Carnot’s own Cycles

Detail from “Il Grande Quadro”, Enrico Baj.

He who refuses to do arithmetic is doomed to talk nonsense.

John McCarthy.


Carnot was not a great mathematician, and not a deep thinker either. Otherwise he would certainly have caught the inconsistencies in his own theory. Had he been a better mathematician he could have derived, from the experimental data available to him, that heat cannot be a conserved quantity.  Had he been a better thinker, he would have understood how his calculation of the amount of work from a given quantity of heat is inconsistent with the principles he laid down for his heat engine to have maximum efficiency. I’ll devote another post to heat as a conserved quantity. In this one I discuss the second point: what is the according to Carnot the maximum amount of work we can get from the heat needed to raise the temperature of one kilogram of water by one degree? And why can’t his calculation possibly be correct?

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Carnot Cycle for the Photon Gas

Light is something like raindrops. Each little lump of light is called a photon and if the light is all one color, all the “raindrops” are the same.

R.P. Feynman.



The photon gas, a box filled with little light lumps, plays a very important role in physics. As a source of black body radiation — collect the photons coming from a small hole in the box and sort them by energy — it put Planck on the road to quantum mechanics, and although his hope that electromagnetic radiation was the origin of irreversibility did not work out as he intended, it is well worth studying this idea.  Here, however, I investigate the photon gas as a medium for a Carnot engine.

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Entropy 1

Prospects are good for laboratory construction and testing of this solid state Maxwell demon in the near future.

D. Sheehan1 (2002).




Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don’t understand it, but by that time you are so used to it, it doesn’t bother you any more.

This quote by Arnold Sommerfeld summarizes my state of mind from the moment I had taught thermodynamics for the third or fourth time. I could do all the exercises and answer almost all student questions (for them it was the first time, so it did not matter what the answer was anyway), but now that I don’t have to do this any more my lack of  understanding does bother me. So I did the stupidest thing you can do: go back to the original literature, Reflections on the Motive Power of Fire, by S. Carnot. Read more

Apparent Paradox?

We demand rigidly defined areas of doubt and uncertainty.

Douglas Adams



In 2001 Bartell published a paper [1] in which he posed, among others, the following question:


Suppose \(N\) photons of frequency \(\nu\) are emitted by a laser aimed at a stationary black body. The black body absorbs the energy \(nh\nu\) and converts it to thermal energy (heat). Compare this with the energy \(N h\nu’\) absorbed if the black body is moving away from the source. By the Doppler effect, \(\nu’\) is less than \(\nu\) and consequently the absorber sees and absorbs photons of lower energy than emitted by the laser. Where did the extra energy go? The answer to this simple question eludes many physical chemistry professors. It does, however, yield some important results

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Gibbs’ Paradox

J. Willard Gibbs

How wonderful that we have met with a paradox. Now we have some hope of making progress.

Niels Bohr


The Gibbs Paradox is one of the topics I discussed in my Finland Lectures as an example of problems that never went away. The paradox was, around 1875, discovered by Gibbs himself, who also proposed a solution. The latest paper I know of claiming a resolution was published in 2014[1] Read more

Another Carnot Mystery

On two occasions I have been asked, “Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?” I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Charles Babbage.

I learned about the Carnot cycle and its consequences teaching thermodynamics. Conscientiously following the textbooks, mainly because as teachers we are no longer allowed to deviate from them in order not to confuse the students. After a number of years I thought I had finally figured out the use and beauty of this cycle to introduce concepts like efficiency and entropy. Almost all textbooks follow the same ritual: introduce the cycle, “prove” how all reversible cycles must have the same efficiency regardless of working substance, and calculate the efficiency using the ideal classical gas isotherms and adiabats. Then finish the exercise by showing that if you integrate heat divided by temperature over the path of the cycle you get zero, and conclude that there is a state quantity, to be called entropy. Then make some remarks about irreversibility and Clausius, and continue on to the next topic. And I like to think that after a few years I could follow that ritual fairly well. Not that I was impressed by the level of student understanding come exam time, but hell, it is thermodynamics, nobody understands that. Fortunately the powers that be, and a lack of others wanting to teach something as arcane and useless as thermodynamics, left me in a position to yearly try to increase my knowledge and understanding of the field.

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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.

Sadi Carnot

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.

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Debunking Jennings

S happens

Bumper sticker

In 2002 Jennings and coworkers wrote a paper [1] in which they claimed that photosynthesis is more efficient than a Carnot heat engine running between the same temperatures. In fact, their final sentence reads: “Thus, \(1-T/T_r\) represents a kind of efficiency horizon beyond which negative entropy is produced and the second law is not obeyed. As this is impossible for a heat machine, it serves to underline the difference between photosynthetic photochemistry and a heat machine.”

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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]. Read more