COMPUTATION OF AQUEOUS METAL SOLUBILITIES USING SPREADSHEET PROGRAMS
David A. Vaccari, Ph.D., P.E.
Dept. of Civil, Environmental
and Coastal Engineering
Stevens Institute of Technology
Hoboken, NJ 07030
NOTE:
This software is given free for copying, provided that the file METALS.TXT is included with each copy, and that the author's name and references not be removed from these versions nor from any versions derived from them. No warranty is expressed or implied as to the accuracy of the software. It is up to each user to validate the results obtained with these programs.
AQUEOUS METAL SOLUBILITY SPREADSHEETS
This software is for computing the aqueous solubilities of several metals, as well as other water quality parameters such as Langelier stability index, hardness, and carbonate system concentrations. They require Excel software or equivalent to run. Shift-Click on each filename to download it:
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This file, with instructions for use of the spreadsheets |
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Spreadsheet program for lead solubility |
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Copper solubility spreadsheet program |
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Ferrous and ferric iron solubility program |
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Computes solubility of calcium and magnesium; and hardness and Langelier Stability index |
Lead Solubility
Oxidized lead is present in water in the +2 valence state. This is capable of forming precipitates of lead hydroxide, Pb(OH)2, lead carbonate, PbCO3 (cerrusite), or basic lead carbonate, Pb3(OH)2(CO3)2 (hydrocerrusite). The solubility of each of these phases must be computed separately. The solid which produces the lowest solubility under given conditions will be the only stable solid phase in that instance.
The conditions which define the system are, fundamentally, the concentrations of the counterions in solution -- hydroxide and carbonate -- and the ionic strength, which affects the activity coefficients of the ions. These can be related to more directly measured parameters: pH, alkalinity, and total dissolved solids. When one of the lead salts dissolves, it releases Pb++ ions. That concentration can be computed using the solubility product constant, Ksp. The Ksp values for lead hydroxide, lead carbonate, and basic lead carbonate used in this study are 2.95e-7, 9.38e-10, and 8.48e-11, respectively. All equilibrium constants for lead used in this study are from Schock and Gardels [1983].
The Pb++ ions also enter into equilibrium reactions with hydroxide and carbonate ions in solution to form a number of stable complex ions. Table 1 gives the complex ions used in the solubility calculations here.
TABLE 1. Soluble lead complexes and formation constants [Schock & Gardels, 1983].
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PbOH+ |
5.89e6 |
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Pb(OH)2 |
1.17e11 |
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Pb(OH)3+ |
7.94e13 |
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Pb2OH+++ |
4.27e7 |
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Pb3(OH)4++ |
1.26e32 |
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Pb4(OH)4++++ |
1.23e35 |
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Pb6(OH)8++++ |
2.04e68 |
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PbCO3 |
1.26e7 |
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Pb(CO3)2-- |
2.14e10 |
The equilibrium for the reaction of each of these species with Pb++ can be expressed by an equilibrium constant called the formation constant (beta). The formation constants can be computed from the standard free energy of formation of each species.
Solubilities of Copper, Iron, Calcium and Magnesium
Similar relationships can be given for other metals, such as copper [Patterson] and iron, calcium and magnesium [Snoeyink and Jenkins]. Spreadsheet programs for computing their solubilities are also given.
The calculations for copper also include complex ion formation with carbonate as a ligand. However, the references used did not propose any stable carbonate complexes for iron. Therefore, while pH has a strong effect on solubility of all the metals, alkalinity would be expected to affect lead and copper, but not iron.
Spreadsheet Programs to Compute Metal Solubilities
The solubility calculations for lead, copper, iron, and hardness (calcium and magnesium) have been incorporated into several spreadsheet programs which are included. The first program is LEAD.XLS. Use of the program is described here, and assumes basic familiarity with the use of the spreadsheet program. The programs COPPER.XLS and IRON.XLS are also included for computation of the solubilities of copper and iron, respectively. The operation of these programs is similar to LEAD.XLS. The program for iron computes the solubilities of both ferrous (II) and ferric (III) iron.
The file HARDNESS.XLS is a spreadsheet program to compute the solubility of calcium and magnesium, and the Langelier Stability Index (LSI). The operation of this program is similar to LEAD.XLS, except as follows: Temperature values can be input and will influence the calcite solubility product [Stumm & Morgan]. Measured values for calcium and magnesium can be input; these are used to compute total hardness as calcium carbonate, and the measured calcium concentration is used to compute the pH of saturation and the LSI.
Only three inputs are required. They are the pH in cell B9 (see Table 2), the alkalinity in milligrams per liter as calcium carbonate in cell B10, and the total dissolved solids in cell B11. The lead solubility under those conditions is automatically computed, with the result appearing in cell B13, in units of micrograms per liter.
Also displayed are the activities of hydrogen, hydroxyl, carbonate, and bicarbonate ions, the concentration (moles per liter) of undissociated H2CO3, the total inorganic carbon concentration, CT (moles per liter), and the acidity, as mg calcium carbonate per liter. The values for species enclosed by parentheses are the activities, whereas the species enclosed by square brackets are given in molar concentrations. The program is useful for examining the relationships among these variables, as well as for solubility computations.
Rows 23 to 45 contain intermediate values used in the computation, including the Ksp values, formation constants, and concentrations of each complex ion assuming each of the possible solid phases. The lead solubility displayed is the lowest solubility produced by any of the three solids: fresh lead hydroxide, lead carbonate, or basic lead carbonate. The solubility of aged lead hydroxide is computed, but not used in the calculation. It is interesting that to compute lead solubility from only three independent variables, over 50 intermediate values are required.
Also included are the activity coefficients (gammas), the alpha values (which give the fraction of total inorganic carbon (CT) composed of each of the forms of carbonate in solution), and the ionic strength. This last value is estimated from the total dissolved solids input at the top of the spreadsheet, but the user can enter it directly if the sheet us first unprotected. A TDS value of 400 mg/L corresponds to an ionic strength of 0.010. The ionic strength is then used via the Davies equation to compute the activity coefficients of the ions [Snoeyink & Jenkins]. It can be seen by experimentation that TDS has a significant effect on lead solubility.
REFERENCES
Patterson, J., "Effect of Carbonate Ion on Precipitation Treatment of Cadmium, Copper, Lead, and Zinc", 30th Purdue Ind. Waste Conf. May 1975.
Schock, M.R. "Response of Lead Solubility to Dissolved Carbonate in Drinking Water", JAWWA, 72, 12, 695 (Dec 1980).
Schock, M.R. and M.C. Gardels, "Plumbosolvency reduction by high pH and low carbonate -- solubility relationships", JAWWA (February 1983).
"Aquatic Chemistry", W. Stumm and J. J. Morgan, Wiley (1970).
Snoeyink, V.L., and D. Jenkins, "Water Chemistry", John Wiley & Sons (1980).
TABLE 2.
Spreadsheet program for computing lead solubility.
A B C D E
1
2 CALCULATION OF
LEAD SOLUBILITY
3 (based on Schock, JAWWA, 75(2), 1987)
4 David A. Vaccari, Ph.D., P.E.
5 Dept. of Civil, Environmental, and Ocean Engg.
6 Stevens Institute of Technology
7 Hoboken, NJ 07030 January 1990
8
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9 pH (units) 8.5
10 Alk (mg CaCO3/L) 150.0
11 TDS (mg/L) 500
12
13 Pb (ug/L): 85.57
14
15 Acidity (mgCaCO3/L) 147.2
16 (H+) 3.16E-09
17 (OH-) 3.16E-06
18 (CO3=) 1.36E-05
19 (HCO3-) 1.30E-03
20 [H2CO3*] 1.03E-05
21 [CT] 1.49E-03
22
23 SPECIES Fresh Aged Carbonate Basic
24 [Pb++] 1.85E-04 2.23E-09 8.97E-09 3.27E-09
25 [PbOH+] 2.45E-03 2.95E-08 1.19E-07 4.33E-08
26 [Pb(OH)2] 1.36E-04 1.64E-09 6.59E-09 2.40E-09
27 [Pb(OH)3-] 3.31E-07 3.98E-12 1.60E-11 5.84E-12
28 [Pb2OH+++] 5.18E-06 7.49E-16 1.21E-14 1.61E-15
29 [Pb3(OH)4++] 3.22E-02 5.59E-17 3.65E-15 1.77E-16
30 [Pb4(OH)4++++] 1.45E-02 3.02E-22 7.95E-20 1.40E-21
31 [Pb6(OH)8++++] 3.31E+01 1.00E-28 4.27E-25 1.00E-27
32 [PbCO3] 1.99E-02 2.39E-07 9.63E-07 3.51E-07
33 [Pb(CO3)2--] 7.37E-04 8.86E-09 3.57E-08 1.30E-08
34 log[Pb total] 2.30 -6.55 -5.95 -6.38
35 Pb (mg/l) 4.12E+07 0.058 0.235 0.086
36 I min calc'd 265.0721 0.00076 0.00076 0.00076
37 (Pb++) 1.17E-04 1.41E-09 5.69E-09 2.07E-09
38
39 Gamma 0 = 1.014 Alpha 0 = 0.0069
40 Gamma 1 = 0.893 Alpha 1 = 0.9786
41 Gamma 2 = 0.635 Alpha 2 = 0.0145
42 Gamma 3 = 0.359
43 Gamma 4 = 0.162
44 I = 0.0125
45 Davies = -0.049