Some Observations

The paper by de Levie (J. Chem. Educ. 1993, 70, 209-217) contains a discussion on the general form of titration curves for a variety of titrations that are obtained without using approximation or segmentation. The following discussion concerning the specific cases of strong or weak acid titrations with strong or weak bases is intended to complement de Levies' excellent article.

Strong Acid with Strong Base

For the addition of a volume V_b of a strong monoprotic base MOH of concentration C_b to a volume V_a of a strong monoprotic acid HA of concentration C_a, de Levie states that the general form of the titration curve in this case gives a quadratic equation for [H⁺]. Specifically this is

(1)

where K_w = [H⁺][OH^-] << 1 is constant; typically K_w = 10^-14. The positive root of eq 1 is given by

(2)

and this is the simple inversion that de Levie refers to. Because K_w <<1, the expression in eq 2 enables the determination of the behavior of the titration curve ([H⁺] or pH against V_b) either side of the equivalence point (where C_aV_a - C_bV_b = 0) as follows:

(3)

(4)

by rationalizing the numerator, together with

for V_b = C_aV_a/C_b. The expression in eq 3 is in agreement with the result obtained by calculating [H⁺] as the number of moles of excess acid C_aV_a-C_bV_b divided by the total solution volume V_a+V_b. Similarly, the expression in eq 4 is in agreement with the result obtained by calculating [OH^-] as the number of moles of excess base C_bV_b-C_aV_a divided by the total solution volume V_a+V_b and then using [H⁺] = K_w/[OH^-]. The two segments given by eqs 3 and 4 are joined by a sharp transition around the equivalence point.

Weak Acid with Weak Base

For the addition of a volume V_b of weak monoprotic base B of concentration C_b to a volume V_a of weak monoprotic acid HA of concentration C_a, it is necessary to consider the equilibria HA <--> H⁺+A^- and B+H₂O <--> BH⁺ + OH^- with corresponding equilibrium constants K_a = [H⁺][A^-]/[HA] and K_b = [OH^-][BH⁺]/[B]. de Levie points out, quite correctly, that in this case it is not possible to perform the inversion required to obtain the titration curve. This is because the equation corresponding to eq 1 is a quartic:

(5)

It is possible, however, to solve eq 5 numerically for [H⁺] for a given volume of added base, V_b, and given values V_a, C_a, C_b, and K_w. Varying V_b and determining the corresponding value of [H⁺] enables the whole titration curve to be determined. A reliable numerical method is required to solve eq 5; my preference is the Newton iteration, which can be written as

(6)

where G₂ is an improvement on an initial guess G₁ and f'([H⁺]) is the derivative of f([H⁺]). This process is repeated iteratively until the desired accuracy is achieved. Bearing in mind that the Newton iteration is sensitive to the initial guess, and that eq 5 has four roots, there remains one crucial question. For what initial guesses will the iteration converge to the required root [H⁺]? Fortunately, the answer is simple.

The figure shows a qualitative plot of f([H⁺]) and four roots r₁, r₂, r₃, r₄ can be identified, while the required positive root is [H⁺] = r₄. Now, the geometrical interpretation of the Newton iteration is that the improvement, G₂, is given by the [H⁺] value at the point of intersection of the tangent at the point (G₁, f(G₁)) [with slope f'(G₁)] with the [H⁺] axis. This process is repeated by "drawing" the tangent at (G₂, f(G₂)). Therefore the figure shows that the iteration will be guaranteed to converge on the root r₄ for any initial guess G₁ to the right of r₄. However, since the required root satisfies 0 < r₄ < C_a (the initial concentration of acid), then [H⁺] = C_a is always to the right of the required root and the Newton iteration is guaranteed to converge on r₄ for the initial guess G₁ = C_a, and this will be true as the volume of base added, V_b, varies.

Similar remarks apply to the cases of strong acid-weak base and weak acid-strong base titrations. In these cases, however, the corresponding equation to 5 is a cubic. Nevertheless, the corresponding Newton iterations will converge on the required root satisfying 0 < [H⁺] < C_a from the initial guess G₁ = C_a for any volume of added base V_b. Again, applying the iteration as V_b varies enables the whole titration curve to be determined.