# Carnot’s Reversible Engine

In bodies employed to realize the motive power of heat there should not occur any change of temperature that may not be due to a change of volume.

Steam engines were very complicated machines by the time Carnot started thinking about their efficiency. More than a century of technological advance had already gone by. He was able to abstract the essentials, and with outdated knowledge, even in his time, derived an expression for the maximum efficiency that we still teach and use today. In the mean time he also laid the ground work for the concept of entropy and the second law of thermodynamics. It is somewhat of a mystery to me how that was possible with all the mistakes he made, and misconceptions he had.

The picture below is of the engine that Carnot designed to get the maximum amount of work from heat in a cyclic process. There is a conspicuous error in this figure, of which I do not understand how Carnot could make it. When discussing his cycle nowadays we use all kinds of knowledge Carnot did not have, and which he needed to derive himself. One of the few important things he does know about gases is that upon rapid expansion they cool and upon rapid compression they heat up. This was a fairly well-established fact, he mentions no less than four different experiments which support it, although he also cites one where the effect is not observed [1]. This phenomenon is very important for his idealized engine since it allows changing the temperature of a gas to a predetermined value by manipulating the volume, essential for the thought he expressed in the quote given at the top of this page.

A is a hot and B a cold plate. Heat can be extracted from or deposited in these plates without changing their temperature. Before the start of the experiment we let the gas in the cylinder equilibrate with the temperature of A, at which point the piston is at position cd. The gas enclosed in the volume abcd has the same temperature as the hot plate. Next we go through four different steps to complete a cycle. At the end of the four steps the piston is again in its initial position cd and the gas has again the temperature of A. It is as if nothing has happened, so the cycle can be repeated indefinitely [2].

First we let the gas expand very slowly. Expansion leads to lowering of the temperature, but if we allow enough time for heat to flow from A into the gas we can manage to keep the temperature constant. The volume increases from piston position cd to ef. During this reversible isothermal expansion all heat entering the gas is converted to work. Nothing is lost [3].

In the second step we take away the heat source A, and let the gas expand further. No heat can enter the system, so the gas cools, to exactly the temperature of plate B. Again nothing is lost, we get a little more work, since some of the ‘heat’ in the system is now converted to work, at the price of cooling the gas [4]. The piston reaches gh.

Now expansion has finished, and we have to bring back the gas to the original state. This can only be done by compressing it, which means performing work on the gas. We put the cylinder on B, and compress slowly. This heats up the gas, the heat flows into B, and the temperature at which this exchange takes place remains the same. All work performed on the gas is converted to heat which is dumped in B. Nothing is lost.

Here Carnot makes a strange mistake, since he claims that the piston will then be at ik. But that cannot be true, it should still be above cd. After all the gas is now at the temperature of B, and in order to heat it up to the temperature of A, we need to adiabatically compress it some more. In this final step the piston returns to its initial position cd, the gas again has the temperature of A, and we can start all over. His writing about the final steps is sloppy to say the least. He “forgets” the final step in his first description of the experiment, and puts piston position ik at the high temperature in the second description.

However, crucial statements are correct: heat is only exchanged at constant temperature, and temperature changes only occur during adiabatic volume changes. No caloric is lost in useless re-establishment of equilibrium.

The question arises how we actually get work from this cycle. According to Carnot the elastic forces of the gas are larger at higher temperatures, so in the expansion step more work is produced than needed in the compression step. We now would say that the pressure is higher. Therefore, upon volume change $$dV$$, $$pdV$$ is larger at the same volume in the expansion phase than in the compression phase. Carnot neglected the work in the adiabatic steps. He argues that we can make these steps arbitrarily small by making the temperature difference between the high and low temperature reservoir small. Here he was lucky again, since for ideal particle gases the work in the two adiabatic processess cancel exactly: the amount of work gained on expansion is lost on compression, so the only relevant stages are isothermal. This is not generally true.

Carnot also realizes that the engine is completely reversible and letting it run in the opposite direction makes it possible to get heat from the low temperature reservoir and deposit it in the high temperature reservoir at the cost of doing work. In other words: to make a refrigerator. Since the engine is reversible, the amount of work needed to bring heat from low to high temperature is exactly the same as the amount of work obtained by running the cycle in the forward direction. This is at the basis of his argument that all reversible engines must have the same efficiency. He does make another serious error in thinking that all gases can take exactly the same amount of heat and convert that to work, but this is the topic of another page. The error is not serious enough to invalidate his final conclusions [5].

The whole basis of the experiment is that heat flows spontaneously from high to low temperature. If we make the temperature of the gas ever so slightly lower than the temperature of A heat will flow into the gas, and if we make it ever so slightly higher than the temperature of B heat will flow out. All versions of the second law of thermodynamics are nothing but restatements of this fundamental law.

If we want heat to flow from cold to warm we need to perform work. That never happens spontaneously.

[1] We now call the rapid expansion and compression processes adiabatic which means no heat is exchanged with the environment. If the change in volume is rapid enough there is just no time for heat to flow in or out of the system. The one experiment Carnot mentions that does not satisfy this law is actually not an adiabatic but a Joule-Thompson process, which indeed gives no temperature change for ideal gases.

[2] Although the steam engine had been in existence for more than 100 years, apparently no one had thought of viewing it as a cyclic process, even though maximizing the amount of work involved maximizing the area of a a cyclic diagram. It reminds me of a story I heard from a student who had worked in Kenya. She told me of a first sentence in a report she had to grade: “In 1848 the missionary Rebmann discovered Mount Kilimanjaro. It is strange that the people who lived there had never noticed it before”.

[3] Complete conversion to work only takes place for the ideal particle gas. If the particles attract each other, some work is needed to pull them apart. For the efficiency this makes no difference.

[4] It is incorrect to state that there is ‘heat’ in the system. But if we replace ‘heat’ with ‘internal energy’ the statement is correct. We’ll need to do that more often.

[5] Obviously all errors Carnot made are in the end irrelevant to the final conclusions. Otherwise we would not talk about his cycle at all.