A Different Look at the Solubility-Product Principle 2. 2. Hugus, Jr. and F. C. Hentz, Jr. North Carolina State University, Raleigh, NC 27695
It is standard practice in general chemistry to present the criterion for precipitationofa salt in terms of the ion-~roduct as opposed t o the solubility-product constant. e he most common exercise required of students, which often represents the extent of rigor to which this principle is pursued, is whether or not precipitation of a salt like AgCl will occur when certain volumes of AgNOa and NaCl solutions with given concentrations are mixed. Recently, and in a quite impromptu fashion, one of us proposed the following opposite approach near the end of a lecture in second-semester general chemistry: At what point (singular!) during the progressive addition of 10-4 molar CI- solution to 10 mL of lo-' molar Ag+ would the ionproduct (cw X e a - ) = KSp= 1.7 X 10-lo?' Some 300 students were dismissed with the following quadratic to consider before the next class meeting:
with u representing the volume of chloride solution added. As one clearly astute student ~ a s s e dout the door. he remarked to the instkctor, "You know, that will be a quadratic with two real roots." After class, the instructor found that this student was indeed correct: two significant roots are obtained-at u = 0.176 mL, where precipitation begins and a t u = 568.059 mL where the precipitate present will redissolve completely. This brief history explains the origin of this paper, and has prompted us to report this interesting extension of an overlooked2 aspect of the solubility-product principle. General Solution for a 1:l Electrolyte We chose to approach the general solution in terms of the ratio, uluo, where uo is the volume of the original sample (analyte) and u is the volume of titrant added. For convenience it was assumed that the analyte and titrant solutions were of equimolar concentrations ( c ) and that volumes were additive. If no precipitation occurs, then the analyte concentration is cuo/(u uo) and the titrant concentration is cu/(u uo). At the points (plural!) of incipient precipitation/dissolution,the product of these must equal K,,:
Now, letting 2 - c2/K., = y , y2+yy+1=0
(5)
which by means of the quadratic formula gives The condition for real rwts in this equation is that y 2 2 4 and that y 5 -2. Therefore, these conditions lead to
with c,,iti,l being the concentration below which incipient precipitation is not possible. Using the data of the original problem proposed in the classroom, y = 56.8235 . . . , giving y values of 56.8059.. . and 0.0176038.. . , and c,,itlesj = 2.60768 X 10-5 molar. I t should be noted that, when y is large in absolute value, the smaller root in eqn. (6) may best be obtained by an expansion of the square root in eqn. (6) and is then approximately lly2. Point of Maximum Precipitation An interesting additional result occurs if we rewrite eqn. (1) to allow a finite amount of precipitation ( a mmol of compound being formed): (9)
or, with the fraction of precipitation, f = a/cuo,
Differentiation of eqn. (11) with respect t o y and setting df/Jy = 0 for a maximum in f , f,. a t y = y,
+
+
Dividing eqn. (11) by eqn. (12) gives
,f
We now define y = u l u ~which , yields
= 1/2(ym- 1)
(13) Now, substituting for f, in eqn. (12) and rearranging shows that for maximum precipitation
'
This value is chosen based on that given at 2S°C by Jonte, J. H., and Martin, D. S., Jr., J. Amer. Chem. Soc., 74,2052 (1952), and is close to the value 1.78 X of Ramene, R., J. MM. E m . , 37,348 (1960).In what follows it is assumed to be exactso that our results can ~ ~ ~ be aoolied without error to other cases. we have not foundany referencein me literatwe mat would indicate recognition of this interesting point in solubility calculations. ~
~
For the original problem, y, 0.872659.. . .
= 2.74532.. . ,
Volume 62 Number 8 August 1985
645
This is, of course, equivalent to determining the value of
compared to K/cmc'". One finds without difficulty that fCy) has a single maximum a t y = nlm, and that the value of f(y) a t this maximum is
-5
6
-4
-2
-3
-1
0
+I
+3
+2
+b
+S
+B
rn Y Reci~ltationcurves for A S 1 liowest curve revresents critical mncenktion. mow above various fractions precipitatedas designat&
Case I. Correspondingly, then if f(y),, is less than Klcmc'", no precipitation will occur.
Case II. If f(.y),= is greater than Klcmc'", precipitation will occur commencing a t y = y l < nlm and redissolution will occur a t y = y2 > nlm, where y~ and yz are the two real, positive roots of the equation,
We have plotted in the figure several curves that summarize relevant information for the case of silver chloride but which are universal curves for any AB solid. (A change in K., would simply translate all n w e s vertically by Yz ~ O ~ ( K , ~ K ~ , , ~ ~ ~ ~ ) ) . The lowest plotted curve represents the critical conceutratiou curve-for any given log c the values of log y l and log There can be a t most two such roots, as may be seen by apyz may he read. Successively higher curves represent the plication of Descartes' rule of signs or by noting that f(y) is fraction precipitated as being 0.50,0.90,0.99, and 0.999. monotone increasing for y < nlm and monotone decreasing There is also plotted a J-shaped curve that represents the for y > n lm. value of logy, (maximum precipitation point) at different log Case 111 is the critical case that occurs when y l = y~ = nfm, c values. This curve is asymptotic toy, = 3 a t high concenand then trations. Summary of Results for AB. Electrolytes
Using the approach given above for the 1:l electrolyte, we have derived expressions for c,itid, y,, and f,. for the AB, case. The table summarizes the results for AB, ABz, AB3, and AB,. I t should he noted that the definitions of E used in the expressions for y, and f, are given in the third column of this table. Treatment of the General Case of the Solubility of A,B,
Let us consider the general case of precipitation of a comwund A,, R..... where the concentration of the orizinal - solutions containing A and B are respectively c and c'. Then the "ion product" is
(3 El"
In this case, if c = c', then the critical concentration mentioned above may be calculated from K. Alternatively, given one of the concentrations the critical value of the other may also be determined. If the concentration ratio, c1/c 0 is given, then the critical value of c, c,,it, is determined from
We now investigate the maximum fraction of A,B, that can he precipitated. If the amount of AmB, precipitated is a (mmol, volumes in mL) when equilibrium has been reached then:
(16)
where as before y = u/uo and we wish t o determine the value of the ion product compared to K ( = K.,).
(22)
Using (3, the concentration ratio c'lc, this equation may be written as
Results for AS. Electrolytes c11l11-l
E
Yet "vx
fmu
AB
2 KBp1'2
5 c2
3-f I+£
1+E
AB2
- (2 K,JV3
3
27 K, --
3
1 - (12
2
2
c'
1+f
ABa
-43 (3 K,)"'
256 Kv -
AB.
n+ 1 -(nKv)"~*'~
--
n
9
(n+l)"'K, "lr'
or
@,
1 - 513 -
1+f
I+€
n(n+2)-1
"(Y)!)"'
646
1+5
15-f
-
='
Journal of Chemical Education
1+f
Next, introducing y ( 2 ufuo) and f(=am/cuo),the latter being the fraction of AmB. precipitated, we find that
Differentiation of this equation with respect t o y with the condition that f be a maximum, i.e., dfldy = 0, leads to
1-f/n l+(
in which y, and f, are the values of y and f a t the point of maximum precipitation. Then suhstitution of this relation into eqn. (24) gives
and The term in square brackets is (cf. eqn. (21)) cCram+" and, if now we use that substitution and define a new quantity, 8, such that flm
m
-n@ + a with
and take the rnth root of eqn. (26), there results
Solving simultaneously eqns. (25) and (28) finally gives
When rn = 1 and P = 1 these expressions yield results identical to those given in the table. Equations (29) and (30) should he used, however, when c' and c differ, that is when 0 # 1.
Volume 62
Number 8
August 1965
647
