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?

Carnot’s book[1] Reflections on the Motivc Power of Fire is generally considered the birth of thermodynamics.  So he must have done a few things right, and to be fair, the theory of heat was still wedded to the idea of caloric,[2] a conserved massless fluid capable of making things hotter or expanding them. Maybe the fact that Lavoisier was an advocate of that concept and he carried some weight in early 19th century France had something to do with that.  On the other hand, in 1824 when the book was published, it had already been almost thirty years since Rumford had shown that he could produce unlimited amounts of caloric by friction, so Carnot could have had some doubts. It does not show [3].

He was correct in a few important things. It is the “fall of caloric” from high to low temperature which makes it possible to create or obtain work[4]. Similar to the fall of water from a height. Therefore you cannot create work from just a hot reservoir, you also need a cold sink. He was probably the first to ask how much work you could obtain from this fall, and answered the question how much water I can lift one meter using the heat necessary to raise the temperature of 1 kg of water by 1o. He established, although he does not call it that, the idea of a reversible cyclic process, and made clear that only such a process can give you the maximum amount of work. He draws a few important conclusions from this, but he could have gone a little further had he given it some more thought.

Here is a brief outline of his ideas that are generally considered correct.

1. There are no perpetual motion machines.
2. Heat engines are devices working in a cyclic fashion between a high and a low temperature heat reservoir. At the end of each cycle the engine is in the same state as when it started.
3. The amount of work you can get from a certain amount of heat is limited; it has a maximum. If not, we had a perpetual motion machine.
4. The maximum amount of work is obtained if there is no useless fall of caloric, and temperature changes are by means of  an adiabatic process.
5. You can also use the work to bring heat from low to high temperature, by running the heat engine in reverse and making it a heat pump.
6. Therefore the maximum amount of work is the same for all heat engines. If not, we had a perpetual motion machine.
7. The maximum amount can only depend on the temperatures of the hot and cold reservoir.
(8. The maximum amount cannot depend in any way on properties of the engine.)

The first of these points was true in his time, and is still true now. Though in the last two hundred years there were,  and are, numerous people trying to construct a perpetual motion machine, so far this has not met with success. The second point is related to the action of steam engines as he knew them. A heat engine is the machine that is able to abstract work from the fall of caloric. After a cycle the only thing that happened is that a certain amount of caloric (or heat)  is taken from the hot reservoir, produces work in the engine, and then is dumped in the low temperature reservoir. The engine is then in the same state, ready for another cycle. How it does this exactly Carnot does not know, he says literally: “in some way it [the caloric] incorporates itself with it the steam…. where it performs some function” (p. 45). Not very helpful, but then again, almost nothing was known at the time of the structure of matter, so I’ll forgive him.  

Point 4 is the one that establishes the process as reversible, as we now call it. No useless fall of caloric means that any existing temperature difference must be used to create work. That means that the flow of heat from the hot reservoir into the engine should take place at (almost) the same temperature. Carnot realizes that you can make the temperature difference between the engine and the hot reservoir infinitesimally small, it has only to be ever so slightly warmer than the engine to let heat flow. This is relevant, since it allows us to run the engine in reverse by making an infinitesimal change, making the heat flow in the opposite direction[5] . He also states that any temperature change in the engine should be due to a volume change. It was known that (rapid) compression could make a gas hot enough to set things on fire. So he uses that to get the engine from low to high temperature, and back. Nowadays we call this an adiabatic process, and since no heat is exchanged with the environment, no heat is wasted in this process either. In his engine heat is only used for changing the volume, which can be construed as producing work because I can lift weights that way. Since no heat is lost or used in ways that do not produce work, this is the way to get the maximum amount.

Running the engine in reverse was also his idea, and leads to an important conclusion. Suppose we had some engine, material, action, whatever is possible, that could create more work than the maximum of a particular machine. Then we could use part of that work to pump all the heat back to the original reservoir, and still have some work left over. In other words, money for nothing: perpetual motion. This brings him to point 6.

Point 6 can be considered a very important conclusion for thermodynamics: all heat engines have the same maximum efficiency.  This has a number of consequences I will come to later, and which leads to the concept of entropy and finally the second law of thermodynamics. But that will also be the topic of another chapter, where I will argue that if there are perpetual motion machines, there is no second law of thermodynamics. 

Although Carnot clearly is aware of the importance of point 6, he does not draw the logical conclusion from that: since any substance whatsoever gives the same maximum, the properties of the substance must be completely irrelevant. I have shown that a gas of light particles (photons) has indeed the same maximum as that of an ordinary gas, even though all its properties are vastly different. The photons have no mass, they are not even permanent. They can be destroyed and created, the specific heats are different, the pressure has a completely different relation with volume and temperature and nevertheless, the maximum (by direct calculation) turns out exactly the same as that for the ideal particle gas.  This means that the only things that can be relevant for the maximum are the temperatures of the hot and cold reservoir.  The conclusion should have been point 8, but apparently Carnot does not come to this conclusion, and in fact in his derivation of the amount of work, his result does depend on for instance the specific heat of the substance of the engine. But he did not see that either.

In fact it is possible to go much further and derive the functional dependence on the temperature. Which is not the temperature difference as Carnot believes. In order not to make this post too long, I will postpone this to another chapter as well , and now continue with the way Carnot derives how much work you can get from caloric.

Although Carnot is famous for his cycle consisting of  isotherms and adiabats, he does not use that cycle himself in his work but he considers two other cycles, of which this is the first one. He needs it to calculate how much caloric is needed for a certain amount of isothermal expansion. For this he uses experimental data, and the law of conservation of caloric.

What did he know?  He knew that if you heat air by one degree under constant (atmospheric) pressure, the volume increases by a factor of 1/267. And he new that if you rapidly compress the air by a factor of 1/115, the temperature also goes up by one degree. The first, isobaric (constant pressure), process is symbolized in the graph by the line from  \(T_i\) to B. The second process, adiabatic (no heat exchange), by the line from \(T_i\) to A. The points A and B can be connected by a line of constant temperature: an isotherm. In the graph the adiabat and isotherm are straight lines, in reality they are curved, but for the small volume changes that I consider here this is not important. 

But he knew more than just that, he knew how much heat was involved in the process. He uses as his unit the amount of heat needed to increase the temperature of water by one degree, something we now call a kcal. (1000 calories). For air it was determined that you needed 0.267 times that amount. In modern parlance we would say that

\[ q = C_p\Delta T\]

in words: the amount of heat (\(q\)) is equal to the heat capacity at constant pressure \(C_p\) multiplied by the temperature difference. For air the modern value of  \(C_p\)  is given as 1.012 J/g K, which gives \(q=0.242\) times that of water. The value is slightly different from that used by Carnot (0.267, p. 100), but considering the accuracy of the available measurements at the time, this is not surprising. 

Since he believes caloric is conserved, he also believes that if you go around the cycle, from  \(T_i\) to B, from B to A and back to \(T_i\), the amount of caloric in the air is the same. No caloric is involved in the adiabatic step from \(T_i\) to B.  If I add caloric going from \(T_i\) to B, then that same amount of caloric must be subtracted if I go from B to A. Therefore, the amount of caloric used to go from  \(T_i\) to B must be exactly the same as the amount of caloric used going from A to B. So now he knows how much caloric is needed to expand air  isothermally by an amount of (1/115 + 1/267) times the original volume. 

We now know this is completely wrong, not just a little bit, but completely and fundamentally. In fact it underestimates the amount of heat needed for the process A to B by a factor of more than three. We can get an estimate from the graph. In modern physics energy \(U\) is conserved, not heat. And energy is the sum of work and heat. The work can be read from the graph. For the isobaric process it is the area of the rectangle under the line \(T_i\)B. For the isothermal process it is the much bigger area under the line AB. So in fact much more heat is needed for the isothermal process than Carnot thinks. 

What Carnot apparently failed to realize is that this cycle could also produce work. Isothermal expansion starting at A, followed by isobaric compression, and finally going back by adiabatic compression to A produces an amount of work equal to the area of the triangle AB\(T_i\). He could have known this. So-called indicator diagrams,  \(pV\) plots as the one above, were used since the 1790’s to estimate the efficiency of steam engines[6].

To calculate the amount of work  you can get from the amount of heat needed to expand the air from A to B Carnot uses a different diagram[7], shown on the left. It consists of an isotherm AB along which expansion takes place and work is produced, two curves of constant volume (called isochores) going straight up or down at A and B, and an isotherm at a slightly lower (0.001 degree) temperature \(T_c\) to bring back the engine to its original state. He makes the difference between the two isotherms very small, because he wants to neglect what happens along the two isochores. The point he wants to bring home is that you need work to bring back the engine to its original state, but this work is at lower temperature, and therefore is less than the work you obtained by the expansion. He uses that the amount of work obtained in this way is the pressure difference times the volume difference[8]. The volume difference is known (see above), as is the volume of 1 kg or air (0.77 m3).  Gay-Lussac measured the pressure difference between the two temperatures as 1/267000 atm.

\[ w = (1/115+1/267)V_i\Delta p = (1/115+1/267)\times0.77\times(10^5/267000)\times4.187= 0.015\,{\rm cal} \]

The factor 105 is to get from atm to Pa, and  4.187 is to get from J to cal. Carnot’s calculation is slightly different, he does not bother to calculate the work itself, but the amount of m3 of water he can lift, given that 1 atm pressure can lift 10.40 m of water. So he does not even get the efficiency, But we can, since we know that 0.267 cal was needed (according to Carnot) to do the expansion. The efficiency is therefore

\[\eta = \frac{\mbox{work obtained}}{\mbox{heat needed}} = \frac{0.015}{0.267} = 0.056\]

which is far too high, the real value is 3×10-6. Carnot does not bother to do this calculation, he merely states that a 1000 units of heat can produce 1.395 units of work, but the units of the two are different. Heat is in calories, but work is related to weight. The acceleration due to gravity is missing. But that is not what I wanted to discuss. I want to discuss the premisses in more detail, and show that the conclusions are in contradiction with point 7: the efficiency should only depend on the temperatures of the hot and cold reservoir.

I’ll start with a quote (p. 94):

It would at first seem natural enough to suppose that for equal differences of temperature the quantities of motive power produced are equal; that is, for example the passage of a given quantity of caloric from a body A, maintained at 100o to a body B, maintained at 50o should give rise to a quantity of motive power equal to that which would be developed in the transfer of the same caloric from a body B, at 50o a body, C, at zero. Such a law would doubtless be very remarkable, but we do not see sufficient reason for admitting it a priori. We will investigate its reality by exact reasoning.

What that exact reasoning is, is unclear. Let me recap. He calculates the amount of heat necessary for the expansion at the  temperature 0.001o by a factor (1/115+1/267) by the method discussed above, and finds that this is 0.267 units (or kcal) for 1 kg  of air, which has a volume of 0.77 m3. He then uses the process of the second cycle, where the low temperature isotherm is at zero degrees, to calculate the work for the same expansion (and contraction) of the gas. Which he finds to be 3.72×10-7 units of work (defined as lifting water, but without the gravitational constant). This can be considered exact reasoning, even though the first calculation is only correct if caloric is conserved.  He then states (p. 101):

For a fall a thousand times greater, for a fall of one degree, the motive power will be very nearly a thousand times the former, or 3.72×10-4.

This is also more or less correct. It would be exact if he had used adiabats connecting the isotherms rather than isochores, but the error is indeed small. But this does not show at all that it only depends on the difference in temperatures as he suggests in the previous quotes. The amount of heat needed to expand a gas at constant pressure at 100o could be different from the amount needed at 50o. It isn’t but he did not know that. The amount of heat needed to expand a gas isothermally is different at different temperatures, however, and he did know that, he actually uses it. Had he used this fact, he would have realized that exact reasoning could not possibly have led to this conjecture. See below for a calculation.

What he did know and what should have worried him no end is that the specific heat at constant pressure of different gases is different (p. 76). He gives values, for instance for hydrogen gas it is 0.903 (compared to air for which  he uses 1.000) and for olefiant gas  (ethylene) 1.553, a considerable difference. He also knows that these gases do conform to Mariotte’s (Boyle’s) law, that pressure times volume is a constant at constant temperature. This leads to a simple inconsistency: for ethylene the amount of heat needed for the same expansion is 1.5 times larger than for air. Nevertheless the amount of work obtained is the same. This goes directly against point 6, mentioned repeatedly in the book, that the working substance can make no difference for the maximum power. The maximum amount of  work you can now get from air from the same amount of caloric is bigger  than that from ethylene. You could build a perpetual motion machine using air to obtain work, and ethylene to pump the heat back to the high temperature reservoir, and still have some work left over.

But it is worse. At another point in his book he argues that the specific heat must depend on the volume, and actually calculates a logarithmic dependence. This is also a consequence of the caloric theory. Why is this worse? Well, now we can build a perpetual motion machine from to engines, one with a volume of 1 m3, and the second with a different volume,for instance 2 m3. So you don’t even need different substances, but you can just use engines of different sizes to get free work from a single heat reservoir.

His assertion that the maximum depends on the difference in temperatures only can also not be correct, but that takes more explaining. Briefly stated: if the maximum can only depend on the temperatures of the hot and cold reservoir, the result should not depend on the temperature scale we use, or the zero point that we assume. If he had measured temperatures in Reamur or Fahrenheit units, his results should have been the same. It does not matter if you believe in caloric or energy as conserved substance. I’ll work this out in a later post, and show that a temperature difference is in fact not even possible.

Since it is sometimes suggested that conservation of caloric can lead to a viable thermodynamics[9] it is useful to work from the correct expressions for adiabats and isotherms, and see where this leads us. Taking the true Carnot cycle, consisting of adiabats and isotherms to calculate the work I get[10]:

\[ w = nR(T_h-T_c)\ln \frac{V_f}{V_i}\]

where the engine is running between temperatures \(T_h\) and \(T_c\), and volumes \(V_i\) and \(V_f\). Using Carnot’s method to calculate the amount of heat needed for the expansion between those two volumes is a little tricky, and we need the functional dependence (which he did not know) between volume and temperature, to calculate where to choose \(T_i\). The adiabat is \(VT^{C_V/nR}\), and calculation of the crossing point (the point marked by \(T_i\) gives

\[ T_i = T_h\left(\frac{V_i}{V_f}\right)^{nR/C_p} \]

so that the amount of heat needed for the expansion is given by

\[q = C_p(T_h-T_i) = C_pT_h\left[1 – \left(\frac{V_i}{V_f}\right)^{nR/C_p} \right] \]

which makes the efficiency equal to:

\[ \eta = \frac{\mbox{work obtained}}{\mbox{heat needed}} = \left(1-\frac{T_c}{T_h}\right) \frac{nR\ln(V_f/V_i)}{C_p\left[1 – \left(\frac{V_i}{V_f}\right)^{nR/C_p} \right] }\]

The temperature dependence is, strangely enough, correct. And shows that Carnot’s assertion that the maximum depends on the temperature difference is incorrect. This is because he neglects to take into account that more heat is needed to expand the gas at higher temperatures, as the above expression shows: it is proportional to \(T_h\).. But worse is that the latter part of the expression contains properties of the medium (the specific heat \(C_p\)), and the initial and final volumes of the expansion. As argued above, this cannot possibly be correct, and leads immediately to the possibility of  perpetual motion. I therefore do not believe that it is possible to build a viable thermodynamics on the principle of conservation of caloric. In a later post I will show that on the basis of what he knew, Carnot could also have derived that caloric cannot be a conserved quantity. It is in contradiction with the ideal gas laws already established, and used by Carnot.


[1] S. Carnot, Reflections on the Motivc Power of Fire, 1824.  The original French and the 1897 English translation can be downloaded by clicking the links. Page references are to the English version.

[2] I’ll use the word caloric  for the conserved substance to distinguish it from the non-conserved heat. It has been suggested that you can often read the word caloric as “entropy”, but that was not Carnot’s intention (he writes that he uses the words interchangeably). I’ll get to that in a later post.

[3] Maybe I am a little too strict. He does write “It is based upon the theory of heat as it is understood today, and it should be said that this foundation does not appear to be of unquestionable solidity. New experiments alone can decide the question.” (p. 107). It did not occur to him that it could also be his theory that could shed light on this question.

[4] Both the amounts of work and heat were fairly well defined in those days. Lifting a weight to a certain height was, and is,  considered work. Heat could be measured in a calorimeter. There was not a lot of experimental data yet, but sufficient to allow Carnot to connect the two. Albeit not in a manner Joule did some 30 years later.

[5] In a true reversible process the system (engine) only goes through an infinite series of equilibrium states.

[6] C. Smith, The Science of Energy. A Cultural History of Energy Physics in Victorian Britain. University of Chicago Press, 1998. p. 34.

[7] He does not use diagrams like the ones I draw at all. But he should have. 

[8] This is of course exactly the area of the parallelogram below AB.

[9] P. Lervig, On the structure of Carnot’s theory of heat, Arch. Hist.  Exact Sci., 9, (1972), 222.

[10] Ask me if you want more details of this calculation.

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