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

This paper recalls the well-known property of different functions represented by the curves with sigmoidal shape (S-shape) [

Generalizing, we refer to a monotonic function y = y(x). The inflection point (x_{inf}, y_{inf}) corresponds to maximal slope

Applying the relation

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

and then at dy/dx ¹ 0 we get

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 [

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).

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

From (8)

Setting d^{2}V/dpH^{2} = 0 and writing

From (11) we obtain for z = z_{inf}

and then for V = V_{inf} [

Analogous result can be obtained for titration of V_{0} mL of C_{0} mol/L NaCl with V mL AgNO_{3} [^{+}][Cl^{–}] = K_{so} we get (13), where [

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.

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 [

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; s_{0}—surface tension of pure water, a, b—constants.

Denoting

we get, by turns:

Putting d^{2}s/dpH^{2} = 0, from Equation (16) we get [H^{+}]∙(1 + b∙C)^{1/2} = K_{1}, and then

From Equation (17) it results that the abscissa (pH_{inf}) corresponding to inflection point does not overlap with pK_{1} value for HL.