In the Classroom
The Ksp–Solubility Conundrum Roy W. Clark and Judith M. Bonicamp Department of Chemistry, Middle Tennessee State University, Murfreesboro, TN 37132
General chemistry texts usually devote a portion of one chapter to the solubility product principle (1). This is commonly in the form of Ksp = f (concentrations of ions raised to some powers). For example, Ksp = [Ag+][Cl], where Ksp is the solubility product constant for the slightly soluble substance silver chloride. [Ag+] and [Cl] are the concentrations in moles per liter of the ions that are in equilibrium with the solid phase of AgCl. Tables of Ksp values are given, and the usual two assumptions made are: the substances are strong electrolytes which ionize 100% in solution; and activities of the ions are close enough to the molar concentrations in these dilute solutions to allow defining the Ksp in terms of concentrations. Reasoning from the equations, which presume that the only forward reaction is the production of the expected ions, the student is asked to calculate solubility from Ksp and vice versa, to calculate the common ion effect on the solubility of substances, and to determine if slightly soluble solids will precipitate under certain concentration suppositions. This Journal contained an excellent article in 1966, by Meites, Pode, and Thomas called “Are Solubilities and Solubility Products Related?” (2). The authors showed clearly that, due to ion pair formation, hydrolysis, complex ion formation, and activity coefficient variations, there are only a few cases in which solubility and Ksp are related in a simple way (Fig. 1). Meites et. al. concluded, “It would be better to confine illustrations of the solubility-product principle to 1-1 salts, like silver bromide and thallium iodide, in which the student’s calculations will yield results close enough to the truth to permit him to feel it is worth his trouble to try to master what he is being taught.” Ignoring the masculine pronoun common in the sixties, this advice is excellent. In the majority of cases solubilities are not simply related to Ksp. Unfortunately, the authors of current general chemistry texts have largely ignored this advice and provide students with large Ksp tables that include
numerous slightly soluble substances. Further, many authors imply by example and by assigned exercises that Ksp is a simple function of solubility. In effect they teach (3) that for AxBy(s) → xAz+ + yBz where solubility is So,
Ksp = [xSo]x[ySo]y
which results in the familiar
Ksp = So2 Ksp = 4So3 Ksp = 27So4 Ksp= 108So5
and
It would seem that authors would restrict their Ksp tables to those slightly soluble compounds (Fig. 1, boxed reaction) that do not suffer from the complications mentioned above. An apparently easy solution to this whole problem goes something like this: Why not simply look up the solubilities of all these compounds and check to see if the solubility as computed from the Ksp is the same as the reported experimental solubility? Then we should know the Ksp values to include in our general chemistry texts. One author of a current chemistry text did try to do just this. Umland calculated solubilities from Ksp values and arrived at a table of 25 compounds (Table 1) whose Ksp values yielded solubilities within an order of magnitude of observed solubilities (4). This is admirable progress, but Umland’s table can be improved upon in several respects. We set out to create an improved table based upon either newer or different data. Problems Encountered A variety of Ksp values are found in different sources; which Ksp value do we use? Solubilities are not easily found for extremely insoluble compounds because the determination of their solubilities is not an easy experiment. Various reference texts give only crude values for these solubilities, Table 1. Solubility Product Constants at 25 ⴗC Compound
Ksp
Fe(OH)2
8 × 10 16
AgBr
5 × 10 13
Hg2Br2
6 × 10 23
Hg2Cl2
1.4 × 10 18
Hg2SO4
7 × 10 7
1.6 × 10
3
12
Ag2C2O4
5 × 10
AgI
9 × 10 17
AgSCN Ag2SO4 BaCrO4
Li2CO3
1.1 × 10 3
1.0 × 10
12
Mn(OH)2
1.7 × 10 13
1.0 × 10
5
NiCO3
1.2 × 10 7
PbBr2
1.2 × 10 5
PbCl2
1.5 × 10 5
PbCO3
1.2 × 10 13
SrSO4
3 × 10 7
SrF2
4 × 10 9
8 × 10
11
BaSO4
1.1 × 10
CaCO3
5 × 10 9
CaF2 Ca(OH)2
1182
Compound
1.6 × 10 10
AgC2H3O2
Figure 1. Complications that affect solubility. Boxed reaction represents usual assumption in general chemistry.
Ksp
AgCl
CaSO4
1.4 × 10 5 × 10
6
7 × 10
5
10
10
NOTE: Reproduced from ref 4 by permission.
Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
In the Classroom Table 2. Ksp Variation in Values Listed in 10 Texts ∆ pKsp Compound Av Ksp Compound Av Ksp Ba(OH)2
5.0 × 10 3 12
0
Co(OH)3
6.0 × 10 44
1.2
0
NiCO3
6.8 × 10 8
1.3
0.1
PbCl2
4.1 × 10 5
1.3
Ag2CO3
8.2 × 10
AgCl
1.7 × 10 10
Hg2I2
4.7 × 10
29
0.1
PbSO4
1.6 × 10 8
0.1
SrSO4
3.1 × 10 7
Ag2SO4
1.4 × 10 5
HgS
2.5 × 10
51
1.3
Ag2S
1.1 × 10 48
1.4
0.2
Cr(OH)3
5.5 × 10 30
1.7
0.2
Al(OH)3
4.1 × 10 33
1.8
10
1.9
PbF2
3.5 × 10
8
0.2
CuCO3
1.7 × 10
CaC2O4
3.0 × 10 9
0.2
PbCO3
7.5 × 10 14
AgI
9.8 × 10 17
0.3
Ni(OH)2
3.1 × 10 15
0.3
BaF2
3.7 × 10 6
9
PbI2
9.5 × 10
BaCrO4
1.3 × 10 10
0.4
Mg(OH)2
9.1 × 10 11
8.8 × 10
23
0.4
AgBr
4.6 × 10 13
1.6 × 10
18
0.5
Sn(OH)2
5.7 × 10 27
CaCO3
4.7 × 10
9
0.5
CaF2
6.5 × 10 10
AgCN
1.3 × 10 16
0.6
CdCO3
3.9 × 10 12
7.8 × 10
16
0.6
Fe(OH)2
3.7 × 10 15
1.7 × 10
13
0.7
MgF2
1.1 × 10 8
BaCO3
4.7 × 10
9
0.7
Fe(OH)3
1.6 × 10 37
Cd(OH)2
1.6 × 10 14
0.7
CoCO3
12.0 × 10 11
3.3 × 10
11
0.7
PbCrO4
1.14 × 10 13
2.5 × 10
12
0.7
Zn(OH)2
2.3 × 10 15
Ca(OH)2
5.5 × 10
6
0.8
FeS
1. 2 × 10 16
Co(OH)2
1.0 × 10 15
Hg2Br2 Hg2Cl2
Pb(OH)2 Mn(OH)2
MnCO3 CuI
0.8
PbS
4.4 × 10 26
5.9 × 10
7
0.8
ZnS
5.4 × 10 22
3.7 × 10
5
0.9
CdS
1.1 × 10 25
1.4 × 10
5
0.9
MnS
3.5 × 10 11
Ag2CrO4
3.4 × 10
12
0.9
CaCrO4
5.7 × 10 4
CuBr
1.3 × 10 8
CuCl CaSO4 PbBr2
0.9
SnS
2.2 × 10 24
3.7 × 10
21
1.1
NiS
1.3 × 10 19
1.0 × 10
9
1.1
Ca3(PO4)2
1.5 × 10 26
BaSO4
2.5 × 10
10
1.1
MgCO3
1.4 × 10 5
SrCO3
6.9 × 10 10
1.2
CuS
1.0 × 10 34
11
1.2
CoS Cu(OH)2
ZnCO3
7.7 × 10
∆ pKsp
most of them determined in the 1930s or earlier. For the very insoluble compounds there are no meaningful values to be found (5). Most of the Ksp values cited were thermodynamic Ksp values. This means that a series of experiments had been done sufficient to extrapolate the measured Ksp values back to zero ionic strength. This, of course, means that the reported Ksp should not agree with the experimental solubility in pure water unless the ionic strength, and thus the activity coefficients, in the saturated water solution are taken into consideration.
Solutions to These Problems We took ten general chemistry textbooks (3, 2.0 4, 6–13) and tabulated the Ksp values in their tables. 2.1 In the majority of these texts, the sources of the val2.1 ues were not stated (and in an eleventh (14) the concept and calculations of Ksp were absent). There 2.2 were 148 compounds listed in the collection, so we 2.2 limited the compounds considered to those which 2.2 appeared in at least five of the texts. This resulted 2.5 in 67 compounds and their supposed Ksp values 2.5 (Table 2). The variation was startling. The last col2.7 umn shows the delta pKsp. There are order-of-magnitude differences in the values for a given com2.8 pound from book to book. We then looked at sev2.9 eral quantitative analysis texts and found that two 3.1 (15, 16 ) had tables based on Martell and Smith’s 3.2 Critical Stability Constants (17 ). Our solution was 3.3 to adopt the Ksp values from the table in the Daniel 3.6 Harris quant text (15) because this table has the advantage of being recent and also of stating the 3.7 ionic strengths of solutions for which the Ksp is re4.4 ported. In most cases these are for zero ionic 4.7 strength and are therefore thermodynamic Ksp val4.9 ues. We shall symbolize thermodynamic Ksp values 5.1 as Kspt and concentration Ksp values as Kspc. For the experimental solubility values we 5.4 searched six sources: Seidell’s Solubilities (18), the 7.7 CRC Handbook (19), Lange’s Handbook (20), The 10.6 Merck Index (21), Stephen and Stephen’s handbook 11.0 (22), and The Practicing Scientist’s Handbook (23). When a compound’s solubility appeared in more than one source, we chose the latest or the value with the most reported significant figures. Solubilities were variously reported in g/100 mL, g/L, g/100 g, and percent; we converted them all to mol/L for the comparison with theoretical values. Finally, to solve the problem that, even without competing reactions, solubilities might not agree with Kspt values because of activity coefficients, we used a computer program called EQUIL1 to calculate solubilities from Kspt values. When this program is given the Kspt value, it calculates the solubility, computes the activity coefficients, and iteratively finds the theoretical solubility assuming no competing reactions (Fig. 2). EQUIL has the option to calculate activity coefficients, γ, using four different equations: simple Debye–Hückel, extended Debye–Hückel, Bokris–Reddy, and Davies. We chose the Davies equation 2.0
ln γ = Az
2
µ1/2 1 + µ1/2
– 0.2
Figure 2. Using EQUIL to correct for activity coefficients.
JChemEd.chem.wisc.edu • Vol. 75 No. 9 September 1998 • Journal of Chemical Education
1183
In the Classroom
where the symbols have their usual significance. A is a well-known constant dependent upon the temperature and the dielectric constant of the solution, whose value is approximately 1.173 at room temperature. The symbol µ stands for ionic strength, defined as
µ = 1 Σ c i z i2 2 i in which c and z are the molarities and charges of the ions, respectively. Our reason for the choice of the Davies equation for the activity coefficient calculation is that according to Martell (17), the Davies equation is often used to extrapolate constants to zero from low-ionic-strength measurements. It seems realistic, then, to use the Davies equation to go back to the actual solubility. The only better solution to obtain the actual solubility would be to go to each of Martell and Smith’s references and locate this datum among all the data there. We chose EQUIL as a reasonable way to avoid this task. The EQUIL program comes with an extensive database of many reactions, including ion pair formations. Since the student cannot generally take these reactions into account, we had to remove from the database all reactions other than the dissolution into ions. Otherwise EQUIL would have incorporated these side reactions into the solubility calculations.
Table 3. Calculated and Observed Solubilities Kspt
Compound
1.8 × 10
S EQUIL 10
1.3 × 10
Exp S
Ref
5
1.36 × 10
5
AgCl
1.8 × 10
AgBr
5.0 × 10 13
5.0 × 10 13
7.1 × 10 7
7.3 × 10 7
21
Ag2C2O4
1.2 × 10 12
1.3 × 10 12
6.9 × 10 5
6.6 × 10 5
21
AgI
8.3 × 10 17
8.3 × 10 17
9.1 × 10 9
1.2 × 10 8
21
12
12
8
25
1.1 × 10
1.04 × 10
6
5.7 × 10
21
AgSCN
1.1 × 10
Ag2SO4
1.5 × 10 5
6.2 × 10 5
2.5 × 10 2
2.7 × 10 2
21
BaCrO4
2.1 × 10 10
2.3 × 10 10
1.5 × 10 5
1.6 × 10 5
21
BaSO4
1.1 × 10 10
1.2 × 10 10
1.1 × 10 5
9.57 × 10 6
21
9
21
3.9 × 10 11
4.6 × 10 11
2.3 × 10 4
2.3 × 10 4
21
Ca(OH)2
6.5 × 10 6
2.3 × 10 5
1.8 × 10 2
1.7 × 10 2
21
CaSO4
2.4 × 10 5
1.0 × 10 4
1.0 × 10 2
1.5 × 10 2
26
16
6
Fe(OH)2
7.9 × 10
5.6 × 10 23
5.5 × 10 23
2.4 × 10 8
7 × 10 8
21
Hg2Cl2
1.2 × 10 18
1.3 × 10 18
6.8 × 10 7
1 × 10 7
21
Hg2SO4
7.4 × 10 7
1.3 × 10 4
3.2 × 10 2
1.3 × 10 3
24
5
5
21
1.6 × 10
NiCO3
1.3 × 10 7
1.9 × 10 7
4.4 × 10 4
1.3 × 10 4
21
PbBr2
2.1 × 10 6
5.3 × 10 6
1.1 × 10 2
2.6 × 10 3
21
PbCl2
1.7 × 10 5
7.0 × 10 5
2.6 × 10 2
1.4 × 10 1
21
6
6
25
12
7.4 × 10
SrSO4
3.2 × 10 7
5.0 × 10 7
7.1 × 10 4
6.0 × 10 5
26
SrF2
2.9 × 10 9
4.0 × 10 9
1.0 × 10 3
3.1 × 10 3
21
AgBrO3
5.5 × 10 5
6.7 × 10 5
8.2 × 10 3
8.5 × 10 3
26
Hg2I2
1.1 × 10 28
1.1 × 10 28
3.0 × 10 10
5.05 × 10 10
21
SrCO3
9.3 × 10
1.0 × 10
2.6 × 10
2.1 × 10
PbCO3
10
6.8 × 10
3.5 × 10
21
Mn(OH)2
14
1.7 × 10
13
8.14 × 10
6
Hg2Br2
13
5.9 × 10
5.6 × 10
4
4.5 × 10
8.2 × 10
7.3 × 10
5
CaF2
16
5.3 × 10
9
CaCO3
9
3.2 × 10
6.4 × 10
5
7.4 × 10
5
25
Note: Compounds in shaded area were added by us because of their acceptably standard solubility behavior.
Conclusions from the Calculations S EQUIL in Table 3 is the solubility calculated from the theoretical Ksp by EQUIL. Exp S is the experimental solubility expressed in moles per liter. Now S EQUIL is compared to Exp S. If the solubility is unaccompanied by side reactions (the boxed equation in Fig. 1), then these should agree. Umland’s criterion for acceptable agreement was “within one order of magnitude”. We think this is too broad a range. We prefer to accept those solubility values that do not exceed EQUIL’s values by a factor of about two and those that are not less than about half EQUIL’s values. This eliminates 10 of 25 compounds from Table 1. We eliminated two more that were not in Harris’ table of Ksp values. We then added three compounds that we discovered to have acceptably standard solubility behavior. These are silver bromate, mercury(I) iodide, and strontium carbonate. Of course, some of the surviving 16 compounds may have complicating reactions whose effects cancel, so that they seem to give agreement with the simple assumptions. We leave them in because they are still appropriate candidates for the calculations we ask of students. We found no compounds with a trivalent ion for which the solubility obeyed the formula 27 S 4 and no compound with 1184
Kspc 10
Table 4. Ksp Values That Predict Solubility γ+ Compound Kspt Kspc 10
γ
AgCl
1.8 × 10
0.996
0.996
AgBr
5.0 × 10 13
5.0 × 10 13
0.999
0.999
AgBrO3
5.5 × 10 5
6.7 × 10 5
0.909
0.909
12
12
1.8 × 10
10
Ag2C2O4
1.2 × 10
0.983
0.935
AgI
8.3 × 10 17
1.3 × 10
8.3 × 10 17
0.999
0.999
BaCrO4
2.1 × 10 10
2.3 × 10 10
0.964
0.964
BaSO4
1.1 × 10 10
1.2 × 10 10
0.970
0.970
11
11
CaF2
3.9 × 10
0.888
0.970
Fe(OH)2
7.9 × 10 16
4.6 × 10
8.2 × 10 16
0.980
0.996
Hg2I2
1.1 × 10 28
1.1 × 10 28
0.998
0.999
Mn(OH)2
1.6 × 10 13
1.7 × 10 13
0.953
0.988
Ag2SO4
1.5 × 10 5
6.2 × 10 5
0.791
0.391
Ca(OH)2
6.5 × 10
6
2.3 × 10 5
0.434
0.812
CaSO4
2.4 × 10 5
1.0 × 10 4
0.467
0.467
Hg2Br2
5.6 × 10 23
5.5 × 10 23
0.998
0.999
SrCO3
9.3 × 10 10
1.0 × 10 9
0.949
0.949
NOTE: Compounds in shaded area need activity coefficients to work well if using Kspt.
Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
In the Classroom
two trivalent ions for which the solubility obeyed the formula 108 S 5. As interesting as these formulas are to derive, they are not practical in cases for which we could find solubilities. Summary Our final table is Table 4. Notice that in this table both Kspt and Kspc values are given. All compounds in this table will give reasonable values of solubility from Kspc. The values will not be as good from Kspt in cases for which the activity coefficients are significantly different from one. Comparison of the Kspt values with the Kspc values illustrates the conditions under which activity coefficients make a difference. What we are suggesting by the construction of this table is not the elimination of all other compounds and their Ksp values from general chemistry texts. Rather, we suggest that these 16 be included in one table of Ksp values that may be used in solubility calculations. Other compounds might be included in a separate table if required to solve problems predicting precipitation formation and common ion effects qualitatively. If authors choose to go into activity coefficients and extrapolation to zero ionic strength in their texts, they will wish to include both Kspt and Kspc. If they prefer not to mention activity coefficients, then the Kspc listing will be appropriate. We hope this is a useful aid to future authors. Acknowledgment Judith Bonicamp acknowledges the support of the MTSU Non-Instructional Assignment Committee. Note 1. EQUIL Version 2.11, Micromath Scientific Software, Salt Lake City, UT 84121-3144. EQUIL is a software program for PCcompatibles.
Literature Cited 1. Nernst, W. Z. Phys. Chem. 1889, 4, 372. 2. Meites, L.; Pode, J. S. F.; Thomas, H. C. J. Chem. Educ. 1966, 43, 667–672. 3. Chang, R. Chemistry, 5th ed.; McGraw Hill: New York, 1994; p 685. 4. Umland, J. B.; Bellama, J. M. General Chemistry, 2nd ed.; West: Minneapolis, MN, 1996; pp 620–628.
5. Computations of solubilities taking account of all competing reactions have been done to yield theoretical solubilities, but we sought experimental values. See De Roo, S; Vermeire, L; Gorller-Walrand, C. J. Chem. Educ. 1995, 72, 419–422. 6. Bodner, G. M.; Pardue, H. L. Chemistry: An Experimental Science, 2nd ed.; Wiley: New York, 1995. 7. Brown, T. L.; LeMay, H. E.; Bursten, B. E. Chemistry: The Central Science, 5th ed.; Prentice Hall: Englewood Cliffs, NJ, 1991. 8. Ebbing, D. D.; General Chemistry, 5th ed.; Houghton Mifflin: Boston, 1996. 9. Hill, J. W.; Petrucci, R. H. General Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1996. 10. Kotz, J. C.; Treichel, P. Chemistry & Chemical Reactivity, 3rd ed.; Saunders: Orlando, FL, 1987. 11. McMurray, J. E.; Fay. R. C. Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1995. 12. Silberberg, M. Chemistry: The Molecular Nature of Matter and Change; Mosby: St. Louis, 1996. 13. Zumdahl, S. S. Chemical Principles, 2nd ed.; Heath: Lexington, KY, 1995. 14. Gillespie, R.; Eaton, D.; Humphreys, D.; Robison, E.; Atoms, Molecules, and Reactions: An Introduction to Chemistry; PrenticeHall: Englewood Cliffs, NJ, 1994. 15. Harris, D. Quantitative Chemical Analysis, 4th ed; Freeman: New York, 1995. 16. Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, 6th ed.; Saunders: Philadelphia, 1994. 17. Martell, A. E.; Smith, R. M. Critical Stability Constants; Plenum: New York, 1976. 18. Seidell, A. Solubilities of Inorganic and Metal Organic Compounds: A Compilation of Quantitative Solubility Data from the Periodical Literature, Vol 1, 3rd ed.; Van Nostrand: New York, 1953. 19. Handbook of Chemistry and Physics, 71st ed.; Lide, D. R., Ed.; CRC: Cleveland, 1990. 20. Lange’s Handbook of Chemistry, 13th ed.; Dean, J. A., Ed.; McGraw-Hill: New York, 1972. 21. The Merck Index: An Encyclopedia of Chemicals and Drugs, 9th ed.; Windholz, M., Ed.; Merck & Co.: Rahway, NJ, 1976. 22. Solubilities of Inorganic and Organic Compounds, Vol 1; Stephen, H.; Stephen, T., Eds.; Pergamon: Oxford, 1963. 23. Moses, A. J. The Practicing Scientist’s Handbook: A Guide for Physical and Terrestrial Scientists and Engineers; Van Nostrand: New York, 1978.
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