Journal of Analytical Sciences, Methods and Instrumentation
Vol.4 No.1(2014), Article ID:44200,4 pages DOI:10.4236/jasmi.2014.41004

Inflection Points on Some S-Shaped Curves

Anna M. Michałowska-Kaczmarczyk1, Tadeusz Michałowski2, Agustin G. Asuero3

1Department of Oncology, The University Hospital in Cracow, Cracow, Poland

2Faculty of Engineering and Chemi­cal Technology, Technical University of Cracow, Cracow, Poland

3Department of Analyti­cal Chemistry, The University of Seville, Seville, Spain


Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 9 December 2013; revised 9 January 2014; accepted 15 January 2014


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 (xinf, yinf) corresponds to maximal slope, where


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 [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 (Veq). 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 = Veq, the fraction titrated


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




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


From (5) and (6) we get


From (8)


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


From (11) we obtain for z = zinf


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


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


At pKso = 9.75 for AgCl, V0 = 100 mL, C0 = 10–4 and C = 10–3, we get Veq – Vinf = 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]


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:


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


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


  1. Sigmoid Function.
  2. Greschonig, H. and Glatter, O. (1986) Determination of Equivalence Points of Sigmoidal Potentiometric Titration Curves. Microchimica Acta, 89, 401-409.
  3. Asuero, A.G. and Michałowski, T. (2011) Comprehensive Formulation of Titration Curves for Complex Acid-Base Systems and Its Analytical Implications. Critical Re­views in Analytical Chemistry, 41, 151-187.
  4. Michałowski, T. (1981) Some Remarks on Acid-Base Titration Curves. Chemia Analityczna, 26, 799-813.
  5. Michałowski, T. and Stępak, R. (1985) Evaluation of the Equivalence Point in Potentiometric Titrations with Appli­cation to Traces of Chloride. Analytica Chimica Acta, 172, 207-214.
  6. von Szyszkowski, B. (1908) Experimentelle Studien über kapillare Eigenschaften der wässerigen Lösungen von Fettsäuren. (Experimental studies of the capillary properties of aqueous solutions of fatty acids), Zeitschrift für Physikalische Chemie, 64, 385-414.