6178
J. Phys. Chem. 1984,88, 6178-6180
Thermodynamics of Critical Points in Two-Component, Three-phase Fluids M. Gitterman Department of Physics, Bar-Ilan University, Ramat Gan, 521 00, Israel (Received: July 2, 1984)
Slopes of three-phase coexistence curves of a two-component system are analyzed near critical points where two of the phases become identical. A general criterion for the upper and lower critical points is obtained.
Many thermodynamic results have been obtained for binary two-phase systems near critical and other special points on the phase diagram.' As examples, one can mention conditions for upper and lower consolute points, their dependence upon pressure, the so-called Gibbs-Konovalov laws for extrema1 points of the temperature or the pressure, among others. Some general results for three-phase systems were obtained very long ago by van der Waals and Konstamm? S~hreinemakers,~ and R u ~ a n o v . ~These authors, however, have not transformed their results into an easily applicable form, possibly because of a lack of experimental data at that time. Critical phenomena in multicomponent fluids have received widespread attention recently, both experimentally5 and theoretically.6 In addition to finding the usual critical lines (surfaces) when two phases become identical, the intersections, the so-called tri- or multicritical points (lines) where three or more phases become identical, have also now been found. Different forms of equations of state, such as the van der Waals equation, might be applied' to the analysis of complicated phase diagrams. It is interesting, however, to find results that can be obtained from general thermodynamic analysis without using a specific model. We will find such a result here for the simplest case of a three-phase fluid containing only two components. Three-component and multicomponent systems, which have more degrees of freedom, can be treated in an analogous way although the calculations become much more cumbersome. The assumption of analyticity at the critical point has been used throughout this article. An analytic equation of state fits the experimental properties of the fluid except for details in the immediate vicinity of the critical point. Probably none of the qualitative results which we obtain below depend on the assumption of analyticity. ten Brinke and Karasz showed6 that the scaling equation of state gives for binary fluids the same qualitative results as an analytic, mean field equation of state.
Equilibrium Regions The system to be considered has at constant pressure and temperaturef = c - p 2 = 2 - 3 2 = 1 degree of freedom. (Here, c and p are the number of components and phases, respectively.) As three independent variables, we choose the pressure P , the temperature T, which are the same in all phases, and the mole fraction of, say, the first component in the @-phase,xl@.(We
+
+
( 1 ) Prigogine, I.; Defay, R. 'Chemical Thermodynamics"; Longmans Green: London, 1954; Chapter 18. (2) van der Waals, J. D.; Konstamm, P. K. 'Lehrbuch der Thermodynamik"; 1912. (3) Schreinemakers, F. A. H. Papers; Pennsylvania State University: University Park, PA, 1965. (4) Rusanov, A. I. Vestn. Leningr. Univ. 1959, N2, 62. (5) Knobler, C. M.; Scott, R. L. In "Phase Transition and Critical Phenomena"; Domb, C., Lebowitz, J., Eds.; Academic Press: New York, 1984, Vol. 9. (6) Griffiths, R. B. J. Chem. Phys. 1974, 60, 195. (7) van Konynenburg, P. H. Philos. Trans. R. SOC.London, Ser. A 1980, 298,495. Scott, R. L.; van Konynenburg, P. H.Discuss. Faraday SOC.1970, 49, 87. (8) ten Brinke, G.; Karasz, F. E. J . Chern. Phys. 1982,77,5249; 1982,78,
4792.
denote the phases by superscripts a,@, and y, and the components by subscripts 1 and 2.) Three-phase equilibrium can be defined as a combination of two two-phase equilibria, say, between phases a-@ and @-y. Writing the equality of the chemical potentials in these phases, we obtain equations describing the equilibrium regions on the phase diagram in the following form (see p 79 in ref 2):
where
v%@ = v" -@, - (X1U - x , @ ) v @ l (2)
sa,@= sa - S@- (xla - x,@)s,l
The values vY,@ and sY.0 are defined analogously. The molar Gibbs free energy g, the molar volume v, and the molar entropy s have been introduced in eq 1 and 2. We have also used the following shortened notations: gS1l ~3~gB/d(x~@)~; up1 dv@/dxlB,etc. Another meaning for the values va-@ and sa,@can be obtained if we recall that, for each molar quantity in each phase, y { u p ) y = ~ 1 + ~(1 1- xl)y2, where y l and y z are the partial molar quantities of the first and second components, respectively. Then, dy/dxl = y1 - yz and eq 2 can be rewritten as
+ ( I - Xla)(V2a - V j ) sa,@ = xla(sla- sl@) + ( 1 - x ~ ~ ) -( s,@) s ~
uai3
= X I q U l a - v,@)
~
(2a)
Equations 2a contain differences of the partial molar volumes and entropies in two phases and are convenient for estimates if the two phases, say, liquid and gas, have significantly different properties. Dividing both sides of eq 1 by dT, one can solve the inhomogeneous system of equations for the slopes of the three-phase equilibrium curves:
-
[UYqXla
- XI@)- V va'5sY.@
q X , Y
- xl@)]g@"
- vY.@sa%@
(3)
The ratio of eq 4 and 3 will give the P-T projection of the three-phase equilibrium region. The latter formula has been listed in the book by van der Waals and Konstamm.2
Upper and Lower Critical End Points Let us now draw in Figure 1 a typical projection of the three-phase equilibrium region onto the T-xl phase diagram. (The P-xl projection looks similar.) Each tie line 00' interposed between AC and BD lines belongs to the three-phase equilibrium. Only one- or two-phase states exist above BD and below AC lines. The boundary stated described by the points B-D (A-C) are those where phases a-@ (@T) become identical. They are called upper and lower critical points, respectively. The tricritical point, if any,
0022-3654/84/2088-6178S01.50/00 1984 American Chemical Societv
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6179
Thermodynamics of Critical Points
T
t
B
The molar volume and the entropy are, in general, smooth functions. Let us, therefore, restrict series 8 to the terms written in eq 8. Substituting eR 8 for y u and y s in eq 7, we can rewrite this equation in the following final form:
Figure 1. Typical temperature (T)-concentration (xI)projection of the phase diagram of a three-phase (a,0, y) system. The line 00' represents equilibrium of the three phases. Only one or two phases pxist above the upper critical and point B-D and below the lower critical end point A-C.
is the point of coincidence of B and C. Notice that according to the Gibbs phase rule one cannot obtain the tricritical point in a two-component liquid system. One has, therefore, to introduce an additional degree of freedom, for example, a set of binary mixtures with different interaction energies (the so-called "quasi-binary" mixtures5). As noted above, we are interested in the thermodynamic behavior near the special points on the phase diagram, namely, near the two critical states B-D and A-C. In order to analyze the form of the equilibrium curves, let us simplify eq 3 near the point B of the phase diagram when phases a and 6 become critical. When the point 0 is approached, xI0 - xl@, and sa@ become small. Dividing each term in eq 2 by xla - xl@and expanding all functions involved around their critical values, we obtain for y (UJ]
From the stability condition of phase /3, it follows that go" > 0. It is clear from Figure 1 that xIY > xl@> xICr.Finally, according to our assumption of smoothness of functions 8, ((~@"')"')(xl~ - X~cr)/(3(U~'1)cr)< 1. Therefore, the sign of the slope of the coexistence curve will be determined by
ua3@,
Hence, the slope will be negative for negative (Uglll)cr
(S@lll)cr
(Ugly
(S@ll)cr
(S~II)~'
if
>O and positive for positive The law of rectilinear diameter (xl@+ x10)/2 a xlCr has been used as we pass to the last equality in eq 5 . Small corrections can be considered if necessary. The superscript cr here and elsewhere means that the value of the appropriate function is taken at the critical point. Substituting eq 5 into eq 3 leads to
The small parameter x,g - xlCrappears in eq 6. We can, therefore, neglect the second term in the numerator of the right-hand side of eq 6. Then, making simple transformations, we obtain
-eq line
1
gall
(xi6 - ~ 1 ~ ~ ) ( ~ g l ~ () ~~g~l l ) ~ S~ ~ , c r
(7)
I---
The right-hand side of eq 7 can be estimated if we know the behavior of the molar volume and the entropy. Alternatively, it can be useful if the slope of the coexistence curve near the critical point is known. Let us, however, make some additional simplifications. Until now, we have used series expansions of the molar and partial molar volume and entropy near the critical point. If we are not too close to the critical point, the assumption of analyticity is correct, and eq 7 can be used. In order to simplify eq 7, let us write the Taylor expansion that binds together values of functions y 3 (u,s]in points D and B with coordinates xIY and xIcr:
( ~ ~ l l if) ~ ~
(U@lll)Cf
(Sp)cr
(Ugly
(Sgll)cr
>O Such a situation is depicted in Figure 1. Of course, the same result can be obtained if one assumes that phases p-y, rather than a-p, become critical; Le., in the vicinity of point C on the phase diagram. All calculations will be similar and, in the final formula (eq 9), index y has to be replaced by a, and all derivatives are taken at the critical point C. Then xIc( < xI@< xlCrand the sign rule (expression 10) will remain unaltered. However, the negative slope will now indicate that C is the minimum point along with a maximum in B, as depicted in Figure 1. Comparison between Two- and Three-phase Systems Let us compare the criteria obtained with that of a binary two-phase system. For coexisting a-p phases, the first of eq 1 describes the equilibrium changes of state. There are now three independnet variables and usually one of them is kept constant. For constant pressure, we obtain from eq 1, also using eq 2a
Near the critical point of the a-p phases, we can expand all the partial molar volumes in eq 11 about the critical point. This yields, finally,' that the critical point of the two-phase system at constant pressure will be a maximum if ( ~ ~ < l 0, ~ and ) ~ will ~ be a minimum if ( ~ ~ l l >) ~0.~ Our result for a critical system, in coexistence with a noncritical system, is similar to that of the two-phase system at constant pressure, except that the higher
6180 The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
replaced by the molar volumes ui. Equation 12 allows immediate experimental verification.
p.50 MPa
Coexistence of Critical and Noncritical Phases Let us now analyze the branches of the equilibrium curve located near D and A in Figure 1. For this purpose, it is convenient to rewrite eq 1-3 by using as an independent variable the mole fraction of the first component in the CY (or y) phase, which does p replacement in eq 3, we not become critical. Making the CY obtain
G+L
1
'0
Sl + s2
0.2
0.4
0.6
0.8
"gH14
Figure 2. Temperature-mole fraction (T,x) phase diagram for the system methane n-hexane at a fixed pressure P = 5.0 MPa. (Reproduced with permission from ref 5 . Copyright 1984, Academic Press.)
+
derivatives of the molar volume and entropy also enter into our criterion.
Phase Diagram of a Three-phase System at Fixed Pressure (Temperature) The T-x phase diagram at constant pressure (P = 5.0 mPa) for the system methane n-hexane is shown in Figure 2. In this figure is shown the three-phase equilibrium line with two critical solution points below it. This line is located near the gas-liquid two-phase region. For the case considered, when the pressure (or the temperature) is held constant, the equilibrium equations 1 allow additional simplification. The system of equations in eq 1 at P = constant has nontrivial solutions only if the condition sa*B (xIa - xlP)-l = sY#(x,y - x1S)-' is met. The latter can be rewritten in the following form:
+
x l q s p - SY)
Gitterman
+ XIB(SY - sa) + XIY(Sa -
SP)
=0
(12)
Analogously, if the temperature, rather than the pressure, is kept constant, the concentrations of the three coexistence phases,x' (i = a, p, y) will satisfy eq 12, where the molar entropies SI are
When the phases p and y are close to becoming identical, we have to expand all the functions appearing in eq 13 near the critical point, in accordance with eq 5. Such a calculation has been performed in ref 4. The final answer is
This formula is useful for the comparison of different experimental data. The interestring "Azeotropic" case is that of x l a = xlCr,when points D and B coincide. Then, however, = = -ScT,*, and both numerator and denominator vanish at the critical point. This uncertainty can be easily a n a l y ~ e d .It~ turns out that, in this case, dT/dxla # 0 at the point where xla = xlCr,unless va = ucr and sa = scr. There is no reason, however, to believe that these equalities are satisfied in the azeotropy point where xla = xiCr. The latter condition is still "random" in the sense that two phases remain distinguishable; for example, they have different thermodynamic derivatives. On the other hand, it can be shown' that, for the two-phase equilibrium described by eq 11, the so-called Gibbs-Konovalov law holds; namely, "along the equilibrium line at constant pressure an extremum of temperature corresponds to equal compositions of two phases." This theorem, and its converse, follows immediately from eq 11. Hence, it turns out that the Gibbs-Konovalov law does not hold in the case considered of two-phase equilibrium of the critical and noncritical phases when this state is approached from the three-phase region. This is not so surprising because, in contrast to the usual two-phase case, the pressure is no longer kept constant. In conclusion, the general criterion for the upper and lower critical point (eq lo), the slopes of the coexistence curves near the critical point (eq 7 and 14), an additional restrictiion (eq 12) on the concentration of three phases coexisting at constant pressure might be compared with experiment.
Acknowledgment. I am grateful to Professors C. Knobler and R. Scott for useful discussions and hospitality during my stay at UCLA, where the main part of this work was done.
