems

One hundred fifty years ago, the two classic laws of thermodynamics were formulated independently by Kelvin and by Clausius, essentially by making the Carnot theorem and the Joule-Mayer-Helmholtz principle of conservation of energy concordant with each other. At first the physicists of the middle 1800s focused primarily on heat engines, in part because of the pressing need for efficient sources of power. At that time, chemists, who are rarely at ease with the calculus, shied away from... 

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... 

If the same reaction (3.13) is realized in an internal combustion engine, i.e., as combustion reaction, the entropy change remains unaltered (because it depends only on the same overall chemical reaction involved in both processes), but not 100% of AG can be converted in useful work, but only a fraction of it, according to Carnot theorem [3]. 

Carnot s principle (Carnot theorem) The efficiency of any heat engine cannot be greater than that of a reversible heat engine operating over the same temperature range. Carnot s principle follows directly from the second law of thermodynamics, and means that all reversible heat engines have the same efficiency, independent of... 

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. 

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). 

In 1879 Lord Kelvin introduced the term nwtivity for the possession, the waste of which is called dissipation at constant temperature this is identical with Maxwell s available energy. He showed in a paper On Thermodynamics founded on Motivity and Energy Phil. Mag., 1898), that all the thermodynamic equations could be derived from the properties of motivity which follow directly from Carnot s theorem, without any explicit introduction of the entropy. 

Fuel cells are electrochemical devices transforming the heat of combustion of a fuel (hydrogen, natural gas, methanol, ethanol, hydrocarbons, etc.) directly into electricity. The fuel is electrochemically oxidized at the anode, whereas the oxidant (oxygen from the air) is reduced at the cathode. This process does not follow Carnot s theorem, so that higher energy efficiencies are expected up to 40-50% in electrical energy and 80-85% in total energy (heat production in addition to electricity). 

Carnot stated that the efficiency of a reversible Camot engine depends only on the temperatures of the heat reservoirs and is independent of the nature of the working substance. This theorem can be proved by showing that the assumption of a reversible engine with any but the known efficiency of a reversible Camot engine leads to a contradiction of the Clausius statement of the second law. 

Equation (6.16), which includes Equation (6.6), is a mathematical statement of Carnot s theorem ... 

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... 

CARNOT S THEOREM EFFICIENCY THERMODYNAMICS CARNOT S THEOREM EFFICIENCY... 

The working of the cell under reversible thermodynamic conditions does not follow Carnot s theorem, so that the theoretical energy efficiency, deflned as the ratio between the electrical energy produced (—AG°) and the heat of combustion (—AH°) at constant pressure, is... 

This theoretical efficiency is much greater (by a factor of about 2) than that of a thermal combustion engine, producing the reversible work, according to Carnot s theorem ... 

Clausius great paper of 1850 can be recognized as a landmark in the development of thermodynamics. As remarked by Thomson in 1851, the merit of first establishing [Carnot s theorem] upon correct principles is entirely due to Clausius. In his 1889 eulogy of Clausius, Gibbs praised the 1850 paper in the following terms ... 

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