I
Louis Meiles, Polytechnic Institute of Brooklyn Brooklyn, New York J. S. F. Pode,
Eton College Windsor, England
I
and Henry C. Thomas
University of North Carolina Chapel Hill
Are Solubilities and Solubility Products Related?
I n a recent critique of the CBA and CHEM Study courses ( I ) , one of us described his quantitative treatments of solubilities and solubility products in the following words: "One factual error that is common to both courses concerns the solubility of calcium sulfate. The literature value is 2.0801 erams CaSOl ~ e liter r at 25'C. or 15.278I millimoles per liter, giving value for K,, of'2.2 X lo-& . . . for the compound CaS04.2H20. The CHElLlS text gives 2.4 X lo-=, whereas CBA gives its solubility as 6.2 millimoles per liter." The latter figure occurs on page 633 of the CBA text and a calculation is based on it, although the correct value is given in a table on page 585. Thus one text cites a solubility that the student could not confirm by experiment, while the other cites a solubility product that he could not obtain by calculation, and then employs it (in Exercise 10.8 and on page 326 of the Teachers Guide) to deduce a solubility that does not agree with experiment. Nevertheless, the value of K,, given in CHEMS is nearly correct. This is not a contradiction. but a reflection of the fact that the relation between sblubilities and solubilitv ~roducts is far less intimate than is stated or impliedAinthese and all other introductory texts. We shall first examine the relation, such as it is, in some detail, using calcium sulfate as an example to show what must be done to compute either of these quantities from the other. We shall conclude that the ideas and calculations that are involved are much too complex for presentation on the elementary level. We shall therefore plead for the illustration of this important topic by examples in which calculations suited to the beginner's ability do not lead to arossly erroneous results. Finally, we shall suggest thai the-solubility product can b e more appropriately and easily introduced in a totally different context without misrepresenting its relation to the solubilitv. here" are three tacit assumptions involved in the familiar equation: K, = 8'
a
in which K,, is the solubility product and s the solubjlity of calcium sulfate. They are: 1. Calcium sulfate is a strong electrolyte, so that its 1 In the original these numbera were erroneously given as 0.2080 and 15.295, respectively.
dissolution can be represented by the equation CaS04(s) = Ca++ SO4-. 2. The solubility product of calcium sulfate can be equated to the product of the concentrations of calcium and sulfate ions. 3. Neither calcium ion nor sulfate ion is consumed bv anv ~rotolvticor other side reaction.
+
The last of these is very nearly, though not quite exactly, true, but both the first and the second traduce the facts. As we shall show, barely over two-thirds of the calcium sulfate is dissociated in a saturated solution, and the solubility product of calcium sulfate is less than one-fourth of the product of the ionic concentrations. Values of the thermodynamic dissociation constants of the ion pairs that exist in solutions of a number of common electrolytes containing polyvalent ions are summarized in the table (3). If it were appropriate to employ concentrations in equilibrium-constant expressions, which is a question that will be discussed later, the value given for calcium sulfate would mean that: Ka = [Ca++][S01-]/[CaSO,(aq)l = 5.2, x (1) in any solution of calcium sulfate, saturated or not. Whether the concentration of the ion pair, which in this case is indistinguishable from undissociated molecular calcium sulfate, is appreciable depends on the value of Ks and on the concentrations of the ions. The value of Kd is largely dependent on thc ionic charges. The values cited in the table are typically about 0.2 when one ion is univalent and the other divalent (BaNOaC, KSOI-, etc.), about 5 X 10WJ when both ions are divalent (CaS04, MgSOn, ctc.), Thermodynamic Dissociation Constants of Ion Pairs in Aqueous Solutions at 25' Ion air
Kd
AgSo,+ BaNOa+ BaOHC CaCn01 Ca[Fe(CN)slCa[Fe(CN)slCaNOa++ CaOH CaSO, CdSOl CoSOn CuSO, FeClt+
Ion pair
Kd
KSOi Ls[Fe(CN)al Laso,++ MgOH MgSO. NaSOdNiSOd PbNOsf TIIFe(CN)slZnOH ZnSO. +
Volume 43, Number 12, December 1966 / 667
when one ion is divalent and the and about 3 X other trivalent (Ca[Fe(CN)&, LaSOa+). The electrostatic attraction between two dissolved ions depends on their charges and on the distance between them. If they can approach each other closely enough, the electrostatic force between them may become so large that they cannot be knocked apart by collision with a solvent molecule having the average thermal energy k T , and the product of their association will then he relatively stable. The stability increases with increasing ionic charge and with decreasing ionic size. According to Bjerrum (S), two ions having charges of 11and zl in an aqueous solution at 25" will form an ion pair of appreciable stability if they can approach each other within 3 . 5 A. ~ ~ The ~ ~radii of most hydrated inorganic ions are between 2 and 6 A under these conditions. I t is unusual for two univalent ions to he able to approach each other within the Bjermm critical clistance of 3.5 A, but it is far more unusual for two cliv$ent ions to be unable to approach each other within 14 A. Hence extensive ion-pair formation is rare for 1-1 electrolytes, hut it is so far the rule for 2-2 electrolytes that no completely dissociated electrolyte of this type is known. Chemical factors may he involved as well as electrostatic ones: the difference between the dissociation constants of ZnOH+ and RaOH+ cannot he explained on a purely electrostatic basis. However, it is interesting to note that the dissociation constants for the seven divalent metal sulfates included in the table all lie in the narrow range (4.4 0.5) X 10W3. The radii of these seven hydrated metal ions are not very diierent, and the differences that do exist have only small effects on the dissociation constants because the size of the anion is fixed. A preliminary calculation employing eqn. (1) yields, if [Cn++] = [SO,=] = c and [CaS04(ay)] = 1.528 X 10W2
*
- c, c =
6.71 X
lo-* M; [CaSO&(ap)l= 5.57 X
lo-= M
According to these figures the dissociation is less than half complete in a saturated solution of calcium sulfate: about 60% of the dissolved salt is present as the undissoriated ion pair. If no other considerations were involved, the solubility product of calcium sulfate would therefore he equal to (6.71 X or 4.50 X lo1. This is only about one-fifth of the square of the solubility. We shall set this result aside temporarily to point out that there are many other exactly similar situations. For example, the acidic dissociation constant of 2,4,6trichlorophenol, C13CsH20H,in water at 25' is given by: Ii.
=
[HaO+][ClrC6H20-I/lC1,C6H20H(aq)] = 1.0 X
(2)
The soluhility of 2,4,6-trichlorophenol in pure water a t this temperature is 4.05 X mole per liter. I n the saturated solution: Recnuse the solution must be electrically neutral: [H3OC] = [ClaCsHzO-I [OH-]
+
On combining eqns. (2)-(4), we obtain:
668 / Journal o f Chemical Educafion
(4)
If we neglect [OH-] on the ground that the solution must he acidic, so that [OH-] must be much smaller than [H30+], we obtain [H30+] = 6.32 X 10-j A f . As this corresponds to [OH-] = K,/[H30+] = 1.58 X 10-'0 M, the approximation is clearly justified. From eqn. (4) we then obtain [C13C6H,0-] = 6.32 X M . On combining this result with eqn. (3), me find that [C13C6H20H(ay)],which is the concentration of the undissociated trichlorophenol that is prescnt in - 0.32 X the saturated solution, is equal to 4.03 X lo-" 3.99 x 10-8 M. This is the equilibrium constant of the reaction:
in water at 25'. No matter what else may be present in an aqueous solution that is saturated with the trichlorophenol at 25', the concentration of the undissociated compound must he constant and equal to 3.99 X lo-= M. We ignore the relatively small variations that will actually result from changes of the activity coefficient with changing ionic strength; for calcium sulfate these will be described below. It may he observed that the undissociated trichlorophenol is analogous to the undissociated calcium-sulfate ion pair. In a saturated solution of calcium sulfate [CaSOa (ay)] is also constant no matter what the concentrations of the dissolved ions may he. We may describe the composition of any saturated solution of 2,4,6-trichlorophenol in water at 2s0 by combining eqns. (2) and (G) : K,,
=
[H~O+llClaCeH~O-I = K.K, = 1.0 X 1W6 X 3.99 X lWa =
4.0 X
(7)
This product of the ionic concentrations in a saturated solution of the trichlorophenol is its soluhility product (neglecting activity coefficients), just as the product of the calcium and sulfate ion concentrations in a saturated solution of calcium sulfate is the solubility product of calcium sulfate. It is not, however, equal to the square of the soluhility of the trichlorophenol in pure water. which is That would be (4.05 X 10-a)2 = 1.64 X almost four orders of magnitude removed from the truth. Nearly 99% of the trichlorophenol that is dissolved in such a solution remains undissociated. To ignore this fact, square the soluhility, and call the result the soluhility product is clearly beyond the hounds of reasonable propriety. The error is smaller for calcium sulfate, because the dissociation constant of the calcium-sulfate ion pair is larger than that of the trichlorophenol, but there is no conceptual difference whatever between the two cases. I t is instructive to consider the information that can he obtained from eqn. (7) about the compositions of saturated solutions of the trichlorophenol in media other than pure water. Two examples will suffice. (1) Suppose that a solution containing 0.001 mole per liter of hydrochloric acid is saturated with the trichlorophenol. The solubility of the latter will be smaller than it is in pure water because of the commonion effect. Taking the concentration of hydrogen ion as 1 X 10-3 M in the saturated solution, we find from eqn. (7) that the concentration of the trichlorophenolate ion will be 4.0 X 10-8/1 X = 4 X lo-' Af.
The solubility s of the trichlorophenol mill be equal to the sum of this value and the concentration of the undissociated con~pound: s = LCIaC&O-I
+ LCIaC&OH(w)I = =
+
4 X 103.99 X lo-= 3.99 X 10-8 A4
This is a thousand times as large as thevalue that would be obtained by neglecting the concentration of the undissociated trichlorophenol. (2) Suppose that a solution containing 0.004 mole per liter of sodium 2,4,6-trichlorophenolate is saturated with the trichlorophenol. The solubility of the latter will again be smaller than it is in pure water. From eqn. (7) we find that the concentration of hydrogen ion in the saturated solution will be 4.0 X 10W9/4 X = 1 X A t . This concentration will result from the dissolution and dissociation of 1 X 10-"ole of the trichlorophenol in each liter of the solution, but in addition 3.99 X mole of the compound will dissolve without dissociating, and the solubility will be s
=
1 X 10-
+ 3.99 X 10-8 = 3.99 X 1 0 - O M
This is four thousand times the value that would be obtained by neglecting the concentration of the undissociated trichlorophenol. It is evident from these examples that the relation between solubility and solubility product may be a very distant one indeed. Ionic Activities
We have thus far estimated the concentrations of calcium and sulfate ions in a saturated solution of calcium sulfate as 6.71 X 10W3 M . Dissolved ions exert electrostatic forces on each other, and these forces produce deviations from ideal behavior. If another electrolyte, with which it has no ion in common, is added to a saturated solution of a salt, the solubility of the salt is altered in ways that cannot be accounted for by chemical reactions among the ions. The solubility of silver chloride in water at 25' is increased about 12% by the presence of 0.01 mole per liter of potassium nitrate (4); that of barium sulfate is increased about 70% (5). The algebraic simphcity of thc soluhilityproduct relationship is most conveniently salvaged by redefining the solubility product as a relation among ionic activities rather than concentrations: The activity of an ion is the product of its concentration and an activity coefficient. Expressing concentrations in moles per liter, we write
f+, may he defined (6) as the geometric mean of the single-ion activity coefficientsappearing in an equation like (9). For calcium sulfate: f* = (fo.+*f8fso,-)'/' (10) while for calcium fluoride eqns. (9) and (10) rrould become: I;,,
=
f~++[Ca++l(j~-[F-])$
(9,)
and
f,
=
(.fc,*+f;-)'/a
(10a)
respectively. The mean ionic molarity activity coefficient may be described by the Debye-Hiickel equation (7) log f, = -0.511 z ~ z 4~ /l ( 1 f 0.329 ad;) (11) for aqueous solutions a t 25", provided that the total concentration of solute is not too high. The quantities z, and zz are the charges on the ions of the electrolyte that is of interest; fi is the ionic strength, defined by the equation (8)
and & is the so-called "distance of closest approach." This appears in the derivation of eqn. (11) as the interionic distance that corresponds to the maximum interionic force. If ionic solutions were as simple as might be wished, &wouldbe simply the sum of the radii of the cation and anion of the electrolyte in question. Because both calcium ion and sulfate ion are divalent, the ionic strength of a saturated solution of calcium sulfate is appreciable even though its concentration is not very high. The above estimate of the ionic concentrations leads to fi = '/z (4cc,++ 4cso,-) = 4 X 6.71 X = 2.6Sa X M. From IO)s++ HIO';
Cs(HIO)s++
K.
=
aoa~ox1~xpl.++ ax,a*/ac.~a,a,.++ (20)
and some hydronium ions are produced by the autopmtolytic reaction: 2 HnO = HsOt OH-; K. = a ~ , o + a a x (21)
+
Hydronium ions are produced in eqns. (20) and (21), and are consumed in eqn. (19). The concentration of hydronium ion a t equilibrium is given by the materialbalance or conservation expression:
+
In using eqn. (22) we shall neglect activity coefficients for the sake of speed in deciding whether a thorough treatment is wort,hwhile. For the same reason we shall assume that neither calcium nor sulfate ion will be consumed to any large extent by the above reactions, and so we shall use the values [Ca++] = [SO,-] = 1.042 X 10-2 M computed above. Making the appropriate substitutions from eqns. (19)-(21) into eqn. (22), we obtain:
Rearrangement and introduction of the appropriate numerical values yield:
Together with eqn. (19), this yields: IHS0.-]/[S04-]
=
Ks[HsO+l
= 9.4X 10-lo
while with eqn. (20) it yields:
These values are so small that they justify the neglect of the tiny fractions of the calcium and sulfate ions consumed by these protolytic reactions. Summary
We may sum up this discussion by saying that the authors of CHEMS and CBA incurred three errors by choosing calcium sulfate as an illustrative example: 1. Because both of these ions are divalent, the electrostatic force between them may become so large as to produce rather extensive ion-pair formation. 2. Because both of these ions are divalent, even a rather dilute solution of calcium sulfate has an a p preciahle ionic strength. 3. Because both of these ions are divalent, the mean ionic activity coefficient is considerably different from 1 even at the ionic strength of the saturated solution. It seems probable that the magnitudes of these errors are much greater than many teachers of beginning courses can have suspected. If the ideas discussed above are phrased in purely thermodynamic language, the conclusions become in principle self-evident. Here "in principle" means that thermodynamics gives no hint as to whether the possibilities are imporlant or not. Experimental chemistry alone can decide this point. If a substance MX in the solid state is in equilibrium with the dissolved and hydrated species R'IX(H20). (aq), which in turn is in equilibrium with its products of dissociation, a condition for minimum free energy can be written:
+
where n m = p q and both ions are written as univalent for simplicity. Equation (27) is valid whether the amount of MX(HzO).(aq) is large, small, or zero. Equation (26) is valid whether MX(HSO). (aq) dissociates completely or not a t all. Equation (25) is valid even if MX(H20), (aq) does not exist or if it accounts for all of the dissolved material. The point was made in 1873 when Gibbs remarked: "The potential for each of the component substances must have a constant value in all parts of the given mass of which that substance is an actual component, and have a value not less than this in all parts of which it is a possible component." One can easily calculate the potential energy of a brick on the parapet of a high building, whether the brick is there or not. If bricks are in fact there and the entire stock of them must be accounted for, a knowledge of this energy may be useful in computing the number of bricks aloft, particularly if a fixed disbursement of wages for work done is the determining factor. It is by no means suggested that high-school students or even college freshmen should be subjected to the rigors of the exact treatment outlined here. On pedagogical grounds the authors of CHEMS and CBA chose to sacrifice chemical refinements where these would have interfered with the continuity of their logical presentation. I n this instance we believe they were right to do so, though it might have been prudent at least to warn teachers that their simple calculations do not agree with experiment in the cases they chose. In addition, we feel that teachers and authors on these levels should be aware of these difficulties and should abstain from elegant variations that ruin the practical point of calculations that their students can be expected to make. It would be better to confine illustrations of the solubility-product principle to 1-1 salts, like silver bromide and thallous iodide, for which the student's calculations will yield results close enough to the truth to permit him to feel that it is worth his trouble to try to master what he is being taught. Finally, we feel that teachers and authors on these levels should seriously consider whether the solubilityproduct principle is in fact closely enough related to solubility to justify its presentation in that context. Too many students somehow acquire the idea that n solubility product is a special kind of equilibrium constant or even the idea that it is not an equlibriuln constant at all. I t would seem to be much better to develop the solubility product as a constraint, imposed by the presence of the solid phase, on the equilihrium constant for a homogeneous equilibrium, in the manner illustrated by the 2,4,6-trichlorophenol example above. Without doing violence to any essential concept, this example could easily be presented to any class that had previously been introduced to the dissociation constants of acids or any other simple equilibrium constants for homogeneous equilibria. The solubility product would then appear in its proper light as a description, not of the solubility, but simply of the ionic concentrations in the saturated solution. We believe that this would provide a sound basis for subsequent presentations of the more esoteric concepts outlined above, whereas the currently customary approach is one that the student must eventually eradicate from his mind if he is to acquire a correct understanding. Volume 43, Number 12, December 7966
/
671
Literature Cifedz (1) PODE,J. S. F., J. CHEM.EDUC.,43,98 (1966). (2) PIRSONS,R., "Handbook of Electmehemic~lConstants," Butterwortha Scientific Publications, London, 1959, pp.
--
54-5fi
(3) BIERRUM,N., Kg1. Damke Vidmk. Selskab., 7 , No. 9 (1926). (4) POPOFF.S., I N D NEUMAN, E.W., J . Ph@. Chem., 34, 1853 (1930). (5) NEUMIN,E. W., J. Am. Chem. Soc., 55, 879 (1933). (6) LEWIS, G. N., AND RANDALL, M., "Thermodynamics,"
672
/
Journal o f Chemical Edumfion
McGmw-Hill Book Company, New York-London, 1923, p. 329. (7) DEBYE,P., AND H ~ ~ C K EE.,L ,Physik. Z., 24, 185 (1923). (8) Dom, M., "Principles of Experimental and Theoretical Electrochemistry," McGraw-Hill Book Co., New York-
.- ~ - ~ . . ~
T.ondon. 1935. o. 278. (9) KIELLAND, J., J. Am. Chem. Soc., 59, 16i5 (1937). L L AND , FIILEY, C . F., C h ~ mRPCS., . 4, 271, 285, (10) R A N ~ ~ M., 291 (1927).
The antiquity of the majority of these references is intentional.
