Inflection Points on Some S-Shaped Curves

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 σ = σ(pH) involved with surface tension, σ.


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 η , where Applying the relation at the inflection point on the curve y = y(x) we have 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 σ = σ(pH) for surface tension (y = σ, x = pH).

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 V 0 mL of C 0 mol/L HCl as titrand (D) with V mL of C mol/L NaOH as titrant (T).At V = V eq , the fraction titrated i.e., CV eq = C 0 V 0 .In this D+T system, the titration curve V = V(pH) has the form where To facilitate the calculations, it is advisable to rewrite (6) into the form ( ) From ( 5) and ( 6) we get 0 0 , where ) ( ) Setting d 2 V/dpH 2 = 0 and writing ( ) From (11) we obtain for z = z inf ( ) ( ) and then for Analogous result can be obtained for titration of V 0 mL of C 0 mol/L NaCl with V mL AgNO 3 [5].Denoting [Ag + ][Cl -] = K so we get (13), where [5] ( ) At pK so = 9.75 for AgCl, V 0 = 100 mL, C 0 = 10 -4 and C = 10 -3 , we get V eq -V inf = 0.16 mL.

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 σ = σ(pH) obtained on the basis of Szyszkowski's empirical formula [6] [ ] ( ) expressing the relationship between surface tension σ and concentration [HL] of uncharged form HL of an aliphatic fatty acid as a surfactant in aqueous media; σ 0 -surface tension of pure water, a, b-constants.
Denoting [ ] we get, by turns: ) ) From Equation (17) it results that the abscissa (pH inf ) corresponding to inflection point does not overlap with pK 1 value for HL.