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Steven D. Gammon University of Idaho Moscow, ID 83844
Spreadsheet Techniques for Evaluating the Solubility of Sparingly Soluble Salts of Weak Acids José L. Guiñón, José García-Antón, and Valentín Pérez-Herranz Departamento de Ingeniería Química y Nuclear, E.T.S.I. Industriales Universidad Politécnica de Valencia, P.O. Box 22012, E-46071 Valencia, Spain
Evaluation of the solubility of salts usually involves solving polynomial equations of degree higher than two, because of the corresponding equilibrium constant expressions and the mass balance and charge balance equations (1, 2). This requires the method of successive approximations, which is tedious and time-consuming. The calculation of solubility of weak acid salts by simplifying assumptions has been reported in this Journal (3–5). With some practice, the equations may be solved using suitable assumptions about the relative importance of one or more species involved in the mass balance and charge balance equations. However in complicated equilibria it is not always obvious which species can be neglected in these equations. The computation of solubility by means of various mathematical programs and algorithms has been proposed (6–8). However, these computing programs can present some difficulties. An initial estimate for the concentration of species and one internal variable of tolerance are required to solve the system of equations. When the solver cannot find solutions to the equations, another set of initial estimates is necessary. Moreover, the solver output for a particular system does not always perform an automatic check. These computations were used only to determine the water solubility of diprotic acid salts, not taking into account other conditions such as the effect of hydrolysis of the cation or the presence of a strong acid in the solution. In this paper we report the use of the program Microsoft Excel (version 5.0) for Windows on a PC-compatible computer for the determining the solubility of sparingly soluble salts of weak acids under different conditions. Nowadays, spreadsheets are widely used and the method described here could be applicable to any common spreadsheet program. In addition, spreadsheets have good graphical tools, making it very easy to prepare the graphical representations. Equilibrium Equations and General Calculation If the solubility of a sparingly soluble salt M m A n is expressed as the molar concentration of a saturated solution, S, the equilibrium concentration of the individual species taking part in the precipitation equilibrium M m A n(s) mM + nA (1) can be expressed by the terms [M]T = mS and [A]T = nS, where the charges of the ions M and A are omitted for simplicity.
At a specified temperature the solubility product is Ksp = [M]m[A]n (2) We consider that the cation of the salt can react with the molecules of water to form hydroxocomplexes of general composition M(OH)i, and the anion may also undergo acid– base reactions to produce species of composition AHj, where i and j are integers ranging from 0 to i and j, respectively. The solubility, S, from the mass balance is
S=
[M]T [A]T = m n
1 [M] + ∑ M OH S= m
i
(3)
= n1 [A] + ∑[AHj ]
(4)
We assume that the formation constants of the hydrolysis species of M and A are known: M(OH)i + iH+
M + iH2O +
A + jH
AH j
Ki
(5)
Kj
(6)
where Kj is the acidity constant of the base, for example, in a triprotic acid,
1 K1 = 1 , K 2 = 1 , K 3 = ,… Ka K a Ka Ka Ka Ka 3
3
2
3
2
(7)
1
K a j being the acidity constants of the acid AH j. Then, the solubility can be expressed as a function of [H+ ]
Ki [M] S= m 1+∑ + H
i
[A] + = n 1 + ∑ Kj H
j
(8)
By substituting the values of [M] and [A] in the solubility product expression, eq 2, we obtain m+n
m mn n S
= K sp 1 + ∑
Ki + i
H
m
n + j
1 + ∑ Kj H
(9)
Equation 9 allows one to calculate the solubility of a sparingly soluble salt as a function of the pH of the solution fixed by a buffer system.
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To determine the salt’s solubility in a system at equilibrium without pH control, we should include the proton condition resulting from the mass balance and change balance equations. In the presence of a strong monoprotic acid of concentration C, the proton condition will be ∑ j[AH j] + [H+] = [OH{] + C + ∑i [M(OH)i]
(10)
By substituting [AH j] and [M(OH)i ] in function of [A] and [M] respectively, we obtain + j
A ∑j Kj H
Ki
{
+ H = OH + C + M ∑ i +
(11)
+ i
H
The values of [A] and [M] from eq 8 are
A =
nS
(12)
+ j
1 + ∑K j H
M =
mS 1+∑
(13)
Ki + i
H
Finally, by substituting these values of [A] and [M] into eq 11, we obtain + {1
+ j
{
+
S = OH – H + C
n ∑ jK j H
+ j
1 + ∑ Kj H
–
m ∑ iK i H
+ {1
1 + ∑ Ki H
{1
(14)
The value of pH that fulfills eqs 9 and 14 simultaneously solves the problem of solubility. It would be possible to substitute S from eq 14 into eq 9 to obtain an equation of high degree in [H+]; this gives the pH in a saturated solution. A much easier method is to obtain a table of values by using the two functions log S vs pH resulting from eqs 9 and 14. The solubility will be obtained from the pH that shows the same value in both log S functions. Alternatively, the solubility can be obtained by the crossing point in the graphical representation of the two functions log S versus pH. The above procedure is easily accomplished using the spreadsheet Microsoft Excel. Practical examples of calculations will be given and discussed in the following sections. Once the solubility of a salt from a given kind of acid is obtained, the worksheet can be used for other salts by simply changing the values of the solubility product and the ionization constants.
the cation and the anion (ferrous sulfide). The third corresponds to the solubility of a salt with extensive hydrolysis of anion (barium carbonate) in the presence of a strong acid. The final example shows the precipitation of a double salt, NH4MgPO4.
Solubility of Barium Oxalate Figure 1 shows the spreadsheet for calculating the solubility of barium oxalate. In this case m = n = 1, j = 2, and, as the hydrolysis of cation is not taken into account, i = 0. The second column shows the values of log S as a function of pH, obtained from eq 9. log S = { 1 ⁄2 pKsp + 1⁄ 2 log(1 + 10 pKa –pH + 10pKa +pKa –2 pH) (15) 2
{
+
log S = log OH – H
2
– (16)
10
pK a 2–pH
1 + 10
All the examples presented are for solutions so dilute that it may be assumed that the ionic strength is zero without substantial error. Hence it is reasonable to use the thermodynamic equilibrium constant, and users must look up or calculate the equilibrium constant for the ionic strength of any real-world example. Four examples are considered, depending upon the behavior of the precipitate in contact with water. The first corresponds to limited hydrolysis of the anion (barium oxalate). The second shows the effect of the hydrolysis of both
1
The third column gives the values of log S obtained from eq 14.
log
Results
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Figure 1. Spreadsheet for calculating the solubility of barium oxalate.
pK a +pK a 2–2pH
+ 2 × 10
pK a 2–pH
1
pK a +pK a 2–2pH
+ 10
1
Note that for pH ≤ 7, the cells are unfilled because the function log([OH{] – [H+]) corresponds to the logarithm of a negative number, which has no physical meaning. One of the most important aspects of the spreadsheet is its efficiency in providing graphical representations of the results. Thus, Figure 1 also shows the plot of log S from eqs 15 and 16. Expression 9 (or 15) has the simple physical interpretation of being the solubility of barium oxalate in a solution of fixed pH. Hence, the solubility of barium oxalate as a function of pH decreases continuously until pH is close
Journal of Chemical Education • Vol. 76 No. 8 August 1999 • JChemEd.chem.wisc.edu
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Figure 2. Spreadsheet for calculating the solubility of ferrous sulfide.
Figure 3. Spreadsheet for calculating the solubility of barium carbonate.
to 6, and then it remains constant. The intersection of the log S curves gives the solubility and pH values of the saturated solution of barium oxalate in water. With the spreadsheet it is very easy and rapid to find the pH value for which the two functions log S, eqs 15 and 16, have the same value. In this example, pH = 7.29 and log S = {3.82, S= 1.51 × 10{4 M, in agreement with the values obtained by other authors (6 ) (pH = 7.30 and S = 1.50 × 10{4 M).
point which gives the pH and the solubility of the saturated solution. In water, C = 0, pH = 9.96, and the solubility is log ~3.89, S = 1.29 × 10{4 M, values in agreement with the literature (6 ) (pH = 10.0, S = 1.30 × 10{4 M). For different values of C the spreadsheet gives the solubility and pH of the saturation points.
Solubility of Ferrous Sulfide Figure 2 shows the spreadsheet for FeS. The results are similar to those of the previous example, but because the hydrolysis of Fe2+ is considered, i = 3. The second column, which presents the solubility of FeS as a function of pH, reaches a minimum at pH ~10. In the third column, which presents log S from eq 14, the solver gives cells labeled as #NUM! at pH > 11 because the term of hydrolysis of the cation (Fe2+) is higher than the term of the hydrolysis of the anion (sulfide), which results in negative values. Analogously to above example, the water solubility of FeS is obtained by the crossing point of log S curves. Here, we obtain pH = 8.57 and log S = {5.40, S = 3.98 × 10{6 M, values slightly different from those given in literature (6 ) (pH = 8.6 and S = 3.7 × 10{6 M,) because of having taken into account the hydrolysis of Fe2+. Solubility of Barium Carbonate Figure 3 shows the spreadsheet for BaCO3 in the presence of a strong acid at several values of concentration, C. The second column gives the solubility of BaCO 3 in a solution of a given acidity. The third through fifth columns give the log S from eq 14 for different values of the concentration of strong acid. The curves intersect with the log S from eq 9 at a
Solubility of MgNH 4PO4 This compound has interest in the separation by precipitation of magnesium cation and in the geological formation of phosphates from natural waters (9). In this case the salt corresponds to the general expression M mP p An , with m = p = n = 1, i = 1, and j = 3. The spreadsheet is shown in Figure 4. The solubility in water is S = 6.02 × 10{4 M (log S = {3.22) and pH = 9.97. The solubility reaches a minimum value at pH ~ 10.7, in agreement with the values reported by Stumm (9). Thus, the precipitation of MgNH 4PO4 is favored in alkaline solutions. Conclusion A Microsoft Excel spreadsheet for determining the solubility of sparingly soluble salts is described. The main interest of this spreadsheet is the efficiency and speed with which the repetition of mathematical expressions and the graphical representations are completed. It takes only a couple of minutes to calculate the pH and solubility of a saturated solution of a specified salt under a specified set of conditions. The chart and worksheet are shown simultaneously on the screen. The worksheet can be used for any salt by simply changing the data for the solubility product constant, dissociation constants of acids, and formation constants of hydroxocomplexes. The results obtained are in agreement with those found in the literature.
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Literature Cited 1. Skoog, D. A.; West, D. M.; Holler; F. J. Analytical Chemistry, 6th ed.; Saunders: Fort Worth, 1992. 2. Harris, D. C. Quantitative Chemical Analysis, 3rd ed.; Freeman: New York, 1991. 3. Cavaleiro, A. M. J. Chem. Educ. 1996, 73, 423–425. 4. Lagier, C.; Olivieri, A. J. Chem. Educ. 1990, 67, 934–936. 5. Cardinali, M. A.; Giomini, G.; Marrosu, G. J. Chem. Educ. 1990, 67, 221–223. 6. De Roo, S.; Vermeire, L.; Gorler-Wairand, C. J. Chem. Educ. 1995, 72, 419–422. 7. Giomini, G.; Marrosu, G.; Cardinali, M. A. Bull. Soc. Chim. Belg. 1995, 104, 691. 8. Holler; F. J. Mathcad ® Applications for Analytical Chemistry; Saunders: Fort Worth, 1994. 9. Stumm, W. S.; Morgan, J. J. Aquatic Chemistry, 2nd ed.; Wiley: New York, 1981.
Figure 4. Spreadsheet for calculating the solubility of MgNH4PO4.
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