Inflection Points on Some S-Shaped Curves

Anna M. Michałowska-Kaczmarczyk, Tadeusz Michałowski, Agustin G. Asuero

Department of Analyti?cal Chemistry, The University of Seville, Seville, Spain.
Department of Oncology, The University Hospital in Cracow, Cracow, Poland.
Faculty of Engineering and Chemi?cal Technology, Technical University of Cracow, Cracow, Poland.

DOI: 10.4236/jasmi.2014.41004                

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

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.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Sigmoid Function. http://en.wikipedia.org/wiki/Sigmoid_function
[2]	Greschonig, H. and Glatter, O. (1986) Determination of Equivalence Points of Sigmoidal Potentiometric Titration Curves. Microchimica Acta, 89, 401-409. http://link.springer.com/article/10.1007%2FBF01207332
[3]	Asuero, A.G. and Michaowski, 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]	Michalowski, T. (1981) Some Remarks on Acid-Base Titration Curves. Chemia Analityczna, 26, 799-813.
[5]	Michalowski, T. and Stepak, R. (1985) Evaluation of the Equivalence Point in Potentiometric Titrations with Application to Traces of Chloride. Analytica Chimica Acta, 172, 207-214. http://dx.doi.org/10.1016/S0003-2670(00)82608-2
[6]	von Szyszkowski, B. (1908) Experimentelle Studien über kapillare Eigenschaften der wasserigen Losungen von Fettsauren. (Experimental studies of the capillary properties of aqueous solutions of fatty acids), Zeitschrift für Physikalische Chemie, 64, 385-414.