The other D.J. Korteweg:thermodynamics of binary mixtures

A.H.M. Levelt, March 2003, www.math.kun.nl/medewerkers/ahml/index.html

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Introduction

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A favorite pastime of mankind: distilling brandy in Charente(France)

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The alambic explained

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In the early 1990s the physicist Paul Meijer (CUA, Washington,DC) draw my attention to large symbolic computations in classicalthermodynamics.

He had noticed several exact computations in J.J. Van Laar’s work,which he and then I confirmed, using Maple.

He also showed me D.J. Korteweg’s forgotten 1891 papers on themathematics underlying Van der Waals theory of binary mixtures.

My sister, J.M.H. Levelt Sengers (NIST, Gaithersburg, MD) and Ihave studied Korteweg’s work from modern physical andmathematical points of view.

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This collaboration has led to a joint publication:

Diederik Korteweg, Pioneer of Criticality, Physics Today,December 2002.

Korteweg’s work is treated in detail in:

J. Levelt Sengers, How fluids unmix: Discoveries by the School ofVan der Waals and Kamerlingh Onnes, Edita, Royal NetherlandsAcademy of Arts and Sciences, Amsterdam (2002).

For all physical and historical details of the subject look at thesepublications and the included references.

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In my presentation emphasis will be on Korteweg’s mathematics,not on the often subtile physical interpretations, though a shortintroduction to the relevant thermodynamics will be given. Cf.[[CAL 1985]] and [[JJK 2001]] for more thermodynamics.

I will follow the general line of Korteweg’s papers:

Sur les points de plissement, Archives neerlandaises, (1), 24, 57-98(1891), (cited: PP)

La theorie generale des plis, Archives neerlandaises, (1), 24,295-368 (1891), (cited: TGP)

I have added geometrical picture, visualizations and animations,based on symbolic-numeric computations.

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Photo gallery of the giants

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Crash course in thermodynamics

Fluid = gas (vapor) and/or liquid (phase)

Fluid inside cylinder with movable piston

The whole au bain-marie, i.e. temperature fixed

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Here we restrict ourselves to one component fluids (= one kind ofmolecules)

Push the piston: the pressure will increase. But not always,because vapor may condense.

Pull the piston: the pressure will decrease. But not always, becauseliquid may vaporize.

The next diagram shows what happens

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Figure 1: liquid, vapor, liquid+vapor

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The graph is an isotherm = curve of constant temperature.

Pulling the piston hard enough only vapor remains, the pressurecontinues to decrease.

When pushing sufficiently, liquid starts to appear. Pushing on,more and more vapor changes into liquid and the pressure remainsconstant. The liquid and vapor phase coexist.

When all vapor has gone, pushing the piston causes pressure toincrease (steeply!).

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Wanted: a good physical/mathematical model of thisphenomenon

In 1873 J.D. van der Waals presented in his doctoral thesis[[VdW 1873]] at the University of Amsterdam his well-knownequation of state

(P +

a

V 2

)(V − b) = R T (1)

an adaptation of P V = R T holding for ideal gases. Here a, b areconstants depending on the fluid under consideration. R is thegeneral gas constant.

Alternative form of (1)

P =R T

V − b− a

V 2(2)

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Graph of Van der Waals equation (7)

The points with positive tangent direction are not stable: thepressure increases with the volume.

Then graph does not contain the horizontal ”coexistence part”.

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Needed: extra rule(s) to describe stable equilibria

In 1875 J.C. Maxwell gave such a rule. It says where the horizontal”green line” must be drawn:

Area I = Area II

in the picture on the next slide.

A nice fast (Maple) algorithm solves this problem

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Maxwell’s equal areas rule

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Maxwell’s rule can be formulated in a different way.

First note that Maxwell’s rule is equivalent to∫ Vg

Vl

P d V = P0(Vg − Vl) (3)

where (Vl, P0) is the left, (Vg, P0) the right endpoint of the greenline segment in figure 1

Let F = F (V ) be such that dF/dV = −P

(e.g. F = −RT log(V − b)− a/V ).

Then the graph of F has a double tangent line having direction−P0. (Check this!)

The next slide shows a picture of the situation.

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F is the free energy, or Helmholz energy, of our system. For anisothermic system it determines the stable phase equilibria. Thefollowing statements follow from the Second Fundamental Law ofThermodynamics:

A point Q on the graph F of F corresponds to a locally stablephase when F is locally convex at Q.

If the tangent line to F at Q lies below F everywhere, then Q is a(pure) globally stable phase.

If a line lies below F with the exception of 2 tangent points Q1, Q2

(as in our case!) then again we have global stability, a mixture ofthe coexisting phases Q1, Q2.

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In the previous slide a number of isotherms in our V P diagramwere drawn. With increasing temperature the green line segmentdecreases till zero length is reached. This is the critical point(O

′′′= D

′′′).

Obviously, the critical isotherm has a point of inflexion at thecritical point and the tangent line is horizontal. Hence, d P/d V = 0and d2P/d V 2 = 0 at the critical point. An easy computation leadsto the following critical values:

Vc = 3b, Pc =127

a

b2, Tc =

827

a

R b(4)

Example H2O: Tc = 374◦ Celsius, Pc = 217 atm.

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Introduce new reduced variables by

Tr = T/Tc, Pr = P/Pc, Vr = V/Vc (5)

In the new variables Van der Waals equation becomes

Pr =8 Tr

3Vr − 1− 3

V 2r

(6)

The same equation for all fluids! This is Van der Waals’ law ofcorresponding states

In the sequel reduced variables will be used frequently. We shallwrite T instead of Tr, etc.

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The curve containing O,O′, O′′, O′′′ = D′′′, D′′, D′, D is thecoexistence curve, the boundary of the coexistence region. To theleft is the liquid phase, to the right the gas phase.

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Systems of 2 components C1, C2 in fluid form

Possible phases: gaseous, liquid (evt. liquid 1 and liquid 2 unmixed)

One-fluid model of Van der Waals [[VdW 1890]]

Recall: Van der Waals equation of state in the case of 1 component:

P =RT

V − b− a

V 2(7)

a, b specific for this component.

For C1, C2 we have a1, b1, resp. a2, b2.

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Van der Waals’s idea: interpolation

a(x) = (1− x)2a1 + 2(1− x) x a12 + x2a2

b(x) = (1− x)b1 + xb2

Here x is the (mole) fraction of C2 (hence 1− x that of C1).

ai is measure of attraction force between pairs of molecules of Ci

a12 is measure of attraction force between molecules of C1 and C2

b1, b2 are the ’sizes’ of the molecules of C1, C2, resp.

Use a = a(x), b = b(x) in (7).

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Van der Waals [[VdW 1890], (26) p. 250] showed that theHelmholtz free energy of the mixture for constant T is

F = −a(x)V−RT log(V−b(x))+RT

((1−x) log(1−x)+x log(x)

)(8)

For T = constant (x, V ) 7→ F (x, V ) represents a surface F in3-space.

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Stable equilibria of phases

Similar to the one component case one has: for Q ∈ F

F convex at Q ⇐⇒ Q represents a locally stable phase

If moreover the tangent plane T at Q to F lies below F everywhere,then Q is a phase in globally stable equilibrium (pure phase).

Let T be a plane tangent to F in two different points Q1, Q2. Suchbitangent planes are our research object. If F lies above Teverywhere, then Q1, Q2 are stable phases which coexist.

More complicate situations are possible (e.g. tritangent planes, etc.)

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The surface in the last picture obviously admits bitangent planes.In general such a situation is not rigid: the bitangent plane rollsover the surface F and the two tangent points, named connodes,describe (branches of) a curve on the surface, called connodal orbinodal

Knowledge of the connodals translates into knowledge on phasesand phase transitions.

On a convex surface connodals don’t exist. The surface must havefolds, plaits in Korteweg’s terminology.

In the picture when the bitangent plane rolls over the surface inone direction the connodes approach each other and finally coincidein what Korteweg calls a plait point (point de plissement), whereascritical point is the name used by physicists.

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Wooden model used by Van der Waals in his courses

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The symmetric case

The symmetric case of the Van der Waals model for binarymixtures is the following one:

a1 = a2, b1 = b2.

It means that the 2 components are identical. The attractionbetween the molecules of the 2 components may be different fromthe attraction between the molecules of component 1, resp.component 2, taken apart.

Korteweg restricts to this case, because it is the only one he canhandle in a rigorous way (as he explains).

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Write b = b1 = b2, a(x) = a(x)/a1, κ = a12/a1 and

F =a1

27bFr, V = bVr, T =

8a1

27bRTr, P =

a1

27b2Pr.

Then (8) becomes

Fr = −8Tr log(Vr− 1)− 27a(x)Vr

+8Tr ((1−x) log(1−x)+x log(x)),

(9)where

a(x) = 1− 2x + 2x2 + 2(1− x)xκ. (10)

(In Fr the term −8Tr log(b) has been omitted. This is allowed,because only the derivatives w.r.t. V and x play a role.)

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References

[BGM 1982] Banchoff, T., Gaffney, T.,McCrory, C. Cusps of GaussMappings, Research Notes in Mathematics, 55,Pittman (1982). Web version by D. Dreibelbis,//www.emis.de/monographs/CGM

[BGT 1996] Bruce, J.W., Giblin, P.J., Tari, F., Parabolic Curves ofEvolving Surfaces, Intern. J. Computer Vision 17,291-306 (1996)

[CAL 1985] Callen, H.B. Thermodynamics and an introduction tothermostatistics, 2nd ed., John Wiley & Sons (1985)

[DD 2001] Dreibelbis, D. A bitangency theorem for surfaces infour-dimensional Euclidean space, Q.J. Math. 52,137-160 (2001)

[JJK 2001] Kelly, J.J. Review of Thermodynamics form Statistical

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Physics using Mathematica,http://www.nscp.umd.edu/ kelly/PHYS603/notebooks.htm(2001)

[KT 1980] Kergosien, Y.L., Thom, R., Sur les points paraboliquesdes surfaces, Comptes Rendues Acad. Sci. Paris 299,705-710 (1980)

[KYR 1996] Kipnis, A. Ya., Yavelov, B. E., and Rowlinson, J.S.,Van der Waals and Molecular Science, ClarendonPress, Oxford (1996)

[KOE 1990] Koenderink, Jan J., Solid Shape, The MIT Press,Cambridge, Mass. (1990)

[KS 1980] Van Konynenburg, P.H., Scott, R.L. Critical lines andphase equilibria in binary Van der Waals mixtures,Phil. Trans. Royal Soc.London, 298, 495-540 (1980)

[PP 1891] Korteweg, D.J. Sur les points de plissement, Archives

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neerlandaises, (1), 24, 57-98 (1891)

[TGP 1891] Korteweg, D.J. La theorie generale des plis, Archivesneerlandaises, (1), 24, 295-368 (1891)

[TK, 2000] Kraska, T. The Internet as Lecture DemonstrationTool,http://van-der-waals.pc.uni-koeln.de/Halifax.html

[VL1 1905] Van Laar, J.J. An exact expression for the course ofthe spinodal curves and their plaitpoints for alltemperatures, in the case of mixtures of normalsubstances, Proc. Kon. Acad. Amsterdam, VIII,646-657 (1905)

[VL2 1905] Van Laar, J.J. On the shape of the plaitpoint curve formixtures of normal substances, Proc. Kon. Acad.Amsterdam, VIII, 33-48 (and table) (1905)

[AL 1995] Levelt, A.H.M. Van der Waals, Korteweg, van Laar: a

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Maple Excursion into the Thermodynamics of BinaryMixtures, Computer Algebra in Industry 2, Edited byA.M. Cohen, L. van Gastel, S.M. Verduyn Lunel. JohnWiley & Sons (1995)

[LS 2002] Levelt Sengers, Johanna M.H. How Fluids Unmix;Discoveries by the School of Van der Waals andKamerlingh Onnes, Edita-KNAW (Royal NetherlandsAcademy of Arts and Sciences) (2002)

[LL 2002] Johanna Levelt Sengers and Antonius H.M. LeveltDiederik Korteweg, Pioneer of Criticality, PhysicsToday, December 2002, 47-54, American Institute ofPhysics (2002)

[PM 1989] Meijer, P.H.E., The Van der Waals equation of statearound the Van Laar point, J. Chem. Phys. 90,448-456 (1989)

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[OK 1993] Okada, K., Catastrophe Theory and Phase Transitions,Trans Tech Publications, Switzerland (1993)

[PLA 1984] Platonova, O.A., Singularities of projections of smoothsurfaces, Russ. Math. Surv. 39:1, 177-178 (1984)

[ROW 1988] Rowlinson, J.S. J.D. van der Waals, On theContinuity of the Gaseous and the Liquid States,Studies in Statistical Mechanics XIV. J.L. Lebowitz,Ed., North Holland, Amsterdam (1988)

[VdW 1873] Van der Waals, J.D. Over de Continuiteit van denGas- en Vloeistoftoestand [On the Continuity of theGaseous and Liquid States], doctoral thesis, Leiden,A.W. Sijthoff (1873)

[VdW 1890] Van der Waals, J.D. Molekulartheorie eines Korpers,der aus zwei verschiedenen Stoffen besteht [Moleculartheory of a substance composed of two different

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species], Z. Physik. Chem. 133-173 (1890). Englishtranslation: cf. Rowlinson 1988.

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