T H E R M O D Y N A M I C S OF S O L U T I O N S W I T H P H Y S I C A L AND CHEMICAL INTERACTIONS Solubility of Acetylene in Organic Solvents H . G . H A R R I S 1 AND J . M . P R A U S N I T Z Department of Chemical Engineering, University of California, Berkeley, Calif. 9.4780
A set of thermodynamically consistent equations has been developed for representing the properties of solvating binary liquid mixtures. Chemical interactions between solute and solvent are taken into account by K, an equilibrium constant for complex formation; physical forces between solute, solvent, and complex are represented by van Laar equations through a single physical interaction parameter, a. Solution of the resulting equations by iterative techniques for typical, physically significant values of K and a demonstrates that a wide range of mixture behavior can be represented. Literature data for solubility of acetylene in a variety of solvents have been analyzed and best values of K and a are reported for 13 solvents. The new set of equations give a better fit of the data than that obtained using common two-parameter equations for the excess Gibbs energy, The method developed here can be extended to a variety of cases with strong specific interactions; it provides a basis for general treatment of strongly nonideal liquid mixtures.
IN LIQUID mixtures, deviations from ideal behavior can be interpreted in terms of intermolecular forces operating within the mixture. Broadly speaking, it is convenient to distinguish between strong attractive (chemical) forces leading to formation of chemical species and weak attractive (physical) forces frequently called van der Waals forces. Accordingly, the traditional theory of liquid solutions has followed two distinct paths: One path, initiated by the work of Dolezalek (Kortum and Buchholz-Meisenheimer, 1952), interprets solution nonideality in terms of chemical forces while neglecting physical forces; the other, represented by the work of van Laar (Hildebrand and Scott, 1964), interprets solution nonideality in terms of physical forces alone. Dolezalek assumed that in a mixture of “apparent” components A and B the “true” molecular species in the mixture may not only be molecules A and B but may also include molecules Az, A3, ., B2, BB, ., and AB, AzB, AsB, ABz, ABa, . ,, etc. Dolezalek also assumed that a mixture of the “true” molecular species is an ideal mixture. Van Laar, on the other hand, denied the existence of any molecular species other than A and B and explained deviations from Raoult’s law in terms of differences among intermolecular forces acting among A-A, A-B, and B-B. While the difference in viewpoint between Dolezalek and van Laar has caused much polemic, we recognize today that both views are extreme representations of the actual situation: The borderline between chemical and physical forces is arbitrary and in many cases designation of a mixture as “chemical” or “physical” is only a matter of taste or convenience. However, while it is reasonable to assume that chemical forces are absent from simple solutions of saturated nonpolar liquids, it is, perhaps, not reasonable to neglect chemical forces in liquid mixtures where hydrogen bonding or charge-transfer complexing is appreciable. Certainly it is not strictly correct to neglect physical forces entirely in “chemical” solutions, although such neglect may, in some cases, provide an excellent approximation (McGlashan and Rastogi, 1958). A physi-
.
1
180
..
..
Present address, Tulane University, New Orleans, La. 70118. IhEC
FUNDAMENTALS
cally more reasonable description of liquid solutions should make allowance for both chemical and physical forces; in the absence of the former, such a description should become identical with van Laar’s treatment and in the absence of the latter, it should become equal to Dolezalek’s theory. Such descriptions necessarily lead to complicated algebra but they have been developed for a few cases, notably for associating solutions of alcohols in saturated nonpolar solvents such as paraffins (Kretschmer and Wiebe, 1954; Renon and Prausnitz, 1967; Scatchard, 1949) ; these descriptions take into account the formation of dimers, trimers, etc., of the hydrogenbonding component. In this work we present a treatment for solvating solutions -i.e., solutions where the unlike components form complexes with one another. Our theoretical formulation takes into account physical forces between the “true” species formed by chemical interactions. We describe in detail our formulation for the case where two components form a complex with 1 to 1 stoichiometry and then, in Appendix 11, we indicate our generalization to cases where the stoichiometry of the complex is arbitrary. Work along similar lines has been reported recently by Kehiaian (1966, 1967, 1968). In view of the common availability of electronic computers, we make no simplifying assumptions in order to achieve an explicit solution to our equations. For practical engineering purposes, however, we attempt to reduce the number of adjustable parameters, since experimental data are rarely sufficiently accurate or plentiful to justify more than two (or a t most three) parameters for a binary mixture a t a fixed temperature. Finally, we illustrate our theoretical development with an analysis of solubility data for acetylene in a variety of organic solvents. Solution Model
Consider a mixture of components A (component 2) and B (component 1) which interact strongly to form a complex AB: A+B$AB
The thermodynamic equilibrium constant is given by
K = - =aAB aAaB
YABZAB
(1)
YAZAYBZB
where activities of A, B and AB = “true” mole fractions of molecular species A, B, and AB
U A , U B , UAB =
zA, ZB, ZAB
activity coefficients of A, B, and AB We use letter subscripts to indicate “true” mole fractions and activity coefficients, and subsequently number subscripts to indicate “apparent” quantities. The terminology “true” and ‘‘apparent” assumes the existence of species AB. Verification of the existence of AB complexes rests both on the ability of the theory to account for observed solubility data and on other independent evidence, such as spectroscopic data (Brand et al., 1960). “True” mole fractions necessarily depend on the particular technique used for their calculation; we have used the quotation marks to emphasize this fact. The true mole fractions of each species are given by stoichiometry, in terms of the normalized extent of complex formation, E (0 5 5 5 1/2) :
Subscripts i, j, and k are understood in this case to range over the three possible species A, B, and AB. For mixtures of components of greatly different molecular size, representation of the activity coefficients may be improved by addition of a term corresponding to the Flory-Huggins entropy of mixing; for VI>> v2, R T In Yk =
vk
(
&j*j
-4
&$‘.p+j)
j
i
j
?A, YB, TAB =
For a system consisting of three species, in this case A, B, and AB, Equation 7 or 9 implies introduction of three physical parameters. Practically, the experimental data are never precise enough to justify three such parameters and it is therefore desirable to relate these parameters to each other. Although this may be done in a variety of ways, we have chosen to relate the parameters through estimation of the solubility parameter of the complex. Details of this calculation are given in Appendix I. With this simplification it is possible to express the “true” activity coefficient of each species as a function of “true” composition and a single physical parameter, CY E LXA-B: ‘YA
= $A
(ZA, ZB, ZAB, a)
YB = f B (ZA, ZB, CAB, a) TAB
=
(10)
AB (%A, ZB, ZAB, a)
The functions represented by f are simply those given in Equation 7 or 9 with each aij calculated from the single binary physical parameter, a. These are given in detail in Appendix where XI,xz = “apparent” mole fraction of solvent and solute. Here species A is taken as the solute, or component 2, and species B as the solvent, component 1. The extent of complex formation, f , is related to the equilibrium constant, “true” activity coefficients, and “apparent” mole fractions:
E=
1
- { 1 - 4[K/
(K
+ K-,)]x~xz]
1.
Finally, the “apparent” activity coefficients of components 1 and 2, which are the experimentally accessible quantities,
are related to “true” quantities by the exact relations 72
l’’
2
(5) 71 =
The “true” activity coefficient ratio, K,, is defined by TAB K, = YAYB TO provide realistically for physical forces we represent “true” activity coefficients by van Laar equations. For each molecular species k : j
i
j
the gas constant absolute temperature “true” mole fraction of species k molar volume of species k volume fraction of species k CY”
=
- van Laar parameter for physical interaction of species i and j
ZA’YA
22
ZBYB
21
These relations are independent of any physical model; they assume only that the “true” species A, B, and AB are in equilibrium (Prigogine and Defay, 1954). For given values of K and a, it is possible to compute “true” compositions and activity coefficients at any specified (‘apparent” composition by simultaneous solution of Equations 2 to 5 and 10; ‘(apparent” activity coefficients are immediately given by Equations 11. Although algebraic solution in closed form is not possible, iterative solution of the equations is straightforward. With the wide availability of electronic computers prevalent today, such iterative calculations can be performed easily. The scheme for solution which we have used consists of initially setting K , equal to unity, and determining “true” mole fractions of all species through Equations 2 to 5. The “true” activity coefficient of each species is then calculated by Equations 10, and the resulting activity coefficient ratio is used for the next iteration. For physically reasonable values of CY, convergence is rapid. After obtaining “true” quantities, apparent activity coefficients are given immediately by Equations 11. This model provides a simple but realistic two-parameter representation of solution behavior for solvating mixtures. VOL.
8
NO. 2
MAY
1969
181
A major advantage of the model is that it can easily be extended to cover a range of temperatures. As suggested by regular solution theory, over modest temperature intervals (Y may be assumed independent of temperature. The change in the equilibrium constant with temperature is given by:
(%J=
AH
-R
where A H = enthalpy of complex formation. For A H constant, Equation 12 may be integrated to give
K = KOexp
(--E)
If data are available over a sufficient temperature range, temperature variation in A H may be evident. If linear variation with reciprocal temperature is assumed for A H ,
then integration of Equation 12 yields
Here AH(o)= enthalpy of complex formation a t a reference temperature To,and AH(’) = enthalpy correction term. Thus, by introducing enthalpy of complex formation, the effect of temperature on 72 and 71 can be taken into account. Application of Model in Typical Cases
To illustrate the main features of the model, we have calculated “apparent” activity coefficients as functions of “apparent” concentration for a variety of typical cases (Figures 1 through 5). Figure 1 gives solute activity coefficients in a system composed of molecules of equal size in which only chemical interactions occur (aij E 0 for all i,j). 31
Figure 1.
I
I
I
I
Activity coefficients in solvated mixtures
Chemical interactions only in a system where m / v z = 1.0
182
IhEC
FUNDAMENTALS
This corresponds to the classical treatment of Doleralek for solvated mixtures. In this case, by assumption, the “true” species form an ideal mixture; thus, for K = 0 the solution follows Raoult’s law, whereas for K > 0 solvation leads to negative deviations from Raoult’s law. Figure 2 shows results for a similar system, except that physical interactions between species are now allowed, as given by Equations 7. In this case, for K = 0 and a > 0 positive deviations from Raoult’s law occur. As R rises, solvation causes a decrease in apparent activity coefficient; for K = 0.5, In 72 exhibits a maximum, and for K = 1, In 7 2 shows a change in sign. This unusual behavior has a simple physical interpretation. For appreciable values of K , the extent of complexing is large. At low solute concentrations the solute encounters predominantly solvent molecules, a large portion of the solute is tied up as complex, and the “apparent” solute activity is much lower than its apparent mole fraction; therefore 72 < 1. As solute concentration rises, however, solute-solute and solute-complex interactions become more frequent, until finally, a t high solute concentrations, the solute “sees” predominantly solute and complex molecules. Thus, a t large $2, physical effects begin to dominate for the solute and positive deviations from Raoult’s law result. The logarithm of the “apparent” activity coefficient is thus initially negative, rises with increasing solute concentration to positive values, and, because of the normalization 72- 1 as x2+ 1, ultimately passes through a maximum. “Apparent” activity coefficients for the solvent exhibit similar behavior in this case; the sets of curves are not completely symmetric, however, as a result of the way in which the (Y& are calculated (see Appendix I ) . Figure 3 shows results for a system in which the solvent molar volume is twice that of the solute. Other parameters remain the same. In this case the van Laar representation leads to highly unsymmetrical curves for solute and solvent activity coefficients (values for y1 are shown by dashed lines for K = 0 and K = 5 ) . Solute activity coefficients exhibit
Figure 2.
Activity coefficients in solvated mixtures
Chemical and physical interactions in a system where vi/vz = 1.0
behavior similar to those in the previous example, although the effects are somewhat more exaggerated as a result of the higher ratio for solvent-to-solute molar volumes. For the results shown in Figures 2 and 3, “true” activity coefficients were calculated according to Equations 7 . For mixtures where u1 >> u2, it is more realistic to incorporate a Flory-Huggins entropy term; for linear molecules the appropriate corrections are given in Equations 9. I n Figure 4 results are presented for the case in which the solvent-solute volume ratio is 12.0; “true” activity coefficients have been calculated with Equations 9. Again other parameters remain constant. In this case the negative contribution to In72 provided by the Flory-Huggins correction strongly influences the results, and positive deviations from Raoult’s law occur only a t relatively high solute concentrations. Stability analysis indicates that the system shown in Figure 4 is thermodynamically unstable a t high solute concentrations, where phase splitting would occur. For K = 0, solute and solvent are miscible over the entire concentration range for (@cu/RT)< 0.83. We have thus repeated the calculations for ( v w / R T ) = 0.5, again for a size ratio of 12.0 (Figure 5 ) . For this case, very slight positive deviations in 72 occur only a t concentrations above 0.95 mole fraction of component 2. From the examples shown, it is apparent that a wide range of solution behavior can be encompassed with this simple two-parameter solution model. The model can be extended to account for simultaneous reactions (self-association of solute and/or solvent) by introduction of appropriate additional equilibrium constants. While such extensions complicate the calculations, an electronic computer can easily handle them. More important, unfortunately, is the need for additional parameters in the form of chemical equilibrium constants. I n the derivation presented above, we have assumed a 1 to 1 stoichiometry for the complex. However, a similar derivation can be made for complexes of arbitrary stoichiom-
1.0
-
-
0
0.2
0.4
0.6
0.8
1.0
x2
Figure 4.
Activity coefficients in solvated mixtures
Chemical and physical interactions in a system where n / v z = 12.0 Flory-Huggins correction included
Va(
-
+-0.5 v2 V(
100 cc/gmol
= 1200 cc/gmol 813 6.5 L ~ / C C ~ ’ ~
Figure 5.
Activity coefficients in solvated mixtures
Chemical and physical interactions in a system where Flory-Huggins correction included
VI/Q
= 12.0
etry, as indicated in Appendix 11. Further, Appendix I1 considers the case where the solute and solvent may form simultaneously several complexes of different stoichiometry. Solubility of Acetylene in Organic Solvents Figure 3.
Activity coefficients in solvated mixtures
Chemical and physical interactions in a system where
VI/Q
= 2.0
The solubility of acetylene has been measured in a variety of liquid solvents (Miller, 1965). Acetylene solubility inVOL.
8
NO.
2
MAY
1969
183
where
P: = saturation (vapor) pressure of acetylene
ea= fugacity coefficient a t Pz8 = saturated liquid molar volume
0.751
I 2
I Figure 6.
I
I
I
4
PRESSURE,
I I I I I 6 810 Atm,
I
I
20
Vapor-phase fugacity coefficients of acetylene
creases markedly in going from nonpolar (physical) solvents to polar solvents with which acetylene interacts chemically through hydrogen bonding. Because of the industrial importance of acetylene, these data provide a useful test for our solution model. R e have therefore reduced available literature data with our equations. Further, we have compared the results with those obtained with other equations used frequently to describe liquid mixture properties. The fugacity of dissolved acetylene is given by
where
T
= absolute temperature
F‘
=
22, y2 =
total pressure “apparent” liquid and vapor mole fractions of acetylene
92
= vapor phase fugacity coefficient
y2
= “apparent” activity coefficient
fiZo
= fugacity of pure liquid acetylene a t
Above the critical temperature this standard state is hypothetical; we have used the corresponding states correlation of Lyckman and Eckert (1965) to extrapolate the standard-state fugacity to temperatures slightly above the critical. As shown in Table I, a t pressures up to 26 atm. the LyckmanEckert correlation gives excellent agreement with fugacities calculated from Equation 17, using the careful vapor pressure measurements of Ambrose (1956) and Ambrose and Townsend (1964) and fugacity coefficients calculated from the virial equation. Liquid volumetric data for acetylene are scarce, and saturated liquid molar volumes given in Table I were calculated from a corresponding-states correlation (Lyckman et al., 1965). To calculate activity coefficients through Equation 16 it is necessary to estimate partial molar volumes of acetylene. For several solvents dilation data are available; these were used to obtain partial molar volumes. For the remainder of the systems partial molar volumes were estimated using a method described by Lyckman and Eckert (1965); the estimated partial molar volumes were assumed to be independent of concentration. At the moderate pressures involved (usually less than 20 atm., in no case greater than 30 atm.) even a large error in estimating partial molar volumes introduces relatively small changes in activity coefficients. Solubility data for 13 systems were reduced by Equation 16. Typical results are presented in Figures 7 and 8; detailed results are tabulated by Harris (1968). For solvents in which solvation occurs, dependence of activity coefficients on concentration follows the pattern described in the preceding section; In 72 has negative (or small positive) values for low acetylene mole fractions, rises with increasing q ,and passes through a maximum. To test the model quantitatively, activity coefficientconcentration data were analyzed to obtain “best” values of K and a for each of the 13 systems. Nonlinear regression techniques were used to find values for K and a which
T and zero
Table 1.
pressure &L
= partial molar liquid volume of acetylene
Over the pressure and temperature ranges covered, y-z is close to unity; therefore, the Lewis fugacity rule holds and the vapor-phase fugacity coefficient, (c2, is that of pure acetylene. Vapor-phase fugacity coefficients were calculated from the virial equation truncated after the second term. Second virial coefficients were obtained from the correlation of Pitzer and Curl (Lewis et al., 1961); they are in excellent agreement with the measurements of Bottomley et al. (1958) over the range 0’ to 40’C. Figure 6 gives calculated vapor-phase fugacity coefficients; they are in substantial agreement with those reported by Miller (1965), based on other data. Below the critical temperature (35.18’ C. ) the standardstate fugacity of acetylene is given by (17)
184
ILEC
FUNDAMENTALS
T. C. -80 70 -60 -50 -40 30 20 - 10
-
-
0
10 20 30 40 50 60
P¶’, Atm.
1.305 2.19 3.48 5.27 7.70 10.87 14.9 20.0 26.3
...
...
.*. ... ... ...
Physical Properties of Acetylene fro, Atm. UP’.
Cc./G.Mole
’pt’
a
b
42.5 43.6 44.8 46.3 47.8 49.6 51.6 54.0 56.8
0.997 0.951 0.933 0.913 0.889 0.863 0.834 0.802 0.766
1.255 2.07 3.22 4.75 6.71 9.13 11.99 15.26 18.84
...
... ... ... ... ...
1.257 2.09 3.25 4.78 6.72 9.10 11.91 15.15 18.82 22.9 27.3 32.1 37.2 42.6 48.2
... ... ... ... *..
...
Critical properties:
... . . I
... ... ...
...
Tc = 35.18 “C. Po = 60.58 atm.
uc = 112.8 cc./gram mole Acentric factor: w = 0.191 a
Calculated from Equation 17. Calculated from correlation of Lyckman and Eckert (1965)
Table II. Chemical Equilibrium Constants and Physical Parameters for Acetylene-Organic Solvent Mixtures Standard % AH. Deviation r = Q, Ref. T," C . K Solvent Oel./Cc. Oal./Gram Mole in yz Vl/VZ f Ryutani, 1959 9.20 0 5.0 n-Hexane 2.12 0 6.7 Ryutani, 1959 10.1 0 n-Octane 2.80 0 Kiyama and Hiraoka, 1956 6.57 4.1 Benzene 1.54 0 0 Ryutani, 1960 5.19 4.3 n-Hexyl ether 0 0.708 -1060 4.05 1 (-200 to 0" C.) Kiyama and Hiraoka, 1956 Tetrahydrofuran 4.15 3.7 25 1.08 - 1270 1.40 1 (10"to 30" C.) Miller, 1965 25 1.13 -1150 15.5 Acetonitrile 3.0 0.90 1 (0" to 40" C.) Holemann and Hasselmann, 1954 2.6 25 1.28 Butyrolactone 11.8 1.32 1 Miller, 1965 15 2.46 Dimethyl 8.34 5.3 1.23 1 sulfoxide 0 4.27 -2850 - (5600/T) 6.7 Holemann and Hasselmann, 1954; 3.77 1.33 1 Dimethyl Miller, 1965;Ryutani, 1960; formamide ( T - 273.2)a Shenderei and IvanovskiI, 1962 Dimethyl 4.28 25 3.85 2.2 Holemann and Hasselmann, 1954 1.60 1 acetamide 25 4.05 4.83 1.1 Holemann and Hasselmann, 1954 N-Methyl 1.67 1 pyrrolidone 25 0.97 9.86 2.3 Miller, 1965 Dioxolane 1.21 2 25 1.23 32 Holemann and Hasselmann, 1954 Ethyl 2.20 2 7.80 acetoacetate
... ... ... ... ...
... ...
... ...
' -60' to
I
80' 0.
minimize the sum of squared per cent deviations of calculated from observed activity coefficients. Computational details are given by Harris (1968). I n calculating activity coefficients of the "true" species, it was desirable to avoid use of the highly expanded molar volume of acetylene as it approaches its critical temperature. As suggested by regular solution theory, over modest temperature intervals, the right-hand side of Equation 7 can be assumed independent of temperature. Molar volumes, volume fractions, and van Laar constants were therefore evaluated a t a reference temperature of 0" C. Activity coefficients
of "true" species were then calculated a t the appropriate temperature by Equation 7 ; a more refined treatment did not appear justified by the data. It was thus necessary to estimate binary van Laar constants (aZl's)for interactions between acetylene, solvent, and complex a t only one reference temperature. These constants were evaluated from the single binary interaction parameter CY by the technique described in Appendix I. For these calculations the solubility parameter of acetylene was estimated (Lyckman et al., 1965) to be 6.5 (cal./cc.)1'2 a t 0" C.; the molar volume a t 0' C. is given in Table I as 56.8 cc. per gram mole. Solvent-solute molar volume ratios (T = u 1 / v 2 ) at 0' C. are given in Table 11. The best values of K and CY for each of the 13 solvents studied are given in Table 11, along with standard per cent
a K (O'CI.
AH
0.2
0.1
Figure 7.
-
3.77 col/cc 4.27 =-2850-?
(T-273.2) col/gmol
I
1
I
1
Calculated and observed activity coefficients
of acetylene in three organic solvents
Figure 8.
Calculated and observed activity coefficients
of acetylene in dimethyl forrnarnide VOL.
8
NO,
2
MAY
1969
185
deviation in 72. I n Figure 7 activity coefficients calculated from these values of K and a for three typical systems are compared with observed values. Although precise assessment of experimental error is difficult, in most cases the fit appears to be within experimental error. Unfortunately, many of the available data are not of high accuracy. With any fitting technique the resulting constants must always be examined critically, since inability of a given model to represent the data precisely, coupled with random experimental error, can give rise to physically unrealistic constants. However, the values for K and a reported in Table I1 are of the expected magnitude. A sensitive test of the model lies in the required temperature dependence of the equilibrium constant. For four systems data were available over a range of temperatures; in these cases AH was fitted simultaneously with K and a by use of Equation 12, assuming a to be temperature-independent. The results, together with the temperature range covered, are included in Table 11. For n-hexyl ether and tetrahydrofuran (both containing ether linkages) AH was found to be -1060 and -1270 cal. per gram mole, respectively. considering the narrow temperature range covered by the data, these values are in gratifying agreement with the spectroscopically determined value of - 1430 cal. per gram mole previously reported for complexing of diethyl ether and phenyl acetylene (Brand et al., 1960). Extensive data, obtained by several workers, are available over a wide temperature range for the solubility of acetylene in dimethyl formamide. For this solvent the data were initially fitted along isotherms at 20’ C. increments between -60’ to 60’ C. A plot of the resulting values of log K us. 1/T gave a line with slight curvature. Therefore, a linear variation of A H with reciprocal temperature was assumed and the appropriate constants were obtained; the entire set of data (84 data points over -60’ to 60’C.) was refitted to give optimum values for K and a. (Again, a was assumed to be temperature-independent. ) Final results are given in Table I1; calculated curves and observed activity coefficients for three isotherms are shown in Figure 8. The last two solvents shown in Table I1 contain bifunctional groups ( f = 2)-that is, we assume that acetylene can form
2t
1
rDIOXOLANE: 25.C a(
9
coI/cc
9.86
b; 0.6
r
ETHYL ACETOACET4TE : 25.C 01. 7.80 collcc I( 1.23
”” 0.2
I
o
I
I
0.2
0.4
I 0.6
I 0.8
I
1 1
1.0
x2
Figure
9. Calculated and observed activity coefficients
of acetylene in two bifunctional solvents f = 2 186
l&EC
FUNDAMENTALS
a 1 to 1 complex and a 2 to 1 complex with the solvent. Reduction of solubility data in these two solvents was performed with equations described in Appendix I1 ; calculated and observed activity coefficients are shown in Figure 9. In both cases fit was improved over that obtained assuming only formation of 1 to 1 complexes. Discussion
For solvents such as butyrolactone, the Dolezalek theory, taking into account solvation alone, clearly will not represent the data. However, for solvents exhibiting stronger chemical interactions, complex formation tends to overshadow physical effects, and solute activity coefficients approach qualitatively those given in Figure 1. To test the influence of a on the quantitative fit in such mixtures, data for several systems were analyzed setting atl E 0 for all i, j pairs. I n most cases the fit was considerably poorer. For N-methyl pyrrolidone at 25’ C., for example, standard deviation increases from 1 to 8%. For dimethyl formamide at the lower temperatures ( K large) the fit was only slightly worse; however, as the temperature increased the fit was much poorer. To provide a comparison of results based on our model with those obtained from other equations, activity coefficientconcentration data for several systems were fitted to four essentially empirical expressions commonly used to represent mixture properties (Table 111). For strongly solvating mixtures the other two-parameter equations generally give considerably poorer data representation. The Wilson equation (1964) gives very poor results; this is not surprising, however, since that equation can give maxima in l n y only under unusual, physically unrealistic circumstances (Prausnitz et al., 1967). Although the two-parameter Redlich-Kister expansion is capable of giving maxima (or minima) in lnyz, deviations are still considerably greater than experimental uncertainty. Very much lower deviations are obtained by use of the nonrandom two-liquid theory (NRTL equation) recently developed by Renon (Renon and Prausnitz, 1968). This equation, which contains two adjustable parameters and a third parameter characteristic of the class of mixture, appears to give excellent results for a wide variety of mixtures which exhibit large deviations from ideality. As might be expected, the additional adjustable parameter in the threeparameter Redlich-Kister expansion results in a n improved fit, essentially within the experimental error. Conclusions
The solution model developed in this work provides representation of a wide range of solution behavior by use of an equilibrium constant for complex formation and a single binary physical interaction parameter. Introduction of enthalpy of complex formation allows calculation of the effect of temperature on activity coefficients. Application of the model to solubility of acetylene in organic liquids shows it to be superior to common two-parameter equations for representation of activity coefficients in solvated solutions. Our model provides a synthesis between the “physical” model of van Laar and the “chemicalJ’ model of Dolezalek. Further, the model can be extended to systems in which simultaneous or consecutive reactions occur; it therefore provides a framework for treating more complex mixtures. Finally, the model contains physically significant parameters which may facilitate correlation, generalization, and prediction of liquid-mixture properties.
Table 111.
Comparison of Equations for Representing Activity Coefficients of Acetylene in Organic Solvents Standard % Deviation in y2 (Acetylene)
Butyrolactone
Dimethyl acetamide
N-Methyl pyrrolidone
2.6
2.2
1.1
This work (Dolezalek-van Laar, two-parameter model) In YZ = f ( K , a,22) Wilson equation (two parameters) r
35
11.4
25
NRTL equation (two parameters)"
Redlich-Kister equation (two parameters) In y2 = r?[a b(1 - 4m)] Redlich-Kister equation (three parameters) In yz = xl*[a b ( 1 - 4x2) c ( 1 - 822
+ +
+
+ 12222)l
1.5
5.5
2.6
5.5
9.8
8.0
2.5
..
1.0
.b
Nonrandomness parameter @ equals 0.2. Only four data points available: three-parameter flt not justifled. Appendix 1. Estimation of Physical Interaction Parameters for Complex
Although independent specification of each of the physical parameters in Equation 7 is possible, it is advantageous to relate the parameters to each other, and thereby reduce the number of required constants. Although this may be done in a variety of ways, we relate the parameters through regular solution theory. The solubility parameter for species k is given by
or, with Equation 1-1, (1-3 1
Physical interaction parameters for species i and J are given by = (6, - 8,)' (1-4) However, rather than attempting to fix SA and SB a priori, SA is initially specified, and 8~ is determined from Equation 1-4 : b B = SA
where AUk = molar energy of complete vaporization of k, and v k = molar liquid volume of k . The energy of vaporization of complex AB is equivalent to (1) liquid-phase dissociation of AB to species A and B, (2) vaporization of A and B, and ( 3 ) gas-phase recombination to give AB. To a rough approximation, for a weak complex, the energy changes in steps 1 and 3 can be assumed equal, and thus
+ '&
(1-5 1
where cy = CXAB = binary interaction parameter of species A and B. The solubility parameter of the complex is next estimated from Equation 1-3, and each cytI is obtained from Equation 1-4. The technique is thus merely a means of relating each of the cyt,'s with one binary constant cy, and a single value for SA. (The results are insensitive to the actual level of SA.) For the work reported here, 8~ (acetylene) was estimated (Lyckman et al., 1965) as 6.5 (cal./cc.)1/2. However, it is necessary to estimate the molar volume of complex AB; examination of volumetric data for a number of reactions involving hydrocarbons indicated that VAB could be taken approximately as: VAB = V B
+0 . 7 5 ~ ~
(1-6)
where VA = molar volume of acetylene. Equation 1-6 was used through the present work. The coefficient 0.75 is not critical; significant changes in this factor resulted in negligible variation in final calculated results. Figure 10 gives values for A-AB interaction parameters as a function of CY and solvent-solute molar volume ratio. Using these techniques for estimating SAB and VAB, "true" activity coefficients are given by:
RT In "/A
= VA[CYA-B@B'
+
(LYA-B
RT In YB = VB[CYA-B@A' o t = ~ ~ cal/cc - ~ , Figure 10. Solute-complex parameter as a function of solute-solvent parameter and molecular size ratio
~A-AB@AB'
- aB-AB ~B-AB@AB' -k
f
+ +
(QA-B
R T In YB =
(VB
LYA-AB
(YB-AB
- CtA-AB)@A@AB]
0.75) [ Q A - A B ~ '
+
(aA-AB
VOL.
8
C~B-AB@B'
aB-AB
NO.
)@B@AB]
2
+
(1-7)
- aA-B)@A@BI M A Y
1969
187
.
if values are available for K and K7(1’),where j = 1, 2, , , ,f. Substitution of n A into Equations I13 and 11-4 thus gives
where, in terms of a,VA, m,and 6 ~ : =ff
CUA-B
immediately “true” moles of each species. Given values for K and a, solution of these equations is similar to that for the 1 to 1 complex case; for a given apparent concentration, each K y ( j ) is initially assumed equal to unity, and the “true” moles of each species are obtained from Equations 11-2 to 11-4. The total number of moles a t equilibrium is given by:
“= + + c f
Appendix II. Generalization of Solution Model
We consider here only one example of generalization of the model, that of a solute dissolved in a polyfunctional solvent, in which complexing can occur with any of several identical sites. Examples of such molecules might be polyethers or polyamides. For this case a series of reactions can occur: A+B
+AB
Here f is taken as the number of functional sites. If the sites are assumed identical, and if complexing at one site does not influence complex formation a t adjacent sites, the equilibrium relations for all of the consecutive reactions can be expressed by a single equilibrium constant, K :
nA
nB
i-1
nA,B
(11-81
Thus “true” mole fraction for each species can be calculated. At this composition activity coefficients are given by Equation 7 or 9, and thus activity coefficient ratios can be calculated from Equations 11-5. Rather than estimate individual solubility parameters for each complex present, we assume that solubility parameters for all complexes are identical to that of the 1 to 1 complex, as given in Appendix I. This simplifies calculations somewhat without seriously affecting the results. Molar volumes of complexes are given by an extension of Equation 1-6. Subsequent iterations converge rapidly on “true” mole fractions and activity coefficients; again, “apparent” activity coefficients are given by Equations 11. Acknowledgment
The authors are grateful to the National Science Foundation for financial support and to the Computer Center, University of California, Berkeley, for the use of its facilities. literature Cited
The stoichiometry for this set of reactions is rather formidable; however, with algebraic rearrangement the following set of equations can be derived relating the “true” number of moles of each species a t equilibrium to the apparent number of moles of components 1 and 2. f nA+
i i(KnA)’/[(n*+
nB
nl)’nK,‘j)]= i-1
bl
?&
+
f
01-21
= 721
(11-3)
i
(KnA)’/
?tg
nZ
[ (nA
i-1
+ nl)’ n
K,(j)]
1B6l
i l
I
where
L i i i a n , E. W., Eckert, C. A., Prausnitz, J. M., Chem. Eng. Sci. 20, 685 (1965). Lyckman, E. W., Eckert, C. A., Prausnitz, J. M., Chem. Eng. Sci. 20, 703 (1965). McGlashan. M. L.. Rastogi. R. P., Trans. Faradav SOC.64, 496 (1958). Miller, S. A., “Acet lene,” Academic Press, New York, 1965. Prausnitz, J. M,,Eciert, C. A,, Orye, R. V., O’Connell, J. P., “Corn uter Calculations for Multicom onent Vapor-Liquid Equiligria,” p. 44, Prentice-Hall, Engiwood Cliffs, N. J., I ,
KJa7 = YA,B/YAYA,-,B
(11-5)
and i
K70)
=
K7(1)Kr@)
. .. K,(i)
(11-6)
j-0
In these equations nA, ng, nAB, etc., represent “true” number of moles of each species at equilibrium; “true” activity coefficients are given as before by Equation 7 or 9. Apparent number of moles are given by nl and m, Without loss of generality we can set nl+ n~ = 1.0 (11-7) Equation 11-3 is linear in n B ; thus substitution for n B in Equation 11-2 gives a single equation with only n A unknown, 188
Ambrose, D., Trans. Faraday SOC.62, 772 (1956). Ambrose, D., Townsend, R., Trans. Faraday Soc. 60, 1025 (1964). Bottomley, G. A., Reeves, C. G., Seiflow, G. H. F., Nature 182, 596 (1958). Brand, J. C. D., Eglinton, G., Morman, J. F., J. Chem. SOC.1960, 2526. Harris, H. G., dissertation, University of California, Berkeley, 1968. Hildebrand, J. H., Scott, R. S., “Solubility of Nonelectrolytes,” 3rd ed., Chap. XI, Dover, New York, 1964. Holemann, P., Hasselmann, R., Forschungsber. Landes Nordrhein-Wkstfalen, No. 109 (1954). Kehiaian. H.. Bull. Acad. Pol. Sci. Ser. Sci. Chim. 14, 891 (1966): 16, 367 (1967);16, 165 (1968). Kiyama, R., Hiraoka, H., Rev. Phys. Chem. Japan. 26, 52 (1955). Kiyama, R., Hiraoka, H., Rev. Phys. Chem. Japan 26, 1 (1956). Kortiim, G.F. A., Buchholz-Meisenheimer, H., “Die,I’heorie der Destillation und Extraktion von Fliissigkeiten, SpringerVerlag, Berlin, 1952. Kretschmer, C. B., Wiebe, R:, J . Chem. Phys. 22, 1697 (1954). Lewis, G.N., Randall, M., Pitzer, K. S., Brewer, L., “Thermodynamics,” 2nd ed., Appendix 1, McGraw-Hill, New York,
I&EC
FUNDAMENTALS
1QA7
Priii‘ine, I., Defay, R., “Chemical Thermodynamics,” p. 410, Wiky, New York, 1954. Renon, H., Prausnitz, J. M., A.Z.Ch.E. J . 14, 135 (1968). Renon, H., Prausnitz, J. M., Chem. Eng. Sci. 22, 299, 1891 (1967). Ryutanj, B., Nippon Kagaku Zassha 80, 1407 (1959). Ryutani, B., Nippon Kagaku Zasshz 81, 1192 (1960). Ryutani, B. N i pon Kagaku Zasshi 81, 1196 (1960). Scatchard, &., &em. Rev. 44,7 (1949). Shenderei, E. R., IvanovskiI, F. P., Gaz. Prom. 7, No. 7, 38 (1962). Wilson, G. M., J. Am. Chem. SOC.86, 127 (1964). RECEIVED for review October 1, 1968 ACCEPTEDMarch 10, 1969
