Evaluating Solubility of Sparingly Soluble Diprotic Acid Salts

In the Classroom

Evaluating Solubility of Sparingly Soluble Diprotic Acid Salts Jean M. Nigretto UFR Sciences et Techniques, Université de Cergy-Pontoise, 5 Mail Gay-Lussac – Neuville sur Oise, 95031 Cergy-Pontoise Cedex, France; [email protected]

The use of logarithmic concentration diagrams to solve problems in ionic equilibria has been explained in a number of textbooks since it was first introduced by Sillen (1). Such representations embody any kind of dissociation reaction, including acid–base reactions, progressive soluble- or insolublecomplex-forming reactions, and redox reactions. The convenience of logarithmic concentration diagrams lies in the ease with which they are constructed and used in approximating numerical solutions of ionic equilibria (2–4 ) or acid–base titration curves (5–7 ). In teaching analytical chemistry to sophomores who had only an average exposure to chemical equilibria as freshmen, precipitation reactions are frequently a pitfall due to an apparent difficulty in dealing with the presence of a new phase in the solution. This is especially true when acid base interactions affect precipitation equilibria. The cation is a Brønsted or Lewis acid and the anion may also be a Brønsted or Lewis base! Most common is the reaction of the anion with water to produce the conjugate acidic forms. When the solution pH is unknown or when it is altered by the dissolution of the precipitate, mathematical evaluation of the solubility is generally felt to be arduous because the knowledge of equilibrium constants involves solving polynomial equations of nth degree in [H+] unless assumptions are made to reduce the difficulty (8). This can be done, for instance, in predicting the extent of the interaction of the salt with water. However, the relative importance of one or more species involved in the mass and charge balance equations depends on this pH. Recent graphing calculator strategies (9, 10) and computations by means of mathematical programing techniques (11) have reduced the difficulty of finding rigorous solutions to these equations but have not necessarily improved comprehension of the analytical concepts. Evaluation of the solubility of salts by using spreadsheet techniques has been proposed (12). However, time and technology to make use of PC-based computational software are sometimes lacking in the chemistry classroom. The purpose of this paper is to reinforce understanding of solubility concepts and give students an insight into the

factors that affect the solid/liquid equilibria. The application described is an obvious extension of the traditional log C diagram method to a situation that has not been presented so far in textbooks. Construction of log S vs pH Diagrams Let us consider a series of salts termed ML, in which L designates a diprotic acid base and M a divalent cation that is assumed to be devoid of any acidic property. This case is conveniently illustrated by carbonate salts, for which the successive acid–base dissociation constants of the acidic form H2CO3 are pKa = 6 and pKa = 10. In fact, data reported for the solubility products of usual carbonates are limited to about pKsp = 13. In the following examples, the pK values of magnesium, calcium, and lead carbonates are 4, 8 and 13, respectively. To facilitate graphical readings, these data have been approximated to integers. Precise data are listed in traditional textbooks (1, 2). An additional illustration of the method is the case of a weaker diprotic acid salt, termed MxLx, provided with pKa = 10, pK a = 13, and pKsp = 15. The constants mentioned constants refer to the following equilibria: 1

2

1

2

ML(s) 2


M2+ + L2


L + H 3O


(Ksp)


+

H2O + HL +

HL + H3O

H 2O + H 2L

(1)

(1/Ka )

(2)

2

(1/Ka )

(3)

1

2+

The solubility of the cationic forms is SM = [M ] and that of the anionic forms is S L = [H2L] + [HL
] + [L2-]. The solubility of non-dissociated ML forms is neglected. According to eq 1, one obtains: S = S M = S L = [M2+] = [L2
] + [HL
] + [H2L]

(4)

Combination with the acid–base constants gives: S = [L2
]关1 + ([H+]/Ka ) + ([H+]2/Ka Ka )兴 2

+

1

+

2

(5)

2

where [H ] represents [H3O ]. Since Ksp = S , the solubility of the salt is linked with pH by S = {Ksp关1 + ([H+]/Ka ) + ([H+]2/Ka K a )兴 }1/2 2

1

2

JChemEd.chem.wisc.edu • Vol. 78 No. 5 May 2001 • Journal of Chemical Education

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0

0

-2

-2

-4

-4

-6

-6

log S

log S

In the Classroom

-8

-8

-10

-10

-12

-12

-14

-14

0

2

4

6

8

10

12

14

0

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6

pH Figure 1. Logarithmic solubility diagram of MgCO3 (p Ksp = 4) taking the total concentration in metal Ctot = 0.1 M. Dissociation constants of H2CO3 are approximated to pKa1 = 6 and pKa2 = 10. Dashed lines refer to non-predominating species, bold line to solubility and predominating forms. (䉬) [H2CO3]; (䉱) [HCO3
]; (䊉) [CO32
]; (䊏) [OH
].

In eq 6, the terms between brackets may reduce to one or two depending on the values of [H+] relative to Ka or Ka . All but the first term are negligible if pH > pK a (conventionally if pH > pKa + 1) or [L2
] >> [HL
] >> [H2L]; the second if pKa – 1 < pH < pKa + 1 or [HL
] >> [H2L] or [L2
]; or the third if pH < pKa – 1 or [H2L] >> [HL
] >> [L2
]. The pH-dependent variation of [L2
] is then obtained by a combination of eqs 4–6 1

2

2

2

1

2

1

[L2
] = {Ksp1/2}/[1 + ([H+]/Ka ) + ([H+]2/Ka Ka )]1/ 2 (7) 2

1

2

wherefrom those of [HL
] and [H2L] derive using the dissociation constants Ka and Ka : 1




HL =

2

2

1

1 + K a / H+ + H+ /K a 2




H2L =

2

2

1

1 + K a / H+ + K a K a H+ 1

1

(8)

1

K sp 1 + H+ /K a + H+ 2/K a K a

1/2

2

(9)

2

2

2

632

2

1

12

14

Figure 2. Logarithmic solubility diagram of CaCO3 (pKsp = 8) taking Ctot = 0.1 M. (䉬) [H2CO3]; (䉱) [HCO3
]; (䊉) [CO32
]; (䊏) [OH
].

becomes smaller than log Ctot . In Figures 2–4, this occurs at pH = 2.5 for PbCO3 or 5 for CaCO3 and MxLx. Since these values are smaller than pKa , the predominant species should be M2+ and H2L. In the case of MgCO3 (Fig. 1), the departure from the horizontal line starts at pH 8; that is, in the pH range in which the predominating free anionic form is now HCO3
. Thereafter, when pH > pKa , the solubility reaches a minimum because S = (Ksp)1/2; that is, 10
2, 10
4, 10
6.5, and 10
7.5 for MgCO3, CaCO3, PbCO3, and MxLx, respectively. 1

2

Graphical Resolutions The charge balance applied to a saturated solution of ML gives (10)

The final pH is necessarily ≥7 owing to the basic behavior of L. In addition, we shall assume that enough anion dissolves to allow the pH to rise above 8 owing to the reactions L2
+ H 2O HL
+ OH
or even L2
+ 2H2O H2L + 2OH
. Finally, subtracting eq 10 from eq 4 to eliminate the concentration of the cation leads to [OH
] = 2[H2L] + [HL
] + [H+]

The slopes of the corresponding logarithmic straight lines vs pH in the three pH domains (pH < pKa , pKa < pH < pKa , and pH > pKa ) are deduced from these equations. One obtains, respectively, (+1), (+1/2), and (0) for log L; (0), (
1/2), and (
1) for log [HL]; and (
1), (
3/2), and (
2) for log H2L. Variations of log S vs pH are plotted in Figures 1–4, using a given total concentration of the cation Ctot = 0.1 M. In acidic medium of pH < 2, extensive hydrolysis of the anion occurs, shifting the equilibria (eqs 1–3) to the right. This displacement tends to remove the anion of the salt, increasing the solubility over the value calculated from the solubility product alone. H2L is then the only anion-containing form that coexists with M2+, both at initial concentration 0.1 M. In buffers of increasing pH, precipitation begins when log S 1

10

[H+] + 2[M2+] = [OH
] + 2[L2
] + [HL
]

1/2

K sp 1 + H+ /K a + H+ 2/K a K a

8

pH

(11)

Relative to other species, the lower ordinates of the log[H] straight line in the diagrams confirms that [H+] can be neglected in this equation. Equation 11 may also be obtained directly by considering the stoichiometry of the possible reactions with water: ML(s) + H2O


HL + H2O

M2+ + HL
+ OH



OH + L

2


(12) (13)

In the numerical resolution of this problem, a fourth power equation in [H+] will be obtained because there are three coexisting equilibria in the solution (eqs 1, 12, 13). Unless simplifying hypotheses are made to evaluate the extent of these reactions, calculations will be rather tedious. The use

Journal of Chemical Education • Vol. 78 No. 5 May 2001 • JChemEd.chem.wisc.edu

0

0

-2

-2

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-4

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log S

log S

In the Classroom

-8

-8

-10

-10

-12

-12

-14

-14 0

2

4

6

8

10

12

14

0

2

4

6

8

10

12

14

pH

pH Figure 3. Logarithmic solubility diagram of PbCO3 (pKsp = 13) taking Ctot = 0.1 M. (䉬) [H2CO3]; (䉱) [HCO3
]; (䊉) [CO32
]; (䊏) [OH
].

Figure 4. Logarithmic solubility diagram of a weak diprotic acid salt MxLx, Ctot = 0.1 M, provided with pKsp = 15, pKa1 = 10, and pKa2 = 13. (䉬) [H2 L]; (䉱) [HL
]; (䊉) [L2
]; (䊏) [OH
]; (ⵧ) 2[H2 L].

of a graphical method proves most convenient because the answer is simply given by the pH at which the log[OH] vs pH line crosses either the log[HL] or the log[2H2L] vs pH line, or both, in critical situations. According to the pH domain in which the intersection falls, the nature of the predominant species is known and the determination of its concentration is graphically straightforward. MgCO3 In the case of MgCO3 (Fig. 1), the abscissa of the intercept falls in the range pH > pKa . HCO3
is the dominant term in eq 11, as this species no longer predominates at this pH. Equation 11 reduces to [OH
] ≈ [HCO3
]. The graphical reading clearly gives pH = 11 (11), S = [Mg 2+] = 10
2 (10
1.98) mol/L. From the less concentrated species, [HCO3
] = 10
3 (10
3.02 ) mol/L and [H2CO3] = 10
8 (10
8.02 ) mol/L, the concentration of the predominant species refines to [CO32
] = 10
2 – 10
3 – 10
8 = 10
2.05 (10
2.02 ) mol/L (calculated values are in italics). Hence, the amount of solid [MgCO3](s) is 10
1 – 10
2 mol/L. CaCO3 In the case of CaCO3 (Fig. 2), pH ≈ 10 (9.91) and S = [Ca2+] = 10
3.85 (10
3.82) mol/L; [H2CO3] = 10
8.15 (10
8) mol/L, whereby [CO32
] = [HCO3
] = 10
4.15 (respectively 10
4.17 and 10
4.09) mol/L and [CaCO3](s) = 10
1 – 10
3.85 mol/L. PbCO3 In the case of PbCO3 (Fig. 3), pH = 8.33 (8.33) and S = [Pb2+] = 10
5.65 (10
5.66) mol/L; [H2CO3] = 10
8 (10
8) mol/L; [CO32] = 10
7.35 (10
7.34) mol/L; whereby [HCO3
] = 10
5.65 – 10
7.35 – 10
8 = 10
5.66 (10
5.66) mol/L and [PbCO3](s) = 10
1 – 10
5.65 mol/L. MxLx In the case of MxLx for which pKa = 10, pKa = 13, and pKsp = 15 (Fig. 4), eq 11 reduces to [OH
] ≈ 2[H2L]. The log[OH
] line crosses the (log[H2L] + log 2) line at pH = 9.15 (9.136), giving log S =
5.15 (5.15) when the log[OH
]

line crosses the solubility line. Hence [M2+] = 10
5.15 (10
5.15) mol/L; [HL
] = 10
6 (10
6.03) mol/L; [L2
] = 10
9.85 (10
9.88 ) mol/L, whereby [H 2L] = 10
5.15 – 10
6 – 10
10 = 10
5.22 (10
5.16) mol/L and [ML](s) = 10
1 – 10
5.15 mol/L.

2

1

2

Conclusion One of the solubility diagrams represented in Figures 1– 4 can be used for any usual diprotic salt by simply changing the data for the dissociation constants and solubility product constants. This can be done by appropriate shifts that affect the ordinates and the abscissas, but the successive slopes should remain unchanged. Clearly, there is more labor involved in setting up solubility diagrams than in their use. But once established, students can feel confident that they have complete and explicit awareness of the basic aspects sustained by simple precipitation phenomena. Literature Cited 1. Sillen, L. B. In Treatise on Analytical Chemistry, Part I, Vol. 1; Kolthoff, I. M.; Elving, P. J., Eds; Wiley: New York, 1959; Chapter 8. 2. Freiser, H. Concepts & Calculations in Analytical Chemistry: A Spreadsheet Approach; CRC Press: Boca Raton, FL, 1992. 3. Skoog, D. A.; West, D. M.; Holler, F. J. Analytical Chemistry, 6th ed.; Saunders: Fort Worth, 1992. 4. Slade, P. W.; Rayner-Canham, G. W. J. Chem. Educ. 1990, 67, 316. 5. Tabbutt, F. D. J. Chem. Educ. 1966, 43, 35. 6. Yingst, A. J. Chem. Educ. 1967, 44, 601. 7. Waser, J. J. Chem. Educ. 1967, 44, 274. 8. Cavaleiro, A. M. J. Chem. Educ. 1996, 73, 423–425. 9. Mertz, C. J. Chem. Educ. 1989, 67, A241. 10. Donato, H. Jr. J. Chem. Educ. 1999, 76, 632–634. 11. Gordus, A. A. J. Chem. Educ. 1991, 68, 291. 12. Guiñón, J. L.; Garcia-Antón, J; Pérez-Herranz, V. J. Chem. Educ. 1999, 76, 1157–1160.

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