Posts Tagged ‘JH van ‘t Hoff’

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The Arrhenius equation explains why chemical reactions generally go much faster when you heat them up. The equation was actually first given by the Dutch physical chemist JH van ‘t Hoff in 1884, but it was the Swedish physical chemist Svante Arrhenius (pictured above) who in 1889 interpreted the equation in terms of activation energy, thereby opening up an important new dimension to the study of reaction rates.

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Temperature and reaction rate

The systematic study of chemical kinetics can be said to have begun in 1850 with Ludwig Wilhelmy’s pioneering work on the kinetics of sucrose inversion. Right from the start, it was realized that reaction rates showed an appreciable dependence on temperature, but it took four decades before real progress was made towards quantitative understanding of the phenomenon.

In 1889, Arrhenius penned a classic paper in which he considered eight sets of published data on the effect of temperature on reaction rates. In each case he showed that the rate constant could be represented as an explicit function of the absolute temperature:

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where both A and C are constants for the particular reaction taking place at temperature T. In his paper, Arrhenius listed the eight sets of published data together with the equations put forward by their respective authors to express the temperature dependence of the rate constant. In one case, the equation – stated in logarithmic form – was identical to that proposed by Arrhenius

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where T is the absolute temperature and a and b are constants. This equation was published five years before Arrhenius’ paper in a book entitled Études de Dynamique Chimique. The author was J. H. van ‘t Hoff.

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Dynamic equilibrium

In the Études of 1884, van ‘t Hoff compiled a contemporary encyclopædia of chemical kinetics. It is an extraordinary work, containing all that was previously known as well as a great deal that was entirely new. At the start of the section on chemical equilibrium he states (without proof) the thermodynamic equation, sometimes called the van ‘t Hoff isochore, which quantifies the displacement of equilibrium with temperature. In modern notation it reads:

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where Kc is the equilibrium constant expressed in terms of concentrations, ΔH is the heat of reaction and T is the absolute temperature. In a footnote to this famous and thermodynamically exact equation, van ‘t Hoff builds a bridge from thermodynamics to kinetics by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions

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Substituting this quotient in the original equation leads immediately to

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van ‘t Hoff then argues that the rate constants will be influenced by two different energy terms E1 and E-1, and splits the above into two equations

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where the two energies are such that E1 – E-1 = ΔH

In the Études, van ‘t Hoff recognized that ΔH might or might not be temperature independent, and considered both possibilities. In the former case, he could integrate the equation to give the solution

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From a starting point in thermodynamics, van ‘t Hoff engineered this kinetic equation through a characteristically self-assured thought process. And it was this equation that the equally self-assured Svante Arrhenius seized upon for his own purposes, expanding its application to explain the results of other researchers, and enriching it with his own idea for how the equation should be interpreted.

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Activation energy

It is a well-known result of the kinetic theory of gases that the average kinetic energy per mole of gas (EK) is given by

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Since the only variable on the RHS is the absolute temperature T, we can conclude that doubling the temperature will double the average kinetic energy of the molecules. This set Arrhenius thinking, because the eight sets of published data in his 1889 paper showed that the effect of temperature on the rates of chemical processes was generally much too large to be explained on the basis of how temperature affects the average kinetic energy of the molecules.

The clue to solving this mystery was provided by James Clerk Maxwell, who in 1860 had worked out the distribution of molecular velocities from the laws of probability. Maxwell’s distribution law enables the fraction of molecules possessing a kinetic energy exceeding some arbitrary value E to be calculated.

It is convenient to consider the distribution of molecular velocities in two dimensions instead of three, since the distribution law so obtained gives very similar results and is much simpler to apply. At absolute temperature T, the proportion of molecules for which the kinetic energy exceeds E is given by

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where n is the number of molecules with kinetic energy greater than E, and N is the total number of molecules. This is exactly the exponential expression which occurs in the velocity constant equation derived by van ‘t Hoff from thermodynamic principles, which Arrhenius showed could be fitted to temperature dependence data from several published sources.

Compared with the average kinetic energy calculation, this exponential expression yields very different results. At 1000K, the fraction of molecules having a greater energy than, say, 80 KJ is 0.0000662, while at 2000K the fraction is 0.00814. So the temperature change which doubles the number of molecules with the average energy will increase the number of molecules with E > 80 KJ by a factor of more than a hundred.

Here was the clue Arrhenius was seeking to explain why increased temperature had such a marked effect on reaction rate. He reasoned it was because molecules needed sufficiently more energy than the average – the activation energy E – to undergo reaction, and that the fraction of these molecules in the reaction mix was an exponential function of temperature.

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The meaning of A

But back to the Arrhenius equation

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I have always thought that calling the constant A the ‘pre-exponential factor’ is a singularly pointless label. One could equally write the equation as

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and call it the ‘post-exponential factor’. The position of A in relation to the exponential factor has no relevance.

A clue to the proper meaning of A is to note that e^(–E/RT) is dimensionless. The units of A are therefore the same as the units of k. But what are the units of k?

The answer depends on whether one’s interest area is kinetics or thermodynamics. In kinetics, the concentration of chemical species present at equilibrium is generally expressed as molar concentration, giving rise to a range of possibilities for the units of the velocity constant k.

In thermodynamics however, the dimensions of k are uniform. This is because the chemical potential of reactants and products in any arbitrarily chosen state is expressed in terms of activity a, which is defined as a ratio in relation to a standard state and is therefore dimensionless.

When the arbitrarily chosen conditions represent those for equilibrium, the equilibrium constant K is expressed in terms of reactant (aA + bB + …) and product (mM + nN + …) activities

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where the subscript e indicates that the activities are those for the system at equilibrium.

As students we often substitute molar concentrations for activities, since in many situations the activity of a chemical species is approximately proportional to its concentration. But if an equation is arrived at from consideration of the thermodynamic equilibrium constant K – as the Arrhenius equation was – it is important to remember that the associated concentration terms are strictly dimensionless and so the reaction rate, and therefore the velocity constant k, and therefore A, has the units of frequency (t^-1).

OK, so back again to the Arrhenius equation

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We have determined the dimensions of A; now let us turn our attention to the role of the dimensionless exponential factor. The values this term may take range between 0 and 1, and specifically when E = 0, e^(–E/RT) = 1. This allows us to assign a physical meaning to A since when E = 0, A = k. We can think of A as the velocity constant when the activation energy is zero – in other words when each collision between reactant molecules results in a reaction taking place.

Since there are zillions of molecular collisions taking place every second just at room temperature, any reaction in these circumstances would be uber-explosive. So the exponential term can be seen as a modifier of A whose value reflects the range of reaction velocity from extremely slow at one end of the scale (high E/low T) to extremely fast at the other (low E/high T).

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© P Mander September 2016

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From the perspective of classical thermodynamics, osmosis has a rather unclassical history. Part of the reason for this, I suspect, is that osmosis was originally categorised under the heading of biology. I can remember witnessing the first practical demonstration of osmosis in a biology class, the phenomenon being explained in terms of pores (think invisible holes) in the membrane that were big enough to let water molecules through, but not big enough to let sucrose molecules through. It was just like a kitchen sieve, we were told. It lets the fine flour pass through but not clumps. This was very much the method of biology in my day, explaining things in terms of imagined mechanism and analogy.

And it wasn’t just in my day. In 1883, JH van ‘t Hoff, an able theoretician and one of the founders of the new discipline of physical chemistry, became suddenly convinced that solutions and gases obeyed the same fundamental law, pv = RT. Imagined mechanism swiftly followed. In van ‘t Hoff’s interpretation, osmotic pressure depended on the impact of solute molecules against the semipermeable membrane because solvent molecules, being present on both sides of the membrane through which they could freely pass, did not enter into consideration.

It all seemed very plausible, especially when van ‘t Hoff used the osmotic pressure measurements of the German botanist Wilhelm Pfeffer to compute the value of R in what became known as the van ‘t Hoff equation

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where Π is the osmotic pressure, and found that the calculated value for R was almost identical with the familiar gas constant. There really did seem to be a parallelism between the properties of solutions and gases.

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JH van ‘t Hoff (1852-1911)

The first sign that there was anything amiss with the so-called gaseous theory of solutions came in 1891 when van ‘t Hoff’s close colleague Wilhelm Ostwald produced unassailable proof that osmotic pressure is independent of the nature of the membrane. This meant that hypothetical arguments as to the cause of osmotic pressure, such as van ‘t Hoff had used as the basis of his theory, were inadmissible.

A year later, in 1892, van ‘t Hoff changed his stance by declaring that the mechanism of osmosis was unimportant. But this did not affect the validity of his osmotic pressure equation ΠV = RT. After all, it had been shown to be in close agreement with experimental data for very dilute solutions.

It would be decades – the 1930s in fact – before the van ‘t Hoff equation’s formal identity with the ideal gas equation was shown to be coincidental, and that the proper thermodynamic explanation of osmotic pressure lay elsewhere.

But long before the 1930s, even before Wilhelm Pfeffer began his osmotic pressure experiments upon which van ‘t Hoff subsequently based his ideas, someone had already published a thermodynamically exact rationale for osmosis that did not rely on any hypothesis as to cause.

That someone was the American physicist Josiah Willard Gibbs. The year was 1875.

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J. Willard Gibbs (1839-1903)

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Osmosis without mechanism

It is a remarkable feature of Gibbs’ On the Equilibrium of Heterogeneous Substances that having introduced the concept of chemical potential, he first considers osmotic forces before moving on to the fundamental equations for which the work is chiefly known. The reason is Gibbs’ insistence on logical order of presentation. The discussion of chemical potential immediately involves equations of condition, among whose different causes are what Gibbs calls a diaphragm, i.e. a semipermeable membrane. Hence the early appearance of the following section

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In equation 77, Gibbs presents a new way of understanding osmotic pressure. He makes no hypotheses about how a semipermeable membrane might work, but simply states the equations of condition which follow from the presence of such a membrane in the kind of system he describes.

This frees osmosis from considerations of mechanism, and explains it solely in terms of differences in chemical potential in components which can pass the diaphragm while other components cannot.

In order to achieve equilibrium between say a solution and its solvent, where only the solvent can pass the diaphragm, the chemical potential of the solvent in the fluid on both sides of the membrane must be the same. This necessitates applying additional pressure to the solution to increase the chemical potential of the solvent in the solution so it equals that of the pure solvent, temperature remaining constant. At equilibrium, the resulting difference in pressure across the membrane is the osmotic pressure.

Note that increasing the pressure always increases the chemical potential since

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is always positive (V1 is the partial molar volume of the solvent in the solution).

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Europe fails to notice (almost)

Gibbs published On the Equilibrium of Heterogeneous Substances in Transactions of the Connecticut Academy. Choosing such an obscure journal (seen from a European perspective) clearly would not attract much attention across the pond, but Gibbs had a secret weapon. He had a mailing list of the world’s greatest scientists to which he sent reprints of his papers.

One of the names on that list was James Clerk Maxwell, who instantly appreciated Gibbs’ work and began to promote it in Europe. On Wednesday 24 May 1876, the year that ‘Equilibrium’ was first published, Maxwell gave an address at the South Kensington Conferences in London on the subject of Gibbs’ development of the doctrine of available energy on the basis of his new concept of the chemical potentials of the constituent substances. But the audience did not share Maxwell’s enthusiasm, or in all likelihood share his grasp of Gibbs’ ideas. When Maxwell tragically died three years later, Gibbs’ powerful ideas lost their only real champion in Europe.

It was not until 1891 that interest in Gibbs masterwork would resurface through the agency of Wilhelm Ostwald, who together with van ‘t Hoff and Arrhenius were the founders of the modern school of physical chemistry.

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Wilhelm Ostwald (1853-1932) He not only translated Gibbs’ masterwork into German, but also produced a profound proof – worthy of Sadi Carnot himself – that osmotic pressure must be independent of the nature of the semipermeable membrane.

Although perhaps overshadowed by his colleagues, Ostwald had a talent for sensing the direction that the future would take and was also a shrewd judge of intellect – he instinctively felt that there were hidden treasures in Gibbs’ magnum opus. After spending an entire year translating ‘Equilibrium’ into German, Ostwald wrote to Gibbs:

“The translation of your main work is nearly complete and I cannot resist repeating here my amazement. If you had published this work over a longer period of time in separate essays in an accessible journal, you would now be regarded as by far the greatest thermodynamicist since Clausius – not only in the small circle of those conversant with your work, but universally—and as one who frequently goes far beyond him in the certainty and scope of your physical judgment. The German translation, hopefully, will more secure for it the general recognition it deserves.”

The following year – 1892 – another respected scientist sent a letter to Gibbs regarding ‘Equilibrium’. This time it was the British physicist, Lord Rayleigh, who asked Gibbs:

“Have you ever thought of bringing out a new edition of, or a treatise founded upon, your “Equilibrium of Het. Substances.” The original version though now attracting the attention it deserves, is too condensed and too difficult for most, I might say all, readers. The result is that as has happened to myself, the idea is not grasped until the subject has come up in one’s own mind more or less independently.”

Rayleigh was probably just being diplomatic when he remarked that Gibbs’ treatise was ‘now attracting the attention it deserves’. The plain fact is that nobody gave it any attention at all. Gibbs and his explanation of osmosis in terms of chemical potential was passed over, while European and especially British theoretical work centered on the more familiar and more easily understood concept of vapor pressure.

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Gibbs tries again

Although van ‘t Hoff’s osmotic pressure equation ΠV = RT soon gained the status of a law, the gaseous theory that lay behind it remained clouded in controversy. In particular, van ‘t Hoff’s deduction of the proportionality between osmotic pressure and concentration was an analogy rather than a proof, since it made use of hypothetical considerations as to the cause of osmotic pressure. Following Ostwald’s proof that these were inadmissible, the gaseous theory began to look hollow. A better theory was needed.

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Lord Kelvin (1824-1907) and Lord Rayleigh (1842-1919)

This was provided in 1896 by the British physicist, Lord Rayleigh, whose proof was free of hypothesis but did make use of Avogadro’s law, thereby continuing to assert a parallelism between the properties of solutions and gases. Heavyweight opposition to this soon materialized from the redoubtable Lord Kelvin. In a letter to Nature (21 January 1897) he charged that the application of Avogadro’s law to solutions had “manifestly no theoretical foundation at present” and further contended that

“No molecular theory can, for sugar or common salt or alcohol, dissolved in water, tell us what is the true osmotic pressure against a membrane permeable to water only, without taking into account laws quite unknown to us at present regarding the three sets of mutual attractions or repulsions: (1) between the molecules of the dissolved substance; (2) between the molecules of water; (3) between the molecules of the dissolved substance and the molecules of water.”

Lord Kelvin’s letter in Nature elicited a prompt response from none other than Josiah Willard Gibbs in America. Twenty-one years had now passed since James Clerk Maxwell first tried to interest Europe in the concept of chemical potentials. In Kelvin’s letter, with its feisty attack on the gaseous theory, Gibbs saw the opportunity to try again.

In his letter to Nature (18 March 1897), Gibbs opined that “Lord Kelvin’s very interesting problem concerning molecules which differ only in their power of passing a diaphragm, seems only to require for its solution the relation between density and pressure”, and highlighted the advantage of using his potentials to express van ‘t Hoff’s law:

“It will be convenient to use certain quantities which may be called the potentials of the solvent and of the solutum, the term being thus defined: – In any sensibly homogeneous mass, the potential of any independently variable component substance is the differential coefficient of the thermodynamic energy of the mass taken with respect to that component, the entropy and volume of the mass and the quantities of its other components remaining constant. The advantage of using such potentials in the theory of semi-permeable diaphragms consists partly in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm, and partly in our ability to express van’t Hoff law as a relation between the quantities characterizing the state of the solution, without reference to any experimental arrangement.”

But once again, Gibbs and his chemical potentials failed to garner interest in Europe. His timing was also unfortunate, since British experimental research into osmosis was soon to be stimulated by the aristocrat-turned-scientist Lord Berkeley, and this in turn would stimulate a new band of British theoreticians, including AW Porter and HL Callendar, who would base their theoretical efforts firmly on vapor pressure.

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Things Come Full Circle

As the new century dawned, van ‘t Hoff cemented his reputation with the award of the very first Nobel Prize for Chemistry “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”.

The osmotic pressure law was held in high esteem, and despite Lord Kelvin’s protestations, Britain was well disposed towards the Gaseous Theory of Solutions. 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 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.”

Lord Berkeley’s landmark experimental studies on the osmotic pressure of concentrated solutions called renewed attention to the subject among theorists, who now had some fresh and very accurate data to work with. Alfred Porter at University College London attempted to make a more complete theory by considering the compressibility of a solution to which osmotic pressure was applied, while Hugh Callendar at Imperial College London combined the vapor pressure interpretation of osmosis with the hypothesis that osmosis could be described as vapor passing through a large number of fine capillaries in the semipermeable membrane. This was in 1908.

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H L Callendar (1863-1930)

So seventeen years after Wilhelm Ostwald conclusively proved that hypothetical arguments as to the cause of osmotic pressure were inadmissible, things came full circle with hypothetical arguments once more being advanced as to the cause of osmotic pressure.

And as for Gibbs, his ideas were as far away as ever from British and European Science. The osmosis papers of both Porter (1907) and Callendar (1908) are substantial in referenced content, but nowhere do either of them make any mention of Gibbs or his explanation of osmosis on the basis of chemical potentials.

There is a special irony in this, since in Callendar’s case at least, the scientific papers of J Willard Gibbs were presumably close at hand. Perhaps even on his office bookshelf. Because that copy of Gibbs’ works shown in the header photo of this post – it’s a 1906 first edition – was Hugh Callendar’s personal copy, which he signed on the front endpaper.

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Hugh Callendar’s signature on the endpaper of his personal copy of Gibbs’ Scientific Papers, Volume 1, Thermodynamics.

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Epilogue

Throughout this post, I have made repeated references to that inspired piece of thinking by Wilhelm Ostwald which conclusively demonstrated that osmotic pressure must be independent of the nature of the membrane.

Ostwald’s reasoning is so lucid and compelling, that one wonders why it didn’t put an end to speculation on osmotic mechanisms. But it didn’t, and hasn’t, and probably won’t.

Here is how Ostwald presented the argument in his own Lehrbuch der allgemeinen Chemie (1891). Enjoy.

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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 realised 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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P Mander July 2015

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8 octobre 1850 – 17 septembre 1936

History

Le Châtelier’s principle is unusual in that it was conceived as a generalization of a principle first stated by someone else.

In 1884, the Dutch theoretician JH van ‘t Hoff published a work entitled Etudes de Dynamique Chimique [Studies in Chemical Dynamics]. In it, he stated a principle drawn from observations of different forms of equilibrium:

“Lowering the temperature displaces the equilibrium between two different conditions of matter (systems) towards the system whose formation produces heat.”

The converse statement was also implied, leading van ‘t Hoff to the realization that application of the principle made it possible “to predict the direction in which any given chemical equilibrium will be displaced at higher or lower temperatures.”

A few months after the publication of the Etudes, the following note appeared on page 786 of volume 99 of Comptes-rendus de l’Academie des Sciences:

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The note covers two pages, but the crucial paragraph is the one shown immediately above, in which Le Châtelier extends van ‘t Hoff’s recently published principle to include pressure and (in modern terms) chemical potential. Rendered in English, the paragraph reads

“Any system in stable chemical equilibrium, subjected to the influence of an external cause which tends to change either its temperature or its condensation (pressure, concentration, number of molecules in unit volume), either as a whole or in some of its parts, can only undergo such internal modifications as would, if produced alone, bring about a change of temperature or of condensation of opposite sign to that resulting from the external cause.”

Just as van ‘t Hoff used inductive reasoning to relate temperature change to displacement of equilibrium, so Le Châtelier adopts the same technique to extend the principle to changes of pressure and potential.

Having arrived at a generalized principle – that systems in stable equilibrium tend to counteract changes imposed on them – Le Châtelier then sought to deduce this result mathematically from equations describing systems in equilibrium. During this quest, he discovered that the American physicist Josiah Willard Gibbs had done a good part of the groundwork in his milestone monograph On The Equilibrium of Heterogeneous Substances (1876-1878). In 1899, Le Châtelier translated this hugely difficult treatise into French, thereby helping many scientists in France and beyond to access Gibbs’ powerful ideas.

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Early misunderstandings

Le Châtelier’s principle, first stated in 1884 and extended as the Le Châtelier-Braun principle in 1887, has stood the test of time. Today we view it as a very useful law, but that was not how it was viewed by some of the academic establishment in the early 20th century. Critics including the illustrious Paul Ehrenfest and Lord Rayleigh regarded the principle as vaguely worded and impossible to apply without ambiguity. As late as 1937, Paul Epstein in his Textbook of Thermodynamics wrote that this criticism “has been generally accepted since”.

This was news to me; when I was taught Le Châtelier’s principle at school, the wording was the same as in Epstein’s day but we had no issues with vagueness or ambiguity. I wondered what this criticism was all about, so I delved into the online archive of ancient journals. And came up with this:

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From J Chem Soc, 1917; vol 111. CarnotCycle hopes that the misspelling of Braun in the title was a genuine typo, and not the deliberate use of irony to mock the authors of the principle.

It is clear from the first paragraph that the charge of ambiguity by Ehrenfest and Rayleigh arose from a failure to distinguish between cause and effect. Perturbations of systems in stable chemical equilibrium are caused by changes in generalized forces which, as Le Châtelier documents, are intensive variables. The ‘response of the system’, or generalized displacements, are the extensive conjugates. This answers Rayleigh’s question as to why we are to choose the one (pressure) rather than the other (volume) as the independent variable.

What surprised me was that this misunderstanding persisted for three decades. It just goes to show that in thermodynamics, even the most perspicacious individuals can have enduring blind spots.

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The Principle behind the Principle

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In the Etudes of 1884, van ‘t Hoff stated his principle on the basis of different observations of equilibrium displacement with temperature. But while reaching his conclusion inductively, he still managed to give a precise mathematical expression of the principle. In modern notation it reads:

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This famous equation, sometimes called the van ‘t Hoff isochore, was stated without proof in the 1884 edition, but in the second edition of 1896 a proof was provided which is based – as with many proofs of that era – on a reversible cycle of operations involving heat and work.

Although thermodynamically exact, the equation provides little insight into why a system in stable equilibrium tends to resist actions which alter that state. Not that this would have bothered van ‘t Hoff, who was much more interested in practicality than philosophical pondering.

But in the early 1900s, physical chemists began to look for an explanation. In A Textbook of Thermodynamics with special reference to Chemistry (1913), J.R. Partington remarked that Le Châtelier’s principle is an expression of “a very general theorem … called the Principle of Least Action. We can state it in the form that, if the system is in stable equilibrium, and if anything is done so as to alter this state, then something occurs in the system itself which tends to resist the change, by partially annulling the action imposed on the system.”

Partington was hinting at a more general notion underlying Le Châtelier’s original description. That notion was more concisely expressed in another volume entitled A Textbook of Thermodynamics, written by Frank Ernest Hoare in 1931, in which he stated “every system in equilibrium is conservative”.

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Interlude : Mapping chemical reactions

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It is one the conditions of stable equilibrium in thermodynamic systems that for a given temperature and pressure, the Gibbs free energy is a minimum. In the context of a chemical reaction, it means that the Gibbs free energy of the reaction mixture will decrease in the manner shown above, where the difference between P (pure products) and R (pure reactants) is the standard free energy of reaction and E is the equilibrium point at the minimum point of the curve.

If the reactants are initially present in stoichiometric proportions, the x-axis represents the mole fraction of products in the reaction mixture. In 1920, a Belgian mathematician and physicist called Théophile de Donder proposed another name for this dimensionless extensive variable. He called it “the degree of advancement of a chemical reaction”, and represented it by the Greek letter ξ (xi).

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Defining conservative behavior

In 1937, Professor Mark Zemansky – at the time an associate professor of physics at what was then called the College of the City of New York – published a textbook entitled Heat and Thermodynamics.

In the last section of the last chapter of the book, Zemansky turns his attention to Le Châtelier’s principle. He considers a heterogeneous chemical reaction which is in phase equilibrium but not chemical equilibrium; under these circumstances the Gibbs free energy G is a function of temperature T, pressure P and degree of advancement ξ.

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When the chemical reaction reaches stable equilibrium at temperature T and pressure P, it follows that ∂G/∂ξ = 0. Zemansky then considers a neighboring equilibrium state at temperature T+dT and pressure P+dP. The new degree of reaction will be ξ+dξ, but the change in the slope of the curve during this process is zero. Therefore

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Zemansky thus arrives at a mathematical definition of conservative behavior for a thermodynamic system consisting of a reaction mixture in stable equilibrium with respect to the reaction to which ξ refers.

The next task is to use the operations of calculus to find expressions for the derivatives ∂ξ/∂T and ∂ξ/∂P in terms of ΔS (=ΔH/T) and ΔV respectively. The first step is to write out fully the condition on dT, dP and dξ required to maintain conservative behavior:

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Zemansky then employs a neat device to introduce S and V into the calculation. The order of differentiation of a state function is immaterial, so he reverses the order of differentiation in the first two terms

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Since (∂G/∂T)P,ξ = –S and (∂G/∂P)T,ξ = V,

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For the sake of brevity, I will introduce at this point a shortcut that Zemansky did not use, but which does not in any way alter the results of his reasoning.

For any extensive property X which varies according to the degree of advancement of a chemical reaction ξ at constant temperature and pressure, the slope of the curve has the following property

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Applying this fact to the above equation, we find that in order to maintain the equilibrium condition ∂G/∂ξ=0, dT, dP and dξ must be such that

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Setting dP=0 yields the result

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When ΔG=0, the denominator is positive. At equilibrium therefore, (∂ξ/∂T)P and ΔH have the same sign. So for an endothermic reaction (positive ΔH) the degree of reaction advancement at equilibrium increases as the temperature increases. This accords with Le Châtelier’s principle.

Setting dT=0 yields the result

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When ΔG=0, the denominator is positive. At equilibrium therefore, (∂ξ/∂P)T and ΔV have opposite signs. For a reaction resulting in a reduction of volume, the degree of reaction advancement at equilibrium increases as the pressure increases. This accords with Le Châtelier’s principle.

Zemansky thus demonstrates that deductions from a mathematical definition of conservative behavior for a thermodynamic system consisting of a reaction mixture in stable equilibrium result in equations which “express in a rigorous form that part of Le Châtelier’s principle which concerns chemical reaction in heterogeneous systems”.

Le Châtelier never got to see this deduction of his principle. He died in 1936, just a year before Zemansky’s book was published.

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P Mander February 2016

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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

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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

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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

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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.

ae08

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

ts01

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.

ts02

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.

ts03

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

ts05

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

ts07

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:

ts08

“… 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

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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P Mander June 2015