Posts Tagged ‘physical chemistry’

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)

– – – –

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).

– – – –

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.

– – – –

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.

– – – –

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.

– – – –

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)

– – – –

P Mander July 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.

– – – –

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.

– – – –

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.

– – – –

Using the formula

afr09

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

– – – –

P Mander March 2015

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Historical background*

It was the American physicist Josiah Willard Gibbs (1839-1903) pictured above who first introduced the thermodynamic potentials ψ, χ, ζ which we today call Helmholtz free energy (A), enthalpy (H) and Gibbs free energy (G).

In his milestone treatise On the Equilibrium of Heterogeneous Substances (1876-1878), Gibbs springs these functions on the reader with no indication of where he got them from. Using an esoteric lexicon of Greek symbols he simply states:

Let
ψ = ε – tη
χ = ε + pv
ζ = ε – tη + pv

As with much of Gibbs’ writings, the clues to his sudden pronouncements need to be sought on other pages or – as in this case – another publication.

In an earlier paper entitled A method of geometrical representation of the thermodynamic properties of substances by means of surfaces, Gibbs shows that the state of a body in terms of its volume, entropy and energy can be represented by a surface:

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Gibbs’ thermodynamic surface of 1873, realized by James Clerk Maxwell in 1874

It can be demonstrated from purely geometrical considerations that the tangent plane at any point on this surface represents the U-related function

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Now this is none other than Gibbs’ zeta (ζ ) function. The question is, did he recognize it for what it was – a Legendre transform? A key feature of On the Equilibrium of Heterogeneous Substances is the business of finding an extremum for a multivariable function subject to various kinds of constraint, and it is known that Gibbs was familiar with Lagrange’s method of multipliers – he mentions the technique by name on page 71, immediately after equation 41. The point here is that the Legendre transformation can be phrased in the same terms – for example, the multiplier expression for finding the stationary value of U when T and P are held constant yields the Legendre transform shown above.

But suggestive though this is, it actually gets us no closer to determining whether or not Gibbs was aware that ψ, χ, ζ  were Legendre transforms. Gibbs gave no indication in his writings either that he knew the transformation trick, or that he had discovered it for himself. We can only estimate likelihoods and have hunches.

*Text revised following input from Bas Mannaerts (see comments below)

– – – –

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In the CarnotCycle thermodynamics library, the first textbook reference to Legendre transformation is by P.S. Epstein in 1937. Epstein was a Russian mathematical physicist who was recruited by Caltech in 1921. He was a renowned commentator on Gibbs’ work, especially in statistical mechanics.

– – – –

Thermodynamics and the Legendre transformation

The fundamental relation of thermodynamics dU = TdS–PdV is an exact differential expression

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where the coefficients Ci are functions of the independent variables Xi. By means of Legendre transformations (ℑ) the above expression generates three new state functions whose natural variables contain one or more Ci in place of the conjugate Xi

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The equation of the tangent plane to the thermodynamic surface generates ℑ3, with ℑ1 and ℑ2 following procedurally from

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– – – –

How the Legendre transformation works

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defines a new Y-related function Z by transforming

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into

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Proof
dZ = dY – d(C1X1)
dZ = dY – C1dX1 – X1dC1
Substitute dY with the original differential expression
dZ = C1dX1 + C2dX2 – C1dX1 – X1dC1
The C1dX1 terms cancel, leaving
dZ = C2dX2 – X1dC1

The independent (natural) variables are transformed from Y(X1,X2) to Z(X2, C1)
The same procedural principle applies to ℑ2 and ℑ3.

– – – –

The Legendre Wheel

Since exact differential expressions in two independent (natural) variables can be written for the internal energy (U), the enthalpy (H), the Gibbs free energy (G) and the Helmholtz free energy (A), each of these state functions can generate the other three via the Legendre transformations ℑ1, ℑ2, ℑ3. This is neatly demonstrated by the Legendre Wheel, which executes the transformation functions

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from any of the four starting points:

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 [click on image to enlarge]

– – – –

Legendre transformations and the Gibbs-Helmholtz equations

For an exact differential expression

leg04

the transforming function

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can be written in terms of the natural variables of Y

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This Legendre transformation is the means by which we obtain the Gibbs-Helmholtz equations. Taking Y=G(T,P) as an example, ℑ1 executes the clockwise transformation

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while the transforming function

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reverses the positions of the natural variables and executes the counterclockwise transformation

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Setting Y=G(T,P) generates six Gibbs-Helmholtz equations, in each of which one of the two natural variables is held constant. Since there are four state functions – U, H, G and A – the total number of Gibbs-Helmholtz equations generated by this procedure is twenty-four. To this can be added a parallel set of twenty-four equations where U, H, G and A are replaced by ΔU, ΔH, ΔG and ΔA.

These equations are particularly useful since they relate a state function’s dependence on either of its natural variables to an adjacent state function on the Legendre Wheel.

– – – –

Who was Legendre?

Adrien Legendre (1752-1833) was a French mathematician. He wrote a popular and influential geometry textbook Éléments de géométrie (1794) and contributed to the development of calculus and mechanics. The Legendre transformation and Legendre polynomials are named for him.

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– – – –

P Mander September 2014

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Volume One of the Scientific Papers of J. Willard Gibbs, published posthumously in 1906, is devoted to Thermodynamics. Chief among its content is the hugely long and desperately difficult “On the equilibrium of heterogeneous substances (1876, 1878)”, with which Gibbs single-handedly laid the theoretical foundations of chemical thermodynamics.

In contrast to James Clerk Maxwell’s textbook Theory of Heat (1871), which uses no calculus at all and hardly any algebra, preferring geometry as the means of demonstrating relationships between quantities, Gibbs’ magnum opus is stuffed with differential equations. Turning the pages of this calculus-laden work, one could easily be drawn to the conclusion that the writer was not a visual thinker.

But in Gibbs’ case, this is far from the truth.

The first two papers on thermodynamics that Gibbs published, in 1873, were in fact visually-led. Paper I deals with indicator diagrams and their comparative properties, while Paper II shows how the relations between the state variables V, P, T, U, S, given in analytical form by dU=TdS – PdV, may be expressed geometrically by means of a surface.

Indeed Gibbs propels the visual argument further by pointing out that analytical formulae are strictly unnecessary for comprehending relationships between thermodynamic state variables, since they can just as easily be understood by applying graphical methods.

Gibbs’ advocacy of the visual approach found instant favor with Maxwell, who in the fourth edition of Theory of Heat devoted no less than 12 pages to an illustrated discussion of Gibbs’ thermodynamic surface, including the wild diagram shown at the head of this post. Maxwell’s enthusiasm was such that he sculpted a clay model of the surface, from which he made a plaster cast and sent it to Gibbs at Yale in 1874.

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Besides his passion for using geometrical constructions to demonstrate connexions between quantities, Maxwell had an influential voice in the scientific world, and it is almost certain that he would have used it to promulgate the geometrical approach to understanding thermodynamic relationships that Gibbs had pioneered. But Maxwell’s death in 1879 at the early age of 48 brought such initiatives to a premature end.

– – – –

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James Clerk Maxwell (1831-1879) striking a pose. The studio backdrop and furnishings confirm that he was quite short in stature, but had large hands with a broad palm and relatively short fingers – the strong, practical hands of a sculptor.

The exposure that Gibbs’ thermodynamic surface gained through the agency of Maxwell proved to be short-lived; no other contemporary scientist followed Maxwell’s lead. One explanation could be that Gibbs’ visual approach lacked appeal because — for reasons best known to himself — he described it in words, not pictures. Another could be that Maxwell’s illustrations of the surface were found too difficult: a joke reportedly circulated at the time that “only one man lived who could understand Gibbs’ papers. That was Maxwell, and now he is dead.”

Whatever the actual truth, the fact remains that none of the milestone literature in the post-Maxwell period took up Gibbs’ visual approach to understanding relationships between thermodynamic properties. Instead, the approach taken in textbooks by Max Planck (1879), GH Bryan (1909), JR Partington (1913) and most importantly by Lewis & Randall (1923) and Guggenheim (1933), was analytical.

Writing in 1936, the American mathematician Edwin Wilson (who had attended Gibbs’ lectures at Yale in 1901-2) argued that Gibbs’ entropy-temperature diagram in Paper I and the thermodynamic surface in Paper II were both victims of the inevitable choices that science makes as it evolves.

He commented: “Science goes on its way, picking and choosing and modifying. The trend of the last fifty years is not towards Papers I and II. Interesting as they are historically, and important because of the preparation they afforded Willard Gibbs for writing his great memoir III [On the Equilibrium of Heterogeneous Substances], there is no present indication that they are in themselves significant for present or future science.”

– – – –

JR Partington’s fascinating Text-book of Thermodynamics (with Special Reference to Chemistry) of 1913, although presenting the subject analytically, nonetheless points out the graphical origins of Gibbs’ early discoveries.

partington

James Riddick Partington (1886-1965), whose Text-book of Thermodynamics was published just before the outbreak of the Great War. It provides a detailed and historically fascinating view of the subject in the decade before Lewis & Randall produced their watershed work.

Commenting on Paper II, Partington writes: “In this very important memoir Gibbs shows that the conditions of equilibrium of two parts of a substance in contact can be expressed geometrically in terms of the position of the tangent planes to the volume-entropy-energy surface of the substance, and he finds that the analytical expression of this property is that the value of this function (U–TS+PV) shall be the same for the two states at the same temperature and pressure.”

For those of us educated in the analytical age, it is indeed remarkable to discover that the free energy function was first obtained by Gibbs using purely graphical methods, and that the pressure-temperature equilibrium relation G(α)=G(β) between two phases of a substance in contact was originally derived from geometrical considerations.

In fact the volume-entropy-energy diagram enabled Gibbs to reach a further conclusion of great importance to his future work in thermodynamics: namely that the volume, entropy and energy of a mixture of portions of a substance in different states (whether in equilibrium or not) are the sums of the volumes, entropies and energies of the separate parts. This suggested to Gibbs that mixtures of substances differing in chemical composition, as well as physical state, might be treated in a similar manner.

It was this clue from Paper II that gave Gibbs the conceptual springboard he needed for investigating chemical equilibrium, the subject matter of Paper III – On the Equilibrium of Heterogeneous Substances.

– – – –

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Ok, so let’s take a closer look at Gibbs’ thermodynamic surface of 1873, realized by Maxwell in 1874. Each point on this surface describes the state of a body (of invariable composition) in terms of its volume, entropy and energy.

Now if we were to slice vertical sections of this surface perpendicular to the energy-volume plane, the curve of section would represent the relation between energy and entropy when the volume is constant; the tangent of the angle of slope of this curve of section is therefore (dU/dS)V. By similar reasoning, the curve of section of the surface perpendicular to the energy-entropy plane represents the relation between energy and volume when the entropy is constant. The tangent of the angle of slope of this curve of section is therefore (dU/dV)S.

From the fundamental thermodynamic relation dU = TdS – PdV, we can identify (dU/dS)V as the absolute temperature T which reckoned from zero is essentially positive, and (dU/dV)S as the pressure P which is reckoned negative when the energy U increases as the volume V increases.

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The first appearance in print of the Gibbs free energy function, as the equation of the tangent plane at any point of the (v,η,ε) surface, in “A method of geometrical representation of the thermodynamic properties of substances by means of surfaces”

The tangent plane therefore represents the same temperature and pressure at all points. Gibbs used this geometrical property of the model to show that if two points in the surface (such as ε’ and ε”) have a common tangent plane, the states they represent can exist permanently in contact. He then gave the analytical expression of this condition – that what we now know as the Gibbs free energy of states ε’ and ε” are equal. But he did not show the geometrical reasoning by which he reached his conclusion.

Maybe he thought we could work it all out in our heads, who knows. Personally I much prefer to see these things drawn – and especially in this case, for it is a rewarding exercise in solid geometry to see how the answer emerges. CarnotCycle is indebted to Ronald Kriz for making available the following explanatory diagram:

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This diagram uses the Greek letters employed by Gibbs to denote internal energy (ε) and entropy (η). Source: Ronald Kriz, private communication

The common tangent plane through states ε’ and ε” cuts the axis of energy at a single point, marked ε. Beginning with the liquid state ε’, the length ε’ε on the axis of energy is the sum of Δε’η (due to the entropy change) and Δε’v (due to the volume change).

Since the tangent plane defines t’ = Δε’η/η’ and –p’ = Δε’v/v’ we have

ε = ε’ – t’η’ + p’v’

The right hand member of this equation is composed entirely of state variables, and thus denotes a state function associated with the point ε’ on the thermodynamic surface.

Turning to the gas state ε”, the length ε”ε on the axis of energy is the sum of Δε”η (due to the entropy change) and Δε”v (due to the volume change).

Since the tangent plane defines t” = Δε”η/η” and –p” = Δε”v/v” we have

ε = ε” – t”η” + p”v”

The right hand member of this equation is composed entirely of state variables, and thus denotes a state function associated with the point ε” on the thermodynamic surface.

An identical result will be obtained for all such pairs of points on the so-called node-couple curve, the branches of which unite at the isopycnic or critical point. Since the magnitude of the state function ε–tη+pv (in modern notation U–TS+PV=G) is the same for each pair, it is demonstrated that G(ε’)=G(ε”) is the analytical expression of the condition of coexistent equilibrium of separate states of a substance of invariable composition at the same temperature and pressure.

– – – –

P Mander May 2014

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Historical background

If you received formal tuition in physical chemistry at school, then it’s likely that among the first things you learned were the 17th/18th century gas laws of Mariotte and Gay-Lussac (Boyle and Charles in the English-speaking world) and the equation that expresses them: PV = kT.

It may be that the historical aspects of what is now known as the ideal (perfect) gas equation were not covered as part of your science education, in which case you may be surprised to learn that it took 174 years to advance from the pressure-volume law PV = k to the combined gas law PV = kT.

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The lengthy timescale indicates that putting together closely associated observations wasn’t regarded as a must-do in this particular era of scientific enquiry. The French physicist and mining engineer Émile Clapeyron eventually created the combined gas equation, not for its own sake, but because he needed an analytical expression for the pressure-volume work done in the cycle of reversible heat engine operations we know today as the Carnot cycle.

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The first appearance in print of the combined gas law, in Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the Motive Power of Heat, 1834) by Émile Clapeyron

Students sometimes get in a muddle about combining the gas laws, so for the sake of completeness I will set out the procedure. Beginning with a quantity of gas at an arbitrary initial pressure P1 and volume V1, we suppose the pressure is changed to P2 while the temperature is maintained at T1. Applying the Mariotte relation (PV)T = k, we write

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The pressure being kept constant at P2 we now suppose the temperature changed to T2; the volume will then change from Vx to the final volume V2. Applying the Gay-Lussac relation (V/T)P = k, we write

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Substituting Vx in the original equation:

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whence

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Differences of opinion

In the mid-19th century, the ideal gas equation – or rather the ideal gas itself – was the cause of no end of trouble among those involved in developing the new science of thermodynamics. The argument went along the lines that since no real gas was ever perfect, was it legitimate to base thermodynamic theory on the use of a perfect gas as the working substance in the Carnot cycle? Joule, Clausius, Rankine, Maxwell and van der Waals said yes it was, while Mach and Thomson said no it wasn’t.

With thermometry on his mind, Thomson actually got quite upset. Here’s a sample outpouring from the Encyclopaedia Britannica:

“… a mere quicksand has been given as a foundation of thermometry, by building from the beginning on an ideal substance called a perfect gas, with none of its properties realized rigorously by any real substance, and with some of them unknown, and utterly unassignable, even by guess.”

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Joule (inset) and Thomson may have had their differences, but it didn’t stop them from becoming the most productive partnership in the history of thermodynamics

It seems strange that the notion of an ideal gas, as a theoretical convenience at least, caused this violent division into believers and disbelievers, when everyone agreed that the behavior of all real gases approaches a limit as the pressure approaches zero. This is indeed how the universal gas constant R was computed – by extrapolation from pressure-volume measurements made on real gases. There is no discontinuity between the measured and limiting state, as the following diagram demonstrates:

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Experiments on real gases show that

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where v is the molar volume and i signifies ice-point. The universal gas constant is defined by the equation

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so for real gases

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The behavior of n moles of any gas as the pressure approaches zero may thus be represented by

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The notion of an ideal gas is founded on this limiting state, and is defined as a gas that obeys this equation at all pressures. The equation of state of an ideal gas is therefore

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– – – –

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William Thomson, later Lord Kelvin, in the 1850s

Testing Mayer’s assumption

The notion of an ideal gas was not the only thing troubling William Thomson at the start of the 1850s. He also had a problem with real gases. This was because he was simultaneously engaged in a quest for a scale of thermodynamic temperature that was independent of the properties of any particular substance.

What he needed was to find a property of a real gas that would enable him to
a) prove by thermodynamic argument that real gases do not obey the ideal gas law
b) calculate the absolute temperature from a temperature measured on a (real) gas scale

And he found such a property, or at least he thought he had found it, in the thermodynamic function (∂U/∂V)T.

In the final part of his landmark paper, On the Dynamical Theory of Heat, which was read before the Royal Society of Edinburgh on Monday 15 December 1851, Thomson presented an equation which served his purpose. In modern notation it reads:

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This is a powerful equation indeed, since it enables any equation of state of a PVT system to be tested by relating the mechanical properties of a gas to a thermodynamic function of state which can be experimentally determined.

If the equation of state is that of an ideal gas (PV = nRT), then

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This defining property of an ideal gas, that its internal energy is independent of volume in an isothermal process, was an assumption made in the early 1840s by Julius Robert Mayer of Heilbronn, Germany in developing what we now call Mayer’s relation (Cp – CV = PΔV). Thomson was keen to disprove this assumption, and with it the notion of the ideal gas, by demonstrating non-zero values for (∂U/∂V)T.

In 1845 James Joule had tried to verify Mayer’s assumption in the famous experiment involving the expansion of air into an evacuated cylinder, but the results Joule obtained – although appearing to support Mayer’s claim – were deemed unreliable due to experimental design weaknesses.

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The equipment with which Joule tried to verify Mayer’s assumption, (∂U/∂V)T = 0. The calorimeter at the rear looks like a solid plate construction but is in fact hollow. This can be ascertained by tapping it – which the author of this blogpost has had the rare opportunity to do.

Thomson had meanwhile been working on an alternative approach to testing Mayer’s assumption. By 1852 he had a design for an apparatus and had arranged with Joule to start work in Manchester in May of that year. This was to be the Joule-Thomson experiment, which for the first time demonstrated decisive differences from ideal behavior in the behavior of real gases.

Mayer’s assumption was eventually shown to be incorrect – to the extent of about 3 parts in a thousand. But this was an insignificant finding in the context of Joule and Thomson’s wider endeavors, which would propel experimental research into the modern era and herald the birth of big science.

Curiously, it was not the fact that (∂U/∂V)T = 0 for an ideal gas that enabled the differences in real gas behavior to be shown in the Joule-Thomson experiment. It was the other defining property of an ideal gas, that its enthalpy H is independent of pressure P in an isothermal process. By parallel reasoning

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If the equation of state is that of an ideal gas (PV = nRT), then

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Since the Joule-Thomson coefficient (μJT) is defined

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and the second term on the right is zero for an ideal gas, μJT must also be zero. Unlike a real gas therefore, an ideal gas cannot exhibit Joule-Thomson cooling or heating.

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Finding a way to define absolute temperature

But to return to Thomson and his quest for a scale of absolute temperature. The equation he arrived at in his 1851 paper,

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besides enabling any equation of state of a PVT system to be tested, also makes it possible to give an exact definition of absolute temperature independently of the behavior of any particular substance.

The argument runs as follows. Given the temperature readings, t, of any arbitrary thermometer (mercury thermometer, bolometer, whatever..) the task is to express the absolute temperature T as a function of t. By direct measurement, it may be found how the behavior of some appropriate substance, e.g. a gas, depends on t and either V or P. Introducing t and V as the independent variables in the above equation instead of T and V, we have

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where (∂U/∂V)t, (∂P/∂t)V and P represent functions of t and V, which can be experimentally determined. Separating the variables so that both terms in T are on the left, the equation can then be integrated:

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Integrating between the ice point and the steam point

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This completely determines T as a function of t.

But as we have already seen, there was a catch to this argumentation – namely that (∂U/∂V) could not be experimentally determined under isothermal conditions with sufficient accuracy.

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The Joule-Thomson coefficient provides the key

Thomson’s means of circumventing this problem was the steady state Joule-Thomson experiment, which measured upstream and downstream temperature and pressure, and enabled the Joule-Thomson coefficient, μJT = (∂T/∂P)H, to be computed.

It should be borne in mind however that when Joule and Thomson began their work in 1852, they were not aware that their cleverly-designed experiment was subject to isenthalpic conditions. It was the Scottish engineer and mathematician William Rankine who first proved in 1854 that the equation of the curve of free expansion in the Joule-Thomson experiment was d(U+PV) = 0.

William John Macquorn Rankine (1820-1872)

William John Macquorn Rankine (1820-1872)

As for the Joule-Thomson coefficient itself, it was the crowning achievement of a decade of collaboration, appearing in an appendix to Joule and Thomson’s final joint paper published in the Philosophical Transactions of the Royal Society in 1862. They wrote it in the form

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where the upper symbol in the derivative denotes “thermal effect”, and K denotes thermal capacity at constant pressure of a unit mass of fluid.

The equation is now usually written

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By the method applied previously, this equation can be expressed in terms of an empirical t-scale and the absolute T-scale:

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where C’P is the heat capacity of the gas as measured on the empirical t-scale, i.e. C’P = CP(dT/dt). Cancelling (dT/dt) and separating the variables so that both terms in T are on the left, the equation becomes:

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Integrating between the ice point and the steam point

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This completely determines T as a function of t, with all the terms under the integral capable of experimental determination to a sufficient level of accuracy.

– – – –

P Mander May 2014