Sadi Carnot

Sadi Carnot

Every one knows that heat can produce motion.

Sadi Carnot

One of the mysteries of science is that an argument or assumption can be completely wrong, but nevertheless lead to a correct result. A case in point is Carnot’s derivation of the efficiency of  a steam engine. In the early days of the 19th century the nature of heat, or caloric as he called it most of the time, was not clear. Carnot obviously considered it some kind of fluid that flows downhill from high to low temperature. In fact he states: “The production of motive power is then due in steam-engines not to an actual consumption of caloric, but to its transportation from a warm body to a cold body, that is, to its re-establishment of equilibrium”. The emphasis in the last sentence is Carnot’s. In his book he uses the words heat and caloric interchangeably [1].

The exact nature of heat was an enigma. Carnot thought it could be produced in a variety of ways. It is not clear if he knew of Count Rumford’s experiments, but he mentions friction and percussion as a possible source, and he does believe that caloric is produced (développé [2]) in the fire converting the water to steam. In the heater the caloric attaches itself to the steam, which upon condensation gives the caloric to the water cooling the condensor. According to the quote given above no caloric is lost during the processes of attachment, moving the piston (“where it performs some function”), or release. He apparently believes caloric is a conserved substance, although he does talk about caloric being developed in the fire and entertains but rejects the possibility that it is consumed during its downhill flow. We now know that heat is not a conserved quantity and is converted to work in the process.

His major idea is correct in an important way. In order to get work from heat you need both a warm and a cold body. It is impossible to get work from just one body, however hot it is. That is why hell must have a uniform temperature. If there were temperature differences condemned physicists would be able extract work to construct a refrigerator and make a nice cool place for themselves.

Having established this principle, he states, almost as an axiom: “wherever there exists a difference in temperature, motive power can be produced“, and then moves on to two very interesting questions: how much useful work (“quantity of motive power”) can we get from a given quantity of caloric; and secondly, does it matter what substance we use for its transfer? Is the amount of work necessarily limited? Could we use alcohol instead of water and get better results? The second question is taken care of by an argument still used in most thermodynamics books. The maximum amount of power that can be obtained by steam must be equal to the maximum power that can be obtained by any means. If substance matters it becomes possible to build a perpetual motion machine, and already in 1824 Carnot is convinced such a machine does not exist. At least not one based on mechanical motion, he is not entirely sure about electricity [3]. The first question: what is the maximum amount of work we can get from a quantity of heat? is harder to answer.

Actually the question is not posed properly. Carnot means: how much work can we get out of a given amount of heat in a cyclic process. There are plenty processes in which all heat can be converted to work. During isothermal expansion of an ideal gas, for instance, all heat entering the gas is converted to work. We end up with expanded gas, and then what? A steam engine that could only do half a stroke would not be of much use. We have to bring the system back to its original state to run the cycle again, and again and again.

So, how do we obtain the maximum amount of work? According to Carnot very little thought is needed to come to the conclusion that you should not have any temperature change that is not due to a volume change. This is based on his insight that a temperature difference can be used to produce work, so if we decrease the temperature difference without the creation of work, it is a “useless re-establishment of equilibrium in the caloric”. This has the important consequence that you should make sure to only have heat flow between bodies of nearly equal temperature. Mathematically this is no problem: we can make the temperature difference as small as we like before we let the heat flow. Nowadays we call such a process reversible since at any point we can revert it. Physically it is impossible to attain full reversibility.

So, based on a completely wrong hypothesis about the nature of heat Carnot reaches two valid conclusions: any temperature gradient can be used to obtain work, and in order to get the maximum amount, you should not waste such a difference: all changes involving the transfer of heat should be done at constant temperature. In fact these conclusions do not depend on the nature of heat at all, the only necessary ingredient is that nature spontaneously diminishes temperature gradients. 

[1] Although E. Mendoza in his introduction states that Carnot may have meant different things with caloric and heat (something along the lines of heat and entropy) I find no evidence for that. Carnot himself states that he uses these terms indifferently.  

[2] I am no expert on early 19th century French (or on any French for that matter), but he may have meant “freed”. There were several problems with the caloric theory, one of them was that sometimes substances released or absorbed heat without change of temperature. Freezing of water and melting of ice are examples. Substances could contain latent heat that could be freed under the right circumstances. 

[3] But he doubts the existence very much. Nowadays magenetism still is occasionally thought to make a perpetual motion machine possible. Not too long ago I saw an item on a Dutch tv program where people can present their invention to a jury, who may award them the possibility to develop the idea further. Two young girls presented something with toiliet rolls and magnets that they thought should work. Which I think is great. What was not so great is that not immediately everyone in the jury realized that this will never lead to anything, kindly explained to the girls that people have been attempting this for centuries and it never works, and recommend the girls for trying.

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