The next level of complexity relative to modeling metal–ligand complexes in aqueous solutions is examination of a system in which we have a significant complexing ligand, along with hydroxide. In general, we would write mole balances to account for the entire set of metal species and to account for the entire set of ligand species.
Certainly, many of the metal species will appear in the ligand balance and vice versa.
Example 11.3 Examine an aqueous solution containing 10–4 M zinc and 10–4 M total phosphate. Produce a plot, similar to that of Example 11.2 depicting the abundances of the various zinc and phosphate species. Consider that the background ionic strength is 0.01 M and is not significantly affected by changes in metal or phosphate speciation.
We retain the applicable portions of the work completed for Example 11.2, include the formation constants for the phosphate ligand and include the effects of ionic strength:
We write mole balances for zinc and phosphate:
When we write the two equations using the two master dependent variables of {Zn+2} and {PO4–3}, we very quickly realize that the two relations are coupled and we can solve neither independently. We therefore arrange a given-find solve block. We write the solve block into a function, that allows the manipulation of the proton activity as one of the arguments along with the activities of free zinc and phosphate:
In order to obtain graphical output, we write our results into several parallel vectors.
The zeroth element of each vector is computed by solving the originally stated solve block for the initial pH desired for the range of values:
We write a short program incrementing pH and successively implementing the solve block to obtain vectors of pH, {Zn+2}, and {PO4–3}, shown in Figure E11.3.1.
Then for a pH range of 5–10, we invoke this set of worksheet programming 26 times to populate the Zn and PO4 vectors. Once the master vectors are populated
we may write relations allowing the population of vectors containing the concentra- tions of six species of interest, shown in Figure E11.3.2. Finally, we may use these six vectors to create six additional vectors each containing the 26 additional values of the abundance fraction for each specie, shown in Figure E11.3.3.
Figure e11.3.1 screen capture of a logical program to compute zinc and phosphate ion activities over a selected ph range.
Figure e11.3.2 screen capture of vector computations for zinc phosphate complex speciation.
A plot of the results shown in Figure E11.3.4 allows us to gain an understanding of the speciation of zinc in aqueous solution with phosphate present. We observe that the zinc ion is predominant at low pH while the polynuclear Zn3(PO4)2 complex is predominant in this system at virtually all pH values between 5.5 and 10, the range of interest for most environmental systems. We might think it obvious that the Zn3(PO4)2 complex would also dominate the abundance diagram for the phosphate ligand. To be sure, however, we have computed the abundance fractions in much the same manner as those for the zinc species (not shown) and plotted them in Figure E11.3.5 to assure ourselves of our hunch. Were we to employ Excel to model
Figure e11.3.3 vector computations for zinc complex abundance fractions.
1×10–65 1×10–5 1×10–4 1×10–3 0.01 0.1 1 10
6 7 8
pH
abundance fraction
9 10
Zn
ZnH2PO4 Zn3(PO4)2 ZnHPO4 Zn(OH)3 Zn(OH)2
Figure e11.3.4 a plot of zinc specie abundance fractions versus ph for zinc hydrolysis and zinc phosphate complex species.
this system we would use {Zn+2} as the master dependent variable, write the mole balance for the phosphate ligand, isolating the activity of the base ligand on the LHS. Then we would write the mole balance on zinc using the LHS of the phos- phate mole balance wherever the phosphate ion would appear. We would algebrai- cally zero this function and use the solver to find the value of zinc ion activity that satisfies the zeroed condition. We would necessarily solve the system for each pH value. We could step into the macroenvironment available from Excel and perhaps create a VBA program that would increment pH and successively solve the system and store the results. The tricky part might be the invocation of the solver from the programming environment. Certainly, the “what you see is what you get” visual interface of the MathCAD worksheet seems to this author to be a better choice.
We move away from Example 11.3 with an understanding that we can quantita- tively model virtually any metal–ligand system, given we have values for the cumulative formation constants and for the acid/base equilibria of the ligand. Were additional ligands present in the system, we would of necessity write a mole balance for each additional ligand and include the appropriate terms from the ligand mole balances in the mole balance written for the metal. Regardless of the complexity or the simplicity of each metal–ligand system, we necessarily must solve the mole balance equations simultaneously. Were we of necessity to include a second or third metal, with interactions among the common ligands, we would write mole balances for the additional metals. Solution of systems with two or more metals would be straightforward using MathCAD. Unfortunately, for modeling with Excel, we would
1×10–65 1×10–5 1×10–4 1×10–3 0.01 0.1 1 10
6 7 8
pH
abundance fraction
9 PO4
ZnHPO4 ZnH2PO4 Zn3PO42 H3PO4 H2PO4 HPO4
Figure e11.3.5 a plot of phosphate specie abundance fractions versus ph for the zinc- phosphate complex formation system.
need to identify the activity of each free metal ion as a master dependent variable, and, unfortunately, Excel’s solver can adjust but one variable at a time. Perhaps we would need to implement the solver with a number of target cells equal to the number of metals and manually iterate the solver among the defined target cells. MathCAD is indeed the more convenient of the two computational software packages for these computations.