University of Cincinnati Cincinnati, Ohio
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In carrying out simple calculations on solution equilibria, approximations of various sorts are frequently made and lead to results which are usually appropriate to the state of the equilibrium under consideration. Nightingale' has explicitly treated the problem of monoprotic and diprotic acids in an extremely useful way, and Kingzhas presented a graphiconumerical approach to what is essentially the same problem. Many other good references exist in this area.= But beyond clearly recognizable inequalities in the algebraic quantities involved, there are no clear-cut "discriminants" available to serve as guides in making these approximations. The use of algebra of inequalities and very simple calculus leads to one such criterion (here presented) applicable both to "simple" equilibria and to equilibria of two "steps." It is offered in the hope of dispensing with a great deal of stumbling and confusion on the part of serious students. Generalization to equilibria involving more than two steps is also easy. Simple Equilibria
Simple, homogeneous equilibria which are governed by equilibrium expressions of the type
are usually treated on the basis of the assumption that "x can be neglected in the denominator since K is very small." (As usual, x is equilibrium concentration or activity of a species, a is its "initial" or formulated concentration, and K is the equilibrium constant.) That this is not a sufficient criterion for neglect of xis generally recognized, but a t present one has recourse only to solving the complete quadratic and then comparing the result with that obtained by neglect of x. This is not only somewhat a waste of time, but is also NIGHTINGALE, E. R.,J. CHEM.EDUC.34, 277 (1957). KING,E. L., J. CAEM.EDUC.31, 183 (1954). For example, see (a) BUTLER,J. N., "Ionic Equilibrium," Addison-Wesley Publishing Co., Reading, Mass., 1964; or (b) RICCI,J. E., "Hydrogen Ion Concentration," Princeton Universit.y Press, ~ r i n c e t o n ,J., ~ ~1954. .
philosophically unattractive. If one is going to have to solve the complete quadratic anyway, why bother to try the approximate method in the first place? The quantitative criterion developed below avoids this logical absurdity, and is useful both in research where homogeneous equilibria are involved and in the teaching of chemical methods at all levels, for while it is simple, it is not obvious. Putting eqn. (1) into quadratic form and solving for x explicitly (the second term on the right-hand side can only be positive in this case), there results z = -K/2 + K(l + 4a/K)'/s/2 (2) When 4a/K is large compared to unity, we have x G (Ka)'/', a not unexpected result, provided the first term on the right-hand side of eqn. (2) can be neglected with respect to the second, i.e., provided that (Ka)"' >> K/2, which means, provided 4a/K is large compared to unity, a condition already employed! Thus, the correct criterion for the neglect of x in the denominator of eqn. (1) is not merely the smallness of K, but rather a. largevalue of 4a/K, but then one might ask, "How large is 'large'?" A more satisfactory approach to this latter problem would be to investigate the sensitivity of x t o the quantity 4a/K, and this can be done most simply by setting 4a/K = R and forming dx/dR for a given value of K. Taking a given value of K involves no loss of generality and avoids the necessity of forming partial derivatives which a longer, but no more useful, treatment would involve. In any case, x and a are the actual free variables in any particular problem a t constant temperature. We find that dx/dR = (K/4)(1 R)-'I2. A plot of dx/dR versus R would then be an indication of the sensitivity of x to R. However, since both a and K are usually written in powers of 10, it is much more convenient to plot dx/dR versus log R. This is given in the accompanying figure. Clearly, x is quite sensitive to R between log R = 0 and log R S 3, since the slope is very steep in that region. Beyond log R Z 3, the slope approaches zero very rapidly. Beyond log R = 4, the slope is for all practical purposes equal to zero, and in this region, neglect of x with respect to a in the denominator of eqn. (1) is justified. In point of fact, a greater accuracy in the computation
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Volume 42, Number I I , November 1965
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of x results than even is justified by the accuracy and reliability of equilibrium constants, a situation which can be easily verified by direct arithmetical experimentation. We then propose the working rule: Neglect of x i n the denominator is justifiable whenever 4a/K is lo4 or larger, or, more roughly, whenever a is larger than K by more than 3 or 4 powers of 10.
complicated. I n fact, it is not, when it is recognized that Kl is by definition always larger than K p (and in fact most usually by several powers of 10 at least), and we shall here show that, for equilibria of the type in scheme (3), the rule set forth above can he applied directly, and that it justifies immediately both approximations to be made in the scheme, namely, that a >> x and x >> y, so that I n order to arrive a t eqns. (4) and (5), it is necessary simply to write and rearrange, and solve first for x and then for y , leaving y as a function of x. The approximations of eqns. (6) arise if, in eqn. (7), x is negligihle with respect to a. Now, using precisely the same reasoning as in the first section, x is negligihle in the denominat,or of eqn. (7) if, in eqn. (4),
We must now consider the role of the second term on the left-hand side of this inequality. Looking back at eqn. ( 5 ) , it is clear that K2 is an upper limit for y; that is, y can never he greater than K2. For if y > K2, then
Diogrornm~ticplot of the derivdive dx/dR vemr log R.
Equilibria of Two "Steps"
Equation (1) might be imagined to apply to an equilibrium such as HA$H++A-
to take a familiar case (however, the entire treatment of this *vauer is hv no means limited to acid-base eqnilih. ria). Suppose the next most complicated equilibrium situation is considered to he a successive dissociation of t,hetype HpA a-z
= z +H y+ + zHA-y
HA-e 2-Y
(3)
H+ +AZ+Y
Y
where the quantities beneath each species indicate equilibrium concentrations (or activities), and where the successive equilibrium constants are K , and KI. In view of the fact that the concentrations derivable fromx and y are given exactly by solutions for x and y in the following forms (with the solution for y left implicit for reasons which will emerge below) z
=
-ZC1/2
+ ( K l / Z ) ( l + 4a/Kt + 4yz/K,')1/2
(4)
+ 2)/2 1 [(K2 + ~
(5)
and
v
=
-(Kz
+
) ~ / Kzzl'l2 4
the mere algebraic problem posed by an analysis of the sort performed in previously would seem prohibitively 610
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Journal o f Chemical Education
which, by transposition of the first term on the righthand side, squaring on both sides, and cancellation of like terms, leads to the impossible result, Kz2 < 0. (Obviously, following similar reasoning with the inequality sign reversed, y can always be less than K2.) Therefore, the maximal value for 4yZ/Klein eqn. (9) is 4K2/KI2. Since the ratio K2/K1is always less than unity, the term 4yZ/K12is always less than four (in actual practice, considerably less). Equation (9) then reads as follows: 4a/K1
+ ( a number much, much less than 4 ) >> 1.
Since we have already used the notion that 4a/K1 2 lo4, and since lo4 >> 4, inclusion of the second term on the left in eqn. (9) is unnecessary and the criterion 4a/K1 is sufficient to provide both the approximations of eqn. (6), a.s will he indicated in the following paragraph. Below this limit, error in x arises from neglect of x with respect to a in the denominator of eqn. (7) and is carried through, although with less impact, into Y. It is now necessary to consider the second approximation in equ. (6). Rewriting eqn. (8) as I f 2 = y(r l)/(r - I), where x = ry, we note immediately that y = K2 to within one or two percent when r 2 lo2. Thus, x 2 102ysuffices to justify this approximation. Replacing x by (K,a)'/' and y by K2, we find as the criterion for the second approximation, (Kla)'/'/K22 10" Rearrangement and use of the limit 4a/K1 2 10" gives K1 2 2K2, an almost automatically filled coudition for any problem, and hence we shall not include it. as a separate criterion from 4a/K1 2 lo4.
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