Thermodynamics of Multicomponent Liquid Mixtures Containing Subcritical and Supercritical Components D. S. Abrams, Fausto Seneci, P. L. Chueh, and J. M. Prausnitz* Chemical Engineering Department University of California, Berkeley, Berkeley, California
In a multicomponent liquid mixture containing several solvents (subcritical components) and several solutes (supercritical components), it is convenient to normalize activity coefficients with the unsymmetric convention; for each solute, a useful standard-state fugacity is Henry’s constant for that solute in the (all-solute-free) solvent rnixure. When that standard-state fugacity is used for each solute, thermodynamic consistency requires that the fugacity of each solvent depends, in part, on the variation of Henry’s constant with the composition of the mixed solvent. This dependence has only a small effect when he solute concentration is low or when the solvents are chemically similar. In general, however, this dependence is not negligible. For thermodynamic consistency, therefore, corrections derived from this dependence should be included in the Chueh-Prausnitz correlation for K factors in hydrocarbon and natural-gas mixtures.
The formal thermodynamics of liquid mixtures has received much ’ attention in the technical literature, but with few exceptions this attention has been confined to those mixtures wherein, at system temperature, every component can exist as a pure liquid, either real or hypothetical. For such mixtures, the activity coefficients are defined in such a manner that for each component its activity coefficient goes to unity as its mole fraction goes to unity. Little attention has been given to the formal thermodynamics of those mixtures, commonly encountered in chemical engineering, wherein some of the components, often called subcritical, can exist as pure stable liquids at the system temperature, while others, often called supercritical, cannot exist as pure stable liquids at the system temperature. While such mixtures are often described with the same thermodynamic formalism i;s that used for mixtures containing only subcritical components, the use of hypothetical standard states, necessary for such description, introduces serious difficulties. To cwercome these difficulties it is convenient to use a thermodynamic formalism based on unsymmetric normalization of activity coefficients; in this formalism a distinction is made between components which, at system temperature, can exist as (real) stable liquids and those which cannot. The former are called solvents and the latter are called solutes. The activity coefficient of each solvent goes to unity as its mole fraction goes to unity but the activity coefficient of each solute goes to unity as the mole fraction of that solute goes to zero in the all-solute-free mixture. This method of representing thermodynamic properties of liquid mixtures is particularly appropriate for high-pressure mixtures containing both subcritical and supercritical components (Prausnitz and Chueh, 1968). However, when using this less-familiar thermodynamic formalism, special care must be exercised to assure thermodynamic consistency. This brief article calls attention to a particular aspect of thermodynamic consistency which appears to have been neglected. A method is suggested for achieving thermodynamic consistency in multicomponent mixtures containing both solvents and solutes. Consider a multicomponent liquid mixture at temperature T containing both solvents and solutes. A solvent component is one whose activity coefficient is normalized by the symmetric convention: the activity coefficient of a solvent goes to unity as its mole fraction goes to unity. A solute component is one whose activity coefficient is nor52
Ind. Eng. Chern., Fundarn..Vol. 1 4 , No. 1, 1975
malized by the unsymmetric convention: the activity coefficient of a solute goes to unity as its mole fraction goes to zero in the solute-free solvent mixture. Let subscript i stand for solvent and sbuscript k for solute. The fugacities of the components are written
where 50 = fugacity of pure liquid (solvent) at system temperature T and zero pressure, y L R = activity coefficient of solvent i at temperature T and zero pressure when all solutes in the mixture obey Henry’s law, and the solute-containing mixture a t T and zero pressure, H k ( M S ) = cient of solvent i in the solute-free mixture a t T and zero pressure, y1 = activity coefficient of solvent i in the solute-containing mixture at T and zero pressure, H k ( M S ) = Henry’s constant for solute k in the solute-free mixed solvent at 7’ and zero pressure, yk* = activity coefficient of solute k at temperature T and zero pressure, n = mole fraction in the liquid phase, y = mole fraction in the vapor phase, and P = total pressure of the mixture. Our concern here is with YR which, contrary to previous assumption (Prausnitz and Chueh, 1968), is not necessarily equal to unity. Since this activity coefficient appears only in eq 1 and not in eq 2, the entire subsequent discussion refers to the fugacities of solvents in mixtures containing at least one solute and at least two solvents. Fugacities in binary systems are not affected. Further, the fugacities of solutes are not affected regardless of the number of components. Activity coefficient Y ~ ( S F is found from differentiation of an excess Gibbs energy function for the solute-free mixture as given previously (Prausnitz and Chueh, 1968). Activity coefficients y i and yk* are obtained from differentiation of an excess function gE* which accounts for solute-solute interactions in the liquid phase at temperature T and at zero pressure. When all solute mole fractions go to zero, g E * goes to zero also. Thus gE* accounts for deviations from Henry’s law for the solutes at constant T and zero pressure. Activity coefficients yc and yh* are related togE* by
I
A useful model for gE* is provided by the dilated van Laar model discussed previously (Prausnitz and Chueh, 1968). The activity coefficient y l R is needed to yield a thermodynamically consistent fugacity for solvent i. When gE* = 0, it does not follow, as previously assumed, that y i R = 1. The superscript R denotes reference solution, i.e., a solution where all solutes obey Henry’s law ( y k * = l), and the solute-free solvent mixture is ideal. The numerical value of ylR depends upon the mixing rule used to find
-06-
I
\
-
\
eb
-08-
-I
I
I
-
\
\
Hk(Ms).
Since activity coefficients Y~(SF),y l , and yl* all follow from differentiation with respect to composition of an excess Gibbs energy, they satisfy the Gibbs-Duhem equations
C
i a l l solvents
xi’d ~n Y ~ ( S F )= o
(5)
and
2
xid
?i
i a l l solvents
+
a3,Volume Fraction Methane Figure 1. Effect of composition on reference activity coefficient of pentane (1) and of hydrogen sulfide (2) in mixtures containing methane.
xkd h Y k * = 0
(6)
k a l l solutes
concentrations of k and 1 are expressed as volumes per volume of solvent (not per volume of solution) as used in the thermodynamics of electrolyte solutions. Equation 12 obeys the boundary conditions, eq 10 and 11. As shown below, the function @ i l k is given by
where the prime denotes solute-free composition. For thermodynamic consistency, therefore, the following Gibbs-Duhem equation must also be satisfied
1
xjd
i a j l solvents
In y i R +
x k d In Ilk,,,,
= 0 (7)
k a l l solutes
Equation 7 shows that, in general, y l R is not unity but depends on how the Henry’s constants in the mixture vary with composition. As previously (Prausnitz and Chueh, 1968), we assume that 2j
In
Hb(MS)
=
‘i i a l l solvents
In
Hk(i)
- -%
2
cc i
cyil’i\El
a
a1 1 sol vents
( 8)
Equation 13 indicates that the function P l l h contains no new parameters. Equation 13 follows directly from the mixing rule used to fine H k ( M S ) (eq 8). Since all = 0111 = 0, we find, as expected, for any solute = 0. Note that while k, Pl&k = = 0111, P l l k # P I L A . Equation 13 shows that whenever solvents i and 1 are chemically similar, PlLk is small. This follows because in ) &(&I. that event all k s m a l l a n d H ~ ( 1= For a mixture containing any number of solvents and solutes, eq 12 can be generalized to yield
where ucR, the critical volume of k , is taken as a molecular-size parameter. The solute-free volume fraction \Ir is defined by Xj’Vc;
where @ k and @l are volume fractions and where j
a l l solvents
Henry’s constant for solute k in pure solvent i (at temperature T and zero pressure) is denoted by H R ( ~ The ). van Laar parameter atl reflects interactions between solvents i and 1. To obtain an expression for -ylR, we consider first a ternary mixture containing one solute and two solvents. For this ternary mixture we note that the following boundary conditions apply In y i R
h yiR
-+
-
0 as xi 0 as
xk
-
1
(10)
0
(11)
Rigorous Derivation of Reference Excess Gibbs Energy We now show that eq 12, 13, 14, and 15 are independent of any physical assumptions for In ylR. We define an extensive excess Gibbs function GR such that
j f i
In the next section it is shown that In -ylR is given by j # k
Euler’s theorem of homogeneous functions yields where n stands for the number of moles and P l l R is a function independent of the concentration of k . Note that the
a l l solutes
a l l solvents
Ind. Eng. Chern., Fundam., Vol. 14, No. 1, 1975
53
Since H k ( M S ) is independent of the concentration of k , it follows from eq 17 and 18 that
1, 2) passes through an extremum (maximum or minimum) a t a value of @[ given by
(19) a l l solvents
Equation 19 holds for all nk. Upon integration, eq 19 yields
E
ai 1n Y i R + f(nk)
(20)
i a l l solvents
-
However, from the normalization of yiR l i m (In y i R ) nk-0
o
(2 1)
As the right-hand side of eq 20 is zero for one value of nk (uiz.,nk = 0), it is zero for all values of n k , thus ni
In Y i R = 0
i a l l solvents
and, upon substituting (22) into (18), we obtain GR _ -
RT
E
nk
In
Hk(MS)
all salutes
Equations 16 and 23 are exact thermodynamic results independent of any molecular model. Upon using eq 8 for f f k ( M S 1 and substituting eq 23 in eq 16, we obtain for In yiR the results shown in eq 12, 13, 14, and 15. Importance of yLR To assess the importance of yLRfor a mixture where O [ l k is not small, we present some illustrative results for a ternary system containing two solvents (n-pentane and hydrogen sulfide) and one solute (methane). Let 1 stand for n-pentane, 2 for hydrogen sulfide, and 3 for methane. The temperature T is 559.7"R. All binary parameters are from Prausnitz and Chueh (1968). Figure 1 shows In y1R and In yzR as a function of @3. The full lines correspond to a solution containing equal volume fractions of the two solvents (@I = %) and the dashed lines correspond to the limiting cases $1 = 0 and = 0. We find, as expected, that when a 3 = 0, ylR = yzR = 1. Further, when @I is large (Le,, close to unity), ylR is also nearly unity and similarly, when is large yzR is near unity. The value of yLRdeviates most from unity when +k is large and @[ is small. When G k is held constant, yLR(i =
54
Ind. Eng. Chem.,
Fundarn.,Vol.
14, No. 1, 1975
Figure 1 shows that in some cases the simplifying assumption yiR = 1 is not reliable. However, the particular system chosen here for illustrative calculations is one where deviations from that simplifying assumption are particularly large. In multicomponent systems containing only hydrocarbons the coefficient b i l k is appreciably smaller than that for the pentane-hydrogen sulfide-methane system. In conclusion, it may be helpful to summarize the main features of this analysis for multicomponent liquid solutions containing at least two solvents and at least one solute. 1. Thermodynamic consistency requires that the expression for the fugacity of a solvent i (eq 1) includes an activity coefficient yiR. This activity coefficient depends on how Henry's constant for a solute varies with solvent composition. 2. Activity coefficient yLRis given by eq 14 and 15. The parameters required are exclusively pure-component and binary parameters, the same ones that are obtained by standard data reduction for binary systems. 3. For a binary system Y~~ = 1for all compositions. 4. The fugacities of solutes are not changed by the results of this analysis. Only the fugacities of solvents are affected in those mixtures where there are at least two solvents and one solute. Thus, in a ternary system containing one solvent and two solutes yiR = 1 for all compositions. 5. In multicomponent systems where there is significant chemical difference between solvents, yiR may deviate appreciably from unity as the concentration of solute rises. 6. Although the specific equations for yiR presented here pertain to the solution model used by Prausnitz and Chueh, the technique for establishing thermodynamic consistency in multi-component solutions is general; that is, eq 16 and 23 can be coupled with any molecular model for gE*. Literature Cited Prausnitz, J. M., Chueh, P. L., "Computer Calculations for High-pressure Vapor-Liquid Equilibria," Chapter 6, Prentice-Hall, Englewood Cliffs, N . J., 1968.
Received for reuieu; May 8, 1974 Accepted September 23,1974 The authors are grateful to the Gas Processors Association, the American Gas Association and the National Science Foundation for financial support, and to J. P. O'Connell and G. Guerreri for their interest in this work.
