Acid-Base Titration and Distribution Curves
JUrg Waser California Institute of Technology Pasadena, California
Although it would appear that everythiig of interest about the computation of acid-base titration curves has already been said, I have not elsewhere seen the approach given below, one I believe to be illuminating. Its two salient features are these: (1) Not the titration curve itself is calculated, hut its inverse. In other words the volumes of base or acid required to reach a series of different pH values are calculated, rather than the other way around. The reason is that the known pH permits the immediate calculation of the ratios of the different species in the solution, while with an unknown p H an equation of second, third, or even higher degree in [ H + ]must be solved first to find the pH. While we wish to know the pH when certain amounts of titrant have been added, we actually want an entire graph, the titration curve, so that it makes no diierence whether we calculate the ordinates for given abscissas or vice versa. (2) The final curve is calculated by adding two intermediate curves, one of which is characteristic of the solvent used (i.e., water) and the other of the (monoprotic) acid-base pair being titrated. The virtue of this procedure is that it reveals the origin of the different features of the titration curve, and exposes portions of the curve that are not explored in the usual titration experiment. We assume that there is no volume change during the titration. This simplifies the calculations and resuits in symmetrical curves that apply equally well to the titration of acids with strongbases and of bases with strong acids. The titrant may, for example, be assumed to be generated directlv in the titration vesselso that there is no volume change, and this is in fact possible in some situations. I n any case, the volume change during most titrations is such that the resulting dilution does not change the characteristic aspects of the titration curve. If experimental titration curves need to be
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Contribution No. 3442 Gates and Crellin Laboratories of Chemistry. 'Assume, e.g., that the titrant is NaOH, whence from charge balance [ H i ] [Net] = [ O H I [A-I. Insertion of [Nat1 = ee yields eqn. (3). [Ht] is used as shorthand for [HaOt] or [HsOlt1. a BUTLER, J. N., "Soh~bilityand pH Cdeulations" and "Ionic Equilibrium, a Mathematical Approach," Addison-Wesley Publishing Co., Inc., Reading, Mass., 1964. FREISER,H., AND FERXINDO, Q., J . CHEM.ED.,42,35 (1965). 3 Note that eqn. (5) is correct even if there is dilution. If the original concentrations and the volumes of acid and base are V,, cam, and Va, we have c. = ca0V./(V, respert,ivelg: c,'. Vs) and c h = es"Vs/(V, Va) so that r is equal to Vss0/ = cs/e. as in eqn. (4). To take dilution explicity into V.c," Va). Solved, account in eqn. (5) we replace c. by cCsV./(V. c." e.g., for Vb/V. we obtain VdV, = ([OH-] - [Htl K.))/(csS lHt1 - [OH-]). K O / ([Ht1
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fitted, the effects of dilution can easily be taken into account. We also assume that the titrant is a strong acid or a strong base, so that it is essentially a solution of H 3 0 + or of OH-. Finally, we treat in detail only the titration of one monoprotic acid or base. Generalizations to titrations of several acids or bases, or to the polyprotic case, are not difficult and will be remarked upon. Mathematical Development
Consider the titration of a monoprotic acid H A with strong base. We denote the total acid concentration by c,,: c, =
LHAl
+ LA-]
(1)
There being no dilution, c. stays constant during the titration. What does happen is that some or all H A is changed into A-. The concentration of base generated is denoted by co. During the titration ca is continually increased. Progress of the titration is measured by the ratio r , 7
=
(2)
c h ,
which will be called the degree of titration. At the beginning of the titration r is 0, while at the equivalence point T is 1.0. To obtain a relationship between [ H + ] and r we note that1 IH+I 12) ,--> +, , -"..= TOW-I, + rn-I, >-, an .. equation that is often called the proton balance emditm~' Solving for ca and using eqns. (1) and (2) we finally obtain L---
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7 =
e,/e. =
. ..-
[OH-] - [Hf1 c,
+
[A-1 [HA1 [A-I
+
(4
The right side can be rearranged by using the dissociation product of water, [ H + ] [ O H - ] = K,, and the acid constant of H A , [ H + ] [ A - ] / [ H A ] = K., so that finallya
The first term on the right of eqns. (4) and (5) describes the behavior of the titration medium (water), while the second term describes the acid being titrated. It represents the fraction of the total original acid present (i.e., of the sum [ H A ] [A-1) that is in the form A-. A graph showing this fraction as function of the pH (Fig. 1) is termed a distributim curve, because it describes the distribution of the total acid concentration over the species A- and H A . The second term in eqn. (5) can also be given the form (1/1 [H+]/K.), which shows that the distribution curve depends on the ratio [H+]/K., or in terms of pH and p K , on the difference
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result of adding the abscissas of corresponding points of the broken curves in the two diagrams. Discussion
Figure 1. Distribution curve for the species A - The function [A-]/[HA] [A-] is plotted ogainrt the pH, relative to the pK of H A The lower vegment of a vertical line drawn through a given pH value or obrcisra reprerenh the fraction of the total, [HA] -I-[A-I, that is in the farm A ; and the upper segment, the fraction that is in the form HA. For eromple, when pH = pK 0.50 these two fraction, arc. rerpec'ively. 0.76 and
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Beyond the points commonly made in discussions of acid-base titrations the figures illustrate particularly well how the pK of the acid being titrated affects the titration curve. As the pK is decreased, the auxiliary diagram must he shifted higher and higher. For a weak acid (Fig. 2) the result is that the pH jump near equivalence is diminished, making titration more and more difficult. For very weak acids the auxiliary diagram is so high that the curve on it no longer afFects the titration curve. Titration is no longer feasible, and it is equally impossible to determine K . from attempts a t titration.
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(pH - pK). All distribution curves of monoprotic acids have, therefore, the same shape when plotted against (pH - pK). The insight afforded by eqn. (5) into the make-up of titration curves is demonstrated in Firmres 2 and 3. which represent the weak acid and the strong acid cases. Both figures consist of an auxiliary diagram on the left and a main diagram on the right. The broken curve in the main diagram shows the first of the two terms in eqn. ( 5 ) , i.e., the expression ([OH-] - [H+])/c., while the auxiliary diagram represents the distribution curve of the species A-. The auxiliary diagram expresses, therefore, a property of the acid-base couple being titrated; and the broken curve in the main diagram, a property of the solvent. The auxiliary diagram must be shifted up or down, until the pK of the acid being titrated is correctly positioned on the pH scale of the diagram on the right. The final titration curve is the '
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Figure 3. Titrotion curve for strong acid b y strong bore. The conrtruction of the titrotion curve i*in principle the some or in the preceding figure but differs in an important debil. Since the acid being titrated is now rtrong IK, = 10.0),i t is practicoily dl dirsocioted and the second term in eqn. ( 5 )is essentially unity. The arrows to be transferred ore therefore all of unit length ond the broken curve on the right is simply rhifhd by one unit to the right. The titrotion reaction is HaO+ OH- = 2HsO. and the ouriliary diagrom is not really needed. The curves are for concentrotions of 0.1 0 M.
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For a strong acid (Fig. 3) our procedure shows why a11 titration curves in an aqueous medium are indistinguishable from each other, no matter what the actual acid constant is. For when the distribution curve on the auxiliary diagram in the figure is positioned so low that it no longer afiects the titration curve, it is immaterial whether the auxiliary diagram is a hit lower or higher. I n other words, the titration reaction is H80+ OH= 2H20. The leveling influence of water on strong acids is a t work. It is also impossible to determine K. from the titration curve of a strong acid.
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Family of Titration Curves Figure 2. Titration curve for weak orid b y strong bare. The ouxiliory diogrom on the left represents the distribution Curve of Figure 1 except thot the pH hor been chosen to be the ordinate and the fraction A- the abscissa. The diogrom is positioned so that the ordinate scales of the two diogrom~match. In the example shown K. i s 1.0 X 10-4 Thetitration curve is obtained by adding the abrcirros of corresponding points of the broken curves, os is indimted by the horirontd arrows thot must be t m n r ferred from the diagram on the left to that on the right. The portion of the titrotion curve with negative r values is urunlly not encountered in titrations and corresponds to strong acid being odded rother than strong base. The titration normally rtortr with T = 0 with no base added or yet. As base is added, r increaser. At r = 1.0 the equivalence point is reached and values of r larger than 1.0 correspond to on excess of bore. The curves pertain to concentrotions of 0.1 0 M.
Figure 4 shows a set of curves that represent the titrations of acids of different strengths with a strong base, neglecting changes of volume. The abscissa shows the degree of titration T,
For an actual titration the region in the figurewith negative values of T is of no concern, because the addition of strong acid rather than of strong base is needed to reach the pH values indicated. But the region is of interest Volume 44, Number 5, May 1967
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Buffer regions: HA/&-
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HeOIOH-
t
Figure 4. Family of titrdion curves for 0.10 M solutions. The curves opply to the titration of an acid of constant K , wi* strong bare lbottorn scale, 71 01 well as l o that of a base of constant Ks = Ku/Kn with strong acid (top scale, r'l.
because it shows the continuity of the curves. The curves also show the pH beyond equivalence. The point r = 0.5 and pH = 7.0 is a symmetry point of Figure 4, and the family of curves can he brought into self-coincidence by rotating the figure 180' about this symmetry point. The reason is the reciprocal relationship between [H+] and [OH-], and this makes the figure also applicable to titrations of bases with strong acid. The degree of titration T' then becomes
,
=
equivalents of acid added total equivalents of base present
This variable is shown a t the top of the figure, and pOH on the right. An aspect of the curves of Figure 4 that is in general discussed incompletely is their relationship to the buffering action that always occurs when both an acid and its conjugate base are present in large amounts. Thus, the relatively flat portions of the curves between r =0.1 and r = 0.9 reflect the presence of both HA and A- in substantial amounts. The steep portions of the curves a t T = 1 and r' = 1 occur because only one of the two species HA and A- is present a t significant levels, so that there is no buffering. However, and this is rarely
276 / Iournol o f Chemical Education
commented upon, buffering is again in evidence in the regions in which T is, respectively, either negative or larger than one. In the first region buffering is caused by the couple HaO+/HzO, because an aqueous solution of strong acid contains significant amounts of HaO+ and H,O. In the second region, the couple H,O/OH- is responsible, because a solution of strong base contains substantial quantities of OH- and H,O. Water may thus exhibit a buffering action, even though it is not useful as buffer in the conventional sense. The effect is related to itsleveling action on strong acids and bases. The magnitudes of the near-vertical steps in the curves in Figure 4 are related to the concentrations of acid and base used. The curves shown apply to concentration of 0.1 M. For smaller concentrations the steps are ~ m a l l e r . ~ The treatment can he extended to titrations of solutions that contain several acid-base couples. Equations (3)-(6) then contain additional terms, one for each new acid-base couple. For each of these an anxiliary diagram is required, and their combination results in a titration curve with one or several steps, depending on the relative positions of the anxiliary diagrams. The case of polyprotic acids is similar, except for the interde pendence of the diierent species that makes for slightly more involved algebra. To illustrate, the formulas for a diprotic acid H2Awith successive acid constants K1 and K z are c. = LKAI
[H+]
+ [HA-I + [A?
(8)
+ cs = 101%-I+ [HA-] + 2[A-] (proton condition)
(9)
The treatment can also be applied to titration curves in nonaqueous systems. Let the solvent be denoted by HSlv and let its autoprotolysis reaction be ZHSlv = HsS1vC
+ Slv-
(11)
with the equilibrium constant KH,. Replacement in eqn. (5) of K, by Ksr., of [H+] by [HzSlv+],and of K. by the acid constant of HA in the solvent used yields a correct expression for r . The titration curves have the same general form as they do in water. In conclusion, the presentation of titration curves suggested here commends itself by its mathematical simplicity and by the insight it provides into the interplay between the properties of the solvent and of the acid-base couple being titrated. --
(LAITINEN,H. A., "Chemical Analysis, an Advanced Text and Reference," MeGraw-Hill Rook, Co., Inc., New York, 1960, pp. 39 and 41.
