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IntroductionA.H.M. Levelt

University of Nijmegen

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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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Distilling in the chemical lab

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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 inJ.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.

Presently, my sister, J.M.H. Levelt Sengers (NIST, Gaithersburg,MD) and I are trying to understand Korteweg’s work from amodern physical and mathematical point of view.

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In the 1970s R.L. Scott and P.H. van Konynenburg studied thegeneral Van der Waals binary mixture model, derived thecomplicate fundamental relations using paper and pencil,performed all further computations numerically and presented theresults graphically.

In my lecture(s) all aspects will be touched upon: introductorythermodynamics, Korteweg’s papers and the results of VanKonynenburg and Scott. The emphasis will be on the mathematics,algorithms and visualizations, not on the often subtile physicalinterpretations. Cf. [[CAL 1985]] and [[JJK 2001]] forthermodynamics.

I will signal open problems. Solutions are welcome

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Photo gallery of the giants:

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James Clerk Maxwell, 1831-1879

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Josiah Willard Gibbs, 1839-1903

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Johannes Diderik van de Waals, 1837-1923, Nobel Prize 1910

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Diederik Johannes Korteweg, 1848-1941

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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. Alsovapor may condense.

Pull the piston: the pressure will decrease. But not always. Alsoliquid 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 (2)

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, thed P/d V = 0 and d2P/d V 2 = 0 at the critical point. A smallcomputation leads to 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

Vr − 1− 27

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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The equal areas algorithm

PmaxP

1P

0

Pmin

Vl Vg

V

P

Difference = Area I - Area II. P0 equal areas lineP0 < P1 < Pmax =⇒ Difference < 0Pmin < P1 < P0 =⇒ Difference > 0

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From the previous picture the algorithm is obvious. Choose errorbound ε > 0. The game is played with two numbers P1, P2:

Pmin ≤ P2 ≤ P1 ≤ Pmax

At the start P1 = Pmax, P2 = Pmin.

If | Difference(P1) |< ε or | Difference(P2) |< ε we are done.

Otherwise, take Q = (P1 + P2)/2. If | Difference(Q) |≤ ε we aredone again.

Otherwise, if Difference(Q) < 0 put P1 := Q, otherwise P2 := Q.

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Computing Difference

For T fixed (below critical temperature) and P1 (resp. P2) asbefore, solve

P1 =8T

V − 1− 27

V 2

for V . Let Vl (resp. Vg) be the smallest (resp. largest) solution.Then

Difference(P1) =∫ Vg

Vl

P dV − P1(Vg − Vl)

Define F (V ) = −8T log(V − 1)− 27/V . Then F ′(V ) = −P and

Difference(P1) = F (Vl)− F (Vg) + P1Vl − P1Vg

Note that

P1Vl =(

8T

Vl − 1+

27V 2

l

)Vl

and similar for P1Vg.

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Finally, an easy computation shows

Difference(P1) = M(Vl)−M(Vg)

where

M(V ) = −8T log(V − 1)− 54V

+8T

V − 1+ 8T

The algorithm is quick: it computes Table 1 in Appendix B of[[KS 1980]] in 75 secs on my notebook (ε = 0.00001). The averagenumber of iterations is 15.

Computing (Vl, P0), (Vg, P0) for a range of values of T , 0 < T < 1,one can draw the boundary of the coexisting region. Cf. next slide

The last slide shows the vapor pressure curve: the dependence ofP0 on T .

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Boundary of the coexistence region

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Vapor pressure curve. Upper endpoint = critical point.

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Short list of books and papers. Extended references in [KYR 1996] and

[LS 2002]

References

[CAL 1985] Callen, H.B. Thermodynamics and an introduction to

thermostatistics, 2nd ed., John Wiley & Sons (1985)

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

using Mathematica,

http://www.nscp.umd.edu/ kelly/PHYS603/notebooks.htm

(2001)

[KYR 1996] Kipnis, A. Ya., Yavelov, B. E., and Rowlinson, J.S., Van

der Waals and Molecular Science, Clarendon Press, Oxford

(1996)

[KS 1980] Van Konynenburg, P.H., Scott, R.L. Critical lines and phase

equilibria in binary Van der Waals mixtures, Phil. Trans.

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Royal Soc.London, 298, 495-540 (1980)

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

neerlandaises des sciences exactes et naturelles (Societe

Hollandaise des Sciences a Haarlem), (1), 24, 57-98 (1891)

[TGP 1891] Korteweg, D.J. La theorie generale des plis, Archives

neerlandaises, (1), 24, 295-368 (1891)

[TK, 2000] Kraska, T. The Internet as Lecture Demonstration Tool,

http://van-der-waals.pc.uni-koeln.de/Halifax.html

[VL1 1905] Van Laar, J.J. An exact expression for the course of the

spinodal curves and their plaitpoints for all temperatures, in

the case of mixtures of normal substances, Proc. Kon. Acad.

Amsterdam, VIII, 646-657 (1905)

[VL2 1905] Van Laar, J.J. On the shape of the plaitpoint curve for

mixtures 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 Maple

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Excursion into the Thermodynamics of Binary Mixtures,

Computer Algebra in Industry 2, Edited by A.M. Cohen, L.

van Gastel, S.M. Verduyn Lunel. John Wiley & Sons (1995)

[LS 2002] Levelt Sengers, Johanna M.H. How Fluids Unmix;

Discoveries by the School of Van der Waals and Kamerlingh

Onnes, Edita-KNAW (Royal Netherlands Academy of Arts

and Sciences) (2002)

[LL 2002] Johanna Levelt Sengers and Antonius H.M. Levelt Diederik

Korteweg, Pioneer of Criticality, Physics Today, December

2002, 47-54, American Institute of Physics (2002)

[PM 1989] Meijer, P.H.E., The Van der Waals equation of state around

the Van Laar point, J. Chem. Phys. 90, 448-456 (1989)

[ROW 1988] Rowlinson, J.S. J.D. van der Waals, On the Continuity of

the Gaseous and the Liquid States, Studies in Statistical

Mechanics XIV. J.L. Lebowitz, Ed., North Holland,

Amsterdam (1988)

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[VdW 1873] Van der Waals, J.D. Over de Continuiteit van den Gas- en

Vloeistoftoestand [On the Continuity of the Gaseous 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 [Molecular theory of a

substance composed of two different species], Z. Physik.

Chem. 133-173 (1890). English translation: cf. Rowlinson

1988.