The Use of Calculated Species Distribution Diagrams to Analyze

Research: Science and Education edited by

Advanced Chemistry Classroom and Laboratory

Joseph J. BelBruno Dartmouth College Hanover, NH 03755

The Use of Calculated Species Distribution Diagrams to Analyze Thermodynamic Selectivity

W

Antonio Bianchi* Department of Chemistry, University of Florence, Via Maragliano 75/77, 50144 Florence, Italy; [email protected] Enrique Garcia-España* Department of Inorganic Chemistry, University of Valencia, C\ Dr. Moliner 50, 46100 Burjassot (Valencia), Spain

Almost all chemical processes, both natural and artificial, proceed through competitive reactions involving concomitant equilibria. Probably the most outstanding examples of these kinds of processes come from living systems. In such systems, according to the environmental conditions (pH, dielectric constant, concentration of reactants, ionic strength, temperature, exposure to light or darkness, etc.), a large number of species can associate to yield different adducts. Nevertheless, in spite of this complexity, high specificity is achieved through selective interaction and recognition of discrete chemical species. Analytical chemists have for a long time been concerned with the study of selective association. The recent upsurge of the new field of supramolecular chemistry, which has its roots in molecular recognition processes, has presented a new impetus for considering selectivity. Indeed, supramolecular chemistry has very much stimulated the search for selective processes in chemical systems of ever-increasing complexity. However, the difficulty of considering all parameters affecting association selectivity has induced many researchers to control environmental variability and, in several cases, even to design systems not influenced by particular environmental parameters. Nevertheless, supramolecular chemistry is a fertile field and it is appealing to students to discover examples of special selectivity patterns. One problem in defining selectivity arises when one is considering systems in which the reactants may participate in multiple protonation equilibria. In these cases, the degree of protonation of the reactants changes with pH and appropriate criteria are required to properly establish selectivity patterns. This paper intends to show how, by invoking simple elements of solution equilibria and by using a computer program for the calculation of equilibrium species concentrations, it is possible to get a prompt analysis of selectivity even in intricate systems. It outlines a didactic itinerary that has proved useful in several undergraduate curricula at the Universities of Florence (Italy) and Valencia (Spain). In consonance with the consensus that a good way to enhance student experience in analyzing scientific problems is to include research problems in existing courses, we have organized this didactic itinerary so as to give shape to the concept of thermodynamic selectivity through the analysis of three concrete examples of increasing complexity. The organization of the paper takes into account the usual steps in the development of a real research project, namely, generating problems, choosing adequate forms of data evaluation, and discussing results. Obviously considerable attention is

dedicated to the preparation and utilization of cumulative species distribution diagrams, which represent the novelty of this approach to the analysis of thermodynamic selectivity. In a separate section we give details for the calculation and visualization of species distribution diagrams. Thermodynamic Selectivity in Solution Equilibria Association equilibria are usually defined by means of thermodynamic parameters, and consequently selectivity is frequently interpreted in a thermodynamic sense, in terms of Gibbs free energy changes of association or equilibrium constants. Let us consider the following equilibria (eqs 1 and 2) where the species A is able to bind both B and C, while B and C do not bind each other. A + B = AB

K1 = [AB]/[A][B]

(1)

A + C = AC

K2 = [AC]/[A][C]

(2)

The K1/K2 ratio accounts for the ability of B to replace C in the AC adduct or, in other words, for the ability of A to discriminate between B and C. AC + B = AB + C

K3 = [AB][C]/[AC][B] = K1/K2 (3)

If K1 > K2, the ratio K1/K2 > 1 and the equilibrium (eq 3) is displaced toward the right side, that is, A binds B in preference to C. The larger the K1/K2 value the greater the binding selectivity. In spite of its simplicity, the use of only this criterion for the analysis of thermodynamic selectivity has often been the source of misleading interpretations, which can be almost exclusively ascribed to inadequate consideration of the influence of environmental conditions on the system equilibria and hence on the association schemes. In this sense, a principal source of uncertainty is the improper consideration of the actual degree of protonation of the reactants. A successful method to establish thermodynamic selectivity is based on the knowledge of the equilibrium constants of all equilibria involved in the system, as well as of environmental conditions. Owing to the complexity of many of these systems, the use of a computer program to calculate the concentration of equilibrium species is normally necessary. Nowadays this is not problematic because many programs for the calculation of species concentration diagrams are available (1–7 ). Let us start with a simple example. EDTA and PENTEN are two classic hexadentate ligands having analogous structures and forming very stable complexes with many metal ions.

JChemEd.chem.wisc.edu • Vol. 76 No. 12 December 1999 • Journal of Chemical Education

1727

Research: Science and Education Table 1. Logarithms of Protonation Constants of EDTA and PENTEN EDTA

PENTEN

Reaction

Log K

Reaction

Log K

EDTA4{ + H+ = HEDTA3{

10.17

PENTEN + H+ = HPENTEN+

10.08

HEDTA3{ + H+ = H2EDTA2{

6.11

HPENTEN+ + H+ = H2PENTEN2+

{

2.68

H2PENTEN2+ + H+ = H3PENTEN3+ 8.99

2.00


1.50


2{

+

H2EDTA + H = H3EDTA H3EDTA{ + H+ = H4EDTA +

H4EDTA + H = H5EDTA

+

9.58

NOTE: Aqueous solutions, 0.1 mol dm{3 ionic strength, 298.1 K (8).

The equilibrium constants for the reactions Cu2+ + EDTA4{ = CuEDTA2{ and Cu2+ + PENTEN = CuPENTEN2+ at 298 K and 0.1 mol dm{3 ionic strength are log K = 18.70 and log K = 22.1, respectively (8). COOH

HOOC N

N

N COOH

HOOC

EDTA

NH2

H2N N

NH2

H2N

PENTEN

Adopting the criterion deriving from eqs 1–3, on the basis of these equilibrium constants one should deduce a marked selectivity of PENTEN over EDTA in the binding of copper. But this is not the situation. In fact, these ligands are involved in proton transfer equilibria (protonation–deprotonation) in solution and thus, since ligand protonation competes with metal ion coordination, the different protonation behaviors of EDTA and PENTEN (Table 1) allow complexation of Cu2+ to occur in different pH ranges. Moreover, the Cu2+ complexes of these ligands also undergo protonation (log K = 3.0 and log K = 8.0 at 298 K and 0.1 mol dm{3 ionic strength, for CuEDTA2{ + H+ = CuHEDTA{ and CuPENTEN2+ + H+ = CuHPENTEN3+, respectively [8]). A clear way to get a quantitative evaluation of the relative binding ability of these ligands toward Cu2+ is to calculate the percentages of all complexes formed as a function of pH in a solution containing the metal ion and both ligands in equimolar amounts. The graphic representation of the computed species concentration curves is given by the distribution diagram in Figure 1. As can be seen, the binding selectivity in this system is

pH-dependent and the coordination of Cu2+ can be switched from EDTA to PENTEN by tuning the solution pH. Under pH conditions where EDTA and PENTEN exist in fully deprotonated forms and only the CuEDTA2{ and CuPENTEN2+ complexes can be formed, the evaluation of selectivity simply made by comparing the formation constants of CuEDTA2{ and CuPENTEN2+ leads to a correct interpretation. There are many complicated systems, however, for which a similar analysis does not lead to a clear evaluation of selectivity, even though particular environmental conditions are selected. Let us continue by considering one of these cases. With the coming of supramolecular chemistry we have seen increasing interest in the formation of organic aggregates. This is an intense field of work for those researchers devoted to speciation and equilibrium constant determinations, owing to the complexity of the systems studied. For the same reason the evaluation of selectivity in such systems is an arduous task, not only from a quantitative point of view but frequently even from a qualitative one. ATP (adenosine triphosphate) gives rise, in its anionic forms, to very stable adducts with polyammonium cations. Table 2 lists the equilibrium constants of the species formed by ATP with [18]aneN6 (L1) and Me4[18]aneN6 (L2), respectively, together with their protonation constants (9–11). H3C

H N

H N NH

HN

N H

N H

H N

N H3C N

N CH3 N

N

H

H3C

L1

L2

To establish whether there is selectivity in the binding of ATP by these ligands we can calculate, according to the previous example, the distribution diagram of the species formed as a function of pH in a solution containing the nucleotide and the two macrocycles in equimolar concentration (Fig. 2a). In the present case, however, the diagram does not furnish much help to our purpose, since it is rather intricate owing to the concomitant formation of many species none of which is clearly prevalent. Table 2. Logarithms of Equilibrium Constants for Protonation of ATP and L and Formation of ATP–L Complexes

Figure 1. Distribution diagram of the complexed species formed as a function of pH in the system Cu2+/EDTA/PENTEN. All reactants are 1 × 10 {3 mol dm{3 in concentration. Percentages are calculated with respect to the analytical concentration of metal ion.

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Reaction

Log K

ATP4{ H+ = HATP3{ HATP3{ + H+ = H2ATP2{

6.24 4.00

L1

L2

L + H+ = LH+ LH+ + H+ = LH22+ LH22+ + H+ = LH33+ LH33+ + H+ = LH44+ LH44+ + H+ = LH55+ LH55+ + H+ = LH66+

10.15 9.48 8.89 4.27 2.21 1.0

9.75 9.11 7.53 2.59 — —

L + 3H+ + ATP4{ = H3LATP{ L + 4H+ + ATP4{ = H4LATP L + 5H+ + ATP4{ = H5LATP+ L + 6H+ + ATP4{ = H6LATP2+

30.99 38.70 43.92 —

29.66 36.41 40.93 44.18

N OTE : Aqueous solutions, 0.15 mol dm{3 ionic strength, 298.1 K (9–11).

Journal of Chemical Education • Vol. 76 No. 12 December 1999 • JChemEd.chem.wisc.edu

Research: Science and Education

Adopting the representation of the species distribution diagram reported in Figure 2b provides a clearer picture. Since thermodynamic selectivity is the preferential binding of one species over another under equivalent conditions, we can disregard the formation of the individual species and consider instead the overall amounts of ATP bound to L1 and L2 as a function of pH. In this way the intricate species distribution diagram shown in Figure 2a is transformed to the simple one shown in Figure 2b, which clearly establishes the complicated pattern of selectivity occurring in the system. As evidenced by Figure 2b, the system presents two pH values (pH 6.7, 8.1) for which the total amount of ATP bound to L1 is equal to the total amount of ATP bound to L2 (isoselectivity points), defining three pH ranges where binding tendencies reverse. This presentation of species distribution diagrams (cumulative distribution diagrams) was first proposed for the analysis of selectivity in supramolecular systems (10, 12), but it was immediately applied also to metal ion coordination in very complicated systems containing highly dentate macrocyclic and linear polyamines (13). Owing to the large number of amino groups, these ligands are able to form mononuclear and polynuclear complexes together with several polyprotonated forms. Hydroxo complexes are also frequently formed. An instructive example is given by the macrocycle [24]aneN8 (L3) and its open chain analogue L4, which were compared in order to analyze the effect of ligand structure on the selectivity in metal ion binding.

a

b

Figure 2. Two representations of the distribution diagram of the complexed species formed as a function of pH in the system EDTA/ L1/L2. All reactants in 1 × 10 {3 mol dm{3 concentration. Concentration curves for individual species are represented in Figure 2a. Figure 2b reports the overall amounts of ATP bound to L1 and L2. Percentages are calculated with respect to the analytical concentration of ATP.

H

N

N

HN

L + H+ = LH+

9.65

10.39

LH+ + H+ = LH22+

9.33

9.77

LH22+ + H+ = LH33+

8.76

9.28

LH33+ + H+ = LH44+

7.87

8.61

LH44+ + H+ = LH55+

4.55

6.68

LH55+ + H+ = LH66+

3.42

4.44

LH66+ + H+ = LH77+

2.71

3.31

LH77+ + H+ = LH88+

1.90

2.93

10.83

10.37

Pb2+ + L = PbL2+ Pb

2+

+

3+

+ L + H = PbLH

19.48

20.328

Pb2+ + L + 2H+ = PbLH24+

26.92

28.912

Pb2+ + L + 3H+ = PbLH35+

31.68

35.845

2Pb2+ + L = Pb2L4+

17.57

18.064

2Pb2+ + L + H+ = Pb2LH5+

23.73

25.39

2Pb2+ + L + H2O = Pb2LOH3+ + H+

7.87

2Pb2+ + L + 2H2O = Pb2L(OH)22+ + 2H+

—

8.68 {2.19

N OTE: Aqueous solutions, 0.15 mol dm{3 ionic strength, 298.1 K ( 13–14).

H

NH N H

N H

L3

Table 3. Logarithms of Equilibrium Constants for Protonation of L3 and L4 and Formation of Their Pb2+ Complexes Log K Reaction L3 L4

H3C

H N

H N

H

CH3

N H

N H N

N

HN

H

NH N H

N H

L4

Table 3 reports the equilibrium constants for the formation of Pb2+ complexes (13) together with the ligand protonation constants (14). Seven and eight complexes of different stoichiometry, both mononuclear and binuclear, are formed by L3 and L4, respectively, and eight protonation equilibria for each ligand compete with complex formation. That is, including the ionic product of water (Kw) and Pb2+ hydrolysis, 33 equilibria define this competitive system. To further complicate matters, the formation of mononuclear and binuclear complexes makes the relative percentages of the species formed strictly dependent on the metal-to-ligand molar ratio (R), the mononuclear complexes being favored by low R values and the binuclear complexes by high values. The analysis of selectivity in such systems could seem impossible, but it does not present more difficulties than the previous example; it is just more laborious, since it requires consideration of different metal concentrations. For instance, the overall distribution diagrams in Figure 3 depict the situation for equimolar concentrations (R = 1) of Pb2+, L3, and L4 (Fig. 3a) and for twofold (R = 2, Fig. 3b) and threefold (R = 3, Fig. 3c) excess of metal ion. For R = 1 there is a general preferential binding of Pb2+ to the macrocyclic ligand L3 (Fig. 3a) over the pH range considered. Increasing the metal ion concentration increases the percentage of complexes formed by the open-chain ligand in alkaline solution and


1729


leads to an inversion of the selectivity trend at high pH (Figs. 3b, 3c). An accurate definition of the system can be obtained, producing a large number of similar distribution diagrams over a wide range of R values. This large set of species concentration curves can be reduced to the simple species distribution surfaces of the three-dimensional distribution diagram in Fig. 4. The intersection of the surfaces corresponding, respectively, to the formation of L3 and L4 complexes is an isoselectivity line defining the regions of pH and R values for which there is preferential binding of the metal ion to each ligand. This isoselectivity line is represented by the 50% contour line in Figure 5; inside the area delimited by this curve there is preferential binding of the metal ion to L4, whereas outside this area L3 complexes prevail.

a

b

Solution Equilibria and Calculation of Species Distribution Diagrams This section contains a general approach to the calculation of species distribution diagrams. For this purpose we consider a chemical system containing four reagents. Four-reagent systems, which are very common in the analysis of thermodynamic selectivity—two such systems were used in the previous section— create a profitable and well-balanced pedagogical background. The same method applies for systems containing smaller, or greater, numbers of reagent species. Let us consider a solution containing the four reagents A, B, C, D forming n complexed species according to the equilibrium reactions a1A + b1B + c1C + d1D = Aa Bb Cc Dd 1

1

1

a2A + b2B + c2C + d2D = Aa Bb Cc Dd A A A A A anA + bnB + cnC + dnD = Aa Bb Cc Dd

c

1

2

2

2

2

n

n

n

n

(4)

The corresponding cumulative equilibrium constants are βa1b1c1d1 = [Aa1Bb1Cc1Dd1]/[A]a1[B]b1[C]c1[D]d1 βa2b2c2d2 = [Aa2Bb2Cc2Dd2]/[A]a2[B]b2[C]c2[D]d2

A

β a n b n cn d n

A A A A A = [Aa Bb Cc Dd ]/[A]a [B]b [C]c [D]d n

n

n

n

n

n

(5)

n

n

If the cumulative equilibrium constants are known and the total (analytical) concentrations (T) of the independent reactants A, B, C, D are fixed, the concentration of all species present in solution can be calculated by solving the set of mass balance equations (eq 6), which furnishes the free reagent concentrations [A], [B], [C], [D]. TA = [A] + ∑ ai βa b c d [A]a [B]b [C]c [D]d i

i

i i i i

i

i i i i

i

i

i

TB = [B] + ∑ bi βa b c d [A]a [B]b [C]c [D]d i

i

i

i

TC = [C] + ∑ ci βa b c d [A]a [B]b [C]c [D]d i

i

i

i

i i i i

i

i

i

i

i i i i

In most cases, such a system is composed of nonlinear equations, whose solution is normally very complicated. For this reason the solution of the nonlinear equations of mass balance is commonly achieved by using iterative methods such as Newton-Raphson’s method (15), which calculates the free concentrations on the base of initial estimates. Estimates 1730

of [A], [B], [C], and [D] are improved by calculating shifts according to the following equation: ∂TA ∂[A] ∂T [A] B ∂[A] ∂T [A] C ∂[A] ∂TD [A] ∂[A] [A]

i

TD = [D] + ∑ di βa b c d [A]a [B]b [C]c [D]d i

(6)

Figure 3. Distribution diagrams of the complexed species formed as a function of pH in the system Pb2+/L3/L4. (a) [L3] = [L4] = [Pb2+] = 1 × 10 {3 mol dm{3; (b) [L3] = [L4] = 1 × 10 {3 mol dm{3, [Pb2+] = 2 × 10 {3 mol dm{3; (c) [L3] = [L4] = 1 × 10 {3 mol dm{3, [Pb2+] = 3 × 10 {3 mol dm{3. Percentages are calculated with respect to the analytical concentration of metal ion.

∂TA ∂[B] ∂T [B] B ∂[B] ∂T [B] C ∂[B] ∂TD [B] ∂[B] [B]

∂TA ∂[C] ∂T [C] B ∂[C] ∂T [C] C ∂[C] ∂T [C] D ∂[C] [C]

∂TA ∂[D] ∂T [D] B ∂[D] ∂T [D] C ∂[D] ∂T [D] D ∂[D] [D]

∆[A] [A] ∆TA ∆[B] ∆TB [B] = ∆[C] ∆TC ∆TD [C] ∆[D] [D]


(7)


Figure 4. 3-D distribution diagrams of the complexed species formed as a function of pH in the system Pb2+/L3/L4. [L3] = [L4] = 1 × 10{3 mol dm{3; R = [Pb2+]/[L]. Percentages are calculated with respect to the analytical concentration of metal ion.

Figure 5. Percentage of ATP bound to L4 in the system Pb2+/L3/L4 as a function of pH and metal-to-ligand molar ratio R. [L3] = [L4] = 1 × 10 {3 mol dm{3. Percentages are calculated with respect to the analytical concentration of metal ion.

ten and reviewed. Other programs are associated with most of the current computer programs for the determination of equilibrium constants. At least two recent programs are available on the Web: http://www.eawag.ch/services_e/software/e_chemeql.html (CHEMEQL program (2), which runs on an Apple MacIntosh) and http://www.chim1.unifi.it/group/vacsab/hyss.htm (HySS program [1e], which runs under either Windows 3.1x or Windows95). The input data are equilibrium constants and analytical concentrations of reagents, and the results are presented both in tabular and in graphical forms. Calculations of species concentration used as examples in this paper were performed by means of the program HySS, which furnishes a variety of data presentations, including tables of concentrations of all species present in solution in the selected pH range and the typical distribution diagrams displayed in Figures 1 and 2a. The preparation of cumulative distribution diagrams such as those in Figures 2b and 3 requires a little additional work, since the concentrations of the individual species have to be summed by using another program and we do not know any program for species distribution diagram calculations that includes this option. Microsoft Excel (16 ) is an easy-to-handle and largely available program that can be used to this purpose. Let us return to the case of competitive pH-dependent binding of ATP by the ligands L1 and L2; data resulting from species concentration calculation performed by the program HySS are collected in tabular form (Table S1 in supplementary material W), corresponding to the distribution diagram in Figure 2a. These data can be transferred to Excel by simple Copy/Paste commands and processed by means of Excel functions to obtain a table of total percentages of unbound ATP and of ATP bound to L1 and to L2, calculated at different pH values (Table S2 in supplementary materialW). A graphical representation of these data, in the form of the cumulative distribution diagram displayed in Figure 2b, can be obtained directly by means of Excel’s tool menu. The three-dimensional distribution diagram shown in Figure 4 was prepared by using Excel to assemble tables of data such as that used to produce Figure 2b, calculated for different concentration of the metal ion. Figures of editorial quality were obtained by using the programs EasyPlot for Windows (17) (Figs. 1–3) and SURF (18) (Fig. 4). The last program also furnished the level curve diagram reported in Figure 5. Conclusion

The derivatives in the symmetrical matrix on the left side can be obtained easily from eqs 6. The diagonal terms are given by expressions such as eq 8, A

∂T A = A + ∂[A]

Σa i

2 i

β a ib ic id i A

ai

B

bi

C

ci

D

di

(8)

and the off-diagonal terms are given by expressions such as eq 9 A

∂T B ∂T A = B = ∂[A] ∂[B]

Σa b β i

i

i

a ib ic id i

A

ai

B

bi

C

ci

D

di

(9)

Many programs for the calculation of species distribution based on the equilibrium-constants approach have been writ-

We have successfully employed this simple procedure for the evaluation of simultaneous equilibria and thermodynamic selectivity as a research tool; it has also been extended to student training as exercises in computational chemistry, thermodynamics, and supramolecular chemistry. The graphic representation of equilibrium data allows a simple visual analysis of speciation and selectivity, even in complicated systems, from both qualitative and quantitative points of view. Students can either use experimental equilibrium constants, to get insight into real systems (a very large amount of equilibrium data is available in books [8], reviews [19], and electronic databases [20, 21]), or invent equilibrium data to build designed systems. In the former case, it is very im-


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Information • Textbooks • Media • Resources

portant to use equilibrium constants determined under the same experimental conditions to obtain a correct representation of the real system. As an exercise, students could be asked to introduce small changes in the values of the equilibrium constants or to use equilibrium constants determined under different experimental conditions, in order to verify this point. Note W

Supplementary materials for this article are available on JCE Online at http://jchemed.chem.wisc.edu/Journal/issues/1999/Dec/ abs1727.html.

Literature Cited 1. (a) Leggett, D. J. Talanta 1977, 24, 535. (b) Garcia, A.; Madariaga, J. M. Comput. Chem. 1984, 8, 193. (c) De Robertis, A.; De Stefano, C.; Sammartano, S.; Rigano, C. Anal. Chim. Acta 1986, 191, 385. (d) De Stefano, C.; Princi, C.; Rigano, C.; Sammartano. S. Comput. Chem. 1988, 12, 305. (e) Alderi, L.; Gans, P.; Ienco, A.; Peters, D.; Sabatini, A.; Vacca, A. Coord. Chem. Rev. 1999, 184, 311–318. 2. Müller, B. chemEQL, version 2.0, User’s Guide to Application; Swiss Federal Institute of Environmental Sciences and Technology (EAWAG): Dübendorf, Switzerland, 1996. 3. Computational Methods for the Determination of Formation Constants; Leggett, D. J., Ed.; Plenum: New York, 1985. 4. Gans, P.; Sabatini, A.; Vacca, A. J. Chem. Soc., Dalton Trans. 1985, 1195–1200. 5. Gans, P.; Sabatini, A.; Vacca, A. Talanta 1996, 43, 1739. 6. Martell, A. E.; Motekaitis, Determination and Use of Stability Constants, 2nd ed.; VCH: New York, 1992. 7. Pettit, L. D. SPECIES, version 3.0; Academic Software: Yorks, UK, 1997.

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8. Martell, A. E.; Smith, R. M. Critical Stability Constants; Plenum: New York, 1977. 9. Bianchi, A.; Micheloni, A.; Paoletti, P. Inorg. Chim. Acta 1988, 151, 269–272. 10. Andrés, A.; Aragó, J.; Bencini, A.; Bianchi, A.; Domenech, A.; Fusi, V.; Garcia-España, E.; Paoletti, P.; Ramírez, J. A. Inorg. Chem. 1993, 32, 3418–3424. 11. Bencini, A.; Bianchi, A.; Garcia-España, E.; Fusi, V.; Micheloni, M.; Paoletti, P.; Ramírez, J. A.; Rodriguez, A.; Valtancoli, B. J. Chem. Soc., Perkin Trans. 2 1992, 1059–1065. 12. Bencini, A.; Bianchi, A.; Burguette, M. I.; Domenech, A.; GarciaEspaña, E.; Luis, S. V.; Ramírez, J. A. In Proceedings of the 1st National Congress on Supramolecular Chemistry, Pavia, Italy, Sep 16–18, 1992; Grafiche G. V.: Milan, 1992; pp 51–52. 13. Andrés, A.; Bencini, A.; Carachalios, A.; Bianchi, A.; Dapporto, P.; Garcia-España, E.; Paoletti, P.; Paoli, P. J. Chem. Soc., Dalton Trans. 1993, 3507–3513. 14. Aragó, J.; Bencini, A.; Bianchi, A.; Garcia-España, E.; Micheloni, M.; Paoletti, P.; Ramírez, J. A; Paoli, P. Inorg. Chem. 1991, 30, 1843–1849. Bencini, A.; Bianchi, A.; Dapporto, P.; GarciaEspaña, E.; Micheloni, M.; Paoletti, P. Inorg. Chem. 1989, 28, 1188–1191. 15. Gans, P. Data Fitting in the Chemical Sciences; Wiley: Chichester, 1992. 16. Excel 97; Microsoft: Redmond, WA, 1997. 17. EasyPlot for Windows, version 2.2-0, release 1; Spiral Software & MIT, 1993. 18. SURF, version 4.10; Program for MS-DOS operating system; Golden Software Inc.: Golden, CO, 1989. 19. Izatt, R. M.; Pawlak, K.; Bradshaw, J. S.; Bruening, R. L. Chem. Rev. 1995, 95, 2529–2586, and reviews therein cited. 20. Smith, R. M.; Martell, A. E. NIST Critical Stability Constants Database, version 2; National Institute of Standards and Technology: Washington, DC, 1995. 21. Pettit, L. D. IUPAC Stability Constants Database; Academic Software: Otley, UK, 1993.


The Use of Calculated Species Distribution Diagrams to Analyze

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