Introduction A.H.M. Levelt University of Nijmegen

Transcription

1 Introduction A.H.M. Levelt University of Nijmegen 1

2 A favorite pastime of mankind: distilling brandy in Charente (France) 2

3 The alambic explained 3

4 Distilling in the chemical lab 4

5 5 In the early 1990s the physicist Paul Meijer (CUA, Washington, DC) draw my attention to large symbolic computations in classical thermodynamics. 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 the mathematics 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 a modern physical and mathematical point of view.

6 6 In the 1970s R.L. Scott and P.H. van Konynenburg studied the general Van der Waals binary mixture model, derived the complicate fundamental relations using paper and pencil, performed all further computations numerically and presented the results graphically. In my lecture(s) all aspects will be touched upon: introductory thermodynamics, Korteweg s papers and the results of Van Konynenburg and Scott. The emphasis will be on the mathematics, algorithms and visualizations, not on the often subtile physical interpretations. Cf. [[CAL 1985]] and [[JJK 2001]] for thermodynamics. I will signal open problems. Solutions are welcome

7 Photo gallery of the giants: 7

8 James Clerk Maxwell,

9 Josiah Willard Gibbs,

10 Johannes Diderik van de Waals, , Nobel Prize

11 Diederik Johannes Korteweg,

12 12 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

13 13 Here we restrict ourselves to one component fluids (= one kind of molecules) Push the piston: the pressure will increase. But not always. Also vapor may condense. Pull the piston: the pressure will decrease. But not always. Also liquid may vaporize. The next diagram shows what happens

14 Figure 1: liquid, vapor, liquid+vapor 14

15 15 The graph is an isotherm = curve of constant temperature. Pulling the piston hard enough only vapor remains, the pressure continues to decrease. When pushing sufficiently, liquid starts to appear. Pushing on, more and more vapor changes into liquid and the pressure remains constant. The liquid and vapor phase coexist. When all vapor has gone, pushing the piston causes pressure to increase (steeply!).

16 16 Wanted: a good physical/mathematical model of this phenomenon In 1873 J.D. van der Waals presented in his doctoral thesis [[VdW 1873]] at the University of Amsterdam his well-known equation 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 are constants depending on the fluid under consideration. R is the general gas constant. Alternative form of (1) P = R T V b a V 2 (2)

17 17 Graph of Van der Waals equation (2) The points with positive tangent direction are not stable: the pressure increases with the volume. Then graph does not contain the horizontal coexistence part.

18 18 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: in the picture on the next slide. Area I = Area II A nice fast (Maple) algorithm solves this problem

19 Maxwell s equal areas rule 19

20 20 Maxwell s rule can be formulated in a different way. First note that Maxwell s rule is equivalent to Vg V l P d V = P 0 (V g V l ) (3) where (V l, P 0 ) is the left, (V g, P 0 ) the right endpoint of the green line 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 P 0. (Check this!) The next slide shows a picture of the situation.

21 21

22 22 F is the free energy, or Helmholz energy, of our system. For an isothermic system it determines the stable phase equilibria. The following statements follow from the Second Fundamental Law of Thermodynamics: A point Q on the graph F of F corresponds to a locally stable phase 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 Q 1, Q 2 (as in our case!) then again we have global stability, a mixture of the coexisting phases Q 1, Q 2.

23 23

24 24 In the previous slide a number of isotherms in our V P diagram were drawn. With increasing temperature the green line segment decreases till zero length is reached. This is the critical point (O = D ). Obviously, the critical isotherm has a point of inflexion at the critical point and the tangent line is horizontal. Hence, the d P/d V = 0 and d 2 P/d V 2 = 0 at the critical point. A small computation leads to the following critical values: V c = 3b, P c = 1 27 a b 2, T c = 8 27 Example H 2 O: T c = 374 Celsius, P c = 217 atm. a R b (4)

25 25 Introduce new reduced variables by T r = T/T c, P r = P/P c, V r = V/V c (5) In the new variables Van der Waals equation becomes P r = 8 T r V r 1 27 Vr 2 The same equation for all fluids! This is Van der Waals law of corresponding states (6) In the sequel reduced variables will be used frequently. We shall write T instead of T r, etc.

26 The curve containing O, O, O, O = D, D, D, D is the coexistence curve, the boundary of the coexistence region. To the left is the liquid phase, to the right the gas phase. 26

27 27 The equal areas algorithm P P max P 1 P 0 P min V l V g V Difference = Area I - Area II. P 0 equal areas line P 0 < P 1 < P max = Difference < 0 P min < P 1 < P 0 = Difference > 0

28 28 From the previous picture the algorithm is obvious. Choose error bound ɛ > 0. The game is played with two numbers P 1, P 2 : P min P 2 P 1 P max At the start P 1 = P max, P 2 = P min. If Difference(P 1 ) < ɛ or Difference(P 2 ) < ɛ we are done. Otherwise, take Q = (P 1 + P 2 )/2. If Difference(Q) ɛ we are done again. Otherwise, if Difference(Q) < 0 put P 1 := Q, otherwise P 2 := Q.

29 29 Computing Difference For T fixed (below critical temperature) and P 1 (resp. P 2 ) as before, solve P 1 = 8T V 1 27 V 2 for V. Let V l (resp. V g ) be the smallest (resp. largest) solution. Then Difference(P 1 ) = Vg V l P dv P 1 (V g V l ) Define F (V ) = 8T log(v 1) 27/V. Then F (V ) = P and Difference(P 1 ) = F (V l ) F (V g ) + P 1 V l P 1 V g Note that and similar for P 1 V g. P 1 V l = ( 8T V l ) Vl 2 V l

30 30 Finally, an easy computation shows Difference(P 1 ) = M(V l ) M(V g ) where M(V ) = 8T log(v 1) 54 V + 8T V 1 + 8T The algorithm is quick: it computes Table 1 in Appendix B of [[KS 1980]] in 75 secs on my notebook (ɛ = ). The average number of iterations is 15. Computing (V l, P 0 ), (V g, P 0 ) 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 of P 0 on T.

31 Boundary of the coexistence region 31

32 Vapor pressure curve. Upper endpoint = critical point. 32

33 33 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, 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.

34 34 Royal Soc.London, 298, (1980) [PP 1891] Korteweg, D.J. Sur les points de plissement, Archives néerlandaises des sciences exactes et naturelles (Société Hollandaise des Sciences à Haarlem), (1), 24, (1891) [TGP 1891] Korteweg, D.J. La théorie générale des plis, Archives néerlandaises, (1), 24, (1891) [TK, 2000] Kraska, T. The Internet as Lecture Demonstration Tool, [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, (1905) [VL2 1905] Van Laar, J.J. On the shape of the plaitpoint curve for mixtures of normal substances, Proc. Kon. Acad. Amsterdam, VIII, (and table) (1905) [AL 1995] Levelt, A.H.M. Van der Waals, Korteweg, van Laar: a Maple

35 35 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] [LL 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) 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, (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)

36 36 [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 Körpers, der aus zwei verschiedenen Stoffen besteht [Molecular theory of a substance composed of two different species], Z. Physik. Chem (1890). English translation: cf. Rowlinson 1988.