Posts Tagged ‘geometry’

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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:

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

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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).

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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, the latter 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.

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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.

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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]

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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

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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

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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.”

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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.”

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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.

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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.

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