Equations of State and Phase Diagrams

Research: Science and Education

L. Glasser Molecular Sciences Institute, School of Chemistry, University of the Witwatersrand, P. O. Wits 2050, South Africa; [email protected]

An equation of state (EoS) is a mathematical summary dimensional) curve within the three-dimensional diagram, and of the (usually thermodynamic) equilibrium properties of a equilibrium among three phases is represented by a straight line material (1). At the very minimum (in the absence of external situated along the constant temperature–pressure equilibrium constraints, such as fixed temperature), such an equation relates condition. Figure 2 shows that the triple “point” of a pT the mechanical, thermal, and volumetric properties of a single projection is actually a misnomer for a tie line in full phase phase of the material; other properties might be needed to fully diagram space; Sandler (11) terms this the “triple-point line”. describe its behavior if the material exists in a particular Thus we see that, although the common practice of environment, such as in a strong electric or magnetic field, depicting only projections suffices for many purposes, it or localized in a surface. also hides some of the features of the relations between the Among the simplest possible EoS’s is that of the ideal gas EoS’s of the different phases. This paper illustrates the phase relations in a unary (single-component) system with a full pV = RT (1) m three-dimensional diagram using authentic data. Such a diagram where p (pressure) represents the mechanical state of the does not seem to have appeared before. system, T (absolute temperature) the thermal state, and Vm To provide this authentic representation of a unary system, (molar volume) the volumetric state. Clearly, there are only we chose carbon dioxide (Fig. 3), for which much (but not two independent variables (which may be arbitrarily selected all) of the required data is available in equation form. Where from among the three) while the remaining variable takes on the equations required could not be found, they were fitted a value dependent on the values assigned to the other two, to experimental data or closely approximated by standard through the EoS. EoS’s may be expressed with other vari- relations, as described in the Appendix. To encompass the ables than p, T, and Vm; for example density, ρ, may replace wide range of values encountered, it is necessary to represent Vm, but the three- (or more)-variable structure remains. the pressure and volume data on logarithmic axes. A more elaborate EoS, such as van der Waals equation The data required are equilibrium temperatures and pres- sures as well as data for RT a p = – (2) Sublimation curve: solid and vapor densities (or molar V – b 2 m Vm volumes) Melting curve: solid and liquid densities retains the same structure of variables, though differently expressed. Furthermore, the van der Waals equation and most Saturation curve: liquid and vapor densities EoS’s for the fluid state generally receive a further thermo- Solid state phase changes: these data are largely uncertain or dynamic interpretation. Such cubic (as well as some other) unknown (10), and so are omitted from the diagram functions display an S-shaped structure with loops between The Appendix lists the equations (whether fitted or theoretical) the liquid and vapor regions. The regions are divided into for carbon dioxide equilibria, or the experimental data used three using Maxwell’s construction (2) to guide the division into gas/vapor, liquid, and phase gap, by recognizing that the liquid and vapor are in equilibrium at a given temperature, so that their molar Gibbs functions are equal. Maxwell’s construction then requires equal areas above and below the tie line, which connects liquid and vapor in equilibrium, passing through the loops of the EoS function. Thus, the full representation of a simple EoS is a three- dimensional image, encompassing the three variables. Three- dimensional models (3–6 ) and diagrams (7) of single-phase systems are not uncommonly found (Fig. 1). However, in graphical representation of multiphase systems, partly because of the difficulty of drawing three-dimensional images (and partly because of the lack of the required data), it is traditional to use projections of the three dimensions onto two-dimensional planes in preference to the full three-dimensional representation (e.g., pV diagrams [1, 2, 8–10] in place of pVT diagrams), while three-dimensional descriptions (3, 11, 12) are usually presented schematically, not to scale. (Petrucci [13] has presented a three-dimensional diagram of p, V, and composition, but for a hypothetical binary system, at a fixed temperature.) Figure 1. Photograph of a model of the ideal gas pVT surface, Equilibria between two phases are represented by a (two- constructed using templates kindly provided by D. B. Hilton (5).

874 Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu Research: Science and Education

to generate the curves. These data were extracted from Gmelin (14) and Landolt-Börnstein (15) for transitions involving the solid state, and from the International Thermodynamic Tables of the Fluid State for carbon dioxide (16 ). The calculated results were prepared on an Excel spreadsheet. For ease of access, a recent EoS for the single-phase fluid region is also listed in the Appendix (17), but was not utilized in preparation of Figure 3. Much thermophysical (and other) data for many materials, including carbon dioxide, is available on the WebBook of the National Institute of Standards and Technology (18).

Literature Cited

1. Atkins, P. W. Physical Chemistry, 6th ed.; Oxford University Press: Oxford, 1998. 2. Wisniak, J.; Golden, M. J. Chem. Educ. 1998, 75, 200. 3. Petrucci, R. H. J. Chem. Educ. 1965, 42, 323. Figure 2. Orthographic (isometric) three-dimensional pVT diagram. 4. Peretti, E. A. J. Chem. Educ. 1966, 43, 253. Adapted from ref 11: Sandler, S. I. Chemical and Engineering 5. Hilton, D. B. J. Chem. Educ. 1991, 68, 496. Thermodynamics, 2nd ed.; Wiley, New York, 1989; p 217; copy- 6. Coch Frugoni, J. A.; Zepka, M.; Rocha Figueira, R.; Coretti, M. right © 1966 by Blaisdell Publishing Co. (John Wiley & Sons, Inc.). J. Chem. Educ. 1984, 61, 1048. Reprinted by permission of John Wiley & Sons, Inc. 7. Remark, J. F. J. Chem. Educ. 1975, 52, 61. 8. Halpern, A. M.; Lin, M.-F. J. Chem. Educ. 1986, 63, 38. 9. Lieu, V. T. J. Chem. Educ. 1996, 73, 837. 10. Gramsch, S. A. J. Chem. Educ. 2000, 77, 718. 11. Sandler, S. I. Chemical and Engineering Thermodynamics, 2nd ed.; Wiley: New York, 1989; p 217. 12. Logo for 14th Russian Conference on Chemical Thermo- dynamics, St. Petersburg, Jun 30–Jul 5, 2002; http://rcct2002. nonel.pu.ru/ (accessed Apr 2002). 13. Petrucci, R. H. J. Chem. Educ. 1970, 47, 825. 14. Gmelins Handbuch der Anorgorganischen Chemie, 8th ed.; Sec- tion C: Part 1, Carbon Dioxide; von Backzo, C., Ed.; Verlag Chemie: Weinheim, 1970. 15. Landolt-Börnstein Zahlenwerte und Funktionen, Vol. 2, 6th ed.; Schäfer, K.; Beggerow, G., Eds.; Springer: Heidelberg, 1971; Part 1, p 723. 16. International Thermodynamic Tables of the Fluid State—3: Carbon Dioxide; Angus, S.; Armstrong, B.; de Reuck, K. M., Eds.; Pergamon: Oxford, 1976. 17. Mäder, U. K.; Berman, R. G. Am. Mineral. 1991, 76, 1547. 18. National Institute of Standards and Technology; NIST WebBook; http://webbook.nist.gov (accessed Mar 2002); at present free of charge. Figure 3. An orthographic pVT diagram for carbon dioxide, with projections onto the pT, pV, and VT planes. To accommodate the full range of data, the logarithms of the pressure and molar volume axes are used. The horizontal lines (constant pT) are tie lines con- Appendix (from ref 16 ) necting phases in mechanical and thermal equilibrium across the phase gaps. The critical point condition and the triple-“point” tie-line Critical point: 3 ᎑1 are labeled. The dot at the end of the liquid–vapor line in the pT Tc = 304.21 K; pc = 73.825 bar; Vm,c = 94.428 cm mol projection represents its termination at the critical point. Volumes in the diagram (corresponding to areas in the projections) are labeled Triple point: 3 ᎑1 solid, liquid, vapor and (above the critical point) gas. The space T3 = 216.58 K; p3 = 5.18 bar; Vm,3(s) = 29.09 cm mol ; 3 ᎑1 3 ᎑1 curves depicted are A: solid sublimation, in equilibrium with vapor; Vm,3( ) = 37.338 cm mol ; Vm,3(g) = 3133.79 cm mol B: vapor condensation, in equilibrium with solid; C: liquid saturation, Saturated vapor pressure/temperature (16 ): in equilibrium with vapor; D: vapor saturation, in equilibrium with liquid; E: solid–liquid melt equilibrium—on the scale of the diagram p 1.935 4 T i ln = a 1– T + a c –1 the two separate curves nearly overlie one another. The small break p 0 Σ i c Tc i=1 T barely discernible at the junction of curves A and C arises from the difference in molar volumes of solid and liquid at the triple “point” where ᎑ ᎑ (cf. Fig. 2). a0 = 11.377371, a1 = 6.8849249, a2 = 9.5924263, a3 = 13.679755,

JChemEd.chem.wisc.edu • Vol. 79 No. 7 July 2002 • Journal of Chemical Education 875 Research: Science and Education

a = ᎑8.6056439 3 ᎑1 4 T/K P/bar ∆Vm,melt(quad) Vm,/cm mol Vm,s(calcd) Saturated liquid density (16 ): Selected Values from Table 8, Reference 16 2 2516.58 5—.18 3—7.3 4 ρ 0.347 i+1 /3 T T 2120 1866.9 67.3 326.7 29.8 –1=c0 1– + Σ ci 1– ρc T i=1 T c c 2725 4912.5 67.1 396.0 29.6 where c0 = 1.9073793, c1 = 0.38225012, c2 = 0.42897885 2930 6169.3 69.0 375.4 29.5 2335 9237.6 52.8 355.0 29.4 Saturated vapor density (16 ): 2237 15,048. 55.7 314.8 29.4 ρ 0.347 2 i+1 /3 g T T References 14 and 15 –1=d0 1– + di 1– ρc Σ Tc i=1 Tc 2067.65 22,942. 40.7 363.8 28.8 2781.65 30,922. 40.3 383.4 28.6 where d0 = ᎑1.7988929, d1 = ᎑0.71728276, d2 =1.7739244 2494.55 43,903. 33.9 373.0 28.5 Melting curve (16 ): 3006.25 52,884. 31.6 302.7 28.5 3717.35 64,864. 31.3 382.4 28.4 p – p3 T ln p + 648.13886 =3ln +ln 648.13886 3428.35 77,845. 36.0 382.1 28.4 3 T3 3038.95 84,826. 22.8 311.9 28.5 3748.55 94,806. 24.6 361.7 28.5 Melting: Volume Changes (refs 14, 15) 3 ᎑1 3757.75 160,78 27.4 341.5 28.6 T/K P /rba ∆Vm,melt /(cm mol ) 3868.45 171,76 23.2 351.4 28.7 2067.65 21,942. 4.7 2781.65 31,922. 4.3 Sublimation curve (applicable down to 90 K) (16 ): 2494.55 44,903. 3.9 3006.25 52,884. 3.6 p T3 T ln p =14.57893 1 – –14.48067 ln + 3717.35 62,864. 3.3 3 T T3 3428.35 77,845. 3.0 2 3 65.35685 T –1 –47.14593 T –1 +14.53922 T –1 3038.95 83,826. 2.8 T3 T3 T3 3748.55 95,806. 2.6 3757.75 180,78 2.4 Sublimation Data (Using the Ideal Gas Equation) 3868.45 141,76 2.3 3 ᎑1 T/K P/bar Vm,v/(cm mol ) ᎑9 12 We have fitted the above data to the following quadratic polyno- 930 6.6 × 10 1.13 × 10 ᎑7 10 mial (given to 10-decimal places—this precision is not justified by 1500 2.1 × 10 3.87 × 10 ᎑ the underlying data, but is required to yield calculated values of 1910 3.7 × 10 6 2.41 × 109 ᎑ sufficient accuracy): 1120 4.2 × 10 5 2.37 × 108 ᎑4 7 ᎑ 1430 3.2 × 10 3.34 × 10 ∆V = 6.9542 × 10 5T 2 – 0.0686277044T + 18.1107164247 m,melt 1640 06.0018 6.2 × 106 Melting results: 1350 08.0084 1.4 × 106 We have fitted the liquid Vm, versus T data below to the following 1460 00.031 424,00 cubic polynomial (given to 10 decimal places—this precision is not 1570 00.099 142,00 justified by the underlying data, but is required to yield calculated 1680 00.27 54,30 values of sufficient accuracy): 1490 00.68 23,10 ᎑6 3 2 Vm, = ᎑3.0331 × 10 T + 0.0028688133T – 2500 10.5 10,70 0.9200958384T + 132.4799710302 2710 30.2 5,34 2916.58 50.1 3,47 ∆Vm,melt(quad) is the calculated value for the volume change at the ᎑ corresponding melting temperature, using the quadratic fit above. NOTE: The van der Waals equation, using a = 3.688 bar cm6 mol 2, 3 ᎑1 Vm,s(calcd) (= Vm, – ∆Vm,melt) is the corresponding calculated molar b = 42.67 cm mol (1), gives essentially the same values. volume of solid CO2 using the difference of the above two equations: ᎑6 3 ᎑3 2 Fluid equation of state (17): ∆Vm,melt(quad) = ᎑3.0331 × 10 T + 2.799268 × 10 T – 0.85146813560T + 114.3692546 A A p = RT – 1 + 2 2 4 B3 TV V V – B1 + B2T – V 3 + C

᎑4 with C = B3/(B1 + B2T ) and B1 = 28.06474, B2 = 1.728712 × 10 , 4 9 9 B3 = 8.365341 × 10 , A1 = 1.094802 × 10 , A2 = 3.374749 × 10 in units of cm3 mol᎑1, K, and bar.

876 Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu