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.

Only in the last couple of years I went to the original source and started to try to understand what Carnot himself actually did[1]. It is slow going. Recently I progressed to pages 22 and 23 of a translation of his original work, published by Dover, which according to the introduction was intended as a popular book for non scientists. Popular science writing probably was not, at least not in France in 1824, what it is today, when authors are warned that your public is effectively halved for every mathematical expression you use. However, according to E. Mendoza, the author of the introduction to this edition, Carnot’s arguments are “mostly verbal statements couched in simple but exact language”. It is true that there is no calculus in the book, but there is no need for that either. It is about the “simple and exact language” on pages 22 and 23 that I want to talk here. Apart from not being simple and exact it is also completely wrong, and for the purpose of his cycle, useless. There is not supposed to be a lesson here or something, that would be far too late anyway. There is also no way to find out what Carnot was thinking, so for me it was just an exercise to try to figure out how he did get an answer close to the correct value using completely wrong methods and ideas. Carnot was still of the opinion vented in the Hermann von Helmholtz quote:

The older view of the nature of heat was that it is a substance, very fine and imponderable indeed, but indestructible, and unchangeable in quantity, which is an essential fundamental property of all matter.

even though in 1824 he should have known better. It had been known for almost thirty years that you could make heat from work, viz. count Rumford’s canon boring experiment. Carnot’s lack of understanding heat and work has as a consequence that he sees heat where it is not, and no heat where it should be.

After introduction of his cycle and establishing two important ingredients for his theory, namely that all reversible engines have the same efficiency and that all gases behave the same in an isothermal process, he starts out to derive a relation between the amount of heat needed to raise the air’s temperature at constant pressure and at constant volume. Nowadays we denote these quantities by \(C_p\) and \(C_V\), and Carnot wants to know \(C_p/C_V\). The way we explain the difference between these two quantities is as follows. If we put heat in a sample of gas at constant volume, all of it goes to raising the temperature. The pressure increases, but since the volume remains the same, no expansion work is done by the gas. If, on the other hand, we put in the same amount of heat at constant pressure, the volume increases as well as the temperature. This means that part of that heat is used to perform expansion work, and only part of it is available to increase the temperature. Therefore more heat is needed to get the same raise in temperature as in the first case, hence \(C_p > C_V\). 

It is not clear to me why he needs this quantity, and, in addition, the ratio \(C_p/C_V\) is not the same for all ideal gases. Not even for the gases Carnot mentions. It is more or less the same for air, nitrogen, and oxygen, with a value of 1.4. But since the efficiency of Carnot’s engines is independent of the medium (it does not even have to be a gas), this value can play no role. An engine with argon as a working medium should be equally efficient, and for that gas \(C_p/C_V = 1.67\). Nevertheless he spends two full pages trying to get a value for air. 

In all fairness, there were few things known in those days. But Carnot knows at least two things: if we rapidly decrease the volume by an amount of 1/116, the temperature goes up by 1\(^\circ\)C. Nowadays we call this adiabatic compression. He also knows that if a gas is heated under constant pressure the volume increases by 1/267 if the temperature is raised by 1\(^\circ\)C. There is a third process he calls “direct heating” by which he apparently means isothermal heating. That means you put heat in the system while keeping the temperature constant. What happens in that case is that all the heat is used to expand the system. We would now say: all the heat is converted to work. Since Carnot believes heat is a conserved quantity, I cannot imagine what image he had in mind when making the statement about direct heating. Reading those passages numerous times I created the figure on the left to understand it better. The figure is constructed with today’s knowledge about the properties of the ideal gas, in Carnot’s book no such diagram appears, end neither does the now famous cycle. Google “carnot cycle images” if you want to see in endless minor variations what it looks like.

All points \(V,p\) in de diagram refer to an ideal gas in equilibrium. Since an ideal gas satisfies the simple law \(pV = nRT\), such a point also determines the temperature of the gas; \(n\)  and \(R\) are constants. The black (hyperbolic) curves are so–called isotherms. Along an isotherm the temperature of the gas remains the same. This means that \(p_fV_f = nRT_f\) but also \(p_iV_f’=nRT_f\). The red curve is an adiabat. Along that curve no heat is added to the system, so if we decrease the volume from \(V_i\) to \(V_f\) the temperature increases. The shape of the isotherms must have been known to Carnot, the shape of the adiabat probably not. We now know that adiabats are steeper than isotherms, but Carnot could have known that as well. The crucial point is that along an adiabat no heat can enter the gas. It is supposed to be isolated from the rest of the universe. 

This figure allows us to see more clearly what Carnot had in mind. The first process he mentions, adiabatic compression of the gas raising its temperature, is the process where we take the gas from point \(a\) to point \(c\). The temperature increases by 1\(^\circ\)C and the volume decreases bij 1/116, so that \(V_f = V_i – V_i/116\). The second process he considers is isobaric (constant pressure) heating, taking the gas from point \(a\) to point \(b\). If we measure the amount of heat needed in that process, we would have \(C_p\) which is, by definition, the amount of heat added at constant pressure divided by the temperature change. In that process the temperature also increases by 1\(^\circ\)C but the final volume becomes \(V_f’=V_i+V_i/267\).  What Carnot wants to know is the amount of heat needed to take the system from point \(a\) to point \(a’\). Along that line the volume is constant, and if we know the amount of heat in that case, we also know \(C_V\).

We then come to a statement that could not be more wrong. He compares the air in points \(b\) and \(c\), and states: “The difference in quantities of heat which the air possesses in both cases is evidently the quantity of heat employed to raise it directly one degree; so then the quantity of heat that the air would absorb in passing from the volume \(V_i-V_i/116\) to the volume \(V_i+V_i/267\) is equal to that required to raise it one degree”. So much for “simple and exact language”. What he means now by “direct heating” is heating under constant pressure, and passing from the first to the second volume means following the isotherm. If heat were a state quantity, that is, a property of the system that does not depend on its history but only on its current state, he would be right. In that case the sum of all the heats going around the curve \(a\to c\to b \to a\) would add up to zero, and since the adiabatic part involves no heat exchange, the statement is correct. However, heat is not a state quantity. And in fact the heat absorbed along the curve \(c\to b\) is completely converted to work.  In reality the two quantities of heat are completely different. The amount of heat absorbed along the line \(a b\) is equal to \(7nR/2\) for air, whereas the amount of heat absorbed going along the isotherm \(cb\) is equal to \(nRT_f\ln (V_f’/V_f)\). There is no conceivable system in which these two quantities are equal.  Carnot’s writing gives no clues on how he thought about heat. Apparently something that could also lose value, like falling water losing potential energy, which is converted to kinetic energy and work. 

The same reasoning can be used to get the amount of heat needed to go from \(a\) to \(a’\). By Carnot’s argument it is the same amount of heat needed to go from \(c\) to \(a’\), so that the ratio of the two specific heats becomes equal to the ratio in lengths along the curve, in other words 

\[ \frac{C_p}{C_V} = \frac{cb}{ca’} \approx \frac{1/267+1/116}{1/116} = 1.43\]

Carnot also makes a final approximation, namely that for the small temperature and volume differences considered the isotherm (he does not call it that) can be approximated by a straight line. This leads to the final equality in the above equation, and the value of 1.43. 

Since the value plays no role in the further development to obtain the efficiency, it could as well have been completely wrong, but the strange thing is that it is correct within the accuracy of the values given and the approximations made. You cannot simply replace “heat” by “internal energy”, which is a state function, since along the isotherm the internal energy does not change, and it cannot be used to calculate the ratio in the way Carnot did. A correct calculation, taking into account both work and heat, where the internal energy change is the sum of heat and work (\(\Delta U = q + w\), the first law of thermodynamics) will give the correct answer, obviously. I leave that as an exercise for the reader. 

It would have made me much happier if the number had been completely wrong, because now I still have to find an explanation. It is not so much feeding wrong numbers to a machine to get a right answer, but feeding right numbers to a wrong machine, and still get it right. I am sure Babbage would not have felt happy about this either.

[1] S. Carnot. Reflections on the Motive Power of Fire, Dover Publications, 1988.

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