Posts Tagged ‘history of science’


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:

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:


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

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: November 2014)

– – – –


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. (see comments below: July 2019)

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Thermodynamics and the Legendre transformation

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

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

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

– – – –

How the Legendre transformation works

defines a new Y-related function Z by transforming


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

from any of the four starting points:


 [click on image to enlarge]

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Legendre transformations and the Gibbs-Helmholtz equations

For an exact differential expression

the transforming function

can be written in terms of the natural variables of Y

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

while the transforming function

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

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.


– – – –

P Mander September 2014


James Clerk Maxwell and the geometrical figure with which he proved his famous thermodynamic relations

Historical background

Every student of thermodynamics sooner or later encounters the Maxwell relations – an extremely useful set of statements of equality among partial derivatives, principally involving the state variables P, V, T and S. They are general thermodynamic relations valid for all systems.

The four relations originally stated by Maxwell are easily derived from the (exact) differential relations of the thermodynamic potentials:

dU = TdS – PdV   ⇒   (∂T/∂V)S = –(∂P/∂S)V
dH = TdS + VdP   ⇒   (∂T/∂P)S = (∂V/∂S)P
dG = –SdT + VdP   ⇒   –(∂S/∂P)T = (∂V/∂T)P
dA = –SdT – PdV   ⇒   (∂S/∂V)T = (∂P/∂T)V

This is how we obtain these Maxwell relations today, but it disguises the history of their discovery. The thermodynamic state functions H, G and A were yet to be created when Maxwell published the above relations in his 1871 textbook Theory of Heat. The startling fact is that Maxwell navigated his way to these relations using nothing more than a diagram of the Carnot cycle, allied to an ingenious exercise in plane geometry.

Another historical truth that modern derivations conceal is that entropy did not feature as the conjugate variable to temperature (θ) in Maxwell’s original relations; instead Maxwell used Rankine’s thermodynamic function (Φ) which is identical with – and predates – the state function entropy (S) introduced by Clausius in 1865.

Maxwell’s use of Φ instead of S was not a matter of personal preference. It could not have been otherwise, because Maxwell misunderstood the term entropy at the time when he wrote his book (1871), believing it to represent the available energy of a system. From a dimensional perspective – and one must remember that Maxwell was one of the founders of dimensional analysis – it was impossible for entropy as he understood it to be the conjugate variable to temperature. By contrast, it was clear to Maxwell that Rankine’s Φ had the requisite dimensions of ML2T-2θ-1.

Two years later, in an 1873 publication entitled A method of geometrical representation of the thermodynamic properties of substances by means of surfaces, the American physicist Josiah Willard Gibbs politely pointed out Maxwell’s error in regard to the units of measurement of entropy:


Maxwell responded in a subsequent edition of Theory of Heat with a contrite apology for misleading his readers:


– – – –

Carnot Cycle revisited

The centrepiece of the geometrical construction with which Maxwell proves his thermodynamic relations is a quadrilateral drawn 37 years earlier by Émile Clapeyron in his 1834 paper Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the motive power of heat).


When Émile Clapeyron drew this PV-plane representation of the Carnot cycle in 1834, heat was believed to be a conserved quantity. By the time Maxwell used the diagram in 1871, heat and work were understood to be interconvertible forms of energy, with energy being the conserved quantity.

This is the first analytical representation of the Carnot cycle, shown as a closed curve on a pressure-volume indicator diagram. The sides ab and cd represent isothermal lines, the sides ad and bc adiabatic lines. By assigning infinitely small values to the variations of volume and pressure during the successive operations of the cycle, Clapeyron renders this quadrilateral a parallelogram.

The area enclosed by the curve equates to the work done in a complete cycle, and Maxwell uses the following contrivance to set this area equal to unity.

Applying Carnot’s principle, Maxwell expresses the work W done as a function of the heat H supplied

W = H(T2 – T1)/T2

with T2 and T1 representing the absolute temperatures of the source and sink respectively.
Maxwell then defines

T2 – T1 = 1
H/T2 = 1

The conversion of heat into work is thus expressed as the product of a unit change in temperature T and a unit change in Rankine’s thermodynamic function Φ, equivalent to entropy S:

W = Δ1T . Δ1S = 1

Maxwell’s definitions also give the parallelogram the property that any line drawn from one isothermal line to the other, or from one adiabatic line to the other, is of unit length when reckoned in the respective dimensions of temperature or entropy. This is of central significance to what follows.

– – – –

Geometrical extensions

Maxwell’s geometric machinations consist in extending the isothermal (T1T2) and adiabatic lines (Φ1Φ2) of the original figure ABCD and adding vertical lines (pressure) and horizontal lines (volume) to create four further parallelograms with the aim of proving their areas also equal to unity, while at the same time enabling each of these areas to be expressed in terms of pressure and volume as a base-altitude product.


As the image from Theory of Heat shown at the head of this article reveals, Maxwell did not fully trace out the perimeters of three (!) of the four added parallelograms. I have extended four lines to the arbitrarily labelled points E, F and H in order to complete the figure.

– parallelogram AKQD stands on the same base AD as ABCD and lies between the same parallels T1T2 so its area is also unity, expressible in terms of volume and pressure as the base-altitude product AK.Ak

– parallelogram ABEL stands on the same base AB as ABCD and lies between the same parallels Φ1Φ2 so its area is also unity, expressible in terms of volume and pressure as the base-altitude product AL.Al

– parallelogram AMFD stands on the same base AD as ABCD and lies between the same parallels T1T2 so its area is also unity, expressible in terms of pressure and volume as the base-altitude product AM.Am

– parallelogram ABHN stands on the same base AB as ABCD and lies between the same parallels Φ1Φ2 so its area is also unity, expressible in terms of pressure and volume as the base-altitude product AN.An

– line AD, which represents a unit rise in entropy at constant temperature, resolves into the vertical (pressure) and horizontal (volume) components Ak and Am

– line AB, which represents a unit rise in temperature at constant entropy, resolves into the vertical (pressure) and horizontal (volume) components Al and An

– in summary: ABCD = AK.Ak = AL.Al = AM.Am = AN.An = 1 [dimensions ML2T-2]

– – – –

Maxwell’s thermodynamic relations

Maxwell’s next step is to interpret the physical meaning of these four pairs of lines.

AK is the volume increase per unit rise in temperature at constant pressure: (∂V/∂T)P
Ak is the pressure decrease per unit rise in entropy at constant temperature: –(∂P/∂S)T

Recalling the property of partial derivatives that given the implicit function f(x,y,z) = 0


Since AK = 1/Ak

(∂V/∂T)P = –(∂S/∂P)T

AL is the volume increase per unit rise in entropy at constant pressure: (∂V/∂S)P
Al is the pressure increase per unit rise in temperature at constant entropy: (∂P/∂T)S

Since AL = 1/Al

(∂V/∂S)P = (∂T/∂P)S

AM is the pressure increase per unit rise in temperature at constant volume: (∂P/∂T)V
Am is the volume increase per unit rise in entropy at constant temperature: (∂V/∂S)T

Since AM = 1/Am

(∂P/∂T)V = (∂S/∂V)T

AN is the pressure increase per unit rise in entropy at constant volume: (∂P/∂S)V
An is the volume decrease per unit rise in temperature at constant entropy: –(∂V/∂T)S

Since AN = 1/An

(∂P/∂S)V = –(∂T/∂V)S

– – – –

In his own words

I leave it to the man himself to conclude this post:

“We have thus obtained four relations among the physical properties of the substance. These four relations are not independent of each other, so as to rank as separate truths. Any one might be deduced from any other. The equality of the products AK, Ak &c., to the parallelogram ABCD and to each other is merely a geometrical truth, and does not depend on thermodynamic principles. What we learn from thermodynamics is that the parallelogram and the four products are each equal to unity, whatever be the nature of the substance or its condition as to pressure and temperature.”


– – – –

P Mander August 2014


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.


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.

– – – –


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.


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.

– – – –


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.


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:


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


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:


– – – –

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

– – – –


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.

– – – –

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.

– – – –

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.

– – – –

P Mander May 2014


Future Nobel Prize winners both. Kamerlingh Onnes and Johannes van der Waals in 1908.

On Friday 10 July 1908, at Leiden in the Netherlands, Kamerlingh Onnes succeeded in liquefying the one remaining gas previously thought to be non-condensable – helium – using a sequential Joule-Thomson cooling technique to drive the temperature down to just 4 degrees above absolute zero. The event brought to a conclusion the race to liquefy the so-called permanent gases, following the revelation that all gases have a critical temperature below which they must be cooled before liquefaction is possible.

This crucial fact was established by Dr. Thomas Andrews, professor of chemistry at Queen’s College Belfast, in his groundbreaking study of the liquefaction of carbon dioxide, “On the Continuity of the Gaseous and Liquid States of Matter”, published in the Philosophical Transactions of the Royal Society of London in 1869.

As described in Part I of this blog post, Andrews’ discovery of the critical temperature (aided and abetted by Joule and Thomson’s earlier discovery of isenthalpic cooling) opened the way to cryotechnological advances of great commercial importance, and gave birth to the industrial gases industry which played such a significant role in shaping the 20th century.

This fact alone was enough to ensure Dr. Andrews’ study a place in the history of physical science. But there was another aspect to his paper – a theoretical one – which had equally far-reaching effects and is the subject of the remainder of this post.

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Dr. Thomas Andrews FRS (1813-1885). Photograph taken in Paris 1875 when Andrews was 62.

Thomas Andrews was a scientist whose experimental skills were evidently comparable to those of the illustrious James Joule. Ten years of care and devotion went into Andrews’ study of the liquefaction of carbon dioxide (called carbonic acid in his day), the essential results of which are contained in this diagram taken from his 1869 paper.


It is a pressure-volume diagram (with the line of no volume to the right) upon which are drawn isothermal carbon dioxide curves for temperatures ranging from 13.1°C to 48.1°C, pressures ranging from 50 to 100 atmospheres. Isothermal air curves are included in the upper left quadrant to illustrate the degree of deviation of the carbon dioxide curves from the rectangular hyperbola associated with ideal gas behavior.

The lowest isothermal curve (13.1°C) shows that at a pressure of around 47 atmospheres, condensation occurs. The compressed gas separates into two distinct coexistent portions – vapor and liquid – along a line of constant pressure, with further compression driving the conversion to the liquid form until finally the whole is converted to liquid, at which point compressibility becomes markedly reduced.

In the next isothermal curve (21.5°C), where condensation takes place at a pressure of about 60 atmospheres, gas and liquid are closer still in density, the liquid occupying nearly a third of the volume of the gas. As James Clerk Maxwell put it in Theory of Heat, written in 1871, “the exceedingly dense gas is approaching in its properties to the exceedingly light liquid”.

These properties eventually coincide at the isopycnic point* (the point of inflexion on the critical isotherm, marked X in the figure below) corresponding to a critical pressure of 72.8 atmospheres and a critical temperature of 31°C.

*isopycnic means ‘of equal density’. The isopycnic point (sometimes called the critical point) is where the densities of vapor and liquid coincide; this occurs under the conditions of critical temperature Tc and critical pressure pc.

Above the critical temperature, isothermals do not show any discontinuity; it is not possible to detect the point at which a liquid becomes a gas or vice versa. If liquid carbon dioxide, represented by point Z in the figure below, is heated at constant pressure until its temperature reaches 48°C, its condition at different temperatures will be represented by the line ZY. At Z the substance concerned is a liquid; at Y it is a gas. The change has taken place smoothly and continuously, representing continuity of state.


It was this revelation provided by Andrews’ data that started theoreticians thinking about how to reconcile the idea of continuity of state with the discontinuous change observed experimentally within the confines of the dotted parabola shown in Fig.12.


The history of science is full of coincidences – Belfast in Northern Ireland was the birthplace of not only Thomas Andrews, but also William Thomson (later Lord Kelvin) and his elder brother James. William frequently discussed thermodynamics with James, who just happened to be professor of civil engineering at Queen’s College Belfast exactly at the time when Thomas Andrews was conducting his famous experiments there.


James Thomson (1822-1892), physicist and engineer, whose achievements were largely overshadowed by his equally hairy brother William Thomson, also known as Lord Kelvin. Photo credit: Wikipedia

Not surprisingly, James Thomson with his practised skills in thermodynamics took an active interest in Andrews’ remarkable results, and in 1871 proposed a highly original solution to the problem of reconciling the discontinuous isotherms below the isopycnic point with the continuous isotherms above it.

Thomson’s thesis was that the gaseous and liquid parts of a discontinuous isotherm (AB and CD in the above diagram) were only apparently discontinuous, and were actually parts of one smooth curve shown in dotted lines in the diagram below. Every isotherm, according to Thomson, was a continuous curve.


Diagram from “Considerations on the Abrupt Change at Boiling or Condensing in reference to the Continuity of the Fluid State of Matter” by Professor James Thomson, LL.D., Queen’s College, Belfast. Communicated to The Royal Society of London by Dr. Andrews. Received July 4, 1871.

The task that now confronted theoreticians was to find a satisfactory mathematical equation for this curve. Thomson’s curve gave some useful clues, as Lewis and Randall subsequently observed in their classic textbook Thermodynamics and the Free Energy of Chemical Substances:

“It is evident that the equation for such a complete curve must be of odd degree in V, for V increases with diminishing P at both ends of the curve. Furthermore the equation must be of at least the third degree in V, since a certain pressure may correspond to more than one volume. At lower temperatures three roots of the equation are real, at the critical point the three coincide, and at higher temperatures two of them become imaginary.”

The first to provide a solution was a physics student in the Netherlands. His name was Johannes van der Waals.

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The title page of Johannes van der Waals’ doctoral thesis

At Leiden University on Saturday 14 June 1873, from 12 noon to 3 pm, Johannes van der Waals defended his doctoral thesis, which sought to explain Thomas Andrews experimental results on the basis of kinetic theory, and whose title “Over de Continuiteit van den Gas en Vloeistoftoestand (On the Continuity of the Gaseous and Liquid State)” was almost exactly the same as Andrews’ 1869 paper.

In his thesis, van der Waals introduced the concepts of molecular attraction and molecular volume, and derived the equation of state which bears his name:

where a and b are gas-specific constants related to molecular attraction and molecular volume respectively; the term a/V2 identifies with the derivative (∂U/∂V)T while b turns out to be equal to a third of the critical volume.

Multiplying out the van der Waals equation gives

Since this expression equals zero, it follows that

Multiplying out and rearranging terms

A cubic equation in V is thus obtained. For any given values of p and T, there will be three values of V, since a, b and R are constants for one mole of a given gas.

The cubic form of van der Waals’ equation produces curves like those shown below. They are very similar to the isothermal curves hypothesized by James Thomson (cf. above), and give three values of V along the line of first-order phase transition where all the roots A,B,C are real; on the critical isotherm the roots are coincident at the isopycnic point. At higher temperatures two of the roots become imaginary as the curves become increasingly hyperbolic.


The van der Waals equation of state modifies the ideal gas equation, and is an improvement on it in accounting for the shape of pressure-volume curves above the critical isotherm. At lower temperatures it is also qualitatively reasonable for the liquid state and the low-pressure gaseous state.

During first-order phase transition (A↔C) however, the equation is clearly at variance with the empirically determined fact that the pressure remains constant. The reason why the van der Waals equation fails to describe the behavior of real substances within the dotted region in the above figure is precisely because it assumes continuity of state. It cannot therefore account for the fact that the substance, by separating into two coexistent phases – liquid and saturated vapor – is rendered more stable than in the homogeneous state.

It should be noted that under certain conditions, states corresponding to the portions AA’ and B’C are respectively realizable as superheated liquid and supersaturated vapor (both portions representing states stable with respect to infinitesimal variations but unstable relative to the coexistent liquid-vapor system). The portion of the curve A’B’, on the other hand, represents states that are absolutely unstable since

and according to the energy test of stability in rational mechanics (not to mention common sense), such states where the volume and pressure increase and diminish together are never realizable.

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Thomas Andrews’ discovery of the critical temperature provided Johannes van der Waals with the starting point for yet further theoretical insight.

It is evident from the van der Waals equation that at the critical temperature, all three values of V are identical. At the isopycnic point (p=pc, V=Vc, T=Tc) the volume V can therefore be set equal to the critical volume Vc, so that


The cubic form of the van der Waals equation gives, on dividing terms by pc:

Identifying like terms

Divide the third term by the second term to get Vc:

Substitute Vc in the third term to get pc:

Substitute Vc and pc in the first term to get Tc:

The term a/V2 in the van der Waals equation identifies with the derivative (∂U/∂V)T, since it follows from the fundamental relation of thermodynamics (dU =TdS – pdV) that

Using the Maxwell relation



Taking the van der Waals equation

and differentiating with respect to temperature at constant volume

Substituting in (1)

The derivative (∂U/∂V)T can be computed from the experimentally determined Joule-Thomson coefficient:

Cp may be obtained calorimetrically or spectroscopically, while (∂V/∂P)T and (∂(PV)/∂P)T can be obtained from data on the compressibility of the gas at constant temperature.

Hence (∂U/∂V)T = a/V2 can be computed for any gas, enabling the constants a and b in the van der Waals equation to be determined. This in turn allows the three critical constants, Vc, pc, Tc, to be calculated.

These critical data – for which van der Waals provided further means of estimation in 1880 with his “principle of corresponding states” – were invaluable in helping Dewar’s determination of the method of liquefying hydrogen (Tc = 33K) in 1898, and Onnes’ determination of the method of liquefying helium (Tc = 5.2K) in 1908.

The van der Waals equation constants a and b also proved useful in the early days of the industrial gases industry. Taking the equation deduced in 1862 by Joule and Thomson for the temperature change ΔT when a gas is subjected to a pressure drop Δp under isenthalpic conditions:

If we apply the van der Waals equation as the equation of state, the Joule-Thomson equation becomes

If a and b become sufficiently small

For most gases the expression in brackets is positive at not-too-high temperatures. A cooling effect is therefore obtained, since Δp is always negative. Carl von Linde, who in 1895 established the first large-scale air liquefaction plant, based the construction of his Joule-Thomson cooling machine on this fact.

It is only when the attractive forces between gas molecules are very small at ordinary temperatures, and thus the constant a becomes minuscule – as is the case for hydrogen and the inert gases helium and neon – that the expression in brackets becomes negative and ΔT becomes positive, i.e. heating occurs. For cooling to occur, the temperature must be lowered below the inversion point, which according to the above equation is

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JD van der Waals 1837 – 1923

The Nobel Prize in Physics 1910 was awarded to Johannes Diderik van der Waals “for his work on the equation of state for gases and liquids”.


HK Onnes 1853 – 1926

The Nobel Prize in Physics 1913 was awarded to Heike Kamerlingh Onnes “for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium”.

photo credits:

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P Mander March 2014