ABSTRACT

This paper refers to inflection point—the fundamental property of S-shaped curves. In this paper, the inflection points are related to pH titration curve pH = pH(V), and to the curve ( = ((pH) involved with surface tension, (.

Keywords:pH Titration; Surface Tension

1. Introduction

This paper recalls the well-known property of different functions represented by the curves with sigmoidal shape (S-shape) [1] , involved with in­flection (inf) point. An inflection point is the point on 2D plane where the curvature of the curve changes direction. The S-shape is characteristic, among others, for potentiometric titration curves [2] . Different methods of equivalence (eq) point determination are based on location of the inflec­tion point on the curves pH = pH(V) or E = E(V), where E—potential, V—volume of titrant added. The inflection points are registered also in different physicochemical studies.

Generalizing, we refer to a monotonic function y = y(x). The inflection point (x_inf, y_inf) corresponds to maximal slope, where

(1)

Applying the relation

(2)

at the inflection point on the curve y = y(x) we have

(3)

and then at dy/dx ¹ 0 we get

(4)

It means that the maximal slope is equivalent with the relation (4) valid for the inverse function x = x(y). This property is important for pH titration curves; namely, the functions V = V(pH) assume relatively simple form [3] .

In this paper, we refer to a simple acid-base ti­tration (y = pH, x = V), and to the relationship s = s(pH) for surface tension (y = s, x = pH).

2. Relation between Equivalence and Inflection Points in pH Titration

The main task of titration made for analytical purposes is the estimation of the equivalence vol­ume (V_eq). Let us consider the simplest case of titration of V₀ mL of C₀ mol/L HCl as titrand (D) with V mL of C mol/L NaOH as titrant (T). At V = V_eq, the fraction titrated

(5)

i.e., CV_eq = C₀V₀. In this D+T system, the titration curve V = V(pH) has the form

(6)

where

(7)

To facilitate the calculations, it is advisable to re­write (6) into the form

(8)

From (5) and (6) we get

(9)

From (8)

(10)

Setting d²V/dpH² = 0 and writing, from (10) we get, by turns,

(11)

From (11) we obtain for z = z_inf

(12)

and then for V = V_inf [3] [4]

(13)

Analogous result can be obtained for titration of V₀ mL of C₀ mol/L NaCl with V mL AgNO₃ [5] . Denoting [Ag⁺][Cl^–] = K_so we get (13), where [5]

(14)

At pK_so = 9.75 for AgCl, V₀ = 100 mL, C₀ = 10^–4 and C = 10^–3, we get V_eq – V_inf = 0.16 mL.

3. A Comment to Szyszkowski Formula

Many physicochemical processes are graphi­cally represented by the curves with the sigmoidal shape. In this section, we refer to the function s = s(pH) obtained on the basis of Szyszkowski’s em­pirical formula [6]

(15)

expressing the relationship between surface tension s and concentration [HL] of uncharged form HL of an aliphatic fatty acid as a surfactant in aqueous media; s₀—surface tension of pure water, a, b—constants.

Denoting and

we get, by turns:

(16)

Putting d²s/dpH² = 0, from Equation (16) we get [H⁺]∙(1 + b∙C)^1/2 = K₁, and then

(17)

From Equation (17) it results that the abscissa (pH_inf) corresponding to inflection point does not overlap with pK₁ value for HL.

References