How did Carnot arrive at his theorem?

[swampwiz]

The part of his theorem to which I am referring is the equation relating the heat & temperature for a reversible between temperature sources.

Q_H / T_H = - Q_C / T_C

Did he derive somehow from the forerunner equations of the ideal gas law (i.e., Charles & Boyle's Laws)? (He could not have taken it from Clapeyron's ideal gas law since he had already died by the time that law got forumlated!) Or did he formulate it from observing real steam engines? Or did he somehow do it another way?

From this equation, it is pretty easy to figure out how to express a property that measures the irreversibility of heat transfer (i.e., entropy) by simply making this property be constant for a reversible process - which this equation does even for a differential amount of heat transfer since such reversible heat transfer is done as a quasi-static process in which the temperature is the same (i.e., the cycle fluid & the temperature source) throughout the process. So the key to figuring out how entropy is derived is understanding how Carnot figured out this equation!

 

[Philip Wood]

I don't think Carnot ever wrote this equation, though he did have a number of deep insights into engine efficiency, which were quite astonishing, bearing in mind that he seems to have been working with a caloric theory of heat, rather than regarding heat as a form of energy.

The equation is probably due to William Thomson (later Lord Kelvin), who defined a temperature scale such that \frac{Q_H}{Q_C}=-\frac{T_H}{T_C}, using Carnot's insights as to the non-dependency on working substance. Thomson was also aware of James Prescott Joule's demonstrations of the equivalence of heat and (irreversible) work.

The notion of entropy, as arising from thermodynamics, was developed mainly by Rudolf Clausius.

 

[swampwiz]

Yes, I know that Clausius came up with the idea of entropy, but it flows completely from this equation, that you say was defined by Thomson. If that is accurate, then Thomson actually had an insight into entropy as well, but didn't know it!

 

[Philip Wood]

I think you may be right. I'm afraid I haven't studied the writings of Carnot, Kelvin or Clausius at first hand; I merely gave a summary of the standard picture obtained from secondary sources.

 

[Darwin123]

The part of his theorem to which I am referring is the equation relating the heat & temperature for a reversible between temperature sources.

Q_H / T_H = - Q_C / T_C

Did he derive somehow from the forerunner equations of the ideal gas law (i.e., Charles & Boyle's Laws)? (He could not have taken it from Clapeyron's ideal gas law since he had already died by the time that law got forumlated!) Or did he formulate it from observing real steam engines? Or did he somehow do it another way?

From this equation, it is pretty easy to figure out how to express a property that measures the irreversibility of heat transfer (i.e., entropy) by simply making this property be constant for a reversible process - which this equation does even for a differential amount of heat transfer since such reversible heat transfer is done as a quasi-static process in which the temperature is the same (i.e., the cycle fluid & the temperature source) throughout the process. So the key to figuring out how entropy is derived is understanding how Carnot figured out this equation!

Sadi Carnot introduced the Carnot cycle on the monograph whose title can be translated “Reflections on the Motive Power of Fire” written 1824. A rigorous analysis based on what he wrote by Rudolf Clausius in “Motive Power of Heat, and on the Laws which can be Deduced from it for the Theory of Heat” written 1850.
I have an English translation of the essay that I will reference. The publication that I am referencing is”
“Reflections on the Motive Power of Fire and other Papers on the Second Law of Thermodnamics by E. Clapeyron and R. Clausius” edited by E. Mendoza (Dover Publications, 1988) ISBN:0-486-59065-8.
The above reference also has a monograph by Clausius, which I suspect is closer to the introductory textbooks. You asked about how the concept of entropy first came up, and I believe it came up with Carnot. However, Carnot used a different word for entropy. He used the word “heat”. So I will discuss how Carnot came up with the concept of entropy, though he didn’t use the word.
Carnot never puts down the equation that you wrote. He gets to the “entropy” formula by a different path. He seems to start out with the idea of entropy as a strange gas, although he doesn’t use the word entropy. He has a quantity called s which he calls “the heat”. The French word for heat is “caloric”, but it doesn’t matter. Mathematically, s is entropy.
Clausius first used the word entropy when describing s. I suspect that formula that you wrote came from Clausius. However, this post focuses on how Carnot came up with the idea of entropy, although he doesn’t use the word.
He uses other equations relating to gases. However, he is rather sparse on equations in general. When it comes to matters regarding heat, he generally states his hypotheses in words and pictures. He then gives arithmetic examples to show precisely what he means. I conjecture this is because the laws concerning gases had already been codified into textbooks as equations. Carnot was making new hypotheses concerning heat. Carnot made statements equivalent to the equations that you learned, but he did not write it down as an equation.
The custom was, and to some extent still is, to write ones physical assumptions in terms of sentences and then codify the assumptions into equations. Note that Newton, in Principia, seldom wrote a complete equation. He states all his assumptions in words and pictures, and then gives arithmetic examples. The formalism that introductory students learn with equations came later. Let me give this example.
Carnot wrote on page 27 of the above reference:
“This theorem may also be expressed as follows. When a gas varies in volume without change in temperature, the quantities of heat absorbed or liberated by this gas are in arithmetical progression, if increments or decrements of volume are found to be in geometrical progression.”

Me, again. Carnot didn’t write down the equation, although it is clear to me what he meant. I read the above sentence as:
RΔV/V=Δs
where R is a constant with units of entropy, V is volume and s is entropy.
Carnot gives lots of arithmetic examples of this. He mathematically defines entropy the first time on page 31-32:
"Since we know, on one hand, the law according to which heat is disengaged in the compression of gases, and on the other, the law according to which the specific heat varies with volume, it will be easy <!> for us to calculate the increase of temperature of a gas that has been compressed without being allowed to use heat. In fact, the compression may be considered as composed of two successive operations: (1) compression at a constant temperature; (2) restoration of the caloric emitted. The temperature will rise through the second operation in inverse ratio with the specific heat acquired by the gas after the reduction of volume-specific heat that we are able to calculate by means of the law demonstrated above. The heat set free by compression, according to the theorem of page 27, ought to be represented by an expression of the form,
s=A+B log V,
s being this heat, v the volume of gas after compression, A and B arbitrary constants dependent on the primitive volume of the gas, and its pressure, and on the units chosen."

Me, again. Carnot basically starts developing everything else from the above equation for s. So entropy actually come first in Carnot’s essay.
It is pretty clear from other statements by Carnot that Carnot thinks of s as a material substance. It is also clear from other statements that s is what we now define as the “entropy”. So historically, the concept of entropy came before your equation!

 

[Philip Wood]

Many thanks for this very interesting post. I need to convince myself that 'entropy' is a better interpretation of s than heat, for certainly the first of your Carnot quotations seems to me to be consistent with either. But you've studied Carnot, and I haven't, so I'll shut up. Thanks, too, for the Dover reference.

Do you have any knowledge about the rôle of Thomson/Kelvin in the development of thermodynamics?

 

[Darwin123]

Many thanks for this very interesting post. I need to convince myself that 'entropy' is a better interpretation of s than heat, for certainly the first of your Carnot quotations seems to me to be consistent with either. But you've studied Carnot, and I haven't, so I'll shut up. Thanks, too, for the Dover reference.

I found something else in the Dover reference. Evidently, Carnot "recanted" a material entropy on his deathbed (practically).
There were some laboratory notes discovered after his death. He wrote these notes after his famous monograph was published. He questioned some of his previous hypotheses, and suggested new experiments. Here is some of what he wrote.
On page 68 of the Dover reference, Carnot states:
"When a hypothesis no longer sufficies to explain phenomena, it should be abandoned. This is the case with the hypothesis which regards caloric as matter, a subtle fluid. The experimental facts tending to destroy this theory are as follows:"
Then he mentions Count Rumford's experiments with friction, air pumps, and adiabatic expansion.
This supports my view that when he wrote his monograph, he was thinking of entropy (i.e., caloric) as a form of matter. My position is that his original hypothesis was partly correct, in the same way that the wave theory of light is partly correct.
Just as light sometimes behaves like a wave rather than a particle, entropy sometimes acts like a fluid instead of a disorderly state of motion. Thus entropy can be characterized by "fluid-disorder duality" !

Do you have any knowledge about the rôle of Thomson/Kelvin in the development of thermodynamics?

Sorry, I don't know much about those fellows. I tracked back through Boltzmann, Gibbs, Clausius, Carnot, Rumford and Lavoisier. I stopped at Lavoisier. Good luck going farther!

 

[Darwin123]

The part of his theorem to which I am referring is the equation relating the heat & temperature for a reversible between temperature sources.

Q_H / T_H = - Q_C / T_C

So the key to figuring out how entropy is derived is understanding how Carnot figured out this equation!

Carnot used caloric theory to derive his equations. So maybe what you really want to know is the origin of caloric theory.
The concept of entropy actually evolved from caloric theory. Entropy is a physical quantity that can flow from hotter to colder objects. Entropy can not flow the other way around. Caloric was defined by the same quantities.
Therefore, what you are asking is actually the origin of caloric theory. Here is a link to the Wiki article on caloric theory.
http://en.wikipedia.org/wiki/Caloric_theory
“One version of the caloric theory was introduced by Antoine Lavoisier. Lavoisier developed the explanation of combustion in terms of oxygen in the 1770s. In his paper "Réflexions sur le phlogistique" (1783), Lavoisier argued that phlogiston theory was inconsistent with his experimental results, and proposed a 'subtle fluid' called caloric as the substance of heat. According to this theory, the quantity of this substance is constant throughout the universe, and it flows from warmer to colder bodies. Indeed, Lavoisier was one of the first to use a calorimeter to measure the heat changes during chemical reaction.
In the 1780s, some believed that cold was a fluid, "frigoric". Pierre Prévost argued that cold was simply a lack of caloric.
Since heat was a material substance in caloric theory, and therefore could neither be created nor destroyed, conservation of heat was a central assumption.[2]
The introduction of the Caloric theory was also influenced by the experiments of Joseph Black related to the thermal properties of materials. Besides the caloric theory, another theory existed in the late eighteenth century that could explain the phenomena of heat: the kinetic theory. The two theories were considered to be equivalent at the time, but kinetic theory was the more modern one, as it used a few ideas from atomic theory and could explain both combustion and calorimetry.
…
Sadi Carnot developed his principle of the Carnot cycle, which still forms the basis of heat engine theory, solely from the caloric viewpoint.”

 

[Darwin123]

The part of his theorem to which I am referring is the equation relating the heat & temperature for a reversible between temperature sources.

Q_H / T_H = - Q_C / T_C

Did he derive somehow from the forerunner equations of the ideal gas law (i.e., Charles & Boyle's Laws)? (He could not have taken it from Clapeyron's ideal gas law since he had already died by the time that law got forumlated!) Or did he formulate it from observing real steam engines? Or did he somehow do it another way?

From this equation, it is pretty easy to figure out how to express a property that measures the irreversibility of heat transfer (i.e., entropy) by simply making this property be constant for a reversible process - which this equation does even for a differential amount of heat transfer since such reversible heat transfer is done as a quasi-static process in which the temperature is the same (i.e., the cycle fluid & the temperature source) throughout the process. So the key to figuring out how entropy is derived is understanding how Carnot figured out this equation!

Carnot was using caloric theory. Caloric is now called entropy. The word entropy was coined by Clausius, but the concept is mathematically equivalent to entropy. Carnot probably became interested in caloric theory by studying Lavosier. Carnot may have been assuming that caloric (i.e., entropy) was conserved. The equations that you wrote can be derived using conservation of entropy.
Entropy can be hypothesized to be an indestructible fluid under a "pressure" that is designated temperature. Fluids can spontaneously move from high pressure areas to low pressure areas, doing work in the process. In analogy, entropy can spontaneously move from high temperature areas to low temperature areas, doing work in the process. Carnot appears to have been working with an analogy between entropy and an ideal gas.
Entropy can spontaneously move from a high temperature reservoir to a low temperature reservoir. If the cycle is reversible, then total entropy is conserved.
The equation that you wrote is equivalent to conservation of entropy. If no entropy is created (i.e., a reversible engine), then the entropy leaving the high temperature reservoir during the entire cycle has to be equal to the entropy entering the low temperature cycle.
So if the total entropy in the reversible entropy is conserved, the change in entropy during a cycle is zero. Therefore,
0=ΔS_H+ΔS_C
where ΔS_H is the entropy that leaves the high temperature reservoir and ΔS_C is the entropy that leaves the cold temperature reservoir. If the mathematical definition of entropy is substituted into the above equation,

0=Q_H/T_H+Q_C/T_C

in your notation.

 

[Philip Wood]

What do you think Carnot would have said about heat flow down a temperature gradient in a conductor (e.g. a lagged metal bar) in the steady state? Would he have claimed that caloric is not conserved? [We know that entropy is gained.] Or would he have said that caloric is conserved? [We know that heat is conserved - as much flows out of the cold end as flows into the hot end.]

 

[Darwin123]

What do you think Carnot would have said about heat flow down a temperature gradient in a conductor (e.g. a lagged metal bar) in the steady state?

You are asking me to speculate. Okay, but please go easy on me. I am not Carnot (unfortunately).

Would he have claimed that caloric is not conserved? [We know that entropy is gained.] Or would he have said that caloric is conserved? [We know that heat is conserved - as much flows out of the cold end as flows into the hot end.]

Carnot based his work mostly on the caloric theory of Lavoisier. According to Lavoisier, caloric was conserved. However, it could be drawn out of materials. Caloric was a fluid that permeated other materials. So caloric could be stored.
Presumably, frictional forces could squeeze caloric out of materials. When one rubbed two sticks together to make fire, one was squeezing the caloric that was already there in the sticks. Based on this picture, here is my speculation on what Carnot would say.
The entropy from the heat source flowed down the conductor. On a submicroscopic scale the conductor is a porous medium. The entropy has to squeeze through the molecules of conductor, the way water has to squeeze through the pores of a charcoal filter.
Carnot would hypothesize that there was already entropy in the conductor before the experiment began. Just as a charcoal filter may already be damp before one starts to use it, the conductor already had a large amount of water.
Carnot may also hypothesize that the conductor is inhomogeneous. There are portions of the conductor that contain "bound entropy". There are also larger pores where entropy can freely flow. Thus, there is "bound entropy" and "free entropy".
There is a frictional force between the entropy and the conductor. The frictional force draws water out of the small pores. As the free entropy flows through the conductor, bound entropy is released. Bound entropy becomes free entropy, because of the frictional force of entropy flow. Therefore, the free entropy density increases as one progresses from the hot end to the cold end.
More entropy is released on the cold end than the hot end because of the release of bound entropy. The bound entropy is being drawn out of the conductor.
The fatal flaw in this hypothesis of a hypothesis is that the process couldn't go on forever. Eventually, the conductor would run out of "bound entropy". Then the conductivity of the conductor would have to decrease. Eventually, the conductor would become an insulator.
Maybe Carnot thought about this. However, I can't find anything Carnot wrote which addresses this issue. Maybe Lavoisier already addressed this. I don't have any references by Lavoisier, so I can't cite anything.
The fact that the metal stays a metal during this process is evidence against the caloric theory. Both Carnot and Lavoisier had to know about this. Rumford had published the results of this type of experiment. Perhaps Carnot and Lavoisier thought that the conductor had LOTS of bound entropy. Maybe they thought that Rumford hadn't waited long enough.
Some notes of Sadi Carnot were found after his death. It turned out that he rejected the caloric theory after he wrote his monograph based on other experiments by Count Rumford. Carnot said that the constant generation of heat while drilling that cannon showed that the caloric theory was wrong.
Carnot did not mention the temperature gradient in a conductor even in his laboratory notes. He was apparently starting a new theory when he died. However, he never had a chance to correct his previous work. So what I just said was speculation.

 

[Philip Wood]

What an interesting speculation! I hadn't wanted you to go to so much trouble, but I did think this was an interesting issue. I suspect that, as you suggest, Carnot didn't pay much attention to the phenomena associated with temperature gradients – unlike his fellow-countryman and (rough) contemporary, Fourier. In today's terms, we might say that Carnot conceived of an ideal process in which entropy is conserved but heat isn't, whereas Fourier studied a process in which heat is conserved but entropy isn't.