## Posts Tagged ‘Amsterdam’

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

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

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

(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

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

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

so that

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

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

(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

(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.

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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P Mander June 2015

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.

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.

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

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

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

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

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:

“… 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.”

(English translation by Matthew Pattison Muir)

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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

Lord Berkeley On some Physical Constants of Saturated Solutions

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P Mander June 2015