Henry Freiser
University of Arizona Tucson, Arizona 85721
Acid-Base Reaction Parameters
Acid-base equilibrium calculations have received a great deal of attention in textbooks, journal articles, and chemical education meetings. Writers and speakers warn us that our presentation of this subject to our students must avoid the Scylla of oversimplification to achieve "clarity" and the Charybdis of "cumbersome" rigorous equations. Computer-based calculatious have been suggested as a means of handling rigorous formulations without the tedium of manual solution of higher order equations. This approach is not only an effective solution for acid-base calculations but it has the added pedagogic advantage of integrating computer techniques in the chemistry curriculum. Although heartily in favor of computer training, I am convinced we should not abandon the combined algebraic graphical approach with which even rather complicated pH calculations have been solvedrigorously and conveniently, and would like to take this opportunity to demonstrate how this approach can be further extended to acid-base titration curve calculations as well as to buffer and sharpness index formulation. The simplicity, utility, and pedagogic value of this approach contrasts sharply with the formidably cumbersome expressions found in texts and papers. By use of dissociation fractions (a's) incornorated in the eauations defining the pH part&ular mixture, &ing a titration curve-or evaluating titration errors and sharpness or buffer indices, such expressions are readily derived, easily used and, importantly, strongly relate to essential "chemical" considerations of the systems under study. Primarily, the approach emphasizes the value of expressing the concentration of all acid-base species in terms of the product of the analytical concentration, C, and an appropriate dissociation fraction, a.' Expressions for or's may bc readily derived for any species, H,,B, in an N-protic acid system, H,B. Thus for a,, the fraction corresponding to the species H,-,B, i.e.
of a
Using for pK values of H1POd,2.23, 7.21, and 12.32, respectively, and noting (with the help of a log a pH graph) that, at pH 4.0, the predominant species is H2POd- so that the only term in the denominator meriting consideration is K1[H+lZ,the value of aa is readily calculated as and [po,a-] = 0.2
x
10-u.6~= I O - I P . * ~ M
Students can easily learn to write a expressions by remembering that ( 1 ) The denominator for each a in rt system is identical. ( 2 ) The denominator will be a decreasing power series in [ H + ] ,starting with [H+INwhere Nis the total number of protons that can dissociate. In each successive term, an additional s t e p wise dissociation constant will be a factor. There will be a total of ( N 1) terms, in which the last is K L K Z ...K N . (3) Since esoh term in the denominator is proportional to the concentration of a particular species, the first term, [H+IN, forms the numerator of ao, the second for at, etc., so that, e.g., a x has K,Ka. . . K Nas its numerator.
+
The variation of a values with pH can best be appreciated from examination of log a versus pH graphs that can be either prepared beforehand or, just as easily, drawn freehand because such curves are largely composed of linear segments of integral number slope^.^ Titration Curve Calculations
Let us consider the calculation of a titration curve in which VA ml of a CA molar solution of an N-protic acid HNAis titrated with a CBmolar solution of NaOH. When VB ml of the NaOH have been added, the following equation, expressing proton balance may be written [Nu+]
+ [H+l
=
[OH-]
+ [HN- ,A1 +
+ . ..N[AI
(1)
+ 2ar + . .. + N ~ N -) [H+l + [OH-I
(3)
~ [ H -NnA1
Substitution gives
which rearranges to we may rewrite
VB = VA C*(ax
Equation (3) lends itself to ready calculation of the
As an illustration, let us find the [POaa-] in a 0.2 M HsPOdsolution whose pH has been adjusted to 4.0.
' FREISER, H., AND FERNANDO, Q.,"Ionic Equilibria in Analytical Chemistry," John Wiley & Sons, New York, 1963. BUTLER, J. N., '(Ionic Equilibrium: A Mathematical Approach," ~R, Addison-Wesley Publishing Co., New York, 1964. F R E I ~H., AND FERNANDO, Q., J. CHEM.EDUC., 42, 35 (1965). FREISER, H., School Science and Mathematics, March, 1967. See last two references in footnote 1. Volume 47, Number 12, December 1970
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809
various points in a titration curve without any "formidable undertaking." Suitable approximations can be made without jeopardizing the reliability of the calculation because of familiarity with the significance of various terms of the expressions at different pH values. Equation (3) may be written in terms of titration fraction, F
for use in calculating titration errors, i.e., titration error a t the first endpoint is F-1, at the second is F-2, etc. Calculations of titration errors are not the best diagnostic indicators of the practicability of a titration. Instead of simply substituting a particular [H+]value in eqn. (4) to obtain F and therefore theL'titrationerror," it is more to the point to see how F varies with pH in the vicinity of the endpoints. For this purpose, the derivative of F with respect to pH dF/dpH, can be seen to be a function of da/ dpH, which in turn are simple functions of the a values themselves. First, it is interesting to note that the change in both [H+] and [OH-] with pH is proportional to [H+] and [OH-], respectively.
Then, since
Similarly
For a monoprotic acid
In each of these cases the da/dpH is proportional to the a in question times a factor which includes all the other a values, each multiplied by the number of protons gained (a minus sign for protons lost). The factor containing the other a values represents the slope of the log ru - pH curve. Thus, the slope of the log [COa2-]versus pH curve will be +2 when H&Os predominates (a, ~ 2 i1, a, and or2 being almost zero), +1 when HC03- predominates (a1 = 1, the other a's 0) and 0 when COa2-predominates (a? 1, the other ru's = 0). From these considerations it is possible to write the value for da/dpH for any a in an N-protic acid system as 2.303 times the product of two factors (a) the a in question and (b) the sum of all the other alphas in the system and the number of protons gained in going from the species corresponding to the a in question to that of each of the other a's. Thus to illustrate with dat/dpH for HnY F;
With the aid of this approach it is now possible to return to questions relating to the sharpness of titration endpoints, buffer indices, and similar characteristics of acid-base mixtures. In effect, the slope of the function of eqn. (4), dF/dpH, is a measure of the buffer capacity, and its reciprocal, dpH/dF, evaluated at equivalence points, measures the sharpness index of the corresponding titration. It is convenient to consider F as consisting of two parts, one solely a function of a values, the second a function of concentration CA and the pH of the point under consideration. Differentiation of the first part
differentiationgives
Using eqn. (5), we have
with respect to pH will of course depend on N, the number of dissociable protons. Thus, for a monoprotic acid
Similarly and for a diprotic acid Equations (8) and (9) can be transformed to express the variation of log a with pH
Z O), Thus, a t high pH (>>pK), when al = 1 (a0 i the slope of the log a. versus pH will be -1 and that of log a1 versus pH will be 0. Conversely at low pH, the slope of the log ao curve is zero and that of log alis+l. For diprotic acids the derivatives of a values with respect to pH can be shown to be 810
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Journal o f Chemical Education
=
2 . 3 (ao(a~4- 4ur) f marl
The regularity of these expressions prompts one to predict that, in general
I n these expressions, it can he seen that, providing there is a reasonable separation of mccessive pK values, only the first term in each of the factors is of significance. Thus, for such a tripmtic acid
dpH = 0 . 4 3 (CA?')/'% S.I.= dF
For a titration of acetic acid (pK molar NaOH (so that CA = CA/2)
=
(29)
4.70) using equi-
5.1. = 10-.5* ( 1 0 - * . 7 ~ / 1 0 - ~ ~ . ~ ) ~ /= ~ . 10'-1JCa3/* CA'/~ (30)
The differential of the other part of F is
d
{w
([OH-]
~ P H
- [H+])/
= 2.3 X
+ [H+])
([OH-]
VaC*
(20)
which naturally retains the same form in all cases. Buffer lndex
The buffer index is a measure of the amount of acid or base that can he added to a solution before the pH changes by a given amount. If eqn. (4) is rewritten as A ' F = CB = CA'F = CA VA VB
+
N
c*'c iai + 1 [H+]
-
[OH-]
The sharpness index of the HOAc-NaOH titration falls to lo3 when the initial concentration of HOAc is 0.013 M or lower. I n general, the sharpness of a titration of a wealc monoprotic acid uaries with the square root of its acid dissociation constant and its initial concentration. This solution is formally analogous to what would he obtained at the second equivalence point of a diprotic acid titration or the last equivalence point of an Nprotic acid titration. I n these cases, the appropriate acid dissociation constant, K2 or KN, must of course he used. It would he interesting to note the factors that determine the sharpness a t an intermediate equivalence point such as that of a half neutralized diprotic acid. From eqns. (16) and (20) we have
(21)
Then
For a monopmtic acid, therefore, the buffer index is
1n thevicinity of the first endpoint, a, = 1, ao = [H+I/ K,, and a2= K2/[H+l. Hence
and that for a triprotic acid is
Sharpness lndex
If on the addition of 1% of the titrant volume at the endpoint the pH changed by 1 unit, the sharpness index would he 1/0.01 or 100. Thus, the sharpness index, dpH/dF, measures the feasibility of conducting a titration within a given limit of error. Usually a value of at least lo3is required. A few illustrative examples will serve to illustrate the usefulness of this concept. First, what factors determine the titration feasibility of a monoprotic acid? For the monoprotic acid case, we have
I n the vicinity of the endpoint, a, = 1, ao so that
=
[H+I/K.,
Usually at the endpoint [OH-] >> [ H f ] and from the approximate solution of the pH at the equivalence point
giving
and finally,the sharpness index, S.I.
Because a t this equivalence point, [H+12 = K I K ~ , we now have
Inasmuch as in any practical titration 2 KIKL'A' >> K,(K&
+ Kw)
the sharpness index here is
which we see to he independent of initial concentration of the titrant and proportional to the ratio of the fGrst and second acid dissociation constants. The sharpness index for carbonic acid (pK1 = 6.35 and pK2 = 10.25) titrated to the first equivalence point is only about 20. This would seem to limit such a titration to a minimum error of 5% or if the pH of the endpoint could be duplicated to 0.5, to 2.5%. I n addition to being very helpful in the detailed analysis of acid-base titrations, the treatment outlined here may he readily adapted to complexometric titrations. The simplicity and regularity of a functions (and their counterparts in metal complex systems) as well as the ready accessibility of their differentials should go a long way to systematize as well as improve the ease of teaching these important aspects of ionic equilibrium calculations. Volume 47, Number 12, December 1970
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