Posts Tagged ‘thermodynamics’

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Tottenham Court Road, London WC2 in 1880

In the study of chemical reactions, thermodynamics enables us to calculate changes in state functions such as enthalpy, entropy and free energy, and determine the direction in which a reaction is spontaneous. But it tells us nothing about the speed of reaction; that is the province of chemical kinetics. Thermodynamics and chemical kinetics can be viewed as complementary disciplines, which together provide the means by which the course of a reaction can be elucidated.

A classic case which exemplifies the dual application of thermodynamics and chemical kinetics is the Tottenham Court Road gas explosion which occurred in July 1880.

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The incident

It was a time of great expansion of the network for gas pipeline transport in London. Gas lighting of streets and buildings was well-established, but now the gas stove was about to become a commercial success, and new gas mains were being laid to supply the anticipated demand.

The Gas Light and Coke Company, which supplied coal gas from a number of gasworks in London, had laid a new 1.2 kilometer (0.75 mile) section of main from Bedford Square to Fitzroy Square, the pipeline crossing Tottenham Court Road at the junction with Bayley Street and running along Percy Street before turning north along the entire length of Charlotte Street.

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On the evening of Monday 5th July 1880, workmen were preparing to connect the new main to the existing network at Bayley Street. Unknown to them however, a faulty valve at the other end of the new main was leaking coal gas, which had mingled with the air in the pipe to form an explosive mixture. In a presumed act of carelessness by one of the workmen at Bayley Street, a flame or other ignition source came in close proximity to the pipe.

The gas mixture detonated and the explosion ripped through the entire length of the new 1.2 kilometer main. A number of people were killed and injured in the blast, and 400 houses were damaged by flying debris. The entire incident lasted about 12 seconds.

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The investigation

A singularly worrying feature of the Tottenham Court Road gas explosion was that it had ripped through over a kilometer of pipeline in a matter of seconds. How could this happen? And how easily could this happen again? For the safety of millions of Londoners, answers had to be found.

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Augustus Vernon Harcourt (1834-1919)

The authorities turned to one the country’s leading chemists, Augustus Vernon Harcourt, who was conducting a program of research in chemical kinetics at Oxford University. Together with his student Harold Baily Dixon (1852-1930), Harcourt began to investigate the rates of propagation of gaseous explosions.

In what sounds like a rather risky experiment, they set up long metal pipes under the Dining Hall of Balliol College Oxford to measure the speed with which explosion waves travel when a mixture of air and coal gas detonates.

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The Dining Hall of Balliol College, Oxford

Twenty three years earlier, the German chemist Robert Bunson (of Bunsen burner fame) had investigated the rate of propagation for the ignition of coal gas and oxygen and concluded that the flame front velocity was less than 1 meter per second. From the experiments at Balliol however, Harcourt and Dixon arrived at a very different answer. In a report to the Board of Trade on the Tottenham Court Road blast, Harcourt concluded that the velocity of a coal gas/air explosion wave exceeded 100 yards per second (91 meters per second).

From the safety point of view, Harcourt and Dixon had shown how absolutely essential it was to prevent air becoming mixed with coal gas in the gas pipeline network. But it would take decades before sufficient theoretical progress was made to allow a detailed understanding of what exactly happened in the great gas explosion of 1880.

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Branching chains

The development of chemical kinetics involved many different contributors in the decades after Harcourt and Dixon’s pioneering work at Oxford. Theories were advanced on several different aspects of the subject, but one piece of theoretical work had particular relevance to the study of explosions.

In 1921, a Danish physical chemist by the name of Jens Anton Christiansen (1888-1969) completed his PhD studies in reaction kinetics at Copenhagen University. In his thesis he incorporated an idea first suggested by Bodenstein in 1913 and introduced the term “kædereaktion”. This term, and the conceptual idea behind it, attracted considerable attention and the equivalent English expression “chain reaction” came into use. Two years later, Christiansen and the Dutch physicist Hendrick Anthony Kramers (1894-1952) published a paper in which they suggested the possibility of branching chains. Their idea was that a chain reaction could involve steps in which one chain carrier (an atom or radical) might not only regenerate itself but also produce an additional chain carrier. If such chain branching occurred, the number of chain carriers could increase extremely rapidly and result in an explosion.

The idea proved to be well-founded, and was further developed by Nikolai Semyonov (1896-1986) and Cyril Norman Hinshelwood (1897-1967). Their work also showed that chain carriers were removed at the walls of the reaction vessel. If the rate of removal of the chain carriers was fast enough to counteract the effect of chain branching, a steady reaction ensued. But if the removal rate could not keep pace with the chain branching rate, an explosion would result.

On the basis of their thinking, the reaction rate expression assumed the form

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where F is a function of the concentrations characteristic of the chain branching step, fa is a function determining the removal of chain carriers, and fb is a function expressing the branching nature of the chain reaction.

In steady reaction conditions, fa is sufficiently greater than fb. But if conditions change so that fa and fb converge, a point will be reached where the difference between them becomes vanishingly small. The reaction rate will soar towards infinity however small F may be, and the evolution of heat in the system will be so great as to cause an explosion.

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Semyonov and Hinshelwood were awarded the Nobel Prize in 1956 for their work on reaction rates

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Piecing the facts together

From the information contained in newspaper reports, and the application of kinetic theory and thermodynamics, it is possible to arrive at a likely explanation of why the great gas explosion of 1880 happened in the way it did.

It is known that coal gas leaked into the newly laid main at its northern end, and that detonation occurred at the other end in Bayley Street. From this it can be inferred that the entire pipeline between these two points contained coal gas admixed with the air that the pipe originally contained. On the assumption that the leaking valve was introducing coal gas at a modest and steady rate, it is likely that the partial pressures of the gases in the pipe were being brought into equilibrium as the coal gas seeped along the pipe.

Newspaper reports stated that the new main between Bayley Street and Fitzroy Square was a metal pipe of fixed (3 ft/0.91 m) diameter. The ratio of the surface area to the enclosed volume or, which is the same thing, the ratio of the circumference to the cross-sectional area

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was therefore constant along its length*.

*assuming the geometry of the bend had no effect on fa. This point is examined later.

At the moment of detonation at Bayley Street, it is a reasonable hypothesis that the function F in the Semyonov-Hinshelwood rate expression was not subject to large variations along the length of the new main. The same can be said of fb, and since the ratio of the circumference to the cross-sectional area of the pipeline was constant, the function fa determining the removal of chain carriers at the walls of the pipe was also constant. In short, the reaction rate expression applying at the end of the pipe – where detonation is known to have occurred – applied at every other point along its length.

At this juncture, it is convenient to recall the combustion reactions of the principal components of coal gas, namely hydrogen, methane and carbon monoxide:

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We observe that from a stoichiometric perspective, none of the reactions involves an increase in volume; in fact two of them result in a decrease. The overall entropy of reaction is negative, and this tells us that the conversion of reactants into products, however rapidly it took place, could not in itself have resulted in any pressure increase under the constant volume conditions of the pipe.

From an enthalpy of reaction perspective however, the situation is very different. The above reactions are all significantly exothermic processes – the calorific value of coal gas is typically around 20 megajoules per cubic meter. In the circumstances of detonation, the virtually instantaneous release of a large amount of heat would result in a similarly rapid rise in temperature, causing sudden compression of the adjacent volume element in the pipe and heating it to the point of detonation. This sequence would be repeated from one volume element to the next, with a wave of adiabatic compression intensifying the pressure as it traversed the pipe. A continuously propagating explosion would then follow the pressure wave along the course of the main as the pipe ruptured.

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The bend in the pipe

The junction of Percy Street with Charlotte Street was the only point along the entire length of the new main which deviated from a straight line. Here the pipeline executed a 90 degree turn, and it raises the question of how a detonation wave can go round corners. The exact construction of the bend is not recorded, but it is likely that an elbow joint was used.

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Geometrically, the bend itself is a quadrant of a torus, whose geometry is such that regardless of whether the elbow has a long or short major radius R, the ratio of the surface area to the enclosed volume is constant

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This is the same ratio as that of the straight pipe. The bend at the junction of Percy Street with Charlotte Street introduced no changes to the fa term in the Semyonov-Hinshelwood rate expression, and thus the conditions for detonation were met at every point of the bend.

So the 90 degree elbow made no difference to the detonation wave. It simply turned sharp right and carried on up to Fitzroy Square, at a velocity of almost 100 meters per second.

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Estimating the power of the explosion

It is known from the analysis of coal gas that one volume of coal gas requires approximately 10 volumes of air for its complete combustion. This means that an explosive mixture with air cannot be formed at coal gas concentrations much above 9%, since there would be insufficient oxygen to support the necessary rate of reaction. Below 7% coal gas concentration, the mixture is also non-explosive, for other reasons.

An average coal gas concentration of 8% throughout the pipeline is therefore a fair estimate, and seems plausible given that the new main contained air when laid and that coal gas was introduced at a modest rate from a leaking valve. We know that the new 1.2 kilometer main had a radius of 0,455 meters, giving a total volume of 780 cubic meters. At the moment of detonation, coal gas is estimated to have filled 8% of this volume i.e. 62 cubic meters. The calorific value of coal gas is typically 20 megajoules per cubic meter, so we can conclude that the Tottenham Court Road gas explosion released around 1,240 MJ in the 12 seconds it took to traverse the pipeline. The power of the explosion was therefore 1240/12 = 103 MW.

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The 3×2 flagstones used on London sidewalks weigh around 70 kg each. The energy released by the Great Gas Explosion of 1880 was sufficient to blast 59,000 flagstones to a height of 30 meters.

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Contemporary accounts

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Charlotte Street after the blast

Newspaper accounts remarked on the rapid progression of the explosion, with one commenting:

“[The main pipe at Bayley Street] burst with a terrific report, and sheets of flame issued suddenly from the earth. Instantly the report seemed to run along Percy Street, which was torn up for sixty or seventy yards (ca. 60 meters), the paving stones flying on each side against the houses.”

“At the corner of Charlotte Street the basements of two houses were shattered. The paving stones were here also sent into the air, falling on and through the roofs of the houses opposite. Further on, the pipe burst again, near the corner of Bennett Street, where there is a large gap in the roadway. Another burst-up occurred near the corner of Howland Street, and at the corner of London Street (now Maple Street) still further on…”

One eye-witness was in Percy Street when the explosion occurred. He experienced the effect of not only the pressure wave from the bursting pipe, but also the decompression wave which followed in its wake:

“I was walking down Percy Street, when I felt the ground shaking under my feet. I immediately saw the centre of the street rising in the air. A tremendous report followed, and then there was a shower of bricks and stones. I felt myself lifted from the ground, and the next moment I was lying among the debris at the bottom of a deep hole in the roadway.”

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P Mander December 2015

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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

William Nicholson and Anthony Carlisle

May 1800: Carlisle (left) and Nicholson discover electrolysis

The two previous posts on this blog concerning the leaking of details about the newly-invented Voltaic pile to Anthony Carlisle and William Nicholson, and their subsequent discovery of electrolysis, are more about the path of temptation and birth of electrochemistry than about classical thermodynamics. In fact there was no thermodynamic content at all.

So by way of steering this set of posts back on track, I thought I would apply contemporary thermodynamic knowledge to Carlisle and Nicholson’s 18th century activities, in order to give another perspective to their famous experiments.

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The Voltaic pile

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Z = zinc, A = silver

In thermodynamic terms, Alessandro Volta’s fabulous invention – an early form of battery – is a system capable of performing additional work other than pressure-volume work. The extra capability can be incorporated into the fundamental equation of thermodynamics by adding a further generalised force-displacement term: the intensive variable is the electrical potential E, whose conjugate extensive variable is the charge Q moved across that potential

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hence

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At constant temperature and pressure, the left hand side identifies with dG. For an appreciable difference therefore

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where E is the electromotive force of the cell, Q is the charge moved across the potential, and ΔGrxn is the free energy change of the reaction taking place in the battery.

For one mole of reaction, Q = nF where n is the number of moles of electrons transferred per mole of reaction, and F is the total charge on a mole of electrons, otherwise known as the Faraday. For a reaction to occur spontaneously at constant temperature and pressure, ΔGrxn must be negative and so the EMF must be positive. Under standard conditions therefore

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The redox reaction which took place in the Voltaic pile constructed by Carlisle and Nicholson was

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ΔG0rxn for this reaction is –146.7 kJ/mole, and n=2, giving an EMF of 0.762 volts.

We know from Nicholson’s published paper that their first Voltaic pile consisted of “17 half crowns, with a like number of pieces of zinc”. We also know that Volta’s method of constructing the pile – which Carlisle and Nicholson followed – resulted in the uppermost and lowest discs acting merely as conductors for the adjoining discs. Thus there were not 17, but 16 cells in Carlisle and Nicholson’s first Voltaic pile, giving a total EMF of 12.192 volts.

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External work

On May 1st, 1800, Carlisle and Nicholson set up their Voltaic pile, gave themselves an obligatory electric shock, and then began experiments with an electrometer which showed “that the action of the instrument was freely transmitted through the usual conductors of electricity, but stopped by glass and other non-conductors.”

Electrical contact with the pile was assisted by placing a drop of water on the uppermost disc, and it was this action which opened the path to discovery. Nicholson records in his paper that at an early stage in these experiments, “Mr. Carlisle observed a disengagement of gas round the touching wire. This gas, though very minute in quantity, evidently seemed to me to have the smell afforded by hydrogen”.

The fact that gas was formed “round the touching wire” indicates that the contact was intermittent: when the wire was in contact with the water drop but not the zinc disc, a miniature electrolytic cell was formed and hydrogen gas was evolved at the wire cathode, while at the anode the zinc conductor was immediately oxidised as soon as the oxygen gas was formed.

In thermodynamic terms, the electrochemical cells in the pile were being used to do external work on the electrolytic cell in which the decomposition of water took place

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ΔG0rxn for this reaction is +237.2 kJ/mole. So it can be seen that the external work done by the pile consists of driving what is in effect the combustion of hydrogen in a backwards direction to recover the reactants.

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Intuitive

Carlisle and Nicholson were intuitive physical chemists. They knew that water was composed of two gases, hydrogen and oxygen, so when bubbles which smelled of hydrogen were observed in their first experiment, it immediately set them thinking. Nicholson wrote of being “led by our reasoning on the first appearance of hydrogen to expect a decomposition of water.”

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William Nicholson (1753-1815)

Nicholson used the term decomposition, so it seems safe to assume they formed the notion that just as water is composed from its constituent gases, it can be decomposed to recover them. That is a powerful conception, the idea that the combustion of hydrogen is a reversible process.

Whether Carlisle and Nicholson extended this thought to other chemical reactions, or even to chemical reactions in general, we do not know. But their demonstration of reversibility, beneath which the principle of chemical equilibrium lies, was an achievement of perhaps even greater moment than the discovery of electrolysis by which they achieved it.

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Anthony Carlisle (1768-1840)

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Redox reactions

Carlisle and Nicholson’s discovery of electrolysis was made possible by the fact that the decomposition of water into hydrogen and oxygen is a redox reaction. In fact every reaction that takes place in an electrolytic cell is a redox reaction, with oxidation taking place at the anode and reduction taking place at the cathode. The overall electrolytic reaction is thus divided into two half-reactions. In the case of the electrolysis of water, we have

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These combined half-reactions are not spontaneous. To facilitate this redox process requires EQ work, which in Carlisle and Nicholson’s case was supplied by the Voltaic pile.

Redox reactions also take place in every voltaic cell, with oxidation at the anode and reduction at the cathode. The difference is that the combined half-reactions are spontaneous, thereby making the cell capable of performing EQ work.

The spontaneous redox reactions in voltaic cells, and the non-spontaneous redox reactions in electrolytic cells, can best be understood by looking at a table of standard oxidation potentials arranged in descending order, such as the one shown below. Using such a list, the EMF of the cell is calculated by subtracting the cathode potential from the anode potential.

[Note that if you use a table of standard reduction potentials, the signs are reversed and the EMF of the cell is calculated by subtracting the anode potential from the cathode potential.]

For voltaic cells, the half-reaction taking place left-to-right at the anode (oxidation) appears higher in the list than the half-reaction taking place right-to-left at the cathode (reduction). The EMF of the cell is positive, and so ΔG will be negative, meaning that the cell reaction is spontaneous and thus capable of performing EQ work.

The situation is reversed for electrolytic cells. The half-reaction taking place left-to-right at the anode (oxidation) appears lower in the list than the half-reaction taking place right-to-left at the cathode (reduction). The EMF of the cell is negative, and so ΔG will be positive, meaning that the cell reaction is non-spontaneous and that EQ work must be performed on the cell to facilitate electrolysis.

The half-reactions of Carlisle and Nicholson’s Voltaic pile, and their platinum-electrode electrolytic cell, are indicated in the table below.

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Table of standard oxidation potentials

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The advent of the fuel cell

Anthony Carlisle and William Nicholson

If Carlisle and Nicholson had disconnected their platinum-wire electrolytic cell after bubbles of hydrogen and oxygen had formed on the respective electrodes, and then connected an electrometer across the wires, they would have added yet another momentous discovery to that of electrolysis. They would have discovered the fuel cell.

From a thermodynamic perspective, it is a fairly straightforward matter to comprehend. Under ordinary temperature and pressure conditions, the decomposition of water is a non-spontaneous process; work is required to drive the reaction shown below in the non-spontaneous direction. This work was provided by the Voltaic pile, the effect of which was to increase the Gibbs free energy of the reaction system.

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Upon disconnection of the Voltaic pile, and the substitution of a circuit wire, the reaction would spontaneously proceed in the reverse direction, decreasing the Gibbs free energy of the reaction system. This system would be capable of performing EQ work.

The reversal of reaction direction transforms the electrolytic cell into a voltaic cell, whose arrangement can be written

H2(g)/Pt | electrolyte | Pt/O2(g)

As can be seen from the above table, the EMF of this voltaic cell is 1.229 volts. We know it today as the hydrogen fuel cell.

Carlisle and Nicholson most surely created the first fuel cell in May 1800. They just didn’t apprehend it, nor did they operate it as a voltaic cell – at least we have no record that they did. So we must classify Carlisle and Nicholson’s fuel cell as an overlooked actuality; an unnoticed birth.

It would take another 42 years before a barrister from the city of Swansea in Wales, William Robert Grove QC, developed the first operational fuel cell, whose essential design features can clearly be traced back to Carlisle and Nicholson’s original.

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Mouse-over link to the original papers mentioned in this post

Nicholson’s paper (begins on page 179)

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P Mander September 2015

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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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P Mander November 2015

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If the man who almost single-handedly invented chemical thermodynamics – the American mathematical physicist Josiah Willard Gibbs – had owned an automobile, he would have had no trouble figuring out the action of antifreeze.

“The problem reduces to consideration of a binary solution in equilibrium with solid solvent,” I can hear old Josiah saying. “Such a thermodynamic system has two degrees of freedom, so at constant pressure there must be a relation between temperature and composition.”

And indeed there is. The relation corresponds to the observed depression of the freezing point of a solvent by a solute. What’s more, its exact form confirms how antifreeze really works.

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Computing chemical potential

We have Josiah Willard Gibbs to thank for introducing the concept of chemical potential (μ) as a sort of generalized force driving the flow of chemical components between coexistent phases.

When the phases are in equilibrium at constant temperature and pressure, the chemical potential of any component has the same value in each phase

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The key point to note here is that μi is the chemical potential of component i in an arbitrary state, i.e. in a mixture of components. In order to compute this potential we need to know two things: the chemical potential of the pure substance μi0 at a pressure p (such as that of the atmosphere), and the mole fraction (xi) of the component in the mixture. Assuming an ideal solution, use can then be made of the textbook formula

dce12 …(1)

With pressure and temperature fixed, this equation has a single variable (xi), from which we can draw the conclusion that the variation in chemical potential of a component in an ideal solution is determined solely by its own mole fraction.

The significance of this fact can be appreciated by considering the following diagrams

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Here is water in equilibrium with ice at 273K. The chemical potentials of the solid and liquid phases are equal; there is no net driving force in either direction. Now consider the effect of adding an antifreeze agent to the liquid phase

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Assuming the temperature held constant at 273K, the addition of antifreeze reduces the mole fraction of water, lowering its chemical potential in accordance with equation 1. The coexistent solid phase now has a higher potential, providing the driving force to transform ice into water. Since the temperature is held constant, this equates to the lowering of the freezing point of water in the mixture.

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Deducing a formula for freezing-point depression

To obtain a formula for the freezing point of water in a solution containing antifreeze, we start with the equilibrium relation

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where the zero superscript indicates a standard potential, i.e. that the solid phase consists of pure ice whose mole fraction x is unity. Substituting the left hand side with

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we obtain

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which after differentiation with respect to temperature at constant pressure and subsequent integration yields the formula for the freezing point of water in a solution containing antifreeze at 1 atmosphere pressure:

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The terms on the right are the molar enthalpy of fusion of water (ΔHf0), the freezing point of pure water (Tf0), the gas constant R and the mole fraction of water (xH2O) in the solution containing antifreeze.

The latter is the only variable, confirming that the freezing point of water in a solution containing antifreeze is determined solely by the mole fraction of water in the mixture – in other words the extent to which the water is diluted by the antifreeze agent.

This is how antifreeze works. There is nothing active about its action. It exerts its effect passively by being miscible and thereby reducing the mole fraction of water in the liquid mixture. There’s really nothing more to it than that.

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Using the formula

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Values for constants

Enthalpy of fusion of water ΔHf0 = 6.02 kJmol-1
Freezing point of pure water Tf0 = 273.15 K
Gas constant R = 0.008314 kJmol-1K-1

Example

651 grams of the antifreeze agent ethylene glycol (molecular weight 62.07) are added to 1.5 kg of water (molecular weight 18.02). What is the freezing point of water in this solution?

Strategy

1. Calculate the mole fraction of water in the solution

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Number of moles of water = 1500/18.02 = 83.2
Number of moles of ethylene glycol = 651/62.07 = 10.5
Mole fraction of water = 83.2/(83.2 + 10.5) = 0.89

2. Calculate the freezing point of water in the solution

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The solution will give antifreeze protection down to 261.65K or –11.5°C

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P Mander March 2015