J . Phys. Chem. 1987, 91, 2428-2432
2428
h 0, L
P C
electron donor) should permit less delocalization than a carbon atom. All acyclic structures were treated as vinylic alcohols and disconnected at the C O bond. Monocycles were disconnected at the two C O bonds. In each case an oxygen fragment and an all-carbon fragment were obtained. We have carried out our modified PMO estimations as we did for carbocycles. As expected, the initial estimated Dewar resonance energy for each oxymonocycle (DREO) produced by this approach is identical with the carbocyclic Dewar resonance energy (DREC)and tends to be too large. The final estimate utilized the Pauling electronegativities for carbon and oxygen as shown in eq 20. Figures 8, 9, and 10
2Bl P
w
P 0
E 0 C
0 Y)
0 (L L
DREO = DREC(X,C/X,O)
0
I
a
Hijckel c a l c u l a t e d value Estimated value-present method Figure 10. Dewar resonance energies for (CH),O- monocycles. component recombination stabilizations to furnish good estimates of Dewar resonance energies. However, they remove the single largest source of error in the original PMO approach as applied to 4n polycycles. iv. Oxymonocycles. We have also undertaken an examination of systems for which one carbon atom has been replaced by an oxygen atom. Trends for Dewar resonance energies of oxaannulenes should mirror those for the corresponding carbocycles. However, the more highly electronegative oxygen atom (two
(20)
present our estimates along with Huckel calculated Dewar resonance energies (see Method section for details). u. Conclusions. In developing an approach to aromaticity estimates for annulenes, which permits one to conclude that 4n monocycles are antiaromatic on the basis of Dewar resonance energies, Dewar and Dougherty have introduced a feature which causes systematic errors in all estimates for Huckel calculated Dewar resonance of 4n polycycles. A modification of their approach has been advanced. The modification which takes into account the stabilizing effect of cyclization on all bonding electrons furnishes more reliable estimates. The proposed modification, while not applicable to polycycles more complex than bicycles, provides a basis for an improvement in the Dewar and Dougherty treatment of 4n polycycles. Cyclization stabilization for 4n polycycles can be estimated more reliably by a simultaneous accounting for both electrons in the HOMO and electrons in the
LOMO. Acknowledgment. We are indebted to the Florida Institute of Technology for financial support.
Computer Calculation of Ionic Equilibrla Using Species- or Reaction-Related Thermodynamic Data Eric L. Cheluget, Ronald W. Missen,* Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1 A4
and William R. Smith Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada NI G 3 2 7 (Received: November 5, 1986; In Final Form: December 24, 1986)
The data available in general for an equilibrium calculation are included in the set {A, p*, N, AGO or K, “restrictions”},where A is the formula matrix for the species involved, p* are the standard chemical potentials or free energies of the species, N is the stoichiometric matrix, AGO and K are the standard free energies and equilibrium constants, respectively,of the reactions contained in N, and “restrictions”represent special constraints. It is shown how any alogrithm (based on free-energy-minimization methods) for computing chemical equilibrium that requires the species-related data set {A, p*) can be used when only the reaction-related data set (N, K)is available (in which case algorithms based on the equilibrium-constantmethod are frequently used). This situation is common in ionic equilibria. Computational advantages of the free-energy-minimization method can only be exploited if species-related data are available. In addition to the usual case (single phase, “no restrictions”), special cases may arise involving stoichiometric restrictions (e.g., restricted equilibrium), compositional constraints (e.g., fixed pH), or additional phases (e.g., solubility). Examples taken from the literature are used to illustrate these situations. The techniques described, which are not restricted to ionic equilibria, involve converting the data given, and if appropriate, the formula vectors of the species, into standard form for use in the algorithm in the usual way.
1. Introduction
The use and classification of general purpose numerical algorithms for computing the composition of a chemical system at equilibrium, an important tool in equilibrium analysis, has been described previously.’ One (common) classification distinguishes 0022-3654/87/2091-2428$01.50/0
between free-energy-minimization methods and equilibriumconstant methods, the latter of which are more familiar to (1) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; Wiley-Interscience: New York, 1982; Chapter 6.
0 1987 American Chemical Society
Calculation of Ionic Equilibria
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2429
chemists, particularly in dealing with ionic equilibriaS2 Although we believe that this classification is often misleading in obscuring similarities among algorithms, there appears to be one important distinguishing feature that may lead to a misunderstanding, since the former uses species-related data (e.g., standard free energies of formation), and the latter uses reaction-related data (i.e,, equilibrium constants): it might wrongly be supposed that the former cannot be applied to situations in which only equilibrium constants are known for a given set of reactions. Hence it may not be commonly used for this r e a ~ o n . ~We . ~ dealt with this briefly in the previous treatmentS but did not explore it sufficiently, nor the applications in ionic equilibria.2 (In the past, other have dealt with some aspects of the topics discussed herein.) There is no comparable apparent difficulty for either type of algorithm, if only species-related data are available. Although use of the methods described here is not restricted to ionic equilibria, the situation addressed is relatively common in that type of system. Both the usual case of a single solution phase and special cases arise. The latter include special compositional constraints such as the specification of a particular pH and solubility calculations. The purpose of this paper is to show more fully that the form of the thermodynamic data is immaterial to the numerical method of solution, and to describe techniques for converting reactionrelated data to species-related data. For this purpose, in what follows, we first outline the usual case and three important special cases, describe the conversion of input data to a standard form, and then compare and discuss our results for several examples taken from the literature. The calculations described here use a general purpose “nonstoichiometric”I algorithm (“the equilibrium algorithm”) that runs on a microcomputer (IBM PC). The details of the algorithm are unnecessary for the purpose here and are to be described elsehwere. For the computations, any general purpose stoichiometric or nonstoichiometric equilibrium algorithm’ that requires species-related free energy data could be used on any type of computer. The procedures for all cases apply whether the system is ideal or nonideal. For simplicity, we restrict the illustrations to ideal solutions or to “pseudoideal” solutions, for reactions in which the given equilibrium constants are not true thermodynamic constants. If necessary, nonideality can be taken into account in the algorithm itself” without altering the treatment of the data described here.
2. Types of Systems/Problems 2.1. The Usual Case. For the usual case, the system is a multispecies single phase (e.g., an aqueous solution) at given temperature and pressure ( T , P ) or at given temperature and volume ( T , V), without any further restrictions; furthermore, information is available for R equilibrium constants (K,) for a set of R linearly independent chemical reactions/equations N
~ u , A =, 0; j = 1, 2,
1=1
..., R
(1)
where A , denotes the molecular formula of species i, N is the number of species, ur, is the stoichiometric coefficient for the ith species in the j t h chemical equation, and N
K, = n a r y ~ J r=l
(2)
where a, is the activity of the ith species. For the usual case also, (2) Leussing, D. L. J . A m . Chem. SOC.1984, 106, 8331. (3) Nordstrom, D. K. et al. In Chemical Modeling in Aqueous Systems; Jenne, E. A., Ed.; ACS Symposium Series No. 93; American Chemical Society: Washington, 1979; p 857. (4) Morin, K. A. Compur. Geosci.1985, 1 1 , 409. (5) Reference 1 Chapter 9. (6) Schott, G . L. J . Chem. Phys. 1964, 40, 2065. (7) Smith, W. R. Ind. Eng. Chem. Fundam. 1976, 15, 227. (8) Krambeck, F. J. Paper presented at A.1.Ch.E. meeting, Miami Beach, . . FL, Nov. 16, 1978. (9) White, C. W., 111; W. D. Seider, AIChEJ 1981, 27, 466. (101 Albertv, R. A. J . Phvs. Chem. 1985, 89. 880. (1 1 ) Reference 1, Chapte; 7
R is equal to the maximum number of linearly independent equations for the system, and hence satisfiesI2 R=N-C where C is the number of components given by
(3)
C = Rank (A) (4) Here A is the ( M X N) formula matrix whose columns are the N formula (subscript) vectors of the species, and M is the number of elements in the N species. The uij form the ( N X R ) stoichiometric matrix N for the given set of reactions. Equation 1 may then be rewritten in matrix form as
AN = 0 (la) Confirmation that R as given does conform to eq 3 may be obtained if desired by using a stoichiometry algorithm which generates a set of R equations from a list of N species in a “canonical” form.12 A modified form of this algorithm is used to convert the given equilibrium-constant data into species-related data in “standard form” for use in the equilibrium algorithm, as described in section 3. 2.2. Stoichiometric Restrictions. Stoichiometric restrictions involve cases for which the number of given linearly independent chemical equations satisfies RCN-C (5) The equilibrium algorithm can be used for such cases as in section 2.1, provided the data are converted as indicated there, and the chemical formulas of the species are modified. The procedure is described in section 3. 2.3. Compositional Constraints. Equilibrium may be subject to one, or more than one, compositional constraint, such as the specification of the concentration of a particular species. An example is the specification of the concentration of H+ in terms of the pH of a solution. The approach for this case builds on the treatment in section 2.2. For each species whose concentration is specified, the activity and (hence) the chemical potential are fixed. The formula vectors of the species, and reactions and corresponding Rs given are modified so as not to involve the species whose activity is fixed. 2.4. Multiphase Equilibria Involving Additional Pure Solids or Liquids. Solubility problems may involve one, or more than one, pure solid in addition to a solution phase. The presence of the additional solid phase is known from the set of equations/ reactions given. Since this problem may be viewed as one in which the activity of a species (Le., that in equilibrium with the solid) is fixed, we may proceed as in section 2.3. The actual presence of the solid at equilibrium can be tested subsequent to the computation.
3. Conversion of Input Data to Standard Form In general the data available for an equilibrium calculation are included in the set (A, p*, N, AGO or K, “restrictions”], where p* are the standard chemical potentials or free energies of the species, AGO are the standard free energies of the given reactions, such that for the j t h reaction -AGo/3?T = In K,
(6)
where 3 is the gas constant and T is temperature, and “restrictions” represent special constraints as in sections 2.2 to 2.4 above. The matrix N satisfies Rank (N) = R (7) the number of reactions. A free-energy-minimization method requires as input a species-related data set ( A , p * ) . In many problems, particularly those involving ionic equilibrium, this set is not directly available, since the reaction-related set (N, AGO or K] is given. It is then necessary, if an algorithm based on such a method is to be used, to obtain a data set (A*, p*) that is equivalent to the given data set (here A* is a “formula matrix” that is compatible with N, as explained (12) Reference 1 , Chapter 2
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Cheluget et al.
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
below). By equivalent we mean that (A*, p * ) satisfies the three equations
NTp* = AGO where superscript T denotes the transpose of a matrix A*N = 0
(8) (9)
and Rank (A*)
+ Rank (N) = N
(10)
Equation 10 combines eq 3, 4, and 7. A p* satisfying eq 8 is said to be consistent with the set (N, AGO) in the sense that it reproduces the given K. We note that p* is not unique, since eq 8 is a set of R linear equations in N unknowns in which N > R. A* and N are said to be compatible if they satisfy eq 9 and 10. If N is given, A* satisfying these equations is not unique (cf. p* above). In this paper, the task is to use the flexibility provided by eq 8-10 to obtain a set (A*, p * ) equivalent to the given set (N, AGO). Any restrictions would be incorporated in this latter set. The key to the approach is the realization that eq 8 and 9 can be regarded as an underdetermined set of linear equations in the N entries of p* and the NC entries of A*’, rewritten as
NT(p*, A*T) = (AGO, 0)
(1 1) 3.1. The Usual Case. From the preliminary discussion of this section, we reinterpret the usual case to mean that the set {A, N, AGO or K) is given, and A and N satisfy eq 9 and 10. For use in the equilibrium algorithm, it is only necessary to obtain a consistent set of standard free energy data ( p * ) from the given K or AGO. This may be done by a modified use of the stoichiometry algorithm,I2 which ordinarily generates N from A by finding a particular solution of the homogeneous eq la. We write eq 8 in the form of a system of homogeneous equations:
(NT, -AGo)(p*,
=0
(12)
By comparison, we see that the matrix (NT,- AGO) in eq 12 takes the place of A in eq l a , and ( p * , 1)’ takes the place of N. We have partially outlined the procedure previ~usly,~ and summarize it here in four steps, since a related procedure is used for the cases to follow: (1) The N species involved in the R chemical equations are arranged in an arbitrary order. (2) The input data are prepared for the stoichiometry algor-AGo/BT) is formed; each ithm;’, the R X ( N 1) matrix (p, row represents a reaction, and each column a species, the last one a fictitious species (called, e.g., DELTAG); the first N entries in each row are the stoichiometric coefficients for the species in the reaction represented by that row (positive for species on right side and negative for those on left side), and the last entry, under DELTAG, is - A G O / R T for the reaction. (3) The stoichiometry algorithm is run using the data in the matrix described in step 2 as input. (4) The output from step 3 is interpreted to provide a set of N - R chemical “equations” such that each represents the “formation” of 1 mol of a particular species (a “noncomponent”), including DELTAG, from a set of R remaining species (the “components”); the value 0 is assigned as the standard free energy of formation (AGo,/%T) of each of the noncomponents, and the value of -AGofl/RTfor each component is the coefficient for that species in the equation for DELTAG; these latter values are then relative to those for the noncomponents chosen as a datum, Le., as components (or elements) in the usual sense. As an internally consistent set, they reproduce the values of the given ICs in accordance with In KJ (= - A G o J / R T ) = ~ - v l J ~ * , / R T
+
I
The values are then used as input for the equilibrium algorithm. 3.2. Stoichiometric Restrictions. For the case of a stoichiometric restriction represented by eq 5 (also referred to as a re-
stricted equilibrium), the formula vectors of the R - 1 noncomponent species are the first R - 1 unit column vectors; Le., they are treated as components, the species to which are assigned zero values for p * i and unit vectors as formula vectors. The formula vector of each of the remaining species is given by the negative values of the stoichiometric coefficients for that species in order of appearance in the set of R - 1 “reactions”. A consistent p* is formed as described in section 3.1, and a compatible A* is formed from the first R - 1 reactions in the output described in step 4 there. These are both then used as input to the equilibrium algorithm. 3.3. Compositional Constraints. For the case of a compositional constraint, e.g., the requirement that the concentration at equilibrium of a particular species be a fixed value, a consistent p* and a compatible A* are formed as follows: (1) The species list is ordered with the species whose concentration is fixed appearing last (Le., as species N, this ensures that this species be a component). (2) The procedure described in section 3.2 is followed to form a preliminary (A*, p*), except that AGO is modified according to AGo,/%T = A.Goj/RT (given) + vN In mN (14) where mNis the fixed molality, say, of species N, this is equivalent to incorporating the constant activity of this species in the equilibrium constant for each reaction in the given (original) set in which it appears. (3) The final A* is obtained by deleting the row corresponding to the species whose composition is fixed; this results in this species (N) having a formula vector of 0. (4) The final P * ~ ’ Sare those given by step 2 except for species N , for which P * N = - R T In
(15)
3.4. Multiphase Equilibria Involving Additional Pure Solids or Liquids. For the case of an additional phase or additional phases, a consistent p* and a compatible A* are obtained as follows: (1) the species list is ordered with the species in the additional phase (e.g., a solid) appearing last; (2) the procedure described in section 3.2 is followed to yield a preliminary A* and a final p * ; (3) the final A* is obtained by deleting the last row (Le., for the solid) in the preliminary A*.
4. Illustrations and Results 4.1. The Usual Case. As an example of the usual case, we consider the system resulting from dissolving FeC13, FeCl,, CuCl,, CuCl, HCI, and NaCl in water at 25 O C . Equilibrium calculations were done for this system by Kimura et al. as part of a study of the leaching of chalcopyrite in chloride s01utions.l~ These calculations were in terms of 22 dissolved species from information on equilibrium constants for a reaction model of 17 chemical equations involving oxidation-reduction (1 equation), formation of chloro complexes (12 equations), hydrolysis of Fe3+ (3 equations), and dissociation of water. They solved iteratively a set of 22 equations comprised of the 17 equilibrium-constant equations and 5 conservation equations. They thus used reaction-related thermodynamic data and not species-related data. They performed calculations for several initial concentrations of the six species listed above and three sets of equilibrium constants for various ionic strengths ( I = 0, 1, and 1 2 m). We have converted their sets of equilibrium constants into internally consistent sets of standard free energies of formation for use with the equilibrium algorithm, as described in section 3. The system, resulting from the (initial) species listed above, is represented’, as ( ( Fe3+, FeC12+, FeC12+, FeCl,, FeCl,-, FeOH2+, Fe(OH)2+, Fe2(OH):+, Fe2+, FeCl+, FeCI2, Cu2+, CuCl’, CuCl,, Cu’, CuCI, CuC12-, C U C ~ , ~CU,CI,~-, -, CI- H+, OH-, H 2 0 ) , (Fe, CI, H, 0, Cu, P ) ) (A) (13) Kimura, R. T.; Haunschild, P. A,; Liddell, K. C. Metall. Trans. B. 1984, I S B , 213.
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987 2431
Calculation of Ionic Equilibria TABLE I: Commwison of Eouilibrium Calculations for System A“ molality species AGof,/R76 ref 13 this paper 76 dev Fe” 0 2.3 X lo4 2.229 X lo4 -0 FeCI2+ -0.5280 8.3 X lo-’ 8.442 X lo-’ -0 -0 FeCI2+ 0 0.1 11 0.1112 0.637 -0 0.6370 FeC1, 1.3609 -40(-) 1.078 X lo-’ 10.8485 7.6 X IO-“ FeCI45.517 X -27(+) 0 7.6 X lo-* FeOH2+ 1.5 X IO-” 6.546 X -56(+) Fe(OH)2+ 0.73514 4.596 X -58(+) -7.3199 1.1 X IO-” Fe2(0H):+ 0.042 0.04194 -0 Fe2+ 0 -0 0.6327 FeCIt 0.3926 0.634 4.56 -0 4.5675 FeCI2 1.5221 0.058 0.05979 CU2+ 0 -3(-) -0.6(+) 0.9019 0.3926 0.907 CUCI+ -0 4.28 4.2804 CUCI, 1.9416
cu+
CUCI CUCI, CuC112cu2Cl42-
c1-
H+ OHH20
22.5429 17.5475 11.1697 12.3909 19.8059 1.2212 0 25.8101 -6.8401
1.5 X 6.1 X 10” 0.0788 0.5215 0.0788 6.60 C
c
c
1.824 X 6.018 X 10” 0.0791 1 0.521 1 0.07886 6.5867 3.0000 1.529 X 0.6939d
Cu(I1)
Cu2’ CuCI’ CUCI~
0.162 0.530 0.308
0.162 0.530 0.308
0.076 0.451 0.473
Fe(I1)
Fe2’ FeCIt FeC12
0.044 0.253 0.703
0.044 0.253 0.703
0.016 0.164 0.820
Fe(II1)
Fe3+ FeCI2+ FeCI,’ FeCI’ FeC14-
0.020 0.213 0.695 0.070 0.002
0.020 0.213 0.695 0.070 0.002
0.007 0.124 0.729 0.133 0.007
-22(-)
-Q -0 -0 -0 -0
“ t = 25 “C, I = 7 m, A t i a l concentration of each dissc ed species = 3.0 m. bValues given represent an internally consistent set for use only in the context of system A. CNotgiven. dMole fraction.
A comparison between results calculated by Kimura et al.I3 and by us is shown in Table I for the conditions indicated. Agreement is essentially obtained for most species, including the major species (rounding of their figures precludes more accurate assessment), but there are significant percentage differences for five of the minor species. We conclude, however, for our purpose here, that the calculated results are the same. The results in column 4 satisfy exactly the equilibrium-constant values given’, for I 1 2. The solution with the equilibrium algorithm used here required 17 iterations and 26 s on the microcomputer. 4.2. Stoichiometric Restrictions. A relatively simple example to illustrate a “stoichiometric restriction” is afforded by the permanganate-peroxide reaction14 used in a standard method for quantitative analysis of peroxide in aqueous solution. (However, since the reaction is virtually complete, equilibrium considerations are not involved in calculating the final composition.) Reaction in the system is normally represented by the (one) chemical equation: 3H2S04+ 2KMn04 + 5 H 2 0 2 = K2SO4 + 2MnS0,
TABLE II: Comparison of Equilibrium Calculations for System B ( t = 25 “C) species fracn within each ion type this paperb restd unrestd metal ion type species ref 15‘ equiId equiIC CU(I) cut 10.001 so.001 10.001 CUCI 10.001 so.001 so.001 CuC120.468 0.468 0.328 C U C I ~ ~ -0.532 0.532 0.672
+ 8 H 2 0 + 502
A reacting system involving these seven species requires two chemical equations for its description, as can be shown by a stoichiometric analysis. If the stoichiometric restriction that (as observed) K M n 0 4 and H202react in a 2:5 ratio in the standard analytical procedure is incorporated as a linear restriction in the stoichiometry algorithm,I2 a modified formula matrix results which generates only one chemical equation, Le., that given above. A somewhat more complicated example which does require equilibrium considerations is provided by the treatment of a modified form of system A by Wilson and Fisher.15 In order to determine the predominant chloro species for each of Cu(I), Cu(II), Fe(II), and Fe(III), considered separately, in aqueous solution, they calculated the equilibrium distributions using stepwise formation constants at 25 OC for a fixed (equilibrium) C1- concentration of 6 M. As pointed out by Kimura et al.,I3 their treatment is equivalent to considering all metal ion types in the same solution but neglecting oxidation-reduction of the copper and iron ions (however, this was apparently not the purpose of (14) Missen, R. W.; Smith, W. R. Chem. I3 [Thirteen] News 1984, Sept.,
2-3. (15) Wilson, J. P.; Fisher, W. W. J. Met. 1981, 33, No. 2, 52.
1 0.0276 Cu(I)/Cu(II) N/A 1 36.2 Fe( II)/ Fe(111) N/A “Recalculated for equilibrium concentration of CI- = 2.525 M. *Initial concentration of each species = 3 M. Redox excluded; equilibrium concentrationof C1- = 2.525 M. dRedox included; equilibrium concentration of CI- = 4.554 M.
Wilson and Fisher). Their treatment is then equivalent to the imposition of a stoichiometric restriction on a system which we represent12 as (cf. system A): ((Fe3+,FeC12+, FeCl,’, FeCl,, FeC14-, Fez+, FeCl+, FeC12, cu2+,c u c i + , c u c i 2 , CU+, c u c i , cuci2-, CUC&~-,ci-), (B) (Fe, C1, Cu, p)I This system does not include the hydrolysis species and Cu2C12of system A. We then used the equilibrium algorithm to calculate the equilibrium distribution of species for system B both on a restricted equilibrium basis and on an unrestricted basis. The latter includes the redox equation Cu+
+ Fe3+ = Cu2+ + Fez+
in addition to the 11 equations for formation of the chloro species. For the calculations we used the 11 stepwise formation constants given by Wilson and Fisher,I5 and the equilibrium constant for the redox reaction (6.17 X lo9 for I L 2) given by Kimura et a1.13 The results of the calculations are shown in Table I1 in the form used by Wilson and Fisher.ls The fractional distribution of species is shown for each metal ion type, Cu(I), Cu(II), etc. We have recalculated the values of Wilson and Fisher for an equilibrium C1- concentration of 2.525 M, the value obtained for the conditions shown for the case of restricted equilibrium. The results in columns 3 and 4, which compare the results obtained by the method used by Wilson and Fisher and the method of section 3 for a stoichiometric restriction (Le., restricted equilibrium), are the same. The results in column 5 show the effect of allowing redox equilibrium in addition to equilibrium with respect to the formation of chloro species. Although the effects on the equilibrium distributions are significant, the main effect, shown at the bottom of Table 11, is on the ratios Cu(I)/Cu(II) and Fe(II)/Fe(III) as a result of redox. With the arbitrary initial concentrations chosen” for Table 11, the former changes from 1 for restricted equilibrium to 0.0276 for unrestricted equilibrium, and the latter changes from 1 to 36.2. The solution for the results in column 4 required 16 iterations and 12 s on the microcomputer, and 14 iterations and 8 s for those in column 5 . 4.3. Compositional Constraints. To illustrate a compositional constraint, we use an aqueous system in which complexing occurs and the pH is fixed. Ginzburg16 has given results of calculations
2432
The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
for the aqueous system containing the three metal ions Zn, Cd, and Cu, and the three ligands NH,, OH, and NTA (the tertiary nitrilotriacetate ion, N(CH2C00)33-)). Typically, in such a case, the starting information consists of the initial concentration of each reactant (in this example, 5 X lo-, M for each of Zn, Cd, Cu, NH,, and NTA), the value of the stability (equilibrium) constant for each complex (in this example, values at 25 “ C for 27 postulated complexes), and the pH (8.0, in this example). For our purpose, the system of 32 dissolved species (taken from Ginzburg’s tableI6) and water is represented’, as (charges on species omitted): ((H, OH, Cd, Cu, Zn, NTA, NH,, HzO, Cd(NH3), Cd(NH312, Cd(NH313, Cd(NHJ4, C ~ ( N H ~ )Cd(NHd6, S, C W H d , C U ( N H ~ )C~U, ( N H ~ )Cu(NH3I4, ~, Zn(NH3), Zn ( N H h , Zn(NH3),, Zn(NH3)4,Cu(OH), C U ( O H ) ~ , Cd(OH), Zn(OH), Zn(OH),, Zn(OH),, Cd(NTA), Cd(NTA),, Cu(NTA) , Zn(NTA), H(NTA) , H ( NH,) , Cd(OH)(NTA)), (H, OH, Cd, Cu, Zn, NTA, NH,)) (C)
For brevity, we do not give further details of the thermodynamic data obtained from the equilibrium constants used by Ginzburg or a detailed comparison of the calculated results. The agreement is essentially complete. The solution with the equilibrium algorithm used here required 17 iterations (cf. Ginzburg: 151) and 32 s on the microcomputer. The calculations done by Wilson and FisherIs for their treatment of the modified system B also illustrate a compositional constraint. They did their calculations for a CI- concentration of 6 M (at equilibrium). Although it is feasible to do this type of calculation, as described by Wilson and Fisher,I5 with a hand calculator, we have obtained their results in each case with, at most, 15 iterations in about 3 s on the microcomputer. 4 . 4 . Multiphase Equilibria Involving a Pure Solid. We consider the solubility of FeS in water at 25 OC to illustrate the case in which a pure solid phase exists at equilibrium with a solution. This system has been treated by Kostrowicki and Liwo” on the basis of values of equilibrium constants for a reaction model of five chemical equations. For our purpose, in terms of the species indicated in these equations, the system is represented’, as ((FeS(s), H+, OH-, Fe2+, Sz-, HS-, H2S, H 2 0 , FeOH’), (H, 0 , Fe, S, p)I
(D)
A set of dimensionless standard free energies of formation (AGofi/RT, obtained from the five equilibrium constants given by Kostrowicki and Liwo is (-27.7958, 16.2346, 20.0181, -1.41508, 13.4366, 0, 0, 0,O). The concentrations at equilibrium for the seven dissolved species calculated here with the equilibrium algorithm from the species-related data are identical with those given by Kostrowicki and LiwoI7 using their algorithm and the reaction-related data. The solution with the equilibrium algorithm used here required 13 iterations (cf. Kostrowicki and Liwo: 9) (16) Ginzburg, G. Talanta 1976, 23, 149. (17) Kostrowicki, J.; Liwo, A. Comput. Chem. 1984, 8, 91
Cheluget et al. and 4 s on the microcomputer. The resulting calculated solubility of FeS is 7.3 X 10” M, and the iron exists in solution mainly as FeOH’, giving a slightly basic solution. We have investigated other systems with similar results. For the usual case (2.1), these include a nonaqueous system (picric acid and triethylamine dissolved in acetonitrile”), a sea-water in water,19 model,Is and the system ~~o~(III)-C~~+-CO~~--PO~which has the potential to form solids and a gas but was not treated as such (it may be necessary to test a postulated system for the presence of additional phases at equilibrium; the means for doing this has been described p r e v i o u ~ l y ~ ~ ’For ~ ) . a compositional constraint (2.3), they include complexing in the Ca-EDTA-H,O system at a fixed PH,~Oand similarly complexing in a biochemical system.21 For a multiphase system (2.4), they include the dissolution of PbO in water, with the additional feature of a fixed pH.22 The execution times for the equilibrium algorithm used here ranged from 4 to 40 s, with an average of about 16 s. 5. Discussion The calculations demonstrate that the techniques described for a number of situations encountered in ionic equilibria can be used to convert reaction-related thermodynamic data (equilibrium constants) into species-related data in standard form: an internally consistent set of standard free energies of formation and appropriate formula vectors. Although this has been demonstrated for ionic equilibria, and is probably most common for this type, the techniques can be used in other types of equilibrium calculations. The overall result is that a general purpose equilibrium algorithm that requires species-related data,’ with its attendant computational advantages, can be used as an alternative to an algorithm requiring reaction-related data. The disadvantages of the procedures currently are the requirements to intervene in the processing of the data in an ad hoc manner to prepare the tables and matrices for the adjustments and modifications described. Although we have not yet attempted it, in principle it should be possible to combine the “front-end’’ manipulation of the data in one algorithm together with the equilibrium calculations. This would allow input of data without further processing either (1) as a set of species together with their standard chemical potentials or free energies of formation, or (2) as a set of chemical equations/reactions together with their equilibrium constants. This may be assisted in part by a symbolic manipulation/computer algebra system, with its list-processing capabilities, as used for a related purpose.23
Acknowledgment. Financial support has been received from the Natural Sciences and Engineering Research Council of Canada. (18) Pytkowicz, R. M.; Hawley, J. E. Limnol. Oceanogr. 1974, 19, 223. (19) Morel, F.; Morgan, J. Enuiron. Sci. Technol. 1972, 6, 58. (20) Stumm, W.; Morgan, J. J. Aquatic Chemistry, 2nd ed.; Wiley: New York, 1981; p 361. (21) Perrin, D. D.; Sayce, I. G. Talanta 1967, 14, 833. (22) Dyrssen, D.; Jagner, D.; Wengelin, F. Computer Calculation of Ionic Equilibria and Titration Procedures, 2nd ed.; Almqvist and Wiksell: Stockholm, 1968; p 166. (23) Farah, N.; Missen, R. W. Can. J . Chem. Eng. 1986, 64, 154.
