S.Paoletti, F. Delben, and V. Crescenzi
2564
Thermodynamics of Polycarboxylate Aqueous Solutions. 3. Binding of Divalent Ions S. Paoletti, F. Delben, and V. Crescenzi* Laboratorio di Chimica delle Macromolecole, lstituto di Chimica, Universitd di Trieste, Trieste, ltaly (Received February 9, 1976) Publication costs assisted by the Universita di Trleste
The binding reactions of copper(II), nickel(II), and barium(I1) by three (1:l)copolymers of maleic acid, namely, maleic acid-ethylene, maleic acid-propylene, and maleic acid-isobutene, respectively, have been studied in dilute aqueous solution on the basis of potentiometric data. The possibility of evaluating the binding constants is critically discussed. The potentiometric data are compared with calorimetric, dilatometric, and spectrophotometric results previously collected by us for the same systems.
Introduction In previous papers from this laboratory a number of calorimetric, dilatometric, and spectrophotometric data on divalent ions (Cu(II),“I), and Ba(I1)) binding by three maleic acid (MA) copolymers in aqueous solution have been reWe present and discuss here the results of potentiometric measurements for the complete series of binding reactions studied, i.e., the interaction between the above mentioned divalent cations and MA-ethylene (MAE), MApropylene (MAP), and MA-isobutene (MAiB) copolymers, for three different values of the metal to polyelectrolyte molar ratio, r . Experimental Section All the potentiometric measurements were performed in a thermostated vessel a t 25 f 0.05 “C using a Radiometer PHM4d pH meter, equipped with a Radiometer combined electrode (GK 2301 C). The polyelectrolyte solution (12-15 ml; polymer concentration, C,, 0.5 X monomolfl; solvent, 0.05 M tetramethylammonium perchlorate) containing the proper amount of metal perchlorate (e.g., Cu(C104)2) was titrated with concentrated standard tetramethylammonium hydroxyde (TMAOH). Each pH was read after a few minutes with stirring of the solution in order to be sure that thermal and chemical equilibrium had been reached in each case. Results and Discussion Analysis of Potentiometric Titration Curves. Typical titration curves for the three polyelectrolytes studied in the presence and in the absence of metal ions are reported in Figure 1.For simplicity, only the data for the highest r values (r is the metal to polymer molar ratio) considered in this work and in previous are given. A critical analysis of the plots of Figure 1allows us to make the following considerations. The presence of the divalent metal ions lowers the potentiometric curves. This is a well-known effect, due to the reduction of the effective negative charge on the polyelectrolytes as a consequence of the interaction between the carboxylic groups and the cation^.^ We can observe that, for all three polyelectrolytes and at each a5value considered, the trend in the ability to lower the pH value is Cu(I1) > Ni(I1) > Ba(I1). This can be ascribed to the fact that passing from Cu(I1) to Ba(I1) the extent of binding may decrease, andlor the mode of binding may change. We suggest, on the basis of experimental evidence and discussions already reported elsewhere,2s3that different modes The Journal of Physical Chemistry, Vo/. 80, No. 23, 1976
of binding prevail going from one counterion to another and that these in turn do influence the extent of binding. The present evidence appears to support this hypothesis. Another important consideration is that for a given divalent cation, the dependence on a of the pH values of the three polyelectrolytes under study is not the same. In order to discuss this point, let us refer to polycarboxylate-copper ions binding, for which we have also relevant spectroscopic evidence.3 Titration curves for MAE and the highest r value (see Figure l a ) show that a t a = 0 there is practically no binding. Binding ensues, however, immediately after the first addition of base. At a = 0.5, for instance (which in our notation corresponds to 25% of the carboxylic functions dissociated), the pH drops from 4.15 (absence of copper ions) to about 3.55. Such a relatively large difference in pH is then nearly constant for the remaining titration curve, being less and less important however passing from a = 1to a = 2. For MAP, the difference in pH between polyelectrolyte solutions with and without added metal ions is not as marked as in the former case for low values of a, but increases with a and reaches a maximum in the neighborhood of a = 1. With MAiB the difference in p H begins to be significant only in the immediate neighborhood of a = 1,and is quite high only beyond this value. This means that with MAiB the overall reaction with formation of chelated species (which is supposed to involve a small change in H+ ion concentration) may proceed only when about 50% of carboxylic groups are dissociated, while with MAP and especially with MAE this limit is sensibly lowered (although we cannot determine it exactly). It is worth pointing out that all these considerations are in qualitative agreement with our spectroscopic evidence on the formation of chelated species between copper ions and MA copolymers as a function of a.3
The titration curve of MAiB in the presence of Cu(1I) ions exhibits two inflection points (see Figure IC).The first one is a normal “equivalence-point”; the second appears after addition of an amount of base equivalent to the metal ions present. In our opinion, this is further evidence that in the case of MAiB binding takes place only when the value of a is close to unity; beyond this value, the addition of base does not produce a significant increase of the pH value, because titration of carboxylic groups formed on binding (vide infra) is to be completed up to a N (1 r). Only after this value, titration of the remaining “weak” carboxylic group starts and produces a relatively high upward shift of the pH curve. The titration curves of the polycarboxylates under study in the presence of nickel and barium ions do not present such
+
2565
Thermodynamics of Polycarboxylate Aqueous Solutions 0.4
0 11.
I
I
1.2
0.8 I
,
I
1
2.0
1.6 I
1
I
11
PH
PH
0
0.4
0.e
1.2
1.6
2.0
o(
Flgure 1. Typical plots from the potentiometric titrations of MAE (a), MAP (b), and MAiB (c) with TMAOH in 0.05 M TMAC104 in absence of added metal ions (0)and in presence of Ba2+ ( r = 0.287; X), Ni2+ ( r = 0.265; O), and Cu2+ ( r = 0.266; 0) Ions, respectively, at 25 O C . monomol/l. Average polymer concentration was 0.5 X
an interesting behavior, showing significant lowering in the pH values only in the range of CY between 1 and 2. This confirms that binding of these two cations by our polyelectrolytes is essentially electrostatic, with the differences between nickel and barium already mentioned2J and pointed out above. Tables of potentiometric data are available as supplementary material (see paragraph a t end of text regarding supplementary material). Determination of Equilibrium Constants. (a)Preliminary Remarks. For the determination of metal binding constants for polyelectrolyte solutions different methods are reported in the l i t e r a t ~ r e .We ~ , tried ~ ~ ~to apply some of these methods to our systems, but we found either that it was not possible, or that absurd results were obtained. Difficulties encountered in the determination of binding constants are, in fact, neither negligible nor always easily overcome. (1) In the evaluation of binding constants, when dealing with polyelectrolytes, a critical step is the correct identification of the reaction stoichiometry. In our case we had to match equilibrium dialysis data with pH measurements evidence, which indicate that, a t a = 1 (on which the whole following discussion is centered), the amount of protons released in solution by the carboxylic groups of the polyelectrolyte as a consequence of binding is not larger than a few (1-2) percent of bound ions. (2) Another problem consists in correctly evaluating first and second proton association constants (KpI and Kpl*, respectively) of the repeating (ligand) units of polydibasic acids. Both Kp values strongly depend on the nature of the polymer, on the ionic strength of the solution, and on the charge density of the polyelectrolyte chain. In the presence of metal ions,
which to some (and sometimes high) extent screen the fixed polymer charges, these proton association constants can in principle be different from those of the pure acid, and therefore cannot be exactly determinable. On the other hand, an incorrect choice of Kp values may lead to a completely misleading evaluation of the extent of binding. (3) Similar difficulties are connected with the demands on the accuracy of p H determinations, not only owing to uncertainties connected with the definition of pH in polyelectrolyte solutions, but also because the experimental uncertainties though small are usually too large for a sufficiently accurate determination of both the amount of metal ions bound and of the stability constants of the complexes. (4) Last but not least, a theoretically insoluble problem is represented by the evaluation of the activity coefficients of the ions (in the presence of the polyelectrolyte) and of the polymer itself. This difficulty is usually overcome just by setting all the activity coefficients equal to unity or by treating them as ~ 0 n s t a n t s . l ~ (b) Binding Constants Calculation Procedure. For all the above said difficulties, we are very sceptical on the possibility of obtaining accurate evaluations of the extent of binding and, moreover, of the related stability constants from potentiometric data alone. Nevertheless, making some assumptions and keeping all these difficulties in mind, we have already published some values of AG of binding, relating them to enthalpy and entropy changes.2 In this paper we present a more complete picture of the situation from the point of view of the stability constants of the complexes, and an attempt to correlate these data with structural parameters. In a first approximation, two binding equilibria could fulfill this need, i.e. M2+ HA- ==MAH+ (a)
+
M2+
+ 2HA- + MA + H2A
(b)
where M2+, HA-, MA, and H2A represent the metal ion, monodissociated monomeric unit, monomeric unit of the complex, and undissociated monomeric unit, respectively. MHA+ represents a (hypothetical) charged (complex) monomeric unit. A third equilibrium
+
M2+ A2- + MA (C) where A2- represents a completely dissociated monomeric unit, can be reasonably excluded a t CY = 1 (unless other concomitant equilibria are invoked) owing to the relatively large difference between the two Kp’s of the polyacids and to the high absolute value of the first proton association constants. It is important to point out that reactions a, b, and c, as written, do not imply any assumption on the mechanism or on the nature of the interactions involved in binding. The latter in fact may be covalent, electrostatic, or “mixed” in nature. Actual p H conditions and what is already well known on divalent metal ion binding by other polycarboxylates and model compounds definitely tend to exclude model a6316 (at least for Cu and Ni ions). Model b is formally the sum of the reactions: HA-
A2-
+ H+
+ M2++ MA HA- + H+ H2A MA + H2A M2+ + 2HAA2-
(bl) (b2) (b3) (b)
The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
S. Paoletti, F. Delben, and V. Crescenzi
2566
TABLE I
Polyelectro- Metal lyte Ion MAE
MAP
-PKb3 PKbl (pKpI1) (pKpI)
Cu2+
4.4
7.1
Ni2+
4.5 f 0.1
7.2
Ba2+
4.8
7.5
Cu2+ 4.4 f 0.1
8.2
Ni2+ 4.8 f 0.1
8.2
Ba2+
5.2
8.2
MAiB Cu2+ 4.3 f 0.6
9.0
Ni2+ 5.0 f 0.2
9.0
Ba2+
9.0
5.7
-
AGb, mb, Avb, - A x , A X , Ami, kcal/mol kcal/mol A X , ml/mol kcal/mol kcal/mol A X , ml/mol a of complex of complex eu of complex of complex of complex eu of complex 1.0 1.2 1.5 1.0 1.5 1.0 1.5 1.0 1.2 1.0
1.5 1.0 1.5 1.0 1.2 1.5 1.0 1.5 1.0 1.5
4.5
4.4 4.4
30
2.9
2.0
16
3.5
0.6
14
4.0
27
2.3
4.0 4.0 2.0
2.5
0.6
10
3.4
4.0 4.0
25
2.1
1.8
13
2.4
0.7
10
It is this model (which will be verified later on) that has been used for the calculation of the total free energy change AGb on addition of divalent ions to our polyelectrolyte solutions (except for MAE and MAP in the presence of barium ions, systems about which more will be said later on) with the following procedure. Using mass and charge balance equations, [HA-], [HzA], and 817 can be expressed as functions of hydrogen and hydroxyl ions, added base, total metal, and polymer concentrations and of the equilibrium constants for processes b l and b3. The expression for Kb is
with
32 41 52 13 36 24 24 26 52 20 51 25 39 31 85 85 23 52 13 40
14
8.2
4.4
42(48)
56
6.6
2.0
29( 34)
37
3.5
0.6
14
24
9.2
4.0
44(48)
52
6.9
2.0
30(32)
46
4.5
0.6
17(18)
39
9.8
4.0
46(46)
59
7.6
1.8
32(31)
51
6.9
0.7
26(26)
41
(3) The true pH values may have a deviation from recorded ones of, at maximum, f 0 . 0 5 units. (4) The activities of the species can be identified with their stoichiometric concentrations. With such limitations and using a computerized iterative method, the Kb values for different molar ratios of metal ions to polyelectrolyte, r, have been calculated. The total free energy change expressed in kilocalories per monomole of polymer, AGb(r), is then plotted against r, for each polymer and counterion considered. The plots are then used to provide: (i) the slope of the curve a t r 0, i.e., the intrinsic differential free energy of binding, AGb (in kilocalories per mole of complex); (ii) by proper interpolation, AGb(r) values, which can be combined with AHb(r) ones (see ref 1, Figure 3; ref 2, Figure 1)to give the points necessary for the construction of A S b vs. r plots. The slopes of the latter curves for r 0 provide the AS), values, i.e., the differential entropy changes accompanying the total binding reactions. The overall A&, M b , Ash, and Avb1,2 values are collected in Table I. ( A v b is the experimental, differential volume change upon binding). It is at this point that we wish to emphasize the great utility of volume change data as a check of the stoichiometry of binding. Under the very reasonable assumption that for large a values (1.2 or 1.5) the amount of bidissociated groups (A2-) available on the polyelectrolyte chains is such that, for the limiting case of r 0, no proton is displaced upon binding (and therefore the observed variations in thermodynamic functions are just those for reaction b2), correction of reaction b for proton dissociation and reassociation can be done a t a = 1 and r 0. Moreover, the volume change of protonation for the MA copolymers in the solvent used appears to be fairly constant for each dissociation step. These copolymers show, in fact, only two distinct values of the differential volume of protonation, one in each region of CY (between 2-1, and 1-0, respectively).1,18 Therefore, corrections to the total A v b of reaction can be done following the scheme
-
-
and
+
[H+] B - [OH-]
- [HA-]
8=
[H+] -k 2/Kb1 [H+l
Z M
(3)
C,, CM,and B being the total concentration of polyelectrolyte, of metal ions, and added base, respectively. Simple substitution in eq 1of experimental concentrations and of the protonation constant (Le., of the Kbl and Kb3 values determined in the absence of divalent counterions) values leads in some cases to absurd results, e.g., values of 8 larger than unity or negative. Therefore, some limitations from independent measurements and some variations of numerical quantities used must be applied in the computation: this has been done with the following criteria. (1)The amount of metal ions bound calculated from potentiometric data must be very close to that found by independent techniques (in this case, equilibrium dialysis data). (2) The Kbl and Kb3 values do not differ from those measured in absence of divalent ions [(KpI)-l and KpII, respectively] by more than a few tenths. This should be true at least for low r values. The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
-
-
Avb2
= Avb
- AVb3 - AVbi
(4)
and hVbl,3 values are from ref 1 and 2. The results for all systems are reported in the last column
Avb
Thermodynamics of Polycarboxylate Aqueous Solutions
2587
of Table I. Comparison of values evaluated according to eq 4 with the A v b values directly measured a t a = 1.5 (or 1.2) shows good agreement thus confirming the essential validity of the hypothesis; only in the case of MAiB/Cu(II) system, a lower value is calculated, a fact which is probably related to particular features of the polyelectrolyte chains. values have then been calculted a t a = 1 along the scheme of eq 4: AGbl and AGb3 values are directly computed from protonation constants, at a = 1,used in Kb calculation (see Table I).23 While eq b2, combined with eq b l and b3, gives a satisfactory value of Avb, (see Table I) at a = 1for the MAiB/Ba(II) system, this is not so for the MAP/Ba(II) (AVb2 should be 51 ml/mol of complex instead of 39) and for the MAE/Ba(II) (50 vs. 24) systems. In view of the comparatively weak interaction of Ba(I1) ion with polycarboxylates, mode of bonding a can be proposed to be present to a partial (MAP) or total (MAE) extent. The essential difference between mode of bonding b2 and a, which for Ba(I1) are probably both electrostatic in nature, lies in the extent of specificity and localized interaction between oppositely charged ions. Hence no correction has been applied for MAE/Ba(II) system, for which the binding constant K ,
K , = [MAH+]/[M2+][AH-]
(5)
is evaluated from potentiometric and dialysis measurements. In the case of MAP/Ba(II), the AVb at a = 1is the mean value between that at a = 1.5 (binding according to equilibrium b2, 39 ml/mol of complex) and the hypothetical value (13 ml/mol of complex) if complete correction would have been made for proton displacement (cf. also MAiB/Ba(II) a t a = 1, A v b = 13 ml/mol of complex). The stoichiometry of the total reaction at a = 1has then been written as
+
M2+ 3/2HA-
lhMA
+ ‘/zMAH+ t lhH2A
(e)
and the thermodynamic functions for the total reactions thus refer to the “mean” of reactions indicated before as reactions a and b. This procedure is quite sketchy, but it must be stressed that it aims essentially a t a comprehension of the binding phenomena. In all cases for the evaluation of the enthalpy change of binding, the algebraic sum of mbl and Affba has been taken to be negligible in comparison with a b and hHb2, a t a = 1.0; to support this assumption there is the fact that the h H b plots, for r small enough, exhibit equal slopes on changing a from 1.0 to 1.2,182 Le., for a situation for which the number of available bidissociated sites per divalent ion largely exceeds unity. Under this hypothesis, ASb2 values from AGbz and mb (set equal to AHbz) have been calculated, and are reported in the first column of Table I headed ASb2. Values of A S b 2 in parentheses (second column) have been directly derived from A S b , A s b l , and ASb, data according to eq 4. The two sets of values are in good agreement, thus confirming the validity of the assumption for both H b 1 , 3 and for the mechanisms of binding (models a, b, e, and eq 4). Concluding Remarks In order to briefly recapitulate a few interesting features which result from our study, we wish to emphasize the following: (1) Evaluation on the basis of potentiometric data of “binding constants” of divalent counterions by weak polyacids involves assumptions on the nature of binding itself and on
the strength of the acidic functions therein involved, which need to be matched with data from independent measurements to avoid meaningless results. (2) In all cases considered both enthalpy and entropy changes on binding are positive, thus making such process entirely entropy driven. (3) For the maleic acid copolymers, binding ability increases with increasing alkyl substitution on the copolymer chains. (4) The thermodynamic parameters for the binding reactions decrease in absolute value on going from Cu(I1) to Ni(I1) and Ba(II), in agreement with analogous features exhibited by the same ions and low molecular weight ligands. (5) The distinctive feature which makes binding of the divalent ions in polycarboxylate aqueous solution so extensive is the particularly favorable entropy change. For instance, for the binding of Cu(l1) and Ni(I1) ions by succinate it is found that H b 2 = 4.6 and H b 2 = 2.5 (kcal/mol of complex), respectively; the two values are quite close to those reported in Table I for the polymeric carboxylates. On the other hand, the A S b 2 values for the latter are nearly twice as large as those reported for succinate with Cu(I1) and Ni(II), 30 and 19 eu, respectively. (6) The noticeable increase in entropy on binding is essentially due to the release of water molecules from the hydration sheaths of the interacting species, as revealed by the associated volume changes. (7) Following Gurney21 and Nancollas,22the temperature dependent contribution to the enthalpy change on binding, AHel,should be related to the amount of solvent molecules bound to the reagents. In the case of the polycarboxylates this term is about 50% larger than for succinic acid (4.2 vs. 3.0 and 3.0 vs. 2.2 kcal/mol for Cu and Ni, respectively), nearly with the same ratio as for the entropic (mainly cratic) term (44 and 30 vs. 30 and 19 eu, respectively). (8) Analysis of the volume and entropy data suggests that the most likely binding equilibria involve displacements of protons and their nearly quantitative reattachment to the available residual charges on the chains. Therefore, the relative value of the two dissociation constants of the polyacids determines the extent of the overall binding reactions.
Acknowledgments. This work has been sponsored by the Italian C.N.R., Rome. The authors are indebted to Professor M. Mandel for helpful discussions. S u p p l e m e n t a r y Material Available: A complete list of potentiometric data (6 pages). Ordering information is available on any current masthead page. References and Notes (1) V. Crescenzi, F. Delben, S.Paoletti, and J. Skerjanc, J. Phys. Chem., 78, 607 (1974). (2) F. Delben and S. Paoletti, J. Phys. Chem., 78, 1486 (1974). (3) S. Paoletti and F. Delben, Eur. Polym. J., 11, 561 (1975). (4) R. W. Armstrong and U.P. Strauss, Encycl. Polym. Sci. rechnol., 10, 806, 825 (1964-1972). See also references there quoted. (5) CY, the degree of neutralization, is the molar ratio of added base to total monomeric (blcarboxylic) units present. (6) B. J. Feiber, E. M. Hodnett, and N. Purdie, J. Phys. Chem., 72, 2496 (1968). (7) B. J. Feiber and N. Purdie, J. Phys. Chem., 75, 1136 (1971). (8) G. Zubay, J. Phys. Chem., 61, 377 (1957). (9) H. Morawetz, A. M. Kotliar, and H. Mark, J. Phys. Chem., 58, 619 (1954). (10) H. P. Gregor, L. P. Luttlnger, and E. M. Loebl, J. Phys. Chem., 59, 34 (1955). (11) M. Mandel and J. C. Leyte, J. Polym. Sci., Part A, 2, 2883 (1974). (12) J. J. O’Neill, E. M. Loebl, A. Y. Kandanian, andH. Morawetz, J. Polym. Sci,, PartA, 3, 4201 (1965). (13) R. L. Gustafson and J. A. Lirio, J. Phys. Chem., 69, 2849 (1965).
The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
Robert H. Lacombe and Isaac C. Sanchez
2568 (14) L. Costantino, V. Crescenzi, F. Quadrlfoglio, and V. Vitagllano, J. Polym. Sci., PartA-2, 5, 771 (1967). (15) R. J. Eldridge and F. E. Treloar, J. Phys. Chem., 74, 1446 (1970). (16) A. Mc Auley and G. H. Nancollas, J. Chem. SOC.,4458 (1961). (17) 6 is the fraction of bound metal ions. (18) A. J. BegalaandU. P.Strauss, J. Phys. Chem., 76, 254 (1972). (19) V. Crescenzi, F.Delben, F. Quadrifoglio, and D. Dolar, J. Phys. Chem., 77, 539 (1973). (20) A. Mc Auley, G. H. Nancollas, and K. Torrance, Inorg. Chem., 6, 136 (1967).
(21) R. W. Gurney, "Ionic Processes in Solution", McGraw-Hill, New York, N.Y., 1953. (22) G. Degischer and 6. H. Nancollas, J. Chem. SOC.A, 1125 (1970). (23) The present results differ only slightly from some already published results from this the difference in A@* between data from the previous paper and the present one are solely due to a different choice of the values of acid dissociation constants of the polyelectrolytes,and slightly different values of the fractions of bound ions, which in a former paper were just taken from dialysis, while in the present one have been adjusted to match potentiometric evidence too.
Statistical Thermodynamics of Fluid Mixtures Robert H. Lacombe and Isaac C. Sanchez' Materials Research Laboratory and Polymer Science and Engineering, University of Massachusetts, Amherst, Massachusetts 0 1002 (Received May 17, 1976)
A unified molecular theory of liquid and gaseous mixtures based on a lattice model description of a fluid is formulated. Pure fluids are completely characterized by three molecular parameters which are known for many fluids. Characterization of a binary mixture requires a knowledge of the pure fluid parameters and an interaction energy; interaction energies have been determined for 18 representative binary mixtures containing nonpolar and polar components. Thermodynamic properties of ternary and higher order mixtures are completely defined in terms of the pure fluid parameters and the binary interaction energies. Quantitative prediction of heats of mixing of a ternary hydrocarbon mixture is demonstrated. Volume changes on mixing are calculable from heats of mixing; good agreement is obtained with experiment for hydrocarbon mixtures. Thermodynamic stability criteria for liquid-liquid (L-L) and liquid-vapor (L-V) phase transitions are shown to be interdependent. The common appearance of a lower critical solution temperature (LCST) in polymer solutions is explained as well as the observation that LCST's usually occur above the normal boiling point of the solvent. Four basic types of L-L phase diagrams are predicted which include upper critical solution temperatures, LCST's, and closed immiscibility loops. Predicted pressure and molecular weight dependences of CST's are in a t least qualitative agreement with experiment. Use of the geometric mean law to estimate interaction energies and to predict mixture miscibility limits is evaluated. Rules for predicting polymer/polymer mixture miscibility are outlined. The second virial coefficient of the chemical potential is evaluated; its temperature dependence compares favorably with experiment. Predicted L-V phase diagrams include those with azeotropes and the unusual critical point phenomena of retrograde condensation. Simultaneous description of L-L and L-V equilibria is demonstrated for nonpolar/polar mixtures.
I. Introduction In a previous publication,' hereafter referred to as I, a statistical mechanical model of pure fluids was presented. It was shown that the model fluid, characterized as an Ising or lattice fluid, was capable of semiquantitatively describing the thermodynamic properties of a wide variety of molecular fluids. The basic simplicity and structure of the theory enables it to be readily extended to fluid mixtures. Formally, the mixture theory is very similar to the wellknown lattice theory of Flory and Huggim2 The essential and important difference is that allowance is made for empty lattice sites or free volume. An equation of state characterizes each of the pure components as well as the mixture. Volume changes upon mixing are now calculable, whereas they are not in the Flory-Huggins theory. More recently, Flory and his co-workers have formulated a new equation of state t h e ~ r y which ~ - ~ resembles the present theory more than the older Flory-Huggins theory. The new Flory theory has emphasized the importance that equation The Journal of Physical Chemistry, Vol. 80, No. 23, 1976
of state contributions can have on solution properties. However, the Flory equation of state, based on a partition function that combines aspects of several models, is limited to the liquid state; it does not reduce to the ideal gas law a t low densities nor does it possess a virial expansion.6 In contrast, the Ising fluid equation of state can semiquantitatively describe both liquid and vapor properties of molecular fluids of arbitrary size and geometry. It is founded on a well-defined statistical mechanical model that is amenable, in principle, to systematic improvement. A complete characterization of the equation of state is given in I. The mixture theory based on the Ising fluid model is versatile and predictive. It is able to qualitatively and often quantitatively account for many, if not all, known types of fluid phase behavior. This includes lower critical solution temperature behavior characteristic of polymer solutions as well as the unusual critical point phenomena of retrograde condensation exhibited by some hydrocarbon mixtures. An important insight on the relationship between liquid-
