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In my previous post JH van ‘t Hoff and the Gaseous Theory of Solutions, I related how van ‘t Hoff deduced a thermodynamically exact relation between osmotic pressure and the vapor pressures of pure solvent and solvent in solution, and then abandoned it in favor of an erroneous idea which seemed to possess greater aesthetic appeal, on account of a chance encounter with a colleague in an Amsterdam street.

Rendered in modern notation, the thermodynamically exact equation van ‘t Hoff deduced in his Studies in Chemical Dynamics (1884), was

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Following his flawed moment of inspiration upon learning the results of osmotic experiments conducted by Wilhelm Pfeffer, he leaped to the conclusion that the law of dilute solutions was formally identical with the ideal gas law

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It would seem van ‘t Hoff was so enamored with the idea of solutions and gases obeying the same fundamental law, that he failed to notice that the latter equation is actually a special case of the former. Viewed from this perspective, the latter’s resemblance to the gas law is entirely coincidental; it arises solely from a sequence of approximations applied to the original equation.

As a footnote to history, CarnotCycle lays out the path by which the latter equation can be reached from the former, and shows how accuracy reduces commensurately with simplification.

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We begin with van ‘t Hoff’s thermodynamically exact equation from the Studies in Chemical Dynamics

ts06 (1)

where Π is the osmotic pressure, V1 is the partial molal volume of the solvent in the solution, p0 is the vapor pressure of the pure solvent and p is the vapor pressure of the solvent in the solution.

Assuming an ideal solution, in the sense that Raoult’s law is obeyed, then

ae02

where x1 and x2 are the mole fractions of solvent and solute respectively. So for an ideal solution, equation 1 becomes

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If the ideal solution is also dilute, the mole fraction of the solute is small and hence

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so that

ae05

For a dilute solution x2 approximates to n2/n1, where n2 and n1 are the moles of solute and solvent, respectively, in the solution. The above equation may therefore be written

ae06

In dilute solution, the partial molal volume of the solvent V1 is generally identical with the ordinary molar volume of the solvent. The product V1n1 is then the total volume of solvent in the solution, and V1n1/n2 is the volume of solvent per mole of solute. Representing this quantity by V’, the above equation becomes

ae07 (2)

which is identical with the empirical equation proposed by HN Morse in 1905. For an extremely dilute solution the volume V’ may be replaced by the volume V of the solution containing 1 mole of solute; under these conditions we have

ts04 (3)

which is the van ‘t Hoff equation.

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It is instructive to compare the osmotic pressures calculated from the numbered equations shown above and those obtained by experiment. It is seen that Eq.1, which involves measured vapor pressures, is in good agreement with experiment at all concentrations. Eq.3 fails in all but the most dilute solutions, while Eq.2 represents only a modest improvement.

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These figures give a measure of van ‘t Hoff’s talent as a theoretician in deducing Eq.1, and the error into which he fell when abandoning it in favor of Eq.3.

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Mouse-over links to works referred to in this post

Jacobus Henricus van ‘t Hoff Studies in Chemical Dynamics

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JH van ‘t Hoff’s laboratory in Amsterdam

The 1880s were important years for the developing discipline of physical chemistry. The gas laws of Mariotte and Gay-Lussac (Boyle and Charles in the English-speaking world) had reached a high point of refinement in Europe following the work of Thomas Andrews and James Thomson in Belfast, and Johannes van der Waals in Leiden. The neophyte science was now poised to discover the laws of solutions.

The need for this advance was clear. As future Nobel Prize winner Wilhelm Ostwald put it in his Lehrbuch der allgemeinen Chemie (1891), “A knowledge of the laws of solutions is important because almost all the chemical processes which occur in nature, whether in animal or vegetable organisms, or in the nonliving surface of the earth, and also those which are carried out in the laboratory, take place between substances in solution. . . . . Solutions are more important than gases, for the latter seldom react together at ordinary temperatures, whereas solutions present the best conditions for the occurrence of all chemical processes.”

In France, important discoveries concerning the vapor pressures exerted by solutions were already being made by François-Marie Raoult. In Germany, the botanist Wilhelm Pfeffer had developed a rigid semipermeable membrane to study the effect of temperature and concentration on the osmotic pressures of solutions. And in the Netherlands, a talented theoretician by the name of Jacobus Henricus van ‘t Hoff (note the space before the apostrophe) was busy writing up his research on chemical kinetics in a work entitled “Studies in Chemical Dynamics”, which contained all that was previously known as well as a great deal that was entirely new.

Then one day in 1883, while van ‘t Hoff was writing the last chapter of the Studies on the subject of chemical affinity, in which he demonstrates an exact relation between osmotic pressure and the vapor pressures of pure solvent and solvent in solution, a chance encounter with a colleague in an Amsterdam street misdirected his thinking and diverted him onto the wrong conceptual road.

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On page 233 of the Studies in Chemical Dynamics, van ‘t Hoff showed that osmotic pressure (D) has a thermodynamic explanation in the difference of vapor pressures of pure solvent and solvent in solution. Yet having discovered this truth, he promptly abandoned it in favor of an idea which seemed to possess greater aesthetic appeal. It was one of those wrong turns we all take in life, but in van ‘t Hoff’s case it seems particularly wayward.

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Jumping to conclusions

Writing in the Journal of Chemical Education (August 1986), the American Nobel Prize winner George Wald relates how van ‘t Hoff had just left his laboratory when he encountered his fellow professor the Dutch botanist Hugo de Vries, who told him about Wilhelm Pfeffer’s experiments with a semipermeable membrane, and Pfeffer’s discovery that for each degree rise in temperature, the osmotic pressure of a dilute solution goes up by about 1/270.

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Hugo de Vries (1848-1935) and Wilhelm Pfeffer (1845-1920)

In an instant, van ‘t Hoff recognized this to be an approximation of the reciprocal of the absolute temperature at 0°C. As he himself put it:

“That was a ray of light, and led at once to the inescapable conclusion that the osmotic pressure of dilute solutions must vary with temperature entirely as does gas pressure, that is, in accord with Gay-Lussac’s Law [pressure directly proportional to temperature]. There followed at once however a second relationship, which Pfeffer had already drawn close to: the osmotic pressure of dilute solutions is proportional also to concentration, i.e., alongside Gay-Lussac’s Law, that of Boyle applies. Without doubt the famous mathematical expression pv = RT holds for both.”

And thus was born, in a moment of flawed inspiration on an Amsterdam street, the Gaseous Theory of Solutions. It even had a mechanism. Osmotic pressure, according to van ‘t Hoff, was caused by one-sided bombardment of a membrane by molecules of solute and was equal to the pressure that would be exerted if the solute occupied the space by itself in the form of an ideal gas. For van ‘t Hoff, this provided the answer to the age-old mystery of why sugar dissolves in water. The answer was simple – it turns into a gas.

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Compounding the error

The law of osmotic pressure, and the gaseous theory that lay behind it, was published by van ‘t Hoff in 1886. Right from the start it was viewed with skepticism in several quarters, and it is not hard to figure out why. As the above quotation shows, van ‘t Hoff had convinced himself in advance that the law of dilute solutions was formally identical with the ideal gas law, and the theoretical support he supplies in his paper seems predicated to a preordained conclusion and shows little regard for stringency.

In particular, the deduction of the proportionality between osmotic pressure and concentration is analogy rather than proof, since it makes use of hypothetical considerations as to the cause of osmotic pressure. Moreover, mechanism is advocated – an anathema to the model-free spirit of classical thermodynamics.

Before long, van ‘t Hoff would distance himself from claims of solute molecules mimicking ideal gases, thanks to a brilliant piece of reasoning from Wilhelm Ostwald – to which I shall return. But van ‘t Hoff’s equation for the osmotic pressure of dilute solutions

ts04

where Π is the osmotic pressure, kept the association with the ideal gas equation firmly in place. And it was this formal identity that led those influenced by van ‘t Hoff along the wrong track for several years.

One such was the wealthy British aristocrat Lord Berkeley, who developed a passion for experimental science at about this time, and furnished a notable example of how one conceptual error can lead to another.

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Misguided research

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Lord Berkeley (1865-1942)

It was known from existing data that the more concentrated the solution, the more the osmotic pressure deviated from the value calculated with van ‘t Hoff’s equation. The idea circulating at the time was that the refinements of the ideal gas law that had been shown to apply to real gases, could equally well be applied to more concentrated solutions. As Lord Berkeley put it in the introduction to a paper, On some Physical Constants of Saturated Solutions, communicated to the Royal Society in London in May 1904:

“The following work was undertaken with a view to obtaining data for the tentative application of van der Waals’ equation to concentrated solutions. It is evidently probable that if the ordinary gas equation be applicable to dilute solutions, then that of van der Waals, or one of analogous form, should apply to concentrated solutions – that is, to solutions having large osmotic pressures.”

And so it was that Lord Berkeley embarked upon a program of research which lasted for more than two decades and failed to deliver any meaningful results because his work was founded on false premises. It is in the highest measure ironic that van ‘t Hoff, just before he was sidetracked, had found his way to the truth in the Studies, in an equation which rendered in modern notation reads

ts06

where Π is the osmotic pressure and V1 is the partial molal volume of the solvent in the solution. This thermodynamic relationship between osmotic pressure and vapor pressure is independent of any theory or mechanism of osmotic pressure. It is also exact, provided that the vapor exhibits ideal gas behavior and that the solution is incompressible.

If van ‘t Hoff had realized this, Lord Berkeley’s research could have taken another, more fruitful path. But history dictated otherwise, and it would have to wait until the publication in 1933 of Edward Guggenheim’s Modern Thermodynamics by the methods of Willard Gibbs before physical chemists in Europe would gain a broader theoretical understanding of colligative properties – of which the osmotic phenomenon is one.

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Brilliant reasoning

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Wilhelm Ostwald (1853-1932)

But to return to van ‘t Hoff’s change of stance regarding mechanism in osmosis. By 1892 he was no longer advocating his membrane bombardment idea, and in stark contrast was voicing the opinion that the actual mechanism of osmotic pressure was not important. It is likely that his change of mind was brought about by a brilliant piece of thinking by his close colleague Wilhelm Ostwald, published in 1891 in the latter’s Lehrbuch der allgemeinen Chemie. Using a thought experiment worthy of Sadi Carnot, Ostwald shows that osmotic pressure must be independent of the nature of the membrane, thereby rendering mechanism unimportant.

Ostwald’s reasoning is so lucid and compelling that one wonders why it didn’t put an end to speculation on osmotic mechanisms. Here is how Ostwald presented his argument:

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“… it may be stated with certainty that the amount of pressure is independent of the nature of the membrane, provided that the membrane is not permeable by the dissolved substance. To understand this, let it be supposed that two separating partitions, A and B, formed of different membranes, are placed in a cylinder (fig. 17). Let the space between the membranes contain a solution and let there be pure water in the space at the ends of the cylinder. Let the membrane A show a higher pressure, P, and the membrane B show a smaller pressure, p. At the outset, water will pass through both membranes into the inner space until the pressure p is attained, when the passage of water through B will cease, but the passage through  A will continue. As soon as the pressure in the inner space has been thus increased above p, water will be pressed out through B. The pressure can never reach the value P; water must enter continuously through A, while a finite difference of pressures is maintained. If this were realized we should have a machine capable of performing infinite work, which is impossible. A similar demonstration holds good if p>P ; it is, therefore, necessary that P=p; in other words, it follows necessarily that osmotic pressure is independent of the nature of the membrane.”

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Epilogue

For van ‘t Hoff, his work on osmosis culminated in triumph. He was awarded the very first Nobel Prize in Chemistry in 1901 for which the citation reads:

“in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”.

But van ‘t Hoff did not have long to enjoy the accolade. “Something seems to have altered my constitution,” he wrote on August 1, 1906, and on March 1, 1911, he died of tuberculosis aged 58.

 

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Mouse-over links to works referred to in this post

Jacobus Henricus van ‘t Hoff Studies in Chemical Dynamics

Wilhelm Ostwald Lehrbuch der allgemeinen Chemie (1891) [English Version – see page 103]

Lord Berkeley On some Physical Constants of Saturated Solutions

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The Honorable Robert Boyle FRS (1627-1691); inset, Richard Towneley (1629-1707)

The fourteenth child of the immensely wealthy Richard Boyle, 1st Earl of Cork, Robert Boyle inherited land and property in England and Ireland which yielded a substantial income. He never had to work for a living, and following three years of travel and study as a teenager in Europe, Boyle decided at the age of 17 to devote his life to scientific research and the cultivation of what was called the “new philosophy”.

In Britain, Boyle was the leading figure in a move away from the Aristotelian view that knowledge was best obtained by the use of reason and logic. Boyle rejected this argument, and insisted that the path to knowledge was through empiricism and experiment. He won over many to his view, notably Isaac Newton, and in 1660 Boyle was a founding member of a society which believed that knowledge should be based first on experiment; we know it today as the Royal Society.

Boyle carried out a wealth of experiments in many areas of physics and chemistry, yet he seems to have been content with obtaining experimental results and generally stopped short of formulating theories to explain them.

Leibniz expressed astonishment that Boyle “who has so many fine experiments, had not come to some theory of chemistry after meditating so long on them”.

But what about Boyle’s law? you ask.

Well, it may surprise you to know that Robert Boyle did not originate the pressure-volume law commonly called Boyle’s law. A description of the reciprocal relation between the volume of air and its pressure does first appear in a book written by Boyle, but he refers to it as “Mr Towneley’s hypothesis”, for reasons we shall see.

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Torricelli leads the way

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Torricelli using a lot more mercury than necessary to demonstrate the barometer.

It was Evangelista Torricelli (1608-1647) in Italy who started it all in the summer of 1644 with the invention of the mercury barometer. It was an impressive device, made all the more impressive by the insight that came with it. For one thing, Torricelli had no problem accepting the space above the mercury as a vacuum, in contrast to the vacuum-denying views of Aristotle and Descartes. For another, he was the first to appreciate the fact that air had weight, and to understand that the column of mercury was supported by the pressure of the atmosphere. As he put it: “We live submerged at the bottom of an ocean of air, which by unquestioned experiments is known to have weight.”

This led Torricelli to surmise that atmospheric pressure should be less in elevated places like mountains, an idea which was put to the test in France by Blaise Pascal, or more precisely by Pascal’s brother-in-law Florin Périer, who happened to live in Clermont-Ferrand, which has Puy de Dôme nearby.

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Puy de Dôme in south-central France, close to Clermont-Ferrand.

On Saturday, September 19, 1648, Florin Périer and some friends performed the Torricelli experiment on the top of Puy de Dôme in south-central France. The height of the mercury column was substantially less – 85 mm less – than the control instrument stationed at the base of the mountain 1,400 metres below.

The Puy de Dôme experiment provided Pascal with convincing evidence that it was the weight of air, and thus atmospheric pressure, that balanced the weight of the mercury column. Torricelli’s instrument provided a convenient means of measuring this pressure. It was a barometer. The news quickly spread to England.

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Reaching new heights

The Torricellian experiment was demonstrated and discussed at the scientific centres of learning in London, Oxford and Cambridge from 1648 onwards. At Cambridge, there was an enthusiastic experimental scientist by the name of Henry Power, who was studying for a degree in medicine at Christ’s College. Power’s home was at Halifax in Yorkshire, and this gave him the opportunity to verify the Puy de Dôme experiment because unlike London, Oxford and Cambridge, the land around Halifax rises to significant heights.

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The hills around Halifax in northern England

On Tuesday, May 6, 1653, Henry Power carried the Torricellian experiment to the summit of Halifax Hill, to the east of the town, where he was able to verify Pascal’s observation. In further experiments, he began to investigate the elasticity of air – i.e. its expansion and compression characteristics. And it was this change of focus that was to characterise England’s contribution to the scientific study of air.

The pioneering work in Italy and France had been concerned with the physical properties of the atmosphere. In England, attention was turning to the physical properties of air itself.

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The Spring of the Air

In 1654, the year after Henry Power’s excursion on Halifax Hill, Robert Boyle arrived in Oxford where he rented rooms to pursue his scientific studies. With the assistance of Robert Hooke, he constructed an air pump in 1659 and began a series of experiments on the properties of air. An account of this work was published in 1660 under the title “New Experiments Physico-Mechanicall, Touching The Spring of the Air, and its Effects”.

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Boyle’s book was a landmark work, in which were reported the first controlled experiments on the effects of reduced air pressure. The experiments are divided into seven groups, the first of which concern “the spring of the air” i.e. the pressure exerted by the air when its volume is changed. It is clear that Boyle had an interest not only in demonstrating the elastic nature of air, but also in finding a quantitative expression of its elasticity. Due to inefficiencies in the air pump and the inherent difficulties of the experiment, Boyle failed in his first attempt. But it was not for want of trying.

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Enter Mr Towneley

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Richard Towneley (1629-1707)

Richard Towneley, whose home was near Burnley in Lancashire, northern England, was curiously similar to Robert Boyle in that the Towneley family estate also generated an income which meant that Richard did not have to work for a living. And just like Boyle, Towneley devoted his time to scientific studies.

Richard Towneley’s home, Towneley Hall in northern England, painted by JMW Turner in 1799

Now it just so happened that the previously-mentioned Henry Power of Halifax was the Towneley family’s physician. This gave Richard and Henry regular opportunities to share their enthusiasm for scientific experiments and discuss the latest scientific news.

In 1660 they both read Robert Boyle’s New Experiments Physico-Mechanicall, and this prompted further interest in the study of air pressure which Power had conducted on Halifax Hill seven years earlier. Not far to the north of Towneley Hall is Pendle Hill, whose summit is 1,827 feet (557 meters) above sea level, and it was here that Power and Towneley conducted an experiment that made history.

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Pendle Hill, photographed by Lee Pilkington

On Wednesday 27th April 1661, they introduced a quantity of air above the mercury in a Torricellian tube. They measured its volume, and then using the tube as a barometer, they measured the air pressure. They then ascended Pendle Hill and at the summit repeated the measurements of volume and air pressure. As expected, there was an increase in volume and a decrease in pressure.

Although their measurements were only roughly accurate and doubtless affected by temperature differences between the base and summit of the hill, the numerical data was sufficient to give them the intuitive realization of a reciprocal relationship between the pressure and volume of air

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On a high hill in northern England, nature revealed one of its secrets to Henry Power and Richard Towneley. Now they needed to communicate their finding.

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Boyle reports the news

It was not until two years after the Pendle Hill experiment that Henry Power eventually got around to publishing the results in Experimental Philosophy (1663). This delay explains why history has never associated the names of Power and Towneley with the gas law they discovered, because the news from Pendle Hill was first given to, and first reported by, Robert Boyle.

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The second edition of 1662

In the appendix to the second edition of his New Experiments Physico-Mechanical published in 1662, Boyle reports Power and Towneley’s experimental activities, which in error he ascribes solely to Richard Towneley. It was on the basis of this second edition of Boyle’s book that the reciprocal relationship between the volume of air and its pressure became known as “Mr. Towneley’s hypothesis” by contemporary authors such as Robert Hooke and Isaac Newton.

A complication was added by the fact that Power and Towneley’s experiment led Boyle to the idea of compressing air in a J-shaped tube by pouring mercury into the long arm and measuring the volume and applied pressure.

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From this experiment, conducted in September 1661, Boyle discovered that the volume of air was halved when the pressure was doubled. Curiously, he translated this finding into the hypothesis that there was a direct proportionality between the density of air and the applied pressure.

It didn’t occur to Boyle that the direct relation between density and pressure was the same thing as the reciprocal relation between volume and pressure discovered earlier by Power and Towneley. Boyle seems to have thought that compression was somehow different to expansion, and that his experiment broke new ground.

For this reason Boyle believed he had discovered something new, and since his name was far better known in scientific circles than that of Henry Power and Richard Towneley, it was inevitable that Boyle’s name would be associated with the newly-published hypothesis, which in time became Boyle’s law.

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Postscript : La loi de Mariotte

The pressure-volume law discovered by Power and Towneley, and confirmed by Boyle, is known as Boyle’s law only in Britain, America, Australia, the West Indies and other parts of what was once called the British Empire. Elsewhere it is called Mariotte’s law, after the French physicist Edme Mariotte.

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Mariotte stated the pressure-volume relationship in his book De la nature de l’air, published in 1679, some 17 years after the second edition of Boyle’s New Experiments Physico-Mechanical. Mariotte made no claim of originality, nor did he make any reference to Boyle. But he was an effective publicist for the law, with the result that his name became widely associated with it.

Some might argue that the attribution is not quite deserved. However, Edme Mariotte stated something immensely important that Robert Boyle neglected to mention. He pointed out that temperature must be held constant for the pressure-volume relationship to hold:

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In the view of CarnotCycle, the pressure-volume law so stated can with justification be called Mariotte’s law.

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Before we begin

Here’s some news. My January 2014 blogpost “Carathéodory: the forgotten pioneer” has been translated into Greek by Giorgos Vachtanidis, and can be seen here.

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Two years on …

Despite the somewhat esoteric nature of Carathéodory’s axiomatic approach to thermodynamics via the geometric behavior of Pfaffians – or perhaps even because of it – my blogpost “Carathéodory: the forgotten pioneer” has received a surprisingly large number of hits, with plenty of brave individuals willing to click on the link to the English version of Carathéodory’s original paper published in 1909 in Mathematische Annalen under the title “Untersuchungen über die Grundlagen der Thermodynamik” [Examination of the foundations of thermodynamics].

Carathéodory’s second axiom “In the neighborhood of any equilibrium state of a system (of any number of thermodynamic coordinates), there exist states that are inaccessible by reversible adiabatic processes”, and the associated theorem giving the condition for dQ to be an integrable differential, constitute the real novelty of his approach.

My original post described Carathéodory’s theorem without going into the proof, since it is rather abstruse and would have appealed only to more avid students of his work. Two years on however, the statistics for this blogpost reveal that there are plenty of avid students wanting to know more. So as a supplementary post, here is a proof of Carathéodory’s theorem, due to Pierre Perrot. Enjoy.

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Carathéodory’s theorem

“If a differential dQ = ΣXidxi, possesses the property that in an arbitrarily close neighborhood of a point P defined by its coordinates (x1, x2,…, xn) there are points which cannot be connected to P along curves satisfying the equation dQ = 0, then dQ is integrable.”

In the following, use is made of a classical result known as the Clausius inequality

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Proof

Cases n = 1 and n = 2 are trivial because a differential function of only one variable is necessarily total whereas a differential function of two variables is necessarily integrable. All points accessible to a given point P form a continuous domain around P. In an n-dimensional space (n≥3) around P, this domain fills a volume [n dimensions], or a surface [(n-l) dimensions], or a curve [≤ (n-2) dimensions].

The first possibility is excluded because it contradicts the hypothesis that around P there are points which are inaccessible. The third possibility is also excluded because the expression dQ = 0 already defines a surface element containing only points accessible to P. Therefore, points close to P and accessible to P define only a surface. If we now consider a point P’ on that surface, it is impossible to go from P to P’ by a curve satisfying the condition ∫dQ = 0 and not situated on this surface, otherwise every point situated within the immediate proximity of P would be accessible, which contradicts the hypothesis.

From a point P1 it is possible to define a surface S1, upon which all points are accessible to P1. Also, from a point P2 not situated on S1, it is possible to define a surface S2. Surfaces S1 and S2 have no common point between them, otherwise it would be possible to go from P1 to P2 by a path such that ∫dQ = 0. Therefore there is a family of surfaces where σ(x1, x2,…, xn) is constant, filling the space and having no common point among them. For this one-parameter family, dσ = 0 implies dQ = 0, from which, between dQ and dσ, there exists a relation of the type:

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where, because dQ = ΣXidxi

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Naturally, the family of surfaces for which σ is constant may also be expressed by S(σ) = constant, where S(σ) is an arbitrary function of σ:

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Hence

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(l/T) is the integrating factor. If a differential dQ has one integrating factor, it has an infinity, S being an arbitrary function of σ.

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Summary

Carathéodory’s theorem shows that if a differential dQ is integrable, the equation dQ = 0 characterizes in a space a family of surfaces sharing no common point. For any point P on one of these surfaces, it is always possible to find, immediately near that point, points which do not belong to the surface and which are therefore inaccessible by a curve solution of the equation dQ = 0. On the other hand, if dQ is not integrable, the equation dQ = 0 does not define any surface in the space and it will always be possible to link any two points with a curve solution of the equation dQ = 0.

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CarnotCycle has been fortunate to acquire the original two-volume set of Gibbs’ scientific papers published by Longmans, Green and Co. in 1906, three years after Gibbs’ death. Volume I is devoted to thermodynamics, while Volume II covers dynamics, vector analysis and multiple algebra, the electromagnetic theory of light, and other miscellaneous topics. Considering that these first edition books are well over a century old, they are in remarkably fine condition.

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