Posts Tagged ‘Gay-Lussac’s Law’


JH van ‘t Hoff’s laboratory in Amsterdam

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

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

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

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


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

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Jumping to conclusions

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


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

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

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

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

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Compounding the error

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

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

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


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

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

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Misguided research


Lord Berkeley (1865-1942)

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

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

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


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

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

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Brilliant reasoning


Wilhelm Ostwald (1853-1932)

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

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


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

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

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

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


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Mouse-over links to works referred to in this post

Jacobus Henricus van ‘t Hoff Studies in Chemical Dynamics

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

Lord Berkeley On some Physical Constants of Saturated Solutions


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.


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.


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


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


Substituting Vx in the original equation:




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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.”


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:


Experiments on real gases show that


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


so for real gases


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


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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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:


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



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.


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


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



Since the Joule-Thomson coefficient (μJT) is defined


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,


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


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:


Integrating between the ice point and the steam point


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


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


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


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:


Integrating between the ice point and the steam point


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.