Archive for August, 2012

Sadi Carnot (1796-1832) One of the great original thinkers.

Sadi Carnot (1796-1832) One of the great original thinkers.

The milestone memoir Réflexions sur la Puissance Motrice du Feu (Reflections on the Motive Power of Fire), published by French engineer Sadi Carnot in 1824, marks the starting point of thermodynamics as a theory-based science. In this work, Carnot developed his powerful ideas with the aid of the caloric theory, which viewed heat as an effect caused by an all-pervading, invisible fluid called caloric. An important tenet of the theory was that caloric was considered to be conserved in all thermal processes.

The fact that Carnot published no other work during his short life led later theoreticians, notably Rudolf Clausius and James Clerk Maxwell, into the error of assuming that Carnot never questioned the validity of the caloric theory. But after Carnot’s death in 1834, a bundle of his papers was found whose contents reveal that he had not only questioned the caloric hypothesis, but had reached the point where he felt compelled to abandon it in favour of its eventual successor, the dynamical theory.

This cannot have been an easy decision for Carnot, since the rejection of the caloric theory in favour of the dynamical theory robbed him of the very principle he had employed in Réflexions sur la Puissance Motrice du Feu to reach his groundbreaking conclusions regarding the motive power of heat.

In this post, I will examine the dilemma that Carnot faced in contemplating this conceptual shift.

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When Carnot began thinking about how steam engines turned heat into work, the theory of machines that produced work through mechanical force supplied by men or animals, the wind or a waterfall was already well understood.

In his memoir, Carnot begins by saying that “a similar theory is evidently needed for heat-engines”, and commences his analysis by noting that the production of motive force in steam engines is always accompanied by the transportation of caloric from a body at elevated temperature to another whose temperature is lower.

It should be noted that in the early 19th century when Carnot was writing his memoir, steam engines had such low thermal efficiency that any consumption of caloric in its passage between the hot and cold reservoirs would barely have been perceptible. So it was quite reasonable for Carnot to think that caloric was conserved by a steam engine in motion.

But how did a steam engine produce motive power? With the aim in mind of ‘a similar theory’ to the mechanical engine, Carnot likened the passage of caloric to the passage of water from a higher reservoir to a lower reservoir as it drives a water wheel – there is no loss of water, and motive power depends on the quantity of water transported and the height of the waterfall.

In an equivalent way, Carnot saw the motive power of a steam engine arising from the quantity of caloric transported, and the ‘height of its fall‘, by which he meant the difference in temperature between the hot and cold bodies. And as with the water wheel analogy, the process involved no loss of caloric.

With this model-based principle in place, Carnot was ready to start finding answers to important questions.

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The first question Carnot poses is one that had been long asked by engineers: Is the motive power of heat invariable or does it vary with the working substance employed to realize it?

To furnish an answer, Carnot conducts a thought experiment involving an imaginary steam engine undergoing a cyclical sequence of four entirely reversible isothermal and adiabatic operations, the hot and cold reservoirs being maintained at given temperatures – a stroke of conceptual genius which we know today as the Carnot cycle.

In this steam engine, Carnot imagines the transfer of a quantity of caloric from a body A to a colder body B, both maintained at a given temperature. Assuming no loss of motive power of caloric, he shows that the motive power produced in a single Carnot cycle can be used to return the same quantity of caloric from body B to body A by operating the cycle in the reverse direction. He then concludes: “An indefinite number of alternative operations of this sort could be carried on without in the end having either produced motive power or transferred caloric from one body to the other.”

He then puts the question “Can there exist a working substance that makes better use of the heat than the steam employed in the cycle just described?” and uses the following argumentation to supply the answer:

If it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, and thus be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without the consumption either of caloric or of any agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible.

To put it in modern parlance, in the first series of operations using steam as the working substance, a quantity C of caloric is transported from body A to body B, producing a quantity M of motive power. Then operating the cycle in reverse, the same quantity M of motive power is used to transport the same quantity C of caloric from body B back to body A, thus restoring the initial conditions. The net result is that no caloric is transferred and no motive power produced.

In the second series of operations using a working substance that is imagined to be more effective than steam, an identical quantity C of caloric is transported from body A to body B, but this time it produces a quantity M+m of motive power. Then operating the previously described cycle (using steam) in reverse, the quantity M of motive power is used, as before, to transport the quantity C of caloric from body B back to body A, thus restoring the initial conditions. The net result is that no caloric is transferred, but a quantity m of motive power is produced – an operation which could be endlessly repeated, producing an unlimited quantity of motive power from nothing. This violates the principle and disproves the initial assertion. Carnot thus concludes: “the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever“.

Carnot has proved that the  motive power of heat is independent of the working substance employed to realize it.

He then asks a second question – whether the motive power of heat is unbounded or subject to an assignable limit – and goes on to prove by further thought experiment that its quantity is fixed solely by the temperatures of the bodies between which the transfer of caloric is effected.

The important point to note is that Carnot arrives at these powerful results by applying the principle that  caloric is a conserved quantity in thermal processes.

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But even as his memoir was going to print (he published it at his own expense), it appears that doubts about the caloric theory were already forming in Carnot’s mind. A sentence in the manuscript which describes the theory as “beyond doubt” was changed at proof stage into less certain language: “the theory … does not appear to be of unquestionable solidity. New experiments alone can decide the question.

The kind of experiments Carnot had in mind are revealed in the bundle of papers found after his death. The 23 loose sheets, containing questions, speculations, fragments of essays, and proposed experiments almost identical to those Joule was later to conduct, chart Carnot’s increasing belief that heat and work are equivalent. He lists Rumford’s experiments on the boring of cannon and the friction of wheels on their spindles among the experimental facts undermining the caloric theory.

One of the sheets contains the following paragraph, in which Carnot identifies heat and work as interconvertible forms of a conserved quantity, and effectively states the first law of thermodynamics in relation to cyclical thermodynamic processes (ΔU = 0, Q – W = 0) :
Heat is simply motive power, or rather motion which has changed its form. It is a movement among the particles of bodies. Wherever there is destruction of motive power, there is at the same time production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power.

On one of the last sheets, Carnot writes:
When a hypothesis no longer suffices to explain phenomena, it should be abandoned. This is the case with the hypothesis which regards caloric as matter, as a subtle fluid.

But as Carnot becomes increasingly convinced that heat and motive power are interconvertible, he is at the same time caught in a dilemma. Because whereas the caloric theory enabled him to prove an assertion by disproving the contrary assertion, the dynamical theory fails to do so.

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As Carnot contemplated the interconvertability of heat and work, he would no doubt have re-run the reversible steam engine thought experiment on the basis of the dynamical theory:

In the first series of operations using steam as the working substance, a quantity H of heat is taken from body A, a quantity M of motive power is produced, and a quantity H-M of heat is transferred to body B. Then operating the cycle in reverse, the quantity H-M of heat is taken from body B, a quantity M of motive power is consumed, and a quantity H of heat is transferred to body A, thus restoring the initial conditions. The net result is that no heat is transferred and no motive power produced.

In the second series of operations using a working substance that is imagined to be more effective than steam, an identical quantity H of heat is taken from body A, but this time a quantity M+m of motive power is produced. Taking h to be the quantity of heat equivalent to the quantity m of motive power, a quantity H-M-h of heat is transferred to body B. Then operating the previously described cycle (using steam) in reverse, the quantity H-M of heat is taken as before from body B, a quantity M of motive power is consumed, and a quantity H of heat is transferred to body A, thus restoring the initial conditions.

The net result is that a quantity h of heat has been taken from the colder body B and a quantity m of motive power has been produced, an operation which could be endlessly repeated. But since we have defined the equivalence relation h=m, there is no violation of principle and the dynamical theory therefore fails to disprove the absurd result of the limitless production of motive power solely by consuming the heat of a body.

This is the core of the dilemma Carnot faced.
He doubted the caloric theory, but it proved his assertions.
He favoured the dynamical theory, but it did not prove his assertions.

One of the loose sheets found after Carnot’s death reveals this difficulty in a passage where he convinces himself that heat can be converted into motive power and vice versa. But while he sees many advantages in this hypothesis, he notes “it would be difficult to explain why, in the development of motive power by heat, a cold body is necessary; why motion cannot be produced by consuming the heat of a body.

The explanation, as we now know, is provided by the second law of thermodynamics. Although Carnot had no time in his short life to grapple with this dilemma, it is not fanciful to suggest that he would have found the solution had he lived. More than twenty years before Clausius, Sadi Carnot had already effectively stated the first law, and as the last quote shows, he only needed to turn his question into an assertion to find the essential statement of the second law.

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An iconic fashion image from 1960: model – Tim Davies, photographer – Terence Donovan

There were three of them: Bailey and Duffy and Donovan. Three British photographers who would turn magazine photography on its head at the start of what was to become known in the UK as the Swinging Sixties. It was Brian Duffy who, with a bold swipe at Cecil Beaton, said – “Before us, fashion photographers were tall, thin and camp. We’re different. We’re short, fat and heterosexual.”

Different they certainly were. Before these blokes burst onto the scene, magazines like Vogue exuded a rarefied atmosphere of day suits, evening gloves and nights at the opera. Bailey, Duffy and Donovan – all in their twenties – brought a new energy and attitude to magazine photography, and changed it.

Terence Donovan came from a working class background in Stepney in east London. As a young man, the backdrop to his life was a stark, post-Blitz industrial landscape, and he used that same backdrop for much of his fashion photography.

When Donovan was commissioned in 1960 for a men’s fashion shoot in Man About Town magazine, he didn’t set up his camera in a Royal park with vintage cars and picnic hampers as props. He chose a grimy power station, with metal ladders and hissing steam providing the atmosphere for a photo set entitled ‘Thermodynamic’.

Donovan’s streetwise documentary style gave a new narrative to the fashion photograph. Together with David Bailey and Brian Duffy, he was instrumental in defining a new era in British fashion photography.

L to R: Bailey, Duffy, Donovan

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The copywriter’s art

Here is the text from the actual magazine, which appeared in January 1961. The abbreviation ‘gns’ stands for guineas, an archaic currency term that was still used in the 1960s for garment pricing. In the days before Britain adopted decimal currency, the pound was divided into 20 shillings. A guinea was 21 shillings, i.e. just over 1 pound.


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Terence Donovan had a studio at 30 Bourdon Street in London’s Mayfair. Just around the corner, in Bourdon Place, sculptor Neal French has created a work entitled Three Figures, featuring Terence Donovan photographing iconic 1960’s model Twiggy (out of shot right) while a passing lady shopper stops momentarily to watch.


In textbooks dealing with thermodynamics, the story is often recounted that Carnot’s groundbreaking memoir Réflexions sur la Puissance Motrice du Feu (Reflections on the Motive Power of Fire) went unnoticed by the scientific world when it was published, and lay forgotten for 24 years until Lord Kelvin seized on its significance and brought Carnot’s seminal work – in which lay the seeds of the second law of thermodynamics – into the spotlight of publicity.

This popular account, although broadly accurate, is incomplete in one important respect. It neglects the vital role played by Clapeyron.

Benoît Émile Clapeyron (1799-1864)

Benoît Émile Clapeyron (1799-1864)

Sadi Carnot (1796-1832) and Émile Clapeyron (1799-1864) were contemporaries. Both were engineers by training, and both had studied at the École Polytechnique in Paris. When Carnot published his memoir (at his own expense) in 1824, Clapeyron had already left Paris and was teaching at an engineering school in St. Petersburg, Russia. He returned to Paris just two years before Carnot’s untimely death during a cholera epidemic in 1832. By that time, Sadi Carnot’s little book had long since disappeared from the booksellers’ shelves. But Clapeyron had evidently obtained a copy, and had clearly recognised the importance of his compatriot’s work. Clapeyron’s own Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the Motive Power of Heat), published in Journal de l’École Polytechnique in 1834, contains a full restatement of Carnot’s theoretical principles – albeit in a more analytical form.

While Carnot’s privately published booklet quickly faded into obscurity, Clapeyron’s memoir published in an academic journal did not. It was read in scientific circles and remained available in scientific libraries, and so kept Carnot’s powerful ideas alive. In 1837, Clapeyron’s memoir was translated into English and formed one of the papers presented in Volume I of Taylor’s Scientific Memoirs. This was how Lord Kelvin became aware of Carnot’s work – we know this because the following footnote appears in the historic paper On an Absolute Thermometric Scale which Lord Kelvin published in Philosophical Magazine in October 1848:

“Carnot’s Theory of the Motive Power of Heat – Published in 1824 in a work entitled Réflexions sur la Puissance Motrice du Feu, by M. S. Carnot. Having never met with the original work, it is only through a paper by M. Clapeyron, on the same subject, published in the Journal de l’École Polytechnique, Vol. XIV, 1834, and translated in the first volume of Taylor’s Scientific Memoirs, that the Author has become acquainted with Carnot’s Theory.”

William Thomson, later Lord Kelvin (1824-1907)

William Thomson, later Lord Kelvin (1824-1907)

At the end of 1848, Lord Kelvin succeeded in obtaining a copy of Carnot’s original work, and the following year published a lengthy paper entitled An account of Carnot’s Theory of the Motive Power of Heat in the Transactions of the Edinburgh Royal Society, XVI, 1849.

Curiously, just as Lord Kelvin came to discover Carnot’s masterwork through the agency of Émile Clapeyron’s memoir, so did another key figure in the development of thermodynamics – Rudolf Clausius.

In his first and most famous paper On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat published in Annalen der Physik  in 1850, Clausius cites Carnot’s memoir on the opening page and adds this footnote:

Réflexions sur la Puissance Motrice du Feu, par S. Carnot. Paris, 1824. I have not been able to obtain a copy of this work, and am acquainted with it only through the work of Clapeyron and Thomson [Lord Kelvin]”

Rudolf Clausius (1822-1888)

Rudolf Clausius (1822-1888)

Clapeyron’s memoir had been translated into German in 1843 and published in the same journal in which Clausius’ 1850 paper appeared – Annalen der Physik – so it likely to be this translation to which Clausius refers. It is doubtful whether Carnot had much inkling of how crucially important his ideas would be to the development of thermodynamics. But there is no doubt that Clapeyron immediately saw value in them. In the introduction to his memoir, Clapeyron describes Carnot’s ideas as ‘both fertile and beyond question‘ and ‘worthy of the attention of theoreticians‘.

He was right. The two foremost theoreticians of the day certainly found Carnot’s fertile ideas worthy of attention. And we have Émile Clapeyron to thank for enabling Lord Kelvin and Rudolf Clausius to discover them.

Our pressure cooker was a Prestige Skyline, very popular in the 1950s. Photo credit

Our pressure cooker was a Prestige Skyline from the 1950s. The regulator is sticking up at the back.

A familiar object in the kitchen of my youth was our pressure cooker. It cooked the vegetables in a fraction of the time, saving significant amounts of energy and therefore cost.

Our pressure cooker was a stainless steel device equipped with a regulator that maintained an internal pressure of 2 atmospheres. I was thinking back to it the other day, and started wondering about the temperature of the superheated water inside the container. This can be calculated using the Clausius-Clapeyron equation


which approximates the slope of the liquid-vapor coexistence curve at vapor pressure P and boiling point temperature T. Using the mathematical identity


the Clausius-Clapeyron equation can also be written in the form


If we further assume that ΔHvap is independent of temperature, integration of the above equation can be performed


If the vapor pressure P1 is known at boiling point temperature T1, this equation can be used to estimate the boiling point temperature T2 at another pressure P2.

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Worked Example

Water boils at 373.15K at a pressure of 1 atmosphere. At what temperature will water boil in a pressure cooker operating at a pressure of 2 atmospheres? The enthalpy of vaporization ΔHvap of water at 373.15K is 40657 Jmol-1 and the gas constant R is 8.3145 JK-1mol-1.


Use the integrated Clausius-Clapeyron equation to solve for T2


The SI units are:
T = K
R = JK-1mol-1
ΔH = Jmol-1

The units of P1 and P2 are immaterial, so long as they are the same.

Note that P1 and P2 are vapor pressures. Since T1 and T2 refer to boiling point temperatures, the vapor pressures P1 and P2 will be the same as the externally applied pressures.



Water in a pressure cooker operating at a pressure of two atmospheres boils at 394K, or 121°C. This explains why vegetables cook so fast in these devices.


One of the devices that college lecturers like to employ when illustrating the concept of entropy is the one pictured above. The narrative goes along these lines: When cream is added to coffee, diffusion and convection produces in the course of time a homogeneous-looking mixture. Compared to its components it is very stable, and never reverts back of its own accord to the initial state of pure cream and black coffee. The diffusion of these two miscible liquids is spontaneous, irreversible and entropy-producing, since the mixture has more microstates, and thus more entropy, than the pure components.

I have never found this illustration particularly helpful, for two reasons.

Firstly, it complicates the picture by mixing together (no pun intended) two distinct concepts. Since the coffee is assumed to be hot and the cream cold, the mixing process involves thermal entropy due to heat exchange as well as configurational entropy due to diffusion.

Secondly, it ignores the spoon.

The spoon has an interesting role, because although stirring doesn’t affect the end point at which configurational entropy attains a maximum, it certainly accelerates the process of reaching it.

In the configurational sense – the sole concern of this article from here on – the spoon represents assisted entropy. The phrase floated into my head one day and remained there, with the consequence that I have developed the habit of noting examples of assisted entropy in the world at large.

The interrelation of work and configurational entropy deserves some mention. One formulation of the second law of thermodynamics is that the direction of spontaneous change in an inanimate system is such that work can be obtained in suitable circumstances. In the coffee and cream example, it is difficult to imagine just how the work accompanying such a spontaneous process would be obtained. But it’s real enough when looked at through the equivalent converse statement, that in order to reverse this mixing process and return the system to its initial configurational state, the amount of work that would need to be provided from the surroundings would be at least the same.

If a process resulting in increased configurational entropy is not spontaneous, but is achieved by doing work on a stable system – mixing together separate piles of black and white peppercorns is an example – then the  amount of work required to return the system to its initial configurational state would again be at least the same.

It’s instructive to apply this latter example of assisted entropy to stable deposits of relatively rare minerals that are sourced, processed, manufactured into products, traded and consumed worldwide, and then discarded.

Think of a lithium ion button battery. The valuable lithium inside it originates perhaps from a concentrated brine on some salt flat in Bolivia, then after a lengthy process of extraction, transportation, processing, manufacture, trading, distribution, retailing and purchasing, it ends up in the pocket calculator of someone in New York. The battery comes to the end of its useful life and then, being a tiny worthless object, it gets tossed without much thought into the trash can. From there it is loaded onto a waste truck that drives it about 1,000 kilometres to Ohio where it is dumped along with countless other lithium ion button batteries into a big hole where it gets mixed in with thousands of tons of other NYC waste.

This is a gigantic exercise in assisted entropy, in which the initial configurational entropy of the lithium in the concentrated brine in Bolivia is vastly increased by the time it reaches the hole in Ohio. The work required simply to recover it, let alone recycle it, is astronomical, as would be the cost. It is commercially irrecoverable. The lithium is lost.

The same applies to any consumer electronics product with a limited lifetime, negligible or zero resale value, and a size small enough to be discarded into unsorted waste that will end up as landfill – mobile phones, MP3 players, pocket cameras, gadgets of that sort. Disposable objects that contain increasing amounts of rare earth elements, that will as a result become more than rare, by becoming irrecoverable.

Market forces are the big spoon of assisted entropy, and the world at large has yet to focus on its irreversible effects. But the need already exists for systems that will enable large-scale recovery of small-scale consumer products containing scarce resources, before they hit the trash can and are lost.