Research: Science and Education
Equations of State and Phase Diagrams 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 of the (usually thermodynamic) equilibrium properties of a material (1). At the very minimum (in the absence of external constraints, such as fixed temperature), such an equation relates the mechanical, thermal, and volumetric properties of a single phase of the material; other properties might be needed to fully describe its behavior if the material exists in a particular environment, such as in a strong electric or magnetic field, or localized in a surface. Among the simplest possible EoS’s is that of the ideal gas pVm = RT
(1)
where p (pressure) represents the mechanical state of the system, T (absolute temperature) the thermal state, and Vm (molar volume) the volumetric state. Clearly, there are only two independent variables (which may be arbitrarily selected from among the three) while the remaining variable takes on a value dependent on the values assigned to the other two, through the EoS. EoS’s may be expressed with other variables than p, T, and Vm; for example density, ρ, may replace Vm, but the three- (or more)-variable structure remains. A more elaborate EoS, such as van der Waals equation
p=
RT – a V m – b V m2
(2)
retains the same structure of variables, though differently expressed. Furthermore, the van der Waals equation and most EoS’s for the fluid state generally receive a further thermodynamic interpretation. Such cubic (as well as some other) functions display an S-shaped structure with loops between the liquid and vapor regions. The regions are divided into 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 threedimensional image, encompassing the three variables. Threedimensional 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.) Equilibria between two phases are represented by a (two874
dimensional) curve within the three-dimensional diagram, and equilibrium among three phases is represented by a straight line situated along the constant temperature–pressure equilibrium condition. Figure 2 shows that the triple “point” of a pT projection is actually a misnomer for a tie line in full phase diagram space; Sandler (11) terms this the “triple-point line”. Thus we see that, although the common practice of depicting only projections suffices for many purposes, it also hides some of the features of the relations between the EoS’s of the different phases. This paper illustrates the phase relations in a unary (single-component) system with a full three-dimensional diagram using authentic data. Such a diagram does not seem to have appeared before. To provide this authentic representation of a unary system, we chose carbon dioxide (Fig. 3), for which much (but not all) of the required data is available in equation form. Where the equations required could not be found, they were fitted to experimental data or closely approximated by standard relations, as described in the Appendix. To encompass the wide range of values encountered, it is necessary to represent the pressure and volume data on logarithmic axes. The data required are equilibrium temperatures and pressures as well as data for Sublimation curve: solid and vapor densities (or molar volumes) Melting curve: solid and liquid densities Saturation curve: liquid and vapor densities Solid state phase changes: these data are largely uncertain or unknown (10), and so are omitted from the diagram
The Appendix lists the equations (whether fitted or theoretical) for carbon dioxide equilibria, or the experimental data used
Figure 1. Photograph of a model of the ideal gas pVT surface, constructed using templates kindly provided by D. B. Hilton (5).
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
Figure 2. Orthographic (isometric) three-dimensional pVT diagram. Adapted from ref 11: Sandler, S. I. Chemical and Engineering Thermodynamics, 2nd ed.; Wiley, New York, 1989; p 217; copyright © 1966 by Blaisdell Publishing Co. (John Wiley & Sons, Inc.). Reprinted by permission of John Wiley & Sons, Inc.
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 connecting phases in mechanical and thermal equilibrium across the phase gaps. The critical point condition and the triple-“point” tie-line are labeled. The dot at the end of the liquid–vapor line in the pT projection represents its termination at the critical point. Volumes in the diagram (corresponding to areas in the projections) are labeled solid, liquid, vapor and (above the critical point) gas. The space curves depicted are A: solid sublimation, in equilibrium with vapor; B: vapor condensation, in equilibrium with solid; C: liquid saturation, in equilibrium with vapor; D: vapor saturation, in equilibrium with liquid; E: solid–liquid melt equilibrium—on the scale of the diagram the two separate curves nearly overlie one another. The small break 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” (cf. Fig. 2).
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. 4. Peretti, E. A. J. Chem. Educ. 1966, 43, 253. 5. Hilton, D. B. J. Chem. Educ. 1991, 68, 496. 6. Coch Frugoni, J. A.; Zepka, M.; Rocha Figueira, R.; Coretti, M. J. Chem. Educ. 1984, 61, 1048. 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 Thermodynamics, 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.; Section 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.
Appendix (from ref 16 ) Critical point: Tc = 304.21 K; pc = 73.825 bar; Vm,c = 94.428 cm3 mol᎑1 Triple point: T3 = 216.58 K; p3 = 5.18 bar; Vm,3(s) = 29.09 cm3 mol᎑1; Vm,3() = 37.338 cm3 mol᎑1; Vm,3(g) = 3133.79 cm3 mol᎑1 Saturated vapor pressure/temperature (16 ):
p ln p = a 0 1 – T Tc c
1.935
4
+ Σ ai i=1
Tc –1 T
i
where 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
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Research: Science and Education a4 = ᎑8.6056439
T/K
Saturated liquid density (16 ): 2 ρ T 0.347+ c 1 – T i ρc – 1 = c 0 1 – T T i=1
Σ
c
i+1 /3
c
where c0 = 1.9073793, c1 = 0.38225012, c2 = 0.42897885 Saturated vapor density (16 ): 2 ρg T 0.347+ d 1 – T i ρc – 1 = d 0 1 – T T i=1
Σ
c
c
p – p3 T p 3 + 648.13886 = 3 ln T + ln 648.13886 3 Melting: Volume Changes (refs 14, 15) P / bar
∆Vm,melt /(cm3 mol᎑1)
267.65
2,942.0
4.71
281.65
3,922.7
4.31
294.55
4,903.4
3.94
306.25
5,884.0
3.62
317.35
6,864.7
3.32
328.35
7,845.4
3.07
338.95
8,826.0
2.83
348.55
9,806.7
2.65
357.75
10,787
2.48
368.45
11,768
2.34
220
5.185 166.91
We have fitted the above data to the following quadratic polynomial (given to 10-decimal places—this precision is not justified by the underlying data, but is required to yield calculated values of sufficient accuracy):
—
37.34
—
6.38
36.77
29.82
225
412.57
6.19
36.07
29.69
230
669.39
6.01
35.49
29.57
937.63
5.82
35.02
29.45
5.75
34.85
29.41
237
1,048.2
267.65
2,942.0
4.72
33.80
28.86
281.65
3,922.7
4.30
33.40
28.68
i+1 /3
Melting curve (16 ):
T/K
216.58
235
where d0 = ᎑1.7988929, d1 = ᎑0.71728276, d2 =1.7739244
ln
∆Vm,melt(quad) Vm,/cm3 mol᎑1 Vm,s(calcd)
P/bar
Selected Values from Table 8, Reference 16
References 14 and 15
294.55
4,903.4
3.93
33.03
28.57
306.25
5,884.0
3.62
32.71
28.50
317.35
6,864.7
3.34
32.41
28.48
328.35
7,845.4
3.07
32.16
28.48
338.95
8,826.0
2.84
31.92
28.51
348.55
9,806.7
2.64
31.74
28.56
357.75
10,787
2.46
31.57
28.64
368.45
11,768
2.27
31.43
28.75
Sublimation curve (applicable down to 90 K) (16 ):
T p ln p = 14.57893 1 – 3 – 14.48067 ln T + T T3 3 2 3 65.35685 T – 1 – 47.14593 T – 1 + 14.53922 T – 1 T3 T3 T3 Sublimation Data (Using the Ideal Gas Equation) T/K
Vm,v/(cm3 mol᎑1)
P/bar
90
6.63 × 10
᎑9
1.13 × 1012
100
2.15 × 10᎑7
3.87 × 1010
110
3.79 × 10᎑6
2.41 × 109
120
4.21 × 10
᎑5
2.37 × 108
᎑4
3.34 × 107
∆Vm,melt = 6.9542 × 10᎑5T 2 – 0.0686277044T + 18.1107164247
130
3.24 × 10
140
0.00186
6.26 × 106
Melting results: We have fitted the liquid Vm, versus T data below to the following cubic polynomial (given to 10 decimal places—this precision is not justified by the underlying data, but is required to yield calculated values of sufficient accuracy): Vm, = ᎑3.0331 × 10᎑6T 3 + 0.0028688133T 2 –
150
0.00843
1.48 × 106
160
0.0314
424,000
170
0.0995
142,000
180
0.276
54,300
190
0.684
23,100
200
1.55
10,700
0.9200958384T + 132.4799710302 ∆Vm,melt(quad) is the calculated value for the volume change at the corresponding melting temperature, using the quadratic fit above. Vm,s(calcd) (= Vm, – ∆Vm,melt) is the corresponding calculated molar volume of solid CO2 using the difference of the above two equations: ∆Vm,melt(quad) = ᎑3.0331 × 10᎑6T 3 + 2.799268 × 10᎑3T 2 –
210
3.27
5,340
216.58
5.19
3,470
0.85146813560T + 114.3692546
NOTE: The van der Waals equation, using a = 3.688 bar cm6 mol᎑2, b = 42.67 cm3 mol᎑1 (1), gives essentially the same values. Fluid equation of state (17 ):
p=
RT V – B 1 + B 2T –
B3
–
A1 TV
2
+
A2 V4
3
V +C
with C = B3/(B1 + B2T ) and B1 = 28.06474, B2 = 1.728712 × 10᎑4, B3 = 8.365341 × 104, A1 = 1.094802 × 109, A2 = 3.374749 × 109 in units of cm3 mol᎑1, K, and bar.
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Journal of Chemical Education • Vol. 79 No. 7 July 2002 • JChemEd.chem.wisc.edu
