I
Robert A. Stalrs Trent University Peterbwough, Ontario K9J 768
Unified Calculation of Titration Curves
Students beginning the study of ionic equilibrium calculations often complain that the methods commonly used are both incomprehensible and dissatisfying. The difficulty consists in large measure in the use of simplifying amumptions involving the neglect of certain small terms, the complaint being that one needs to know the answer before beginning the calculation. The dissatisfaction arises, in the calculation of titration curves, from the piecemeal approach: separate calculations for the initial solution, the stoichiometric point, and the regions before and after the stoichiometric point; four different calculations, using four different sets of simplifying assumptions. If the result is to be a single curve, cannot it be calculated from a single algebraic expression? Of course it can, and the systematic method of tackling problems of multiple equilibria, now commonly described in textbooks (I) can easily be extended (24) to the calculatiou of complete titration curves for acids and bases. A cursory survey of those analytical textbooksmost easily available to me (7-11) suggests that the application of this systematic method to acid-base titration curves is not common. In what follows I have illustrated the method by the calculation of a general curve for the titration of a monoprotic weak base by a monoprotic weak acid. The method may easily be extended to polyprotic acids or bases. (see especially Freiser (3)) and to cases involving precipitation. (It loses its simplicity when polymerization can occur.) The method involves solving simultaneously a set of equations which includes equilibrium-constant equations, a statement of electrical neutrality, and a sufficient number of statements of the consewation of appropriate atoms. It always leads, except in the simplest case, in which both acid and base are strong, to a cubic or higher order equation in [H+] (12). The difficulty has led at least two authors (13,14) to write an iterative computer program. I t is easily avoided, however, by treating the hydrogen-ion concentration as the independent variable ( 2 4 ) . If the whole curve is to be calculated, it is just as useful to have an expression for the volume of titrant as a function of pH as vice versa. The Weak-Acid, Weak-base Titration Let us consider ua ml of a solution containing Cb moleil of the monoprotic weak base BOH, with constant Kb, to which we add u, ml of a C. molar solution of the monoprotic weak acid HA, constant KO.I imply that the solvent is water, though it need not he, and since I ignore activity coefficients, the constants are "apparent." The equations to be solved are: Equilibria K, = [H+][OH-]
K.
=
(1)
lH+lIA-]/[HA1
Kb = [Bt] [OH-]/[BOH]
(2)
(3)
Electrical neutrality [H+] - [OH-]
+ [Bf] - [A-] = 0
(4)
Conservation of A and B atoms [A-I [B+]
+ [HA] = u,C./(u,
+up)
+ [BOH] = ubCbl(u. + ub)
From these mav be obtained
(5)
(6)
A,B
!"
20 v.ml 30 100 ml of 0.1 N base with 0.5 Nsolutionsof acid. Curve ArpK. = -10, pKb = -10:CuNe B:pK. = -10, pKa = 2: Curve C: pK. = 4, pKb = 6; Curve D: pK. = 4, pKb = 9. OO
Titration c u ~ e calculated s for tihation o f
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-
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The figure shows the result of using eqn. (7)to calculate pH cuwes for a strong hase-strong acid titration, and for an assortment of weak hase-weakacid titrations including one highlsunsatisfadory one (Dl.One was worked out with pencil and paper, the others with a programmable calculator, which was helpful but not essential. Familiar shapes of curves will be recognized, but the power of the method is evident in the easy calculation of the case involving extremely weak base, Kb = which is a tricky one to cope with by approximate methods. (Note that the pH in the post-stoichiometric region is not independent of the value of Kb.) In presenting this method of calculation, I do not wish to suaaest that it should s u ~ ~ l a nhut t , rather that it should su&ement the tra(iitionbi methods hased on appropriate a ~ ~ r o x i m a t i o nThe s . truth surely is that the very. ubierrions . c&d at the beginning of this note are proof of the correctness of the usual approach. Literature Cited
p. 53. (3) Reiaer, H.. J. CHEM. EDUC., 47.809-811 (1970). (4) Kaluek, W. J.,sndFernando,Q.,J. CHEM.EDUC.,49.202 (19721. 151 Flelser. H.. and Femando.Q.. "IonieEauilibria inhalyticalChemiatry." Wiley, New ~ m k1, k . pp. 111-124 &d 223-246 (6) Walton,H. F,"Prineipler andMethalsof Cbemicalhalysis,"2ndEd.,Pmntia-HdL, Englcwoad Cliffs, N.J.. 1964, pp. 249-260. da la chimie ana1ytiqw:o~lysequanfilotiwminemh," (7) Charlot. G.,"L~~Methgdes Mason. Paris. 1966, pp. 271-3. (8) Fischer, R. B., and Peter., D. G.. "Quantitative Chemical Analysis: 3rd Ed., Saunden, Phils.. 1968,pp. 481t191. (9) Kolthoff. 1. M, Sandell, E. B., Meehan. E. J.. and Bruckenstein. S.,"Quantitative Chemical Analysis,"4th Ed., Maemillan, New York, 1369, pp. 47-92. ilOI Pidnvk.D. h el r n ~ t r. v . " A c a d e m i c P ~ NelsYork ~~~. ~, L a n d Rank. C. W.. " A ~ ~ l v l i ~Cm and London. 1974. pp, 35536b. 1111 voecl. A. I.. "A Tert-Bmk af Quantitative lnoresnle Analvsis." 3rd Ed., Lonmans. . . io&o, i g ~ ipp. . 9~99. (12) Bruckenstein. S., and Kolfhoff, I. M., Interacience Reprint, N.Y., 1959, pp. 448ff. (Reprioted from Kolthoff I. M, and Elving, P. J., '"Treatiseon Analytical Chemistry: l.B.Ch. 12.) (13) Breneman. G.L.,J. CHEM. EDUC..Sl. 812 (1974). (1976). (14) Brsnd,M. J. D.,J.CHEM.EDUC..53,771-2 ~~~~
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Volume 55, Number 2,February 1978 / 99
