Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4© Springer-Verlag Berlin Heidelberg 2013

Heike Kahlert and Fritz Scholz

Acid-Base Diagrams

Heike Kahlert

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Fritz Scholz

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

ISBN 978-3-642-37901-7e-ISBN 978-3-642-37902-4

Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013943686

© Springer-Verlag Berlin Heidelberg 2013

English edition of “Säure-Base-Diagramme” (Springer Spektrum), 2013, ISBN 978-3-642-37703-7

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This book is dedicated to Prof. Dr. Günter Henrion on the occasion of his 80th birthday. It is an expression of esteem for an inspirational teacher, who first sparked our love of pH-logc i diagrams during our chemistry studies at Humboldt University, BerlinHeike Kahlert and Fritz Scholz

Heike Kahlert and Fritz Scholz

Preface

The understanding of acid–base equilibria is of basic importance for chemistry, and thus also for biochemistry, biology, environmental sciences, etc. Hardly any chemical technique, any biochemical process, any environmental phenomenon can be understood without a profound knowledge of the underlying acid–base equilibria on a quantitative basis. However, even for the simplest chemical systems the mathematical calculations can be rather complicated, and the desire will arise to describe the systems by well-chosen approximations. For all these requirements there is one powerful means, the so-called acid–base diagrams (pH-log c i diagrams), which permit a simple presentation of the dependencies of the concentrations of all participating species as a function of pH of the aqueous solutions. With their help, it is easy to find the possible simplified equation which permits a straightforward calculation of special cases. These diagrams also permit the construction of titration diagrams. The present book is the result of the many years of teaching experience of the authors, during which they have learned what the usual problems of understanding are which students have in using these diagrams. The book has been written because there was no other textbook which presented the fundamentals and applications of pH-log c i diagrams in the necessary depth and with the desired simplicity. It was not the goal to describe these diagrams comprehensively with all imaginable special cases, but the authors had the aim of giving clear and straightforward instruction on how to construct and use these tools for problem-solving. We hope that this book will guide students of chemistry, biochemistry, biotechnology, biology, pharmacy, physics, environmental sciences, geosciences, hydrology, medicine, etc. in their attempts to handle acid–base equilibria.

Heike Kahlert

Fritz Scholz

Greifswald

January 2013

Contents

1 Introduction 1

Literature 2

2 The Math Behind the pH-log c i Diagrams 5

Literature 18

3 Constructing pH-log c i Diagrams 19

3.​1 Monobasic Acids 19

3.​2 Dibasic Acids 22

3.​3 Tribasic Acids 25

3.​4 Tetrabasic Acids 29

4 The Application of pH-log c i Diagrams for Graphical Estimation of the pH of Solutions and for the Derivation of Useful Simplified Equations 33

4.​1 Monobasic Acids and Their Corresponding Bases 36

4.​1.​1 Very Strong Acids and Their Corresponding Very Weak Bases 36

4.​1.​2 Strong Acids and Their Corresponding Weak Bases 40

4.​1.​3 Weak Acids and Their Corresponding Strong Bases 46

4.​1.​4 Very Weak Acids and Their Corresponding Very Strong Bases 51

4.​1.​5 Ranges of Validity of the Simplified Equations for Monobasic Acids 55

4.​2 Dibasic Acids, Ampholytes, and Diacidic Bases 56

4.​2.​1 Dibasic Acids 56

4.​2.​2 Simplified Equation for Ampholytes 62

4.​2.​3 Diacidic Bases 68

4.​3 Salt Solutions with Protolyzing Anions and Cations 72

4.​4 Examples 76

4.4.1 The pH-log c i Diagram of Water 76

4.​4.​2 Acetic Acid/​Acetate 77

4.​4.​3 Hydrogen Sulfide 78

4.​4.​4 Phosphoric Acid 79

4.​4.​5 Ascorbic Acid 80

4.​4.​6 Acetylsalicylic Acid 81

4.​4.​7 Benzoic Acid 83

4.​4.​8 Glycine 84

4.​4.​9 Aspartic Acid 85

4.​4.​10 Ethylenediaminet​etraacetic Acid (EDTA) 86

Literature 88

5 The Use of pH-log c i Diagrams for the Construction of Titration Diagrams 89

5.​1 Titration of Hydrochloric Acid of Various Concentrations with Sodium Hydroxide Solution 91

5.​2 Titration of Sodium Hydroxide Solution with Hydrochloric Acid 93

5.​3 Titration of Different Concentrations of Acetic Acid with Sodium Hydroxide 94

5.4 Titration of Moderately Strong Acids Having Different p K a Values with Sodium Hydroxide 96

5.​5 Titration of Sulfuric Acid 98

5.​6 Titration of Dibasic Amino Acids 99

5.​7 Comparison of the Titrations of an Acid and its Corresponding Base:​ (a) Ammonium Ions with Sodium Hydroxide, (b) Ammonia with Hydrochloric Acid 100

6 Titration Errors 103

6.​1 Systematic Titration Errors 103

6.1.1 Systematic Titration Errors in Titrations of Acids with p K a Values Between 0 and 14 with Very Strong Bases (e.g., NaOH) 104

6.1.2 Systematic Titration Errors of Titrations of Bases with p K b Values Between 0 and 14 with a Very Strong Acid (e.g., Hydrochloric Acid) 106

6.​1.​3 Systematic Titration Errors of Titrations of Very Strong Acids with Very Strong Bases and Vice Versa 108

6.​2 Random Titration Errors 110

Literature 111

Appendix A: Derivation of the Exact Functions and the Equations of the Asymptotes for Multibasic Acids113

Index135

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4_1© Springer-Verlag Berlin Heidelberg 2013

1. Introduction

Heike Kahlert1  and Fritz Scholz¹

(1)

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Abstract

Acid–base diagrams, better called pH-logc i diagrams, are the most beneficial tools which have been developed to describe acid–base equilibria. They present the interrelations between the logarithm of all equilibrium concentrations c of the species i and the pH value of the solution. They make it possible, without any math calculations, to find the approximate pH of solutions of acids and bases and buffers. They can be also applied to find out what approximations in the math calculations of the pH (or concentration of any species) are allowed under certain conditions. These acid–base diagrams can also be used to construct titration curves in order to extract the most important data, such as pH at the equilibrium point, etc. A good training in using these diagrams is necessary for a wide range of science studies. The present book provides the basics as well as a good number of applications.

Acid–base diagrams, better called pH-logc i diagrams, are the most beneficial tools that have been developed to describe acid–base equilibria. They present the interrelations between the logarithm of all equilibrium concentrations c of the species i and the pH value of the solution. They make it possible to find the approximate pH of solutions of acids and bases and buffers without any math calculations. They can be applied to find out what approximations in the math calculations of the pH (or concentration of any species) are allowed under certain conditions. These acid–base diagrams can also be used to construct titration curves to extract the most important data, such as pH at the equilibrium point. Good training in the use of these diagrams is necessary for a wide range of science studies. This book provides the basics as well as a good number of applications.

pH-logc i diagrams have several fathers (Fig. 1). The Danish chemist Niels Bjerrum introduced the coordination system in 1915 [1], the Swedish chemists H. Arnfelt [2] and G. A. Ölander [3] further developed them, and finally the Swedish chemists G. Hägg [4] and L. G. Sillén [5, 6] popularized them on an international scale. The latter led to these diagrams being called Hägg diagrams in Germany and Sillén diagrams in the USA [7].

Fig. 1

a Niels Bjerrum (March 11, 1879, Copenhagen, Denmark – September 30, 1958, Copenhagen, Denmark). Bjerrum studied chemistry in Leipzig, Zurich, Paris and Berlin. He obtained a Ph.D. from the University of Copenhagen in 1908. From 1914 to 1949 he was Professor at the Royal Veterinary and Agricultural College in Copenhagen. His research concerned acid–base equilibria and electrolyte solutions. (Copyright Morten J. Bjerrum Reproduced from [8]). b Gunnar Hägg (December 14, 1903, Stockholm, Sweden – May 28, 1986, Uppsala, Sweden) studied chemistry in Stockholm, and in 1936 he became Professor in Uppsala. His main research area was X-ray diffraction. (Archive, University of Uppsala). c Lars Gunnar Sillén (July 11, 1916, Stockholm, Sweden – July 23, 1970, Danderyd/Stockholm) studied chemistry in Stockholm, and was a Professor of Inorganic Chemistry at the Royal Institute of Technology, Stockholm from 1950. At the beginning of his career he also did research in X-ray diffraction, but later he turned his interest to chemical solution equilibria and sea chemistry. Sillén was a pioneer in applying computer programs to the calculation of chemical equilibria, and his research group was most productive in providing new estimations of highly reliable equilibrium constants (Svenskt biografiskt lexikon)

This book provides a consistent presentation of material, and the reader can use it without consulting additional literature. Of course, there are a number of textbooks and monographs that can be consulted [7, 9–16], as well as a number of original journal papers.

The complete book is based on the acid–base theory of Brønsted (Brönsted) and Lowry, according to whom acids are proton donators and bases are proton acceptors. All other acid–base theories are not discussed here, and we refer to the respective literature [9, 17].

Literature

1.

Bjerrum N (1915) Sammlung chemisch und chemisch-technischer Vorträge 21:1

2.

Arnfeld H (1937) Svensk Kemisk Tidskrift 49:96 (cited from: Hägg G (1950) Die theoretischen Grundlagen der analytischen Chemie. Birkhäuser, Basel)

3.

Ölander A (1939) Kompendium i teoretisk kemi [Tekniska Högskolan Stockholm] (cited from: Hägg G (1950) Die theoretischen Grundlagen der analytischen Chemie. Birkhäuser, Basel)

4.

Hägg G (1950) Die theoretischen Grundlagen der analytischen Chemie. Birkhäuser, Basel

5.

Sillén LG, Lange PW, Gabrielson CO (1952) Problems in physical chemistry. Prentice Hall, New York

6.

Sillén LG (1959) Graphical presentation of equilibrium data. Kapitel 8. In: Kolthoff IM, Elving PJ, Sandell EB (eds) Treatise on analytical chemistry. The Interscience Encyclopedia, New York

7.

Butler JN (1998) Ionic equilibrium. Solubility and pH calculations. Wiley, New York

8.

Scholz F (2011) Bjerrum, Niels. In: Bard AJ, Inzelt G, Scholz F (eds) Electrochemical dictionary. Springer, Berlin

9.

Bliefert C (1978) pH-Wert-Berechnungen. Verlag Chemie, Weinheim

10.

Seel F (1976) Grundlagen der Analytischen Chemie, 6th edn. Verlag Chemie, Weinheim

11.

Blumenthal G (1993) Grundlagen der quantitativen chemischen Analyse. In: Kolditz L (ed) Anorganikum, 13th edn. Barth, Leipzig

12.

Otto M (2011) Analytische Chemie, 4th edn. Wiley-VCH, Weinheim

13.

Kotrlý S, Šůcha L (1985) Handbook of chemical equilibria in analytical chemistry. E. Horwood, Chichester

14.

Šůcha L, Kotrlý S (1972) Solution equilibria in analytical chemistry. Van Nostrand Reinhold Comp, London

15.

Inczedy J (1976) Analytical applications of complex equilibria. E. Horwood/Akadémiai Kiadó, Chichester/Budapest

16.

Christian GD (2004) Analytical chemistry. Wiley, New York

17.

Kahlert H (2012) Acid–base theories. In: Bard AJ, Inzelt G, Scholz F (eds) Electrochemical dictionary, 2nd edn. Springer, Berlin, pp 4–6

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4_2© Springer-Verlag Berlin Heidelberg 2013

2. The Math Behind the pH-logc i Diagrams

Heike Kahlert1  and Fritz Scholz¹

(1)

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Abstract

Figure 2 shows the coordination system as used for the construction of pH-logc i diagrams of aqueous solutions.

Figure 2 shows the coordination system as used for the construction of pH-logc i diagrams of aqueous solutions.

Fig. 2

Coordination system of pH-logc i diagrams: normally, only the second quadrant (the colored part) is used, and constructed as a quadratic diagram because only that range of pH and logc i is of significance in aqueous solutions

In an aqueous solution of the acid $$ \mathrm{ HB} $$, the following equilibria are established:

(1)

(2)

The equilibrium (1) is characterized by the equilibrium constant K a:

(3)

This equilibrium constant is called acidity constant. It has no unit. The quantities $$ a_{\mathrm{ i}}^{\mathrm{ c}} $$ are the activities (to be more exact the concentration activities, hence the superscript c). They are related to the concentrations $$ {c_{\mathrm{ i}}} $$ in mol L⁻¹ by the relationship:

(4)

The activity coefficients $$ {\gamma_{\mathrm{ i}}} $$ can be understood as correction coefficients by which the concentrations have to be multiplied to fulfill the law of mass action. The quantity $$ c^* $$ is the concentration of the species in its reference state, that is, for concentration activities this is the concentration 1 mol L⁻¹. That quantity leads to unit-free activities. In fact, Eq. (3) also contains the activity of water; however, the molar ratio activity is $$ a_{{{{\mathrm{ H}}_2}\mathrm{ O}}}^{\mathrm{ mb}} $$ = 1, and it is not necessary to write this term. Because the activity coefficients approach 1 for diluted solutions, it is customary (and no bad approximation) to formulate the acidity constants on the basis of equilibrium concentrations, and Eq. (3) can be written as follows:

(5)

The so defined acidity constant has a unit, here mol L⁻¹, which in calculations has the advantage that the units can be easily used for checking the general correctness of derived equations. For the sake of simplicity, in this book concentrations are used throughout, and the pH-logc i diagrams are generally not applicable to activities. At concentrations higher than 10⁻³ mol L⁻¹, one needs to be always alert to the possibility that the activity coefficients may deviate from 1.0, which may considerably affect the pH.

The purpose of constructing a pH-logc i diagram is to display the logarithm of the concentrations of all chemical species as a function of pH. In the case of a mono basic acid $$ \mathrm{ HB} $$ in water, the following species have to be considered: $$ \mathrm{ HB} $$, $$ {{\mathrm{ B}}^{-}} $$, $$ {{\mathrm{ H}}_2}\mathrm{ O} $$, $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$. In this book, the solvated proton will be always written as $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$, although more complex hydrates such as $$ {{\mathrm{ H}}_5}\mathrm{ O}_2^{+} $$, $$ {{\mathrm{ H}}_7}\mathrm{ O}_3^{+} $$ and $$ {{\mathrm{ H}}_9}\mathrm{ O}_4^{+} $$ may even more precisely describe it. Because water is the main constituent of aqueous solutions, and the concentration of water is practically constant, it is not necessary to display it in the diagrams (see Sect. 4.​4.​1). The concentration activity of $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ determines the pH of a solution:

(6)

The symbol p stands for “–log.” It was introduced in 1909 by the Danish chemist Søren Peter Lauritz Sørensen (Fig. 3) [1]. Initially, Sørensen used to write $$ \mathrm{ p}_{\mathrm{ H}}^{+} $$; however, this was not generally accepted, and people wrote pH instead. In line with the suggestions of the International Union of Pure and Applied Chemistry, pH is not written in italics, although it is a symbol of a physical quantity [2]. For a long time, it was believed that Sørensen had chosen “p” as a symbol for “power” (“Potenz” in German); however, a recent historical study has revealed that he used it simply as a symbol for a variable [3], standing for the $$ {{\mathrm{ H}}^{+}} $$ concentration C p or 10−p. In modern style this is: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=1{0^{{-\mathrm{ pH}}}} $$! Nowadays, the definition $$ \mathrm{ pX}=- \log {c_{\mathrm{ X}}} $$ is generally used in chemistry for a species X, and also for various equilibrium constants, e.g., $$ \mathrm{ p}{K_{\mathrm{ a}}}=- \log {K_{\mathrm{ a}}} $$ (see further down; here, K has to be written in italics). The equilibria (1) and (2) connect the activities (concentrations) of all species ($$ \mathrm{ HB} $$, $$ {{\mathrm{ B}}^{-}} $$, $$ {{\mathrm{ H}}_2}\mathrm{ O} $$, $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$, and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$) with each other, i.e., it is impossible to change the activity of one species without affecting all others. Hence, it makes sense to display these relations in a diagram. In aqueous solutions the range of pH values from 0 to 14 is of special importance, as the pH of real solutions usually falls within these limits. Although pH values below 0 and above 14 are in principle possible, such limitation of the diagrams is well reasoned. Similarly, the range of concentrations $$ {c_{\mathrm{ i}}} $$ can be limited, because only those between 1 mol L⁻¹ and 1.0 × 10⁻¹⁴ mol L⁻¹ are of any importance in real solutions (of course, there may be special cases with concentrations above 1 mol L⁻¹, but then the deviations of the activity coefficient from 1.0 will be very remarkable, and special calculations are necessary). Hence, on the logc i axis, the range from 0 to −14 will be plotted in the diagrams. This logc i range and the mentioned pH range (0–14) are situated in the 2nd quadrant of the coordination system shown in Fig. 2. Since usually data in this quadrant will be of interest, it is normal to construct a quadratic diagram as displayed in Fig. 4.

Fig. 3

Søren Peter Lauritz Sørensen (January 9, 1868, Havrebjerg, Denmark – February 12, 1939, Charlottenlund, Denmark)

Fig. 4

This is the quadratic diagram with the range of pH and logc i values important in aqueous solution

The relation between $$ \log {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}} $$ and pH, as well as that between $$ \log {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$ and pH, are the same for all acid–base systems, because these quantities are related to each other by the equilibrium (2), the autoprotolysis of water. Also here, the pH will be approximately calculated using the concentration:

(7)

Although this equation is not completely correct from the point of view of physical chemistry, it suffices for our purpose. The equilibrium constant of the autoprotolysis (2) is

(8)

and it can be approximately described by concentrations as follows:

(9)

$$ {K_{\mathrm{ w}}} $$ is called the ion product of water. It is a temperature-dependent quantity. At 25 °C it holds that: $$ - \log {K_{\mathrm{ w}}}=\mathrm{ p}{K_{\mathrm{ w}}}=13.995 $$. Equation (7) defines a straight line in the pH-logc i diagram (cf. Fig. 5).

Fig. 5

The pH-logc i diagram with the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ line of water

Equation (9) can be rearranged by using 10⁻¹⁴ mol² L⁻² for $$ {K_{\mathrm{ w}}} $$:

(10)

(11)

(12)

(13)

(14)

Equation (14) also describes a straight line in the pH-logc i diagram. This is the so-called $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ line (cf. Fig. 6).

Fig. 6

The pH-logc i diagram with the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ lines of water

Figure 6 displays only the components $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$. These species are part of the two acid–base systems $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$/ $$ {{\mathrm{ H}}_2}\mathrm{ O} $$ and $$ {{\mathrm{ H}}_2}\mathrm{ O} $$/$$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$. $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ is the strongest acid which can exist in water, and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ is the strongest base in water. In the case of all acids $$ \mathrm{ HB} $$ which are stronger than $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ in water, the equilibrium (1) is so much shifted to the right side that one can claim a complete conversion to $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ {{\mathrm{ B}}^{-}} $$, although in chemical equilibria “complete” never means 100 % and, at least theoretically, a certain concentration of HB will be always present. Bases which are much stronger than $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ in water will be almost “completely” converted to $$ \mathrm{ HB} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ according to the reaction

(15)

This leads to the effect that all acids which are much stronger than $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ appear to be of the same strength of acidity, i.e., concentration of $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ (provided they have equal analytical concentrations of HB). All bases which are much stronger than $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ will appear also to be of the same strength, i.e., exhibiting the same concentrations of $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ (at equal analytical concentrations of $$ {{\mathrm{ B}}^{-}} $$). This phenomenon is known as the levelling effect of water. It is customary to classify acids and bases according to their strength; however, one should remember that any such classification is arbitrary. It certainly makes sense to designate acids which are stronger than $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ in water as very strong acids, and their corresponding bases as very weak bases. It is impossible to define a strict border line above which acids are very strong: the reason is that there is no way to define the $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ value of $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ in water on a strict thermodynamic basis [4]. Similarly, no $$ \mathrm{ p}{K_{\mathrm{ b}}} $$ value can be defined for $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$. Nevertheless it is reasonable to assume that the $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ of $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ is near to 0 (more precisely, slightly below zero) and the $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ of $$ {{\mathrm{ H}}_2}\mathrm{ O} $$ near to 14 (more precisely, slightly above) (see Sect. 4.​4.​1). Thus, distinguishing very strong, strong, moderately strong, weak and very weak acids is only possible with soft borders (see Fig. 7):

Fig. 7

Classification of acids and bases according to their strength in water

Since the calculation of $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ values according to Eq. (3) is based on the transfer of a proton from HB to $$ {{\mathrm{ H}}_2}\mathrm{ O} $$, it is the basicity of $$ {{\mathrm{ H}}_2}\mathrm{ O} $$ that determines the strength of an acid in water. Hence, all $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ values defined by Eq. (3) are valid for water only, and in other solvents different $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ values have to be used [5]. Table 1 gives a number of $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ values of acids in water.

Table 1

$$ {\rm p}{K_{{\rm a}}} $$ values of acids in water. Most data are from Ref. [10]

Acid–base pair

pK a values [ionic strength]a (25 °C)

ca. –10

ca. –10

ca. –9

ca. –7

−3 [0]

−1.34 [0]

$$ {{\mathrm{ C}}_2}{{\mathrm{ O}}_4}{{\mathrm{ H}}_2}/{{\mathrm{ C}}_2}{{\mathrm{ O}}_4}{{\mathrm{ H}}^{-}} $$ (oxalic acid/hydrogenoxalate)

1.271

1.99 [0], 1.55 [0.1]

2.148 [0], 2.0 [0.1]

2.19 [0], 2.83 [0.1]

3.17 [0], 2.92 [01], 2.96 [1.0]

$$ \mathrm{ HCOOH}/\mathrm{ HCO}{{\mathrm{ O}}^{-}} $$ (formic acid/formate)

3.752

$$ {{\mathrm{ C}}_2}{{\mathrm{ O}}_4}{{\mathrm{ H}}^{-}}/{{\mathrm{ C}}_2}\mathrm{ O}_2^{2- } $$ ( hydrogenoxalate/oxalate)

4.266

$$ {{\mathrm{ H}}_3}\mathrm{ CCO}\mathrm{ OH}/{{\mathrm{ H}}_3}\mathrm{ CCO}{{\mathrm{ O}}^{-}} $$ (acetic acid/acetate)

4.756

4.99 [0], 5.69 [0.1]

6.35 [0], 6.16 [0.1], 6.02 [1.0]

7.02 [0], 6.83 [0.1], 6.61 [1.0]

7.199[0], 6.72 [0.1], 6.46 [1.0]

9.244 [0], 9.29 [0.1], 9.40 [1.0]

9.21 [0], 9.01 [0.1], 8.95 [1.0]

9.5 [0]

10.33 [0], 10.0 [0.1], 9.57 [1.0]

12.35 [0], 11.74 [0.1], 10.79 [3.0]

13.9 [0], 13.8 [1.0]b

ca. 23

aThe ionic strength can affect the $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ values. Normally those data are tabulated which are extrapolated for an ionic strength of zero

bFor the second protolysis stage of hydrogen sulfide, even $$ \mathrm{ p}{K_{\mathrm{ a}}} $$ values of 16 and 18 are given in the literature

A closer look at the data in Table 1 shows that the pK a values of very strong acids (pK a below 0) and very weak acids (pK a larger than 14) are not given with the high precision of acids with pK a values between 0 and 14. The reason is that for the very strong acids, the activities of HB and for the very weak acids, the activities of $$ {{{\rm B}}^{-}} $$ are extremely small, and a precise experimental determination is difficult. Thus, these pK a values are not known with very high precision.

Another interesting feature is exhibited by hydrofluoric acid (HF): in comparison to HCl, HBr, and HI the pK a value of HF is surprisingly large (−4 would probably be a better understandable value). This can be easily explained with the following equilibria:

(a)

(b)

The species $$ \left[ {{{\rm {F}}^{-}}{{{\rm H}}_3}{{{\rm O}}^{+}}} \right] $$ is a very stable ion pair having no net charge. The pK a value is calculated according to $$ {K_{{\rm a}}}=\frac{{{a_{{{{{ \rm F}}{-}}}}}{a_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}}}{{{a_{{\rm H}{\rm F}}}}} $$ using the activities of fluoride ions, hydronium ions, and HF. For the latter, however, the classical experimental data (potentiometry, conductometry) do not distinguish between HF and the ion pair $$ \left[ {{{{\rm F}}^{-}}{{{\rm H}}_3}{{{\rm O}}^{+}}} \right] $$, and thus a too large pK a value results. This value does not reflect the real tendency of HF to transfer a proton to water [6]. The reason for the high stability of the ion pair is the small radius of the fluoride ion, which leads to a strong coulomb interaction with the hydronium ion.

The table also shows pK a values of metal-aqua ions. At first glance, it may be surprising that metal-aqua ions are Brønsted acids. However, it is easy to understand that the coordinated water molecules are affected by the positively charged metal ions. This weakens the O–H bonds in the coordinated water, and thus the proton of coordinated water molecules is more easily transferred to a free water molecule than a proton from a free water molecule to another free water molecule. The formation of hydroxo complexes from metal-aqua complexes is of immense importance: the hydroxo complexes have a strong tendency to undergo condensation reactions leading to dimeric, trimeric, oligomeric complexes, finally even to the formation of insoluble metal oxide hydrates. The driving force for these condensation reactions is the entropy gain by releasing water molecules. These reactions have shaped the world. Before the rise of photosyntheses on earth, the atmosphere did not contain free oxygen, and the ocean held large amounts of dissolved iron(II) ions. Oxidation by the oxygen formed in photosyntheses oxidized iron(II) to iron(III), and the iron(III)-aqua ions formed hydroxo complexes which then underwent condensations until finally iron(III) oxide hydrates, and later iron(III) oxide were formed and deposited. Fifty-eight percent of all oxygen ever formed on earth was thus used for iron(III) deposition! The interesting relationships between pK a values of metal-aqua ions and metal ion radii, etc. are described elsewhere [7].

For depicting the functions

(16)

and

(17)

in the pH-logc i diagram (Fig. 6), it is necessary to derive appropriate equations. This requires the law of mass action (5) $$ {K_{\mathrm{ a}}}=\frac{{{c_{{{{\mathrm{ B}}^{-}}}}}{c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}}}{{{c_{\mathrm{ H}\mathrm{ B}}}}} $$, and an equation connecting the quantities $$ {c_{\mathrm{ HB}}} $$ and $$ {c_{{{{\mathrm{ B}}^{-}}}}} $$. According to the reaction (1), $$ \mathrm{ HB} + {{\mathrm{ H}}_2}\mathrm{ O}\ \rightleftarrows\ {{\mathrm{ B}}^{-}} + {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$, the acid can be present in solution either as $$ \mathrm{ HB} $$ or as $$ {{\mathrm{ B}}^{-}} $$. Using concentrations in the unit mol L⁻¹, i.e., number of particles (amount in mol) per volume, the following equation must hold for the overall concentration $$ C_{\mathrm{ HB}}^\mathrm{ \circ } $$:

(18)

Relations of that kind will be called here “amount balances.” The overall concentration $$ C_{\mathrm{ HB}}^{\circ } $$ has to be carefully distinguished from equilibrium concentrations, such as $$ {c_{\mathrm{ HB}}} $$ and $$ {c_{{{{\mathrm{ B}}^{-}}}}} $$: equilibrium concentrations give the concentration of certain species following establishment of equilibrium (in the case of acid–base reactions, that establishment is almost instantaneous, i.e., extremely fast). Thus, equilibrium concentrations are the real concentrations of species. Overall concentrations are often also called analytical concentrations, because they can be determined with classical methods of analysis, such as titrations. It is important to understand that Eq. (18) is only correct when using amount concentrations (molarities), i.e., when giving the concentrations in mol L⁻¹. When using mass concentrations, e.g., in g l⁻¹, Eq. (18) would be wrong, as $$ \mathrm{ HB} $$ has a larger mass than $$ {{\mathrm{ B}}^{-}} $$.

Here follows the derivation of the two functions (16) and (17):

$$ {K_{\mathrm{ a}}}=\frac{{{c_{{{{\mathrm{ B}}^{-}}}}}{c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}}}{{{c_{\mathrm{ H}\mathrm{ B}}}}} $$ (5)

$$ C_{\mathrm{ HB}}^{\circ }={c_{\mathrm{ HB}}}+{c_{{{{\mathrm{ B}}^{-}}}}} $$ (18)

Equations (31) and (32) describe curves in the pH-logc i diagram which strongly resemble that of a hyperbola. In the following discussions the curves will be called hyperbolas, although this is mathematically not fully correct. The two hyperbolas will be designated as the $$ \mathrm{ HB} $$ line and the $$ {{\mathrm{ B}}^{-}} $$ line. It is also good to write the species names at the hyperbola in the diagram. Giving there $$ {c_{\mathrm{ HB}}} $$ and $$ {c_{{{{\mathrm{ B}}^{-}}}}} $$ would be wrong, as the logarithms of concentrations are plotted there.

As can be seen from Fig. 8, each hyperbola has two branches which quickly convert to almost linear lines. Only in the vicinity of $$ \mathrm{ p}\mathrm{ H} = \mathrm{ p}{K_{\mathrm{ a}}} $$ do the hyperbola have a pronounced curvature. This makes it possible to use the asymptotes instead of the hyperbola. In order to do so, the equations of the asymptotes need to be derived. In order to find the asymptotes of the hyperbola described by Eq. (31), it is necessary to study what term in $$ \left( {1+{10^{{\mathrm{ pH}-\mathrm{ p}{K_{\mathrm{ a}}}}}}} \right) $$ will dominate in the two pH ranges, below and above pK a :

Fig. 8

The pH-logc i diagram with the lines of $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$, $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$, HB and $$ {{\mathrm{ B}}^{-}} $$ for acetic acid with $$ C_{\mathrm{ HB}}^{\circ } $$ = 0.1 mol L⁻¹ and pK a = 4.75

(31)

Analogously, it is possible to find the asymptotes of the hyperbola described by Eq. (32) by anlysis of the two terms in $$ \left( {1+{10^{{-\mathrm{ pH}+\mathrm{ p}{K_{\mathrm{ a}}}}}}} \right) $$:

(32)

In most cases it is not necessary to plot the hyperbolas, but it is sufficient to display the asymptotes, i.e., the simple line Eqs. (33), (34), (35) and (36) within the limits of their validity! Figure 9 shows such a pH-logc i diagram.

Fig. 9

The pH-logc i diagram with the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ line, the $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ line and the asymptotes of HB and $$ {{\mathrm{ B}}^{-}} $$ for acetic acid with $$ C_{\mathrm{ HB}}^{\circ } $$ = 0.1 mol L⁻¹ and pK a = 4.75

Looking at Fig. 9 can prompt the following question: how is it possible that the concentration of $$ {{\mathrm{ B}}^{-}} $$ so strongly varies in the range $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}}} $$ whereas that of $$ \mathrm{ HB} $$ is constant, if Eq. (18) $$ C_{\mathrm{ HB}}^{\circ }={c_{\mathrm{ HB}}}+{c_{{{{\mathrm{ B}}^{-}}}}} $$ holds true? The answer is: (a) in the entire range $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}}} $$, both concentrations vary, and that of $$ \mathrm{ HB} $$ appears to be constant only because the asymptote (33) is displayed, and (b) because the logarithm of $$ {c_{{{{\mathrm{ B}}^{-}}}}} $$ is plotted, a straight line with the slope +1 follows, but — with the exception of the vicinity of pH = pK a — very small concentrations $$ {c_{{{{\mathrm{ B}}^{-}}}}} $$ are involved, and thus it follows that $$ \log {c_{\mathrm{ HB}}}\approx \log C_{\mathrm{ HB}}^{\circ } $$ is a good approximation.

The philosophy of pH-logc i diagrams

:

1.

The diagrams show the single acid–base systems, e.g., those of water and HB, independent of each other and no interaction is taken into consideration.

2.

From the diagrams one can find (a) the approximate equilibrium concentrations of all species at any pH, and, the other way round, (b) the approximate pH of solutions having specified equilibrium concentrations of certain species. The determined data are approximate, because the interactions between the single acid–base systems are not taken into account.

The crossing point of the HB and $$ {{{\rm B}}^{-}} $$ lines is the so-called buffer point: an acid–base buffer is a solution which has the capability of keeping its pH value rather unaffected by additions of acids or bases. The pH of buffer solutions can be calculated as follows. Finding the logarithm of Eq. (5) $$ {K_{{\rm a}}}=\frac{{{c_{{{{{\rm B}}^{-}}}}}{c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}}}{{{c_{{\rm H}{\rm B}}}}} $$ gives $$ \log {K_{{\rm a}}}= \log {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}+ \log \frac{{{c_{{{{{\rm B}}^{-}}}}}}}{{{c_{{\rm H}{\rm B}}}}} $$, i.e., $$ \log {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}= \log {K_{{\rm a}}}+ \log \frac{{{c_{{\rm H}{\rm B}}}}}{{{c_{{{{{\rm B}}^{-}}}}}}} $$ or $$ {\rm pH}={\rm p}{K_{{\rm a}}}-\lg \frac{{{c_{{\rm HB}}}}}{{{c_{{{{{\rm B}}^{-}}}}}}} $$. This equation is frequently called buffer equation or also the Henderson–Hasselbalch Equation because the physiologists L. J. Henderson [8] and K. A. Hasselbalch [9] derived it. Clearly, it is nothing else but a form of the law of mass action (Eq. 5). At a concentration ratio $$ \frac{{{c_{{{{{\rm B}}^{-}}}}}}}{{{c_{{\rm HB}}}}}=1, $$ the buffer has a pH equal to the pK a of the involved acid. At that point, the buffer has also his largest capacity to minimize the effect of added acids and bases, i.e., the largest buffer capacity $$ \beta =\frac{{{\rm d}n}}{{\rm d}({\rm pH})} $$. Here, $$ \frac{{{\rm d}n}}{{\rm d}({\rm pH})} $$ is the first derivative of the function $$ n=f({\rm pH}) $$ in which n is the amount of added $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ or $$ {\rm O}{{{\rm H}}^{-}} $$. Thus, the buffer capacity is the reciprocal value of the slope of the dependence of the pH as function of added amounts of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ or $$ {\rm O}{{{\rm H}}^{-}} $$. The buffer capacity decreases for $$ \frac{{{c_{{{{{\rm B}}^{-}}}}}}}{{{c_{{\rm HB}}}}}< 1 $$ and $$ 1< \frac{{{\it c_{{{{{\rm B}}^{-}}}}}}}{{{\it c_{{\rm HB}}}}} $$, and an acceptable buffer capacity can be realized only in the range $$ 0.1\leq \frac{{{c_{{{{{\rm B}}^{-}}}}}}}{{{c_{{\rm HB}}}}}\leq 10 $$, i.e., in the range $$ \left( {{\rm p}{K_{{\rm a}}}-1} \right)\leq {\rm pH}\leq \left( {{\rm p}{K_{{\rm a}}}+1} \right) $$. The pH of buffers depends not only on the ratio $$ \frac{{{c_{{{{{\rm B}}^{-}}}}}}}{{{c_{{\rm HB}}}}} $$, but the buffer capacity depends also on the overall concentration of the acid and base forms $$ {C^{\circ }}={c_{{{{{\rm B}}^{-}}}}}+{c_{{\rm HB}}} $$.

Literature

1.

Sørensen SPL (1909) Compt Rend Lab Carlsberg 8:1 and 396

2.

Cohen ER, Cvitas T, Frey JG et al (eds) (2007) IUPAC quantities, units and symbols in physical chemistry, 3rd edn. RSC, Cambridge, UK, p 70

3.

Nørby JG (2000) Trends Biochem Sci 25:36–37CrossRef

4.

Henrion G, Scholz F, Schmidt W, Zettler M (1982) Z phys Chem (Leipzig) 263:634–636

5.

Himmel D, Goll SK, Leito I, Krossing I (2010) Angew Chem 263:7037–7040CrossRef

6.

Giguere PA (1979) J Chem Educ 56:571–575CrossRef

7.

Tytko KH (1979) Chem unserer Zeit 13:184–194CrossRef

8.

Scholz F (2012) Henderson, Lawrence Joseph. In: Bard AJ, Inzelt G, Scholz F (eds) Electrochemical dictionary, 2nd edn. Springer, Berlin, p 451

9.

Scholz F (2012) Hasselbalch, Karl Albert. In: Bard AJ, Inzelt G, Scholz F (eds) Electrochemical dictionary, 2nd edn. Springer, Berlin, p 446

10.

Kotrlý S, Šůcha L (1985) Handbook of chemical equilibria in analytical chemistry. E. Horwood Ltd., Chichester

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4_3© Springer-Verlag Berlin Heidelberg 2013

3. Constructing pH-logc i Diagrams

Heike Kahlert1  and Fritz Scholz¹

(1)

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Abstract

The construction of the pH-logc i diagram of a monobasic acid HB is stepwise shown in Figs. 10, 11, 12, 13, and 14. The acid has an overall concentration $$ C_{\mathrm{ HB}}^{\bigcirc } $$ and a given $$ \mathrm{ p}K_{\mathrm{ a}}^{\mathrm{ HB}} $$ value:

3.1 Monobasic Acids

The construction of the pH-logc i diagram of a monobasic acid HB is shown stepwise in Figs. 10, 11, 12, 13, and 14. The acid has an overall concentration $$ C_{\mathrm{ HB}}^{\circ } $$ and a given $$ \mathrm{ p}K_{\mathrm{ a}}^{\mathrm{ HB}} $$ value:

Fig. 10

The coordination system of the pH-logc i diagram

Fig. 11

Plotting the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ lines of the water system

Fig. 12

Marking the coordinates $$ \log {c_{\mathrm{ i}}}= \log C_{\mathrm{ HB}}^{\circ }=-1 $$ and $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}}}=-4.75 $$, using here the example of acetic acid with $$ C_{\mathrm{ HB}}^{\circ } = {10^{-1 }}\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$

Fig. 13

Plotting the two asymptotes of $$ \mathrm{ HB} $$ with Eq. (33) for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}}} $$ and Eq. (34) for $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}}} $$. Example: acetic acid with $$ C_{\mathrm{ HB}}^{\circ } $$ = 0.1 mol L⁻¹ and pK a = 4.75

Fig. 14

Plotting the two asymptotes of $$ {{\mathrm{ B}}^{-}} $$ with Eq. (35) for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}}} $$ and Eq. (36) for $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}}} $$. Example: acetic acid with $$ C_{\mathrm{ HB}}^{\circ } $$ = 0.1 mol L⁻¹ and pK a = 4.75

3.2 Dibasic Acids

In the case of dibasic acids $$ {{\mathrm{ H}}_2}\mathrm{ B}, $$ the following two protolysis reactions have to be considered:

(37)

(38)

The equilibrium equation (37) is characterized by the following law of mass action:

(39)

and equilibrium equation (38) by:

(40)

As an analogy to Eq. (18), the following amount balance has to be formulated:

(41)

The mathematical equations of the functions are given in the Appendix. The pH-logc i diagram can be easily constructed by plotting the line equations for the pH ranges separated by the pK a values (see Appendix). The construction is analogous to that of monobasic acids, with the following specificity. The asymptote of $$ {{\mathrm{ H}}_2}\mathrm{ B} $$ has the slope −2 in the range $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}2}} $$, that is, in the range where $$ {{\mathrm{ B}}^{2- }} $$ is the dominating species. The asymptote of $$ {{\mathrm{ B}}^{2- }} $$ has the slope +2 in the range $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, that is, where $$ {{\mathrm{ H}}_2}\mathrm{ B} $$ is dominating. The stepwise construction of the diagram is illustrated in Figs. 15, 16, 17, 18, and 19.

Fig. 15

Plotting the coordination system with the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}}$$ lines of water

Fig. 16

Marking the coordinates, here for the example of an acid with $$ \log {c_{\mathrm{ i}}}= \log C_{{{{\mathrm{ H}}_2}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}1}}=3 $$ and $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}2}}=8 $$

Fig. 17

Plotting the three lines of $$ {{\mathrm{ H}}_2}\mathrm{ B} $$ in the ranges $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}2}} $$. Example: acid with $$ \log C_{{{{\mathrm{ H}}_2}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=3 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}2}}=8 $$

Fig. 18

Plotting the three lines of $$ \mathrm{ H}{{\mathrm{ B}}^{-}} $$ in the ranges $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}2}} $$. Example: acid with $$ \operatorname{l}\mathrm{ o}gC_{{{{\mathrm{ H}}_2}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=3 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}2}}=8 $$

Fig. 19

Plotting the three lines of $$ {{\mathrm{ B}}^{2- }} $$ in the ranges $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$ and $$ \mathrm{ pH} > \mathrm{ p}{K_{\mathrm{ a}2}} $$. Example: acid with $$ \log C_{{{{\mathrm{ H}}_2}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=3 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}2}}=8 $$

3.3 Tribasic Acids

For $$ {{\mathrm{ H}}_3}\mathrm{ B}, $$ the following protolysis equilibria have to be written:

(42)

(43)

(44)

The mathematical equations of the functions are given in the Appendix. The diagram can be constructed in analogy to the previous examples (Figs 15 to 19).

Fig. 20

Plotting the coordination system with the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ lines of water

Fig. 21

Marking the coordinates, here for an acid with $$ \log {c_{\mathrm{ i}}}= \log C_{{{{\mathrm{ H}}_3}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}2}}=5 $$ and $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}3}}=12 $$

Figures 20, 21, 22, 23, 24 and 25 show the construction of the diagram and the lines of the different species. The line of $$ {{\mathrm{ H}}_3}\mathrm{ B} $$ has the slope −2 in the range $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ ($$ \mathrm{ H}{{\mathrm{ B}}^{2- }} $$ is dominating here). For $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}3}} $$ (B³⁻ is dominating) the slope is −3, which is not visible in the diagram because of the limited range of $$ \log {c_{\mathrm{ i}}} $$ and pH (it is outside the used quadrant).

Fig. 22

Plotting the three lines of $$ {{\mathrm{ H}}_3}\mathrm{ B} $$ in the ranges $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$ and $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$. The line of $$ {{\mathrm{ H}}_3}\mathrm{ B} $$ for $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}3}} $$ is situated outside the range of the diagram. Example: $$ \log C_{{{{\mathrm{ H}}_3}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}= 5 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}3}}=12 $$

Fig. 23

Plotting the lines of $$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{-}} $$ in the ranges $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}3}} $$. Example: $$ \log C_{{{{\mathrm{ H}}_3}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}= 5 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}3}}=12 $$

Fig. 24

Plotting the lines of $$ \mathrm{ H}{{\mathrm{ B}}^{2- }} $$ in the ranges $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}3}} $$. Example: $$ \log C_{{{{\mathrm{ H}}_3}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}= 5 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}3}}=12 $$

Fig. 25

Plotting the lines of $$ {{\mathrm{ B}}^{3- }} $$ in the ranges $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}3}} $$. The line for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$ is situated outside the range of the coordination system. Example: $$ \log C_{{{{\mathrm{ H}}_3}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}= 5 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}3}}=12 $$

The line of $$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{-}} $$ has the slope −2 in the range $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}3}} $$ (where $$ {{\mathrm{ B}}^{3- }} $$ is dominating).

The line of $$ \mathrm{ H}{{\mathrm{ B}}^{2- }} $$ has the slope +2 in the range $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$ (where $$ {{\mathrm{ H}}_3}\mathrm{ B} $$ is dominating).

The line of $$ {{\mathrm{ B}}^{3- }} $$ has the slope +3 for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$ ($$ {{\mathrm{ H}}_3}\mathrm{ B} $$ is dominating). In the given example, this line is situated outside the range of the diagram. In the range $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$ the slope is +2 ($$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{-}} $$ is dominating).

3.4 Tetrabasic Acids

In case of a tetrabasic acid $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ the following protolysis equilibria are established:

(45)

(46)

(47)

(48)

The mathematical equations of the functions are given in the Appendix. The plotting of all species in the diagram can be made with the help of the line equations given in the Appendix. Following the procedures for the mono- to tribasic acids, this is stepwise made in Figs. 26, 27, 28, 29, 30, and 31.

Fig. 26

Plotting the coordination system with the $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ and $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ lines of water, and marking the coordinates, here for an example of an acid with $$ \log {c_{\mathrm{ i}}}= \log C_{{{{\mathrm{ H}}_4}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}2}}=4 $$, $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}3}}=7 $$ and $$ \mathrm{ pH}=\mathrm{ p}{K_{\mathrm{ a}4}}=12 $$

Fig. 27

Plotting the lines of $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ and $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$. The line of $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ for $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}} $$ is situated outside the plotted range. Example: acid with $$ \log C_{{{{\mathrm{ H}}_4}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=4 $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}=7 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}4}}=12 $$

Fig. 28

Plotting the lines of $$ {{\mathrm{ H}}_3}{{\mathrm{ B}}^{-}} $$ for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ and $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$. Example: acid with $$ \log C_{{{{\mathrm{ H}}_4}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=4 $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}=7 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}4}}=12 $$

Fig. 29

Plotting the lines of $$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{2- }} $$ for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}} $$. Example: acid with $$ \log C_{{{{\mathrm{ H}}_4}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=4 $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}=7 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}4}}=12 $$

Fig. 30

Plotting the lines of $$ \mathrm{ H}{{\mathrm{ B}}^{3- }} $$ for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}} $$. Example: acid with $$ \log C_{{{{\mathrm{ H}}_4}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=4 $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}=7 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}4}}=12 $$

Fig. 31

Plotting the lines of $$ {{\mathrm{ B}}^{4- }} $$ for $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$ and $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}} $$. Example: acid with $$ \log C_{{{{\mathrm{ H}}_4}\mathrm{ B}}}^{\circ }=-1 $$, $$ \mathrm{ p}{K_{\mathrm{ a}1}}=2 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=4 $$, $$ \mathrm{ p}{K_{\mathrm{ a}3}}=7 $$ and $$ \mathrm{ p}{K_{\mathrm{ a}4}}=12 $$

The line of $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ has the slope −2 in the range $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ ($$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{2- }} $$ is dominating). The slope is −3 in the range $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$ ($$ \mathrm{ H}{{\mathrm{ B}}^{3- }} $$ is dominating). Sometimes, also in the example shown in Fig. 27, the concentrations of $$ {{\mathrm{ B}}^{4- }} $$ are so small in the range $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}} $$ that the line is situated outside the range of the diagram. There the line has a slope of −4.

The line of $$ {{\mathrm{ H}}_3}{{\mathrm{ B}}^{-}} $$ has a slope of −2 for $$ \mathrm{ p}{K_{\mathrm{ a}3}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}4}} $$ ($$ \mathrm{ H}{{\mathrm{ B}}^{3- }} $$ is dominating). For $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}} $$ ($$ {{\mathrm{ B}}^{4- }} $$ is dominating) the slope is −3. In some cases (e.g., example in Fig. 28) that line is already outside the range of the diagram.

The line of $$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{2- }} $$ has a slope of +2 in for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$ where $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ is dominating. For $$ \mathrm{ pH}>\mathrm{ p}{K_{\mathrm{ a}4}}, $$ the slope is −2 ($$ {{\mathrm{ B}}^{4- }} $$ is dominating).

The line of $$ \mathrm{ H}{{\mathrm{ B}}^{3- }} $$ has a slope of +3 for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$ where $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ is the dominating species. In the range $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}}, $$ the slope is +2. Here, $$ {{\mathrm{ H}}_3}{{\mathrm{ B}}^{-}} $$ is dominating.

The line of $$ {{\mathrm{ B}}^{4- }} $$ has a slope of +4 for $$ \mathrm{ pH}<\mathrm{ p}{K_{\mathrm{ a}1}} $$ where $$ {{\mathrm{ H}}_4}\mathrm{ B} $$ is dominating. However, the concentrations of $$ {{\mathrm{ B}}^{4- }} $$ are so small in that range that they are outside the range of the diagram. In the range $$ \mathrm{ p}{K_{\mathrm{ a}1}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}2}} $$, i.e., the range where $$ {{\mathrm{ H}}_3}{{\mathrm{ B}}^{-}} $$ dominates, the slope is +3, and for $$ \mathrm{ p}{K_{\mathrm{ a}2}}<\mathrm{ p}\mathrm{ H}<\mathrm{ p}{K_{\mathrm{ a}3}} $$ the slope is +2 ($$ {{\mathrm{ H}}_2}{{\mathrm{ B}}^{2- }} $$ is dominating).

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4_4© Springer-Verlag Berlin Heidelberg 2013

4. The Application of pH-logc i Diagrams for Graphical Estimation of the pH of Solutions and for the Derivation of Useful Simplified Equations

Heike Kahlert1  and Fritz Scholz¹

(1)

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Abstract

For the derivation of equations which allow an approximative calculation of pH values, it is again necessary to refer to the reactions of a monobasic acid HB in water:

For the derivation of equations that allow an approximative calculation of pH values, it is again necessary to refer to the reactions of a monobasic acid HB in water:

(1)

(2)

The following quantities have to be considered: $$ {c_{{\rm HB}}}$$, $$ {c_{{{{{\rm B}}^{-}}}}} $$, $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$ and $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$. Mathematics demand for the determination of n variables to have a system of n independent equations that do not contradict each other. Hence, four such equations are needed here. These are the already introduced Eq. (9) for the ion product of water, Eq. (18) for the amount balance, and Eq. (49) for the charge balance:

(5)

(9)

(18)

(49)

In the case of the reaction of a cation acid, for example, $$ {\rm NH}_4^{+} $$, with water, the following reactions have to be written:

(1*)

(2)

(The asterisk marks equations that are only slightly different from the form without asterisk.)

From these two equations follows:

(49*)

because, according to Eq. (1*), one neutral particle B is formed per one $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$, and one $$ {\rm H}{{{\rm B}}^{+}} $$ cation is lost.

Combining Eqs. (5), (9), (18), and (49) and rearranging them gives a polynomial equation of degree 3 (cubic equation) with respect to $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$:

In the case of polybasic acids, the number of equilibria increases, as does the number of mathematical equations. Finally, this leads to equations of the degree 4 for a dibasic acid, degree 5 for a tribasic acid, etc.

The equations for a simple base $$ {{{\rm B}}^{-}} $$ are analogously:

(15)

(2)

The following four variables have to be considered: $$ {c_{{\rm HB}}} $$, $$ {c_{{{{{\rm B}}^{-}}}}} $$, $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$ and $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$. The four equations necessary for further derivations are Eq. (60) for the base constant of B⁻, Eq. (9) for the ionic product of water, and the equations for amount and charge balances:

(60)

(9)

(61)

(62)

Equation (62) follows from reaction equation (15), as one HB is formed per $$ {\rm O}{{{\rm H}}^{-}} $$ on account of losing one $$ {{{\rm B}}^{-}} $$ anion (see the box below Eq. (49)). The system of Eqs. (60), (9), (61), and (62) results in a cubic equation with respect to $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$:

(63)

In the following section, it will be shown that pH-logc i diagrams make it possible to find the approximate pH of various solutions in the diagram, as well as to find out what approximations are permitted in deriving simplified equations for pH calculations, to avoid the necessity to solving Eqs. (59) or (63), or even more complex equations.

For the derivation of simplified equations, it is necessary to consider what summand can be ignored in Eqs. (v), (49), (61) or (62): whether a summand can be neglected (taken as zero) depends only on the precision of the desired calculations. Normally, neglecting a summand is permitted when it contributes less than 1 % to the sum. This means that the smaller summand can be neglected when it is two orders of magnitude smaller than the other on the logc i-scale of the pH-logc i diagram.

When in this book equations of the form x ≈ y are used to derive simplified equations, it would be strictly considered necessary to write in all the follow-up equations the symbol “≈”. However, since the thus-derived equations are always denoted as simplified equations (approximations), this is not done, and the equality symbol (=) is used. This is in agreement with common practice.

4.1 Monobasic Acids and Their Corresponding Bases

4.1.1 Very Strong Acids and Their Corresponding Very Weak Bases

In the following derivations, the symbols HB and B⁻ are used throughout, even when specific examples are displayed in the figures. Solutions of HB are called acid solutions, and solutions of B⁻ are called base solutions, although the latter are in fact salt solutions (e.g., Na⁺ B⁻). Later it will be shown that there also exist cation acids that are part of salts, for example, $$ {\rm NH}_4^{+}{\rm C}{{{\rm l}}^{-}} $$ and amino acid hydrochlorides (Sect. 4.2.1.2).

4.1.1.1 Very Strong Acids in High Concentration

Very strong acids possess pK a values smaller than zero. Figure 32 shows the example of the very strong acid HCl (pK a = −7) in rather high concentration (e.g., $$ \log C_{{\rm HCl}}^{ \circ }=-1 $$).

Fig. 32

pH-logc i diagram of hydrochloric acid (pK a = −7) at 0.1 mol L⁻¹ concentration. The range for pH < 0 is only shown for didactic reasons. Normally this is not done

Of course, Eq. (49) must hold:

(49)

This excludes the fact that the real pH is situated on the right side of the crossing point P₁ in Fig. 32, as there the sum of $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}+{c_{{{{{\rm B}}^{-}}}}} $$ is always larger than $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$. The real pH must be situated slightly left to P₁, as the following reason shows. The overall $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ concentration is the sum of what was formed in reaction equation (1) and reaction equation (2). The concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ which has to be considered in Eq. (49) can at most be 10⁻¹³ mol L⁻¹ (see the crossing point P₃). Since 10⁻¹³ mol L⁻¹ is negligible compared with 10⁻¹ mol L⁻¹ (the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ concentration formed in the reaction of HCl with water), the crossing point P₁ gives accurately the pH of the solution, and it holds that

(64)

Hence, the pH of the solution can be found on the pH scale at the crossing point of the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and the $$ {\rm C}{{{\rm l}}^{-}} $$ lines, because the concentration of HCl is negligible in the pH range 0–14. Hence, it is possible to neglect $$ {c_{{\rm HB}}} $$ relative to $$ {c_{{{{{\rm B}}^{-}}}}} $$ in Eq. (18):

(18)

From this follows the relationship:

(65)

Taking Eqs. (64) and (65), the following approximation can be derived:

Simplified equation for a very strong acid in high concentration:

(66)

4.1.1.2 Very Strong Acid in Low Concentration

A hydrochloric acid solution of an overall concentration of 10⁻⁶ mol L⁻¹ is presented in Fig. 33:

Fig. 33

pH-logc i diagram of hydrochloric acid (pK a = −7) at 10⁻⁶ mol L⁻¹ concentration. The range for pH < 0 is only shown for didactic reasons. Normally this is not done

Since Eq. (49) holds here also, the real pH must be situated on the pH scale left of P₁. Furthermore, it holds that the concentration of HCl is negligible compared to that of the chloride ions. Hence, Eq. (18) can be simplified as follows:

(65)

However, the equilibrium concentration of OH⁻ from the autoprotolysis of water is no longer negligible in the charge balance (Eq. (49)), and the pH of the solution will be slightly left to the crossing point P₁. Some rearrangement of the equations makes it possible to derive a simplified equation for this case:

This quadratic equation can be solved and yields:

Simplified equation for a very strong acid in low concentration:

(70)

(The mathematical solution $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}=\frac{{C_{{\rm H}{\rm B}}^{ \circ }}}{2}-\sqrt{{\frac{{{{{\left( {C_{{\rm H}{\rm B}}^{ \circ }} \right)}}^2}}}{4}+{K_{{\rm w}}}}} $$ does not correspond to a real concentration, and therefore the equation is usually given without the minus sign.)

For very low overall concentrations, e.g., 10⁻¹⁰ mol L⁻¹ (cf. Fig. 34) the diagram shows that the simplification equation (65) is still valid; however, in the equation of the charge balance, the concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ from the autoprotolysis of water is much larger than the equilibrium concentration of the Cl⁻ ions. From this follows:

Fig. 34

pH-logc i diagram of a hydrochloric acid (pK a = −7) at 10⁻¹⁰ mol L⁻¹ concentration. The range for pH < 0 is only shown for didactic reasons. Normally this is not done

Simplified equation for a very strong acid in very low concentration:

(71)

Of course, the real pH should be – no matter how little – left to the pH-coordinate of the neutrality point (P₁ in Fig. 34), as an acid solution can never have a neutral or basic pH.

4.1.1.3 Very Weak Bases

As an example, a solution of sodium chloride is chosen: the chloride anion is the very weak base corresponding to the very strong acid HCl. The aqua-complex of sodium ions is a very weak acid in terms of the Brønsted acid–base definition, and hence it does not affect the pH of the solution (see Sect. 4.1.4).

In case of a NaCl solution, the Eqs. (61) and (62), i.e., the amount balance and the charge balance, have to be fulfilled:

(61)

(62)

Equation (62) indicates that the real pH of the NaCl solution has to be – no matter how little – right to the point P₁ in Fig. 34. However, according to Figs. 32, 33, and 34 the degree of protonation of the chloride ions is in the entire pH range from 0 to 14, and for whatever overall concentrations of chloride, always negligible, and hence it follows as a very good approximation that:

(72)

Thus it holds for all NaCl solutions the pH is practically 7 (provided carbon dioxide is absent):

Simplified equation for a very weak base in any concentration:

(71)

The frequently given reasoning is that a NaCl solution is neutral because it is the salt of a very strong acid, and a very strong base is rather formal and does not allow a detailed understanding. A better reasoning is as follows: a NaCl solution is neutral because the chloride ion is a very weak base and the sodium-aqua ion is a very weak acid.

4.1.2 Strong Acids and Their Corresponding Weak Bases

4.1.2.1 Strong Acids in High Concentration

Formic acid has a pK a value of 3.75 and thus it belongs to the strong acids. Figure 35 shows the pH-logc i diagram for the overall concentration 10⁻¹ mol L⁻¹.

Fig. 35

pH-logc i diagram for formic acid (pK a = 3.75) at $$ C_{\mathrm{ HB}}^{ \circ } $$ = 10⁻¹ mol L⁻¹

As always, the charge balance must hold:

(49)

This means that the real pH will be somewhat left of the pH-coordinate of P₁, as to the right of this, the sum $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}+{c_{{{{{\rm B}}^{-}}}}} $$ is always larger than $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$ (in other words, in addition to the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ concentration produced by the protolysis of HB, there is also some $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ from the autoprotolysis of water). The diagram shows that the concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ is at the point P₁, and also, to the left to it, much smaller than the concentration of B⁻ (cf. point P₂). Hence it is a good approximation to write:

(64)

This means that the pH-coordinate of the crossing point of the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and B⁻ lines gives in very good approximation the pH of the solution.

A simplified equation can be easily derived when the amount balance (Eq. (18)) is taken in to account:

Solving this equation leads to:

Simplified equation for a strong acid in high concentration:

(77)

(The mathematical solution $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}=-\frac{{{K_{{\rm a}}}}}{2}-\sqrt{{\frac{{K_{{\rm a}}^2}}{4}+{K_{{\rm a}}}C_{{\rm H}{\rm B}}^{ \circ }}} $$ does not correspond to a real concentration, and therefore the equation is usually given without the minus sign.)

4.1.2.2 Strong Acids in Low Concentration

The considerations made in Sect. 4.1.2.1 are equally valid for such low concentrations that (i) the lines of HB are situated left of the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ line, and (ii) the concentration of B⁻ is still much larger than that of $$ {\rm O}{{{\rm H}}^{-}} $$ at P₁ and left of that point (cf. Fig. 36). Also in this case, the pH is approximately given by the pH-coordinate of the crossing point of the B⁻ and $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ lines, and it can be calculated with the simplified equation (77).

Fig. 36

pH-logc i diagram of formic acid (pK a = 3.75) at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻⁴ mol L⁻¹

In cases of such low concentrations that the approximation $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{{{\rm B}}^{-}}}}} $$ (64) does not hold anymore (see Fig. 37) the cubic equation (59) has to be used. The equation $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm O}{{{\rm H}}^{-}}}}}=\sqrt{{{K_{{\rm w}}}}} $$ (Eq. (71)) is only valid for extremely low concentrations, as such solutions are practically neutral (see Fig. 38).

Fig. 37

pH-logc i diagram of formic acid (pK a = 3.75) at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻⁶ mol L⁻¹

Fig. 38

pH-logc i diagram of formic acid (pK a = 3.75) at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻¹⁰ mol L⁻¹

4.1.2.3 Weak Bases in High Concentration

Figure 39 depicts the situation of a solution of sodium formate at a concentration of 0.1 mol L⁻¹: the aqua-complex of sodium is a very weak Brønsted acid, and therefore only the protonation of formate ions can affect the pH.

Fig. 39

pH-logc i diagram of sodium formate solution at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻¹ mol L⁻¹ (Formic acid: pK a = 3.75)

The charge balance gives for this base solution:

(62)

This equation indicates that the pH must be higher than the pH of point P₁ in Fig. 39, as otherwise the sum of $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}+{c_{{\rm H}{\rm B}}} $$ would be larger than $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$. It also holds that the equilibrium concentration of HB is much larger than the equilibrium concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$, and thus Eq. (62) can be simplified and gives:

(78)

The pH of that sodium formate solution can be taken from the diagram as the pH-coordinate of the crossing point of the $$ {\rm O}{{{\rm H}}^{-}} $$ and HB lines (point P₁ in Fig. 39).

For the derivation of a simplified equation, the equilibrium concentrations HB and B⁻ have to be compared at the points P₁ and P₃: $$ {c_{{\rm HB}}} $$ is 2 × 10⁻⁶ mol L⁻¹ at P₁, and thus it is negligible compared to $$ {c_{{{{{\rm B}}^{-}}}}} $$ ≈ 0.1 mol L⁻¹. This makes it possible to simplify the amount balance in Eq. (61) to give:

(72)

The following rearrangements yield the desired simplified equation:

Simplified equation for a weak base in high concentration:

(84)

4.1.2.4 Weak Bases in Low Concentration

When the concentration of the base is so low that the simplification equation (78) does not hold, i.e., when the line of HB is situated to the left of the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ line (cf. Fig. 40), the pH of the solution must still be basic. The relationship equation (72) is still a good simplification, and the desired simplified equation follows as given here.

Fig. 40

pH-logc i diagram of sodium formate solution at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻⁴ mol L⁻¹ (Formic acid: pK a = 3.75)

Simplified equation for a weak base in low concentration:

(90)

For very low concentrations of the base, Eq. (71) can be used.

4.1.3 Weak Acids and Their Corresponding Strong Bases

$$ {\rm NH}_4^{+} $$/NH₃ is a good example for a weak acid with a corresponding strong base. At first, an ammonium chloride solution is considered. In Sect. 4.1.1.3 it was discussed that chloride ions are not significantly protonated in the pH range of 0–14, i.e., the concentration of HCl is negligible in this range. Hence, only the ammonium ions (pK a = 9.25) will affect the pH.

4.1.3.1 Weak Acids in High Concentration

Figure 41 depicts the pH-logc i diagram of an ammonium chloride solution at $$ C_{{\rm HB}}^{ \circ } $$ = 10⁻¹ mol L⁻¹.

Fig. 41

pH-logc i diagram of an ammonium chloride solution at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻¹ mol L⁻¹ (ammonium: pK a = 9.25)

The charge balance

(49)

shows that the pH is somewhat left of the pH-coordinate of P₁ in Fig. 41, because on the right side of P₁ the sum $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}+{c_{{{{{\rm B}}^{-}}}}} $$ is always larger than $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$. The equilibrium concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ is caused by the protolysis of HB and also by the autoprotolysis of water. However, the concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ is at point P₁ 1.3 × 10⁻⁹ mol L⁻¹ (see point P₂). This is a negligible concentration compared to that of B⁻ (8 × 10⁻⁶ mol L⁻¹), and hence one can write

(64)

This means that the pH-coordinate of the crossing point of the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and B⁻ lines gives the pH of the solution with a very satisfying precision. For deriving a simplified equation, the amount balance equation (18) of a weak acid has to be considered:

(18)

On the left side of P₁, it is a well-based simplification to neglect the equilibrium concentration of B⁻ relative to that of HB:

(91)

The simplified equation follows after some rearrangements:

Simplified equation for a weak acid in high concentration:

(93)

4.1.3.2 Weak Acids in Low Concentration

Figure 42 depicts the situation of an ammonium chloride solution at a concentration of 10⁻⁴ mol L⁻¹.

Fig. 42

pH-logc i diagram of an ammonium chloride solution at $$ C_{\mathrm{ HB}}^{\circ } $$ = 10⁻⁴ mol L⁻¹ (ammonium: pK a = 9.25)

The charge balance (Eq. (49)) demands again that the pH must be slightly below the pH-coordinate of point P₁ in Fig. 42. However, here it is not possible to neglect the concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ relative to that of B⁻, i.e., the simplified relationship equation (64) is no longer allowed. In the amount balance equation (18), it is still appropriate to neglect the concentration of B⁻ with respect to that of HB, since – left of P₁ – the equilibrium concentration of B⁻ is always much smaller than the concentration of HB. Hence it is appropriate to write Eq. (91):

(91)

Rearrangements give a useful simplified equation:

Simplified equation for a weak acid in low concentration:

(98)

In the case of very low concentrations, Eq. (71) can be used.

4.1.3.3 Strong Bases in High Concentration

Figure 43 shows the diagram of an ammonia solution with an overall concentration of $$ C_{{\rm ammonia}}^{ \circ } $$ = 10⁻¹ mol L⁻¹. The diagram is of course identical with that shown in Fig. 41, but the information which needs to be extracted is now different: For the solution of a base, the charge balance is given by Eq. (62):

Fig. 43

pH-logc i diagram of an ammonia solution, $$ C_{\mathrm{ ammonia}}^{\circ } $$ = 10⁻¹ mol L⁻¹ (Ammonium: pK a = 9.25)

(62)

From this follows that the pH of the solution must be somewhat higher than the pH-coordinate given by P₁ in Fig. 43. Otherwise, the sum $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}+{c_{{\rm H}{\rm B}}} $$ would exceed the concentration of $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$. The diagram shows that the concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ is equal or less than 6.8 × 10⁻¹² mol L⁻¹ (cf. P₂), i.e., it is negligible with respect to the concentration of HB at the point P₁. Hence, the Eq. (62) can be simplified to yield the relationship equation (78):

(78)

From this follows that the pH-coordinate of the crossing point of the $$ {\rm O}{{{\rm H}}^{-}} $$ and HB lines gives a very good estimate of the real pH of the solution. A simplified equation can be derived as follows:

Simplified equation for a strong base in high concentration:

(105)

(The mathematical solution does not correspond to a real concentration, and therefore the equation is usually given without the minus sign.)

In cases where the concentration of the base is so small that the relationship equation (78) does not hold anymore, the cubic equation (63) has to be solved. For very small concentrations, Eq. (71) can be used.

4.1.4 Very Weak Acids and Their Corresponding Very Strong Bases

4.1.4.1 Very Weak Acids

Ethanol has a pK a value of 16, and thus it is a very weak acid. Figure 44 shows the pH-logc i diagram of an aqueous solution of 10⁻¹ mol L⁻¹ ethanol. From the diagram, it is obvious that the concentration of B⁻ (ethanolate ions) is very small in the entire pH range from 0 to 14, i.e., ethanol reacts only in a negligible amount as an acid. The charge balance gives:

Fig. 44

pH-logc i diagram of an aqueous solution of 10⁻¹ mol L⁻¹ ethanol (pK a − 16). The range of pH > 14 is shown for didactic reasons only. Normally it is not shown

(49)

and thus the pH can only be lower than the pH-coordinate of point P₁ in Fig. 44, i.e., left of the neutrality point of water. There, however, the concentration of ethanolate (B⁻) is so small that the relationship equation (71) holds:

Simplified equation for a very weak acid in (almost) any concentration:

(71)

4.1.4.2 Very Strong Bases in High Concentration

As discussed in Chap.​ 2, in water the strength of those bases which are stronger than $$ {\rm O}{{{\rm H}}^{-}} $$ ions are leveled off. As an example, ethanolate ions react with water to yield ethanol and $$ {\rm O}{{{\rm H}}^{-}} $$ ions. If one would try to prepare an aqueous solution of 10⁻¹ mol L⁻¹ of sodium ethanolate (cf. Fig. 45), the following charge balance must hold:

Fig. 45

pH-logc i diagram of an aqueous solution prepared by dissolving 10⁻¹ mol of sodium ethanolate in water so that 1 l of solution results. The range pH > 14 is shown for didactic reasons only. Normally it is not shown

(62)

This means that the pH of the solution must be situated slightly to the right of the pH-coordinate of P₁ in Fig. 45, since on the left side of P₁ the sum $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}+{c_{{\rm H}{\rm B}}} $$ is always larger than the concentration $$ {\rm O}{{{\rm H}}^{-}} $$ (in other words, the overall concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ results from the protolysis of B⁻and autoprotolysis of water). The concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ at P₁ is about 10⁻¹³ mol L⁻¹, and so it can be neglected in the charge balance, and the pH is given by the pH-coordinate of the crossing point of the $$ {\rm O}{{{\rm H}}^{-}} $$ and ethanol (HB) lines:

(78)

The diagram also reveals that the equilibrium concentration of ethanolate ions can be neglected in relation to the ethanol concentration, so that the amount balance can be approximated by the relationship:

(106)

This makes it possible to derive a simplified equation by using the Eqs. (78), (106), and (11):

Simplified equation for a very strong base in high concentration:

(107)

(108)

4.1.4.3 Very Strong Bases in Low Concentration

Figure 46 shows the pH-logc i diagram of a solution resulting from dissolution of 10⁻⁶ mol sodium ethanolate in 1 L of water. The charge balance is as follows:

Fig. 46

pH-logc i diagram of an aqueous solution prepared by dissolving 10⁻⁶ mol of sodium ethanolate in water so that 1 l of solution results. The range pH > 14 is shown for didactic reasons only. Normally it is not shown

(62)

In this case also, the pH must be situated on the right side of the pH-coordinate of P₁ in Fig. 46, as on the left side the sum $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}+{c_{{\rm H}{\rm B}}} $$ is always larger than the concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ (in other words, the overall concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ results from the protolysis of B⁻ and autoprotolysis of water). Here it is no longer permitted to use the simplified equation (78). The equilibrium concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ is not negligible in comparison to the ethanol concentration. However, the concentration of ethanolate ions can still be ignored in comparison to the ethanol concentration, so that the amount balance can be written as follows:

(106)

Using Eq. (11) follows:

Solving the quadratic equation yields:

Simplified equation for a strong base in low concentration:

(111)

(The mathematical solution $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}=-\frac{{C_{{{{{\rm B}}^{-}}}}^{\circ }}}{2}-\sqrt{{\frac{{{{{\left( {C_{{{{{\rm B}}^{-}}}}^{\circ }} \right)}}^2}}}{4}+{K_{{\rm w}}}}} $$ does not describe a real concentration, and therefore the minus sign before the square root is usually not written.)

For solutions of very low concentrations, Eq. (71) can be used as a very good simplified equation.

4.1.5 Ranges of Validity of the Simplified Equations for Monobasic Acids

The simplified equations (66), (70), (77), (93), and (98) have been derived for some special cases; however, they are valid for larger ranges of $$ \log C_{{\rm HB}}^{ \circ } $$ and pK a. To display the ranges of validity, diagrams with the coordinates $$ \log C_{{\rm HB}}^{ \circ } $$ and pK a are very useful [1–6, 7]. For the determination of validity ranges, a maximum error needs to be defined: that maximum error is the maximum allowable deviation between the results using (a) the cubic equation (59) and (b) the respective simplified equation. Figure 47 shows the ranges of validity of the simplified equations (66), (70), (77), (93), and (98) with respect to $$ \log C_{{\rm HB}}^{ \circ } $$ and pK a when the maximum allowable error is 0.02 pH units. Should one accept larger errors, the validity ranges will expand, while for smaller errors they will shrink. This figure demonstrates that there is only a very small area in which no simplified equation can be used, and the cubic equation (59) has to be solved.

Fig. 47

(a) Validity ranges of the simplified equations (70), (77), (93), and (98). (b) Validity range of the simplified equation (66). For the calculation of the borderlines (validity ranges), a maximum allowable error of 0.02 pH units has been fixed

4.2 Dibasic Acids, Ampholytes, and Diacidic Bases

4.2.1 Dibasic Acids

When a dibasic acid H₂B is dissolved in water, the following equilibria have to be considered for the derivation of simplified equations for pH calculations:

(37)

(38)

(2)

The five variables $$ {c_{{{{{\rm H}}_2}{\rm B}}}} $$, $$ {c_{{{\rm H}{{{\rm B}}^{-}}}}} $$, $$ {c_{{{{{\rm B}}^{2- }}}}} $$, $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$ and $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$ require the formulation of five independent equations. These are the equations of the laws of mass action of the protolysis equilibria (equations of the acidity constants of the acids H₂B, $$ {\rm H}{{{\rm B}}^{-}}, $$ and the ion product of water) and the amount and charge balances. Hence one needs the already introduced Eqs. (39) and (40) for the acidity constants, Eq. (9) for the ion product of water, Eq. (41) for the amount balance and Eq. (112) for the charge balance:

(39)

(40)

(9)

(41)

(112)

This system of equations leads to an equation of the degree 4, which is not derived here. Instead, a number of practically important cases will be treated here.

4.2.1.1 Dibasic Acids with One Very Strong and One Strong Protolysis Reaction: $$ {\rm\bf p}{\boldsymbol K}_{{\rm\bf a}\bf1}<< {\bf0},\ {\rm\bf p}{\boldsymbol K}_{{\rm\bf a}\bf2}\ > {\bf0} $$

4.2.1.1.1 High Concentration

An important example for such acid is sulfuric acid having the following acidity constants: $$ {\rm p}{K_{{\rm a}1}}=-3 $$ and $$ {\rm p}{K_{{\rm a}2}}=1.99 $$. Figure 48 shows the pH-logc i diagram for a concentration of 10⁻¹ mol L⁻¹ of sulfuric acid.

Fig. 48

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ of sulfuric acid. The range pH < 0 is shown for didactic reasons only. Normally it is not shown

The charge balance gives in the case of a dibasic acid:

(112)

From Fig. 48, one can deduce that the pH must be left of the pH-coordinate of P₁, as on the right side the sum $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}+{c_{{{\rm H}{{{\rm B}}^{-}}}}}+2{c_{{{{{\rm B}}^{2- }}}}} $$ is always larger than the equilibrium concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ (in other words, the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ results from the protolysis of H₂B and HB, and also from the autoprotolysis of water). The pH is mainly given by the reaction equation (37), and the other two reactions have only a smaller contribution to the overall $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ concentration. To the left of the pH-coordinate of P₁, the equilibrium concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ ions (cf. P₂) is so small that it can be neglected in the charge balance, and one can write:

(113)

In the case of the amount balance, the equilibrium concentration of H₂B can be neglected in relation to the equilibrium concentrations of HB⁻ and B²⁻ (cf. P₄), and so it holds as a good approximation:

(114)

For deriving a simplified equation, the following rearrangements have to be made:

Using the formula for solving quadratic equations allows deriving the following simplified equation:

Simplified equation for a dibasic acid with $$ \mathbf{p}{{\boldsymbol K}_{\mathbf{a}\bf1}}<< \bf0\ \mathbf{and}\ \mathbf{p}{{\boldsymbol K}_{\mathbf{a}\bf2}} > 0 $$ in high concentration:

(122)

(The mathematical solution does not describe a real concentration and therefore the minus sign before the square root is usually not written.)

4.2.1.1.2 Low Concentration

Figure 49 shows the pH-logc i diagram of sulfuric acid at a concentration of 10⁻⁵ mol L⁻¹. From the diagram one can conclude that the charge balance

Fig. 49

pH-logc i diagram of sulfuric acid at 10⁻⁵ mol L⁻¹. The range pH < 0 is shown for didactic reasons only. Normally it is not shown

(112)

can be simplified, as the pH will be slightly left of the pH-coordinate of P₁. There, the concentrations of $$ {\rm O}{{{\rm H}}^{-}} $$ and HB⁻ are negligible compared to that of B²⁻:

(123)

A closer look at the amount balance

(41)

reveals that –left of P₁– the concentrations of HB⁻ (see P₃) and H₂B are very small in comparison to that of B²⁻ and so one can write:

(124)

Combining Eqs. (123) and (124) gives the following simplified equation:

Simplified equation for a dibasic acid with $$ \mathbf{p}{{\boldsymbol K}_{\mathbf{a}\bf1}}<<\bf0\ \mathbf{and}\ \mathbf{p}{{\boldsymbol K}_{\mathbf{a}\bf2}}>0 $$ in low concentration:

(125)

4.2.1.2 Dibasic Acid with a Large Difference in pK a Values: $$ {\rm\bf p}{{\boldsymbol K}_{{\rm\bf a}\bf1}} >\bf0,\ {\rm\bf p}{{\boldsymbol K}_{{\rm\bf a}\bf2}}>\bf0,\ {\rm\bf p}{{\boldsymbol K}_{{\rm\bf a}\bf1}}<< {\rm\bf p}{{\boldsymbol K}_{{\rm\bf a}2}} $$

4.2.1.2.1 High Concentration

Many amino acids are dibasic acids with pK a values between 0 and 14. The K a values usually differ by a factor of 10⁵. A good example is alanine, which can exist in aqueous solutions in the following forms:

The species H₂Al⁺, HAl+/−, and Al⁻ correspond to the forms H₂B, HB⁻ and B²⁻ of “ordinary” dibasic acids of the formula H₂B. Certainly, there is also an equilibrium of “intramolecular” proton transfer, i.e., a tautomeric equilibrium: $$ {\rm HAl}\ \rightleftarrows\ {\rm HA}{{{\rm l}}^{+/- }} $$. However, its equilibrium constant is so large that the concentration of HAl is very small. In ethanol–water mixtures as solvent, the respective equilibrium constants of glycine, l-alanine, l-valine, l-leucine, l-phenylalanine, l-isoleucine, l-methionine, and l-serine are all between 10³ and 10⁵ [8].

Figure 50 depicts the pH-logc i diagram of an alanine hydrochloride (H₂Al⁺Cl⁻) solution of 0.1 mol L⁻¹ ($$ {\rm p}{K_{{\rm a}1}}=2.3 $$, $$ {\rm p}{K_{{\rm a}2}}=9.9 $$).

Fig. 50

pH-logc i diagram of an alanine hydrochloride (H₂Al⁺Cl⁻) solution of 0.1 mol L⁻¹ ($$ \mathrm{ p}{K_{\mathrm{ a}1}}=2.3 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=9.9 $$)

The charge balance of a dibasic acid is:

(112)

According to Fig. 50, the pH of this solution must be found slightly left to the pH-coordinate of P₁ because on the right the sum $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}+{c_{{{\rm H}{{{\rm B}}^{-}}}}}+2{c_{{{{{\rm B}}^{2- }}}}} $$ is always larger than the equilibrium concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ (in other words, the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ concentration results from the protolysis of H₂B, the protolysis of HB, and the autoprotolysis of water). However, on the left side of P₁, the equilibrium concentrations of $$ {\rm O}{{{\rm H}}^{-}} $$ (see P₃) and B²⁻ are negligible, so that the charge balance can be simplified as follows:

(126)

Hence, the pH-coordinate of P₁ gives in a very good approximation the pH of the solution. The amount balance is:

(41)

According to the diagram, the equilibrium concentration of B²⁻ is negligible in comparison to the concentrations of H₂B and HB⁻ (cf. the points P₁, P₂, and P₄ in Fig. 50), and so one can simplify this balance as follows:

(127)

The following rearrangements show how a simplified equation can be derived:

This quadratic equation can be solved in the usual way, and one obtains the following simplified equation for the concentration of hydronium ions:

Simplified equation for a dibasic acid with large differences between the pK a values ($$ \mathbf{p}{{\boldsymbol K}_{\mathbf{a}\bf1}},\mathbf{p}{{\boldsymbol K}_{\mathbf{a}\bf2}}>\bf0,\mathbf{p}{{\boldsymbol K}_{\mathbf{a}1}}<< \mathbf{p}{{\boldsymbol K}_{\mathbf{a}2}} $$) in high concentration:

(132)

(The mathematical solution $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}=-\frac{{{K_{{\rm a}1}}}}{2}-\sqrt{{\frac{{K_{{\rm a}1}^2}}{4}+{K_{{\rm a}1}}C_{{{{{\rm H}}_2}{\rm B}}}^{ \circ }}} $$ does not describe a real concentration, and therefore the minus sign before the square root is usually not written.)

This formula shows that such acid appears as a strong monobasic acid, and only the first protolysis step determines the pH.

4.2.1.2.2 Low Concentration

Figure 51 shows the pH-logc i diagram of an alanine hydrochloride (H₂Al⁺Cl⁻) solution at 10⁻⁴ mol L⁻¹. Even at this low concentration, the acid appears as a strong monobasic acid, and the second protolysis step can be neglected, i.e., the equilibrium concentration of B²⁻ is very small compared to those of HB⁻, H₂B and $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$. Thus, the pH can be approximately read as the pH-coordinate of P₁, completely as in the case of a strong monobasic acid, and the pH can be approximately calculated with Eq. (132).

Fig. 51

pH-logc i diagram of an alanine hydrochloride (H₂Al⁺Cl⁻) solution of 10⁻⁴ mol L⁻¹ ($$ \mathrm{ p}{K_{\mathrm{ a}1}}=2.3 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=9.9 $$)

4.2.2 Simplified Equation for Ampholytes

An ampholyte reacts in solution both as acid and as base. The following protolysis equilibria are established in water:

(133)

(134)

(2)

For the calculation of the five variables $$ {c_{{{{{\rm H}}_2}{\rm B}}}} $$, $$ {c_{{{\rm H}{{{\rm B}}^{-}}}}} $$, $$ {c_{{{{{\rm B}}^{2- }}}}} $$, $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$ and $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$, it is necessary to write down five independent equations. These are the equations of the law of mass action of the protolysis equilibria and the charge and mass balances:

(40)

(135)

(9)

(136)

(137)

Equation 137 is based on the following consideration: The reaction Eqs. (133), (134), and (2) produce cations, here only $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$, and the anions $$ {{{\rm B}}^{2- }} $$ and $$ {\rm O}{{{\rm H}}^{-}} $$, and anions are neutralized by the formation of $$ {{{\rm H}}_2}{\rm B} $$. By consuming the anions $$ {\rm H}{{{\rm B}}^{-}} $$, reaction equation (134) forms as many uncharged molecules $$ {{{\rm H}}_2}{\rm B} $$ as it has consumed $$ {\rm H}{{{\rm B}}^{-}} $$. From this follows:

(137)

For an uncharged ampholyte HB the following equilibria have to be formulated:

(133*)

(134*)

(2)

Then the charge balance gives:

(137*)

4.2.2.1 An Ampholyte That Is Strongly Acidic and Weakly Basic

A good example is here sodium hydrogen sulfate NaHSO₄. Figure 52 gives the pH-logc i diagram of a solution containing 10⁻¹ mol L⁻¹ NaHSO₄. The following pK values need to be considered: $$ {\rm p}{K_{{\rm a}2}}=1.99 $$ for the reaction $$ {\rm HSO}_4^{-}+{{{\rm H}}_2}{\rm O}\ \overset{{}}{\rightleftarrows}\ {\rm SO}_4^{2- }+{{{\rm H}}_3}{{{\rm O}}^{+}}, $$ and $$ {\rm p}{K_{{\rm b}2}}=17 $$ for the reaction $$ {\rm HSO}_4^{-}+{{{\rm H}}_2}{\rm O}\ {\rightleftarrows}\ {{{\rm H}}_2}{\rm S}{{{\rm O}}_4}+{\rm O}{{{\rm H}}^{-}} $$.

Fig. 52

pH-logc i diagram of a NaHSO₄ solution at 10⁻¹ mol L⁻¹ ($$ \mathrm{ p}{K_{\mathrm{ a}2}}=1.99 $$ and $$ \mathrm{ p}{K_{\mathrm{ b}2}}=17 $$). The range pH < 0 is shown for didactic reasons only. Normally it is not shown

The charge balance is as follows:

(137)

Comparing the concentrations of H₂B and $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ in the entire pH range from 0 to 14 reveals that the concentration of H₂B is always much smaller than that of $$ {{{\rm H}}_3}{{{\rm O}}^{+}}, $$ and thus one can simplify the charge balance:

(138)

This means that pH of the solution can be read slightly left of the pH-coordinate of P₁ in Fig. 52, as right of this the sum of the concentrations of $$ {\rm O}{{{\rm H}}^{-}} $$ and B²⁻ are always larger than that of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ (in other words, the $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ concentration results from the protolysis of H₂B, that of HB, and the autoprotolysis of water). However, one can also see from the diagram that left of P₁ in Fig. 52 the equilibrium concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ is very small (see P₃) in comparison to that of B²⁻, and the charge balance can be further simplified to give:

(139)

and the pH of the solution can be approximately read as the pH-coordinate of P₁.

For deriving a simplified equation, one needs to compare the concentrations of H₂B, HB⁻, and B²⁻ (see P₁, P₂ and P₄). The equilibrium concentration of H₂B is much smaller than those of HB⁻ and B²⁻, and hence the amount balance can be simplified as follows:

(140)

The following rearrangements lead to a quadratic equation which can be easily solved:

Solving this quadratic equation makes it possible to calculate the equilibrium concentration of the hydronium ions:

Simplified equation for ampholytes which are strongly acidic and weakly basic:

(146)

(The mathematical solution $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}=-\frac{{{K_{{\rm a}2}}}}{2}-\sqrt{{\frac{{K_{{\rm a}2}^2}}{4}+{K_{{\rm a}2}}C_{{{\rm H}{{{\rm B}}^{-}}}}^{\circ }}} $$ does not describe a real concentration, and therefore the minus sign before the square root is usually not written.)

This shows that an ampholyte which is strongly acidic and weakly basic can be treated like a strong monobasic acid. The acidity character dominates over the basic character. This is also true at low concentrations.

4.2.2.2 An Ampholyte that is Weakly Acidic and Weakly Basic; pH Values of Amino Acid Solutions

As explained in Sect. 4.2.1.2, amino acids have a carboxyl group and an amine group, and the carboxyl group is a stronger acid than the protonated amino group, i.e., the carboxyl group protonates the amino group. Thus the nonionic form undergoes a rearrangement as follows $$ {\rm HAl}\ \rightleftarrows\ {\rm HA}{{{\rm l}}^{+/- }} $$ (this is a tautomeric equilibrium) and a zwitterion is formed:

(zwitterion of alanine).

The term zwitterion is derived from the German word “Zwitter” for hybrid and hermaphrodite. Figure 53 shows the pH-logc i diagram of solution containing 10⁻¹ mol L⁻¹ alanine ($$ {\rm p}{K_{{\rm a}1}}=2.3 $$, $$ {\rm p}{K_{{\rm a}2}}=9.9 $$). In contrast to Figs. 50 and 51, here a solution of HAl is considered, not a solution of alanine hydrochloride (H₂Al⁺Cl⁻)!

Fig. 53

pH-logc i diagram of a solution containing 10⁻¹ mol L⁻¹ alanine ($$ \mathrm{ p}{K_{\mathrm{ a}1}}=2.3 $$, $$ \mathrm{ p}{K_{\mathrm{ a}2}}=9.9 $$) (H₂Al⁺ corresponds to H₂B, $$ \mathrm{ HA}{{\mathrm{ l}}^{+/- }} $$ corresponds to HB⁻, and Al⁻ corresponds to B²⁻ of the dibasic acids H₂B)

The charge balance gives:

(137)

As a first approximation, one can assume that HAl+/− will react to equal concentrations of the corresponding acid H₂Al⁺ and the corresponding base Al⁻, i.e., that the pH-coordinate of P₁ gives the pH of the solution. Although there the concentration $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$ is not really two orders of magnitude smaller than $$ {c_{{{{{\rm H}}_2}{\rm B}}}} $$, the concentration $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$ is even more than two orders of magnitude smaller than $$ {c_{{{{{\rm B}}^{2- }}}}} $$: thus one can simplify Eq. (137) to give: $$ {c_{{{{{\rm H}}_2}{\rm B}}}}\approx {c_{{{{{\rm B}}^{2- }}}}} $$. This means that the pH of the solution can be read as the pH-coordinate of P₁. The accuracy of that approximation certainly depends on the two pK a values.

For the derivation of a simplified equation, one needs to consider the amount balance of an ampholyte solution (Eq. (136)):

(136)

The diagram (Fig. 53) shows that the concentrations of H₂B and B²⁻ in P₁ are negligibly small in comparison to that of HB⁻, so that balance can be simplified as follows:

(147)

The following rearrangements lead to a quadratic equation:

The quadratic equation can be solved to yield:

Simplified equation for an ampholyte which is weakly acidic and weakly basic:

(155)

For the case that $$ {K_{{\rm a}1}}{K_{{\rm w}}}\ll {K_{{\rm a}1}}{K_{{\rm a}2}}C_{{{\rm H}{{{\rm B}}^{-}}}}^{\circ } $$ and $$ C_{{{\rm H}{{{\rm B}}^{-}}}}^{\circ}\gg {K_{{\rm a}1}} $$, Eq. (155) can be further simplified to give:

(156)

This equation is based on the assumption that $$ {c_{{{{{\rm H}}_2}{\rm B}}}}\approx {c_{{{{{\rm B}}^{2- }}}}} $$, whereas Eq. (155) is based on the exact balance $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}={c_{{{\rm O}{{{\rm H}}^{-}}}}}+{c_{{{{{\rm B}}^{2- }}}}}-{c_{{{{{\rm H}}_2}{\rm B}}}} $$.

Equation 156 is a good simplified equation for the amino acid alanine at a concentration of 10⁻¹ mol L⁻¹: Eq. (155) gives pH = 6.01, and Eq. (156) gives pH = 6.10, i.e., the difference is smaller than 2 %.

4.2.3 Diacidic Bases

In a solution of a diacidic base, the following equilibria are established:

(157)

(134)

(2)

The determination of the five variables $$ {c_{{{{{\rm H}}_2}{\rm B}}}} $$, $$ {c_{{{\rm H}{{{\rm B}}^{-}}}}} $$, $$ {c_{{{{{\rm B}}^{2- }}}}} $$, $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}} $$, and $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$ requires five independent equations. These are the equations of the law of mass actions of all protolysis reactions and the equations for charge and amount balances:

(158)

(135)

(9)

(159)

(160)

4.2.3.1 Diacidic Base with a Strong and a Weak Protolysis Reaction

A good example for such base is the sulfide ion ($$ {\rm p}{K_{{\rm b}1}}=0.1 $$ and $$ {\rm p}{K_{{\rm b}2}}=7.0 $$). First, a high concentration is considered: Fig. 54 depicts the pH-logc i diagram of a solution containing 0.1 mol L⁻¹ Na₂S.

Fig. 54

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ Na₂S

The charge balance gives:

(160)

From the diagram follows that the pH should be read slightly to the right of the pH-coordinate of P₁, as on the left side of P₁ the sum of HB⁻, $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$, and H₂B is always larger than the concentration of $$ {\rm O}{{{\rm H}}^{-}} $$ ions. The diagram further tells that in this range the concentrations of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and H₂B are much smaller than the concentration of HB⁻, so that it is a good approximation to write:

(161)

and the pH-coordinate of P₁ indeed gives the pH of the solution in a very good approximation. The amount balance is:

(159)

Since the concentrations of HB⁻ and B²⁻ are much larger than that of H₂B (see P₁, P₃, and P₃ in Fig. 54) the amount balance simplifies as follows:

(162)

The following rearrangements make it possible to derive a quadratic equation:

The quadratic equation can be solved with respect to the concentration of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ ions:

Simplified equation for a diacidic base with a strong and a weak protolysis reaction, and in high concentration:

(169)

(The mathematical solution does not describe a real concentration, and therefore the minus sign before the square root is usually not written.)

In this case, the second protolysis reaction can be disregarded. The base behaves like a monoacidic base (see Eq. (105) in Sect. 4.1.3.3). For low concentrations of a diacidic base, a cubic equation has to be solved (see Eq. (63)).

4.2.3.2 Diacidic Base with Two Weak Protolysis Reactions

As an example for such base, the oxalate ion can be considered: $$ {\rm p}{K_{{\rm b}1}}=9.73 $$ and $$ {\rm p}{K_{{\rm b}2}}=12.73 $$. Figure 55 shows the pH-logc i diagram of a solution containing 0.1 mol L⁻¹ of sodium oxalate.

Fig. 55

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ of sodium oxalate (oxalate: $$ \mathrm{ p}{K_{\mathrm{ b}1}}=9.73 $$ and $$ \mathrm{ p}{K_{\mathrm{ b}2}}=12.73 $$)

Considering the charge balance is

(160)

one can see that the pH of the solution should be slightly right of the pH-coordinate of P₁, since left of this point the sum of the concentrations of HB⁻, $$ {{{\rm H}}_3}{{{\rm O}}^{+}}, $$ and H₂B are always larger than that of the $$ {\rm O}{{{\rm H}}^{-}} $$ ions. Further, in this range, the concentrations of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and H₂B are much smaller than that of HB⁻ (see P₁, P₂, and P₃), so that the simplified charge balance

(161)

follows. Hence, the pH-coordinate of P₁ gives the pH of the solution to a very good approximation. The amount balance

(159)

can also be simplified in P₁ because the concentration of B²⁻ is much larger than those of HB⁻ and H₂B:

(170)

The next rearrangements give a quadratic equation:

Simplified equation for a diacidic base with two weak protolysis reactions and in a high concentration:

(174)

This shows that this base can be treated like a weak monoacidic base because the second protolysis reaction can be disregarded (see Eq. (84) in Sect. 4.1.2.3).

4.3 Salt Solutions with Protolyzing Anions and Cations

As mentioned in the previous chapters, for the pH of a salt solution it is important whether the anion or the cation or both ions are undergoing protolysis reactions. If both ions exist in aqueous solutions as hydrated ions which do not undergo protolysis to any significant extent, the solution will be neutral, as in the case of sodium chloride NaCl (see Sect. 4.1.1.3).

When the hydrated cation does not react with water, but the anion does, the solutions are alkaline (the anions are bases which are protonated by water yielding OH⁻ ions). The pH-logc i diagrams of such cases have been described in the previous chapters.

When only the hydrated cation reacts with water, the solution will be acidic. Such solutions can be treated like those of all other Brønsted acids.

Salts with protolysing anions and cations may give acidic, neutral, or basic solutions, or they may even completely decompose. The pH of such solutions can be rather easily found with the help of pH-logc i diagrams. As first examples of such salts, we shall consider those having a monobasic acid and a monoacidic base, e.g., ammonium formate, ammonium acetate, and ammonium cyanide. Ammonium ion is a weak acid $$ {\rm p}K_{{\rm a}}^1 = 9.25 $$. The following Table 2 gives the pK a values of the acids corresponding to the bases (the anions of the salt).

Table 2

$$ {\rm p}K_{{\rm a}}^2 $$-values of the acids corresponding to the anions

Anion

Corresponding acid

Formate

Formic acid

3.75

Acetate

Acetic acid

4.75

Cyanide

Hydrogen cyanide

9.21

The pH-logc i diagrams can be constructed in the usual way. The anions and cations exist in equal overall concentrations. Dissolving 0.1 mol of these salts in water so that one liter of solution results, the overall concentration of the pairs ammonium/ammonia and anion/corresponding acid is also 0.1 mol L⁻¹. In the Figs. 56, 57, and 58, the respective diagrams are shown for solutions containing 0.1 mol L⁻¹ of the ammonium salts.

Fig. 56

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ ammonium formate

Fig. 57

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ ammonium acetate

Fig. 58

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ ammonium cyanide

In the aqueous solution, the following equilibria are established:

(175)

(176)

i.e., water acts as a proton-transferring agent for the following net reaction:

(177)

From this follows that the equilibrium concentrations of B¹ and A² should be equal. This condition is met in the crossing point of the acid line of the acid corresponding to the anion and the base line of the cation, i.e., in P₁ in Figs. 56, 57, and 58. The pH-coordinate of this point is the pH of the solutions. This graphical way of finding the pH is equivalent to the calculations derived for ampholytes which are weakly acidic and weakly basic (see Sect. 4.2.2.2):

(156)

In the case that the anions and cations have different charges, as, for example, in the case of ammonium oxalate, the approach is similar: The only difference to be taken into consideration is the different overall concentration of anions and cations. Figure 59 depicts the pH-logc i diagram of a solution containing 0.1 mol L⁻¹ (NH₄)₂C₂O₄. The total concentration of oxalate species is then 0.1 mol L⁻¹, however, the total concentration of ammonia/ammonium is 0.2 mol L⁻¹. The following proton transfer reactions take place in the solution:

Fig. 59

pH-logc i diagram of a solution containing 0.1 mol L⁻¹ ammonium oxalate

(178)

(179)

The net equilibrium is:

(180)

The pH of the solution can be read as the pH-coordinate of P₁ in Fig. 59, as that is the crossing point of the lines of NH₃ and $$ {\rm H}{{{\rm C}}_2}{\rm O}_4^{-} $$, i.e., the point where their concentrations are equal. Please note that the pH cannot be calculated using Eq. (156)!

4.4 Examples

4.4.1 The pH-logc i Diagram of Water

It is also possible to construct a pH-logc i diagram for water. With the density of water at 20 °C (0.998203 g cm⁻³) and its molar mass (18.0153 g mol⁻¹) one can calculate the concentration of pure water: 55.408 mol L⁻¹ ($$ \log C_{{{{{\rm H}}_2}{\rm O}}}^{ \circ } $$ = 1.74). Water is an ampholyte:

(2)

As mentioned at the very beginning, it is impossible to give the pK a value of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and the pK b value of $$ {\rm O}{{{\rm H}}^{-}} $$ on a strictly thermodynamic basis [9]. However, using the concentration of water in pure water ($$ \log C_{{{{{\rm H}}_2}{\rm O}}}^{ \circ } $$ = 1.74), one can construct the diagram shown in Fig. 60. From this diagram, one can read the apparent pK a values of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ (−1.74) and H₂O (15.74). However, both values are based on the surely wrong assumption that the activity of water in water is equal to its molar concentration! Therefore, one should regard Fig. 60 simply as a formal presentation, just to show that there is also a function of H₂O, corresponding to the lines of $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ and $$ {\rm O}{{{\rm H}}^{-}} $$.

Fig. 60

pH-logc i diagram of water with $$ \log C_{{{{\mathrm{ H}}_2}\mathrm{ O}}}^{\circ } $$ = 1.74 and the two not entirely correct values pK a($$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$) = −1.74 and pK a($$ {{\mathrm{ H}}_2}\mathrm{ O} $$) = 15.74

4.4.2 Acetic Acid/Acetate

Figure 61 shows the pH-logc i diagram of the corresponding acid–base pair acetic acid/acetate ions.

Fig. 61

pH-logc i diagram of the corresponding acid–base pair acetic acid/acetate ions $$ \log C_{\mathrm{ i}}^{\circ } $$ = −1.00, pK a = 4.75

In this diagram, one can read the pH values of the following solutions:

P 1 : aqueous solution containing 0.1 mol L⁻¹ of sodium acetate: $$ {c_{{\rm H}{\rm Ac}}}\approx {c_{{{\rm O}{{{\rm H}}^{-}}}}} $$, pH = 8.86.

P 2 : aqueous solution of 0.1 mol L⁻¹ acetic acid: $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm A}{{{\rm c}}^{-}}}}} $$, pH = 2.88.

P 3 : acetic acid/acetate buffer with maximum buffer capacity (see Chap.​ 2): 0.05 mol L⁻¹ acetic acid + 0.05 mol L⁻¹ sodium acetate; pH = pK S = 4.75.

P 4 : acetic acid/acetate buffer with lower buffer capacity: 0.1 mol L⁻¹ acetic acid, 0.01 mol L⁻¹ sodium acetate; pH = 3.75.

P 5 : acetic acid/acetate buffer with lower buffer capacity: 0.01 mol L⁻¹ acetic acid, 0.1 mol L⁻¹ sodium acetate; pH = 5.75.

4.4.3 Hydrogen Sulfide

In classical chemical separations, hydrogen sulfide is frequently used to separate and identify metal ions. Metal sulfides also play an important role in the environment. With the help of the pH-logc i diagram of hydrogen sulfide, one can easily find what sulfide species are present in a solution, and what metal sulfides can be precipitated under certain pH conditions. At 20 °C, a saturated solution of this gas in water contains 0.197 mol L⁻¹ H₂S following establishment of the equilibrium:

(181)

Figure 62 shows the pH-logc i diagram of hydrogen sulfide for this overall concentration ($$ \log C_{{{{{\rm H}}_2}{\rm S}}}^{ \circ } $$ = −0.71) and the acidity constants pK a1 = 7.00 and pK a2 = 13.9.

Fig. 62

pH-logc i diagram of hydrogen sulfide ($$ \log C_{{{{\mathrm{ H}}_2}\mathrm{ S}}}^{\circ } $$ = −0.71 and pK a1 = 7.00, pK a2 = 13.9)

The diagram makes it possible to determine the following pH data:

P 1 : saturated aqueous solution of hydrogen sulfide: $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm H}{{{\rm S}}^{-}}}}} $$, pH = 3.87.

P 2 : aqueous solution containing 0.197 mol L⁻¹ Na₂S: $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}\approx {c_{{{\rm H}{{{\rm S}}^{-}}}}} $$, pH = 13.3.

To tell at what pH the precipitation of a certain metal sulfide will take place, the following approach can be chosen. Assuming a manganese solution containing 10⁻² mol L⁻¹ Mn²⁺ ions, one needs to find the solubility product of MnS in a reference book. The solubility product K s is 10⁻¹⁵ mol² L⁻² (at 20 °C):

(182)

Manganese sulfide will precipitate when its solubility product is exceeded, i.e., when the S²⁻ concentration becomes larger than 10⁻¹³ mol L⁻¹ (for the given Mn²⁺ concentration of 10⁻² mol L⁻¹). According to the pH-logc i diagram (Fig. 62) this is the case at P₃, i.e., the precipitation will take place for pH values above 4.3. At lower pH values, the sulfide ion concentration is insufficient to produce a precipitate.

4.4.4 Phosphoric Acid

Phosphoric acid is a tribasic acid (see Fig. 63).

Fig. 63

pH-logc i diagram of phosphoric acid ($$ \log C_{\mathrm{ i}}^{\circ } $$ = −1.00 and pK a1 = 2.15, pK a2 = 7.2 and pK a3 = 12.35)

The diagram allows reading the following pH values:

P 1 : aqueous solution of 0.1 mol L⁻¹ H₃PO₄: $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{{{\rm H}}_2}{\rm PO}_4^{-}}}} $$, pH = 1.58.

P 2 : aqueous solution containing equimolar concentrations (0.05 mol L⁻¹) of H₃PO₄ and $$ {\rm Na}{{{\rm H}}_2}{\rm P}{{{\rm O}}_4} $$: pH = pK a1 = 2.15.

P 3 : aqueous solution containing 0.1 mol L⁻¹ NaH₂PO₄: $$ {c_{{{{{\rm H}}_3}{\rm P}{{{\rm O}}_4}}}}\approx {c_{{{\rm H}{\rm P}{\rm O}_4^{2- }}}} $$, pH = 4.68.

P 4 : aqueous solution containing 0.1 mol L⁻¹ NaH₂PO₄ and 0.01 mol L⁻¹ Na₂HPO₄ (NaH₂PO₄ and Na₂HPO₄ in the ratio of 10:1): pH = 6.2.

P 5 : aqueous solution containing equimolar concentrations (0.05 mol L⁻¹) of NaH₂PO₄ and Na₂HPO₄: pH = pK a2 = 7.2.

P 6 : aqueous solution containing NaH₂PO₄ and Na₂HPO₄ in the ratio 1:10 (0.01 mol L⁻¹ NaH₂PO₄, 0.1 mol L⁻¹ Na₂HPO₄): pH = 8.2.

P 7 : aqueous solution containing 0.1 mol L⁻¹ Na₂HPO₄: $$ {c_{{{{{\rm H}}_2}{\rm PO}_{{_4}}^{-}}}}\approx {c_{{{\rm PO}_4^{3- }}}} $$, pH =9.77.

P 8 : aqueous solution containing equimolar concentrations (0.05 mol L⁻¹) of Na₂HPO₄ and Na₃PO₄: pH = pK a3 = 12.35.

P 9 : aqueous solution containing 0.1 mol L⁻¹ Na₃PO₄: $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}\approx {c_{{{\rm H}{\rm PO}{{{_4^2}}^{-}}}}} $$, pH = 12.67.

4.4.5 Ascorbic Acid

In aqueous solutions ascorbic acid can be present in the following forms:

For simplicity, only one mesomeric formula is given for HA⁻ and A²⁻, although the different resonance structures contribute much to the stability of the bases, and thus also explain the rather high acidity of the OH groups (which are normally less acidic). Hydrogen ascorbate is frequently called simply ascorbate, which is strictly speaking wrong, but this custom results from the fact that ascorbate A²⁻ does not play any role in real, especially biochemically relevant, solutions. In food products, ascorbic acid, i.e., vitamin C, is present especially in fruits and vegetables. Citrus fruits in particular, such as oranges and lemons, contain large amounts after harvesting. Since ascorbic acid is easily oxidized, it is frequently used as an antioxidant to preserve food products. As the structural formula shows, ascorbic acid has four hydroxyl groups; however, only those at the C-2 and C-3 atoms can be deprotonated in water. The pK a values are pK a1 = 4.17 and pK a2 = 11.57. Ascorbic acid is often added to fruit juices and beverages. Thus, there are apple juices which contain – according to the producers – 55 mg vitamin C in 100 mL juice. This is a concentration of 3 × 10⁻³ mol L⁻¹ ($$ \log C_{{\rm i}}^{ \circ } $$ = −2.52). Figure 64 shows the respective pH-logc i diagram.

Fig. 64

pH-logc i diagram of ascorbic acid ($$ \log C_{\mathrm{ i}}^{\circ } $$ = −2.52, pK a1 = 4.17, pK a2 = 11.57)

The pH-coordinate of P₁ gives the pH of an aqueous solution of vitamin C of that example: $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm H}{{{\rm A}}^{-}}}}} $$, pH = 3.33. Since fruit juices also contain other acids with similar pK a values, their pH is usually in the range of 3 to 4. A solution of sodium hydrogen ascorbate would have at the same concentration a pH of 7.7 (see P₂ in Fig. 64, $$ {c_{{{{{\rm H}}_2}{\rm A}}}}\approx {c_{{{\rm A}^{2- }}}} $$). For parenteral delivery of vitamin C (injections, infusions), hydrogen ascorbate is used to make the solution pH compatible with blood.

4.4.6 Acetylsalicylic Acid

Acetylsalicylic acid (ASA, Aspirin®) is a common analgetic, anti-inflammatory, and antipyretic drug. Because of its acidity (pK a = 3.49) it can also cause some irritations which can prompt side reaction such as heartburn and vomiting when orally administered. Moreover, when regularly taken, it may lead to mucosal irritation and bleeding in the gastrointestinal tract, and also may lead to gastric ulcers. Typical dosage forms are tablets and effervescent tablets. One tablet may contain 500 mg ASA. This tablet dissolved in 150 mL of water will give an overall concentration of 0.018 mol L⁻¹. Figure 65 depicts the pH-logc i diagram of this solution ($$ \log C_{{\rm i}}^{ \circ } $$ = −1.73).

Fig. 65

pH-logc i diagram of an ASA solution ($$ \log C_{\mathrm{ i}}^{\circ } $$ = −1.73, pK a = 3.48)

P₁ ($$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm A}{\rm S}{{{\rm A}}^{-}}}}} $$) allows reading the pH which is 2.63. In effervescent tablets ASA is usually mixed with ascorbic acid, sodium dihydrogencitrate and citric acid, as well as with sodium hydrogen carbonate and disodium carbonate. If a tablet containing only 400 mg ASA and 240 mg vitamin C is dissolved in 150 mL water (this corresponds to 0.0148 mol L⁻¹ ASA and 0.009 mol L⁻¹ vitamin C) one can describe the solution with the help of the pH-logc i diagram shown in Fig. 66.

Fig. 66

pH-logc i diagram for a solution containing 400 mg ASA and 240 mg vitamin C in 150 mL water

The pH at P₁ is 2.63. Because of the slightly less acidic ascorbic acid, some additional $$ {{{\rm H}}_3}{{{\rm O}}^{+}} $$ is produced and the pH will be below 2.63. In commercial effervescent tablets, the above-mentioned added buffer systems keep the pH much higher, as the acids will be partially neutralized by the carbonates.

4.4.7 Benzoic Acid

In the food industry, benzoic acid (E 210 according to European codification), sodium benzoate (E 211), potassium benozoate (E 212), and calcium benzoate (E 213) are used as preservatives in ketchup, sausages, mustard, sauces, margarine, fish salads, and pickled foods. Benzoic acid is a moderately strong acid with pK a = 4.19. However, the acid is only sparingly soluble in water (2.9 g l⁻¹, i.e., 0.024 mol L⁻¹ at 25 °C). Fig. 67 shows the pH-logc i diagram of benzoic acid for an overall concentration of 0.01 mol L⁻¹. The pH of a solution containing 10⁻² mol L⁻¹ benzoic acid can be read at P₁ ($$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm benzoat}{{{\rm e}}^{-}}}}} $$): it is 3.10. The pH of a solution containing 10⁻² mol L⁻¹ sodium benzoate can be read at P₂ ($$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}\approx {c_{{\rm benzoic}\ {\rm acid}}} $$): it is 8.1.

Fig. 67

pH-logc i diagram of benzoic acid at an overall concentration of 0.01 mol L⁻¹

Fig. 68

pH-logc i diagram of glycine at an overall concentration of 0.1 mol L⁻¹

4.4.8 Glycine

Glycine is the most simple α-amino acid. It is an essential amino acid and a constituent of most proteins. It cannot be synthesized by the human organism. In aqueous solution, it exists mainly in the zwitterionic form HG+/− because the carboxylate group is so acidic (pK a1 (H₂G⁺) = 2.4) that it protonates the amino group (pK a2 (HG+/−) = 9.8). In glycine hydrochloride, it is in the cationic form (H₂G⁺), i.e., both the amino group and the carboxylate group are protonated (Fig. 68).

The diagram makes it possible to read the following pH values:

P 1 : aqueous solution containing 0.1 mol L⁻¹ glycine hydrochloride: $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{\rm H}{{{\rm G}}^{+/- }}}}} $$, pH = 1.71.

P 2 : aqueous solution containing 0.1 mol L⁻¹ glycine: $$ {c_{{{{{\rm H}}_2}{{{\rm G}}^{+}}}}}\approx {c_{{{\rm G}^{-}}}} $$, pH = 6.1.

P 3 : aqueous solution containing 0.1 mol L⁻¹ sodium glycinate: $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}\approx {c_{{{\rm H}{{{\rm G}}^{+/- }}}}} $$, pH = 11.41.

Comment: P₂ is called the isoelectric point the amino acid exists there mainly in the zwitterionic form, and only in very small and equal concentrations of the cationic and anionic forms. In an electric field, amino acids do not move at that pH, which is important in electrophoresis for isoelectric focussing.

4.4.9 Aspartic Acid

Frequently, hydrogen aspartate is called aspartate because the real aspartate (A²⁻) does not exist in significant concentrations under physiological conditions. Aspartic acid has two carboxyl groups. The carboxyl group at the α-C atom (pK a1 = 2.0) is more acidic than that at the β-C atom (pK a2 = 3.9). pK a3, i.e., the pK a value of the ammonium group is 8.8. Figure 69 shows the pH-logc i diagram of aspartic acid at an overall concentration of 0.01 mol L⁻¹. The isoelectric point (P₁: $$ {c_{{{{{\rm H}}_3}{{{\rm A}}^{+}}}}}\approx {c_{{{\rm H}{\rm A}^{-}}}} $$) is at pH = 2.95. Under physiological conditions (pH = 7.4), hydrogen aspartate is the dominating species (P₂ in Fig. 69).

Fig. 69

pH-logc i diagram of aspartic acid at an overall concentration of 0.01 mol L⁻¹

4.4.10 Ethylenediaminetetraacetic Acid (EDTA)

Ethylenediaminetetraacetic acid (H₄Y) plays an important role as a complex-forming reagent because it forms chelate complexes. For the metal complex formation of the ligand, i.e., the anion base Y⁴⁻, the protonation equilibria of the ligand are very important. The acid H₄Y is a tetrabasic acid. Instead of the sparingly soluble acid H₄Y, the disodium or the trisodium salts are normally used in complexometric titrations. In a pH-logc i diagram, one can easily read the pH values of these solutions: given an overall concentration of EDTA of 0.02 mol L⁻¹ ($$ \log C_{{\rm i}}^{ \circ } $$ = −1.70), which is a common concentration for complexometric titrations, Fig. 70 depicts the pH-logc i diagram.

Fig. 70

pH-logc i of an EDTA solution ($$ \log C_{\mathrm{ i}}^{ \circ } $$ = −1.70, pK a1 = 2.07, pK a2 = 2.75, pK a3 = 6.24, pK a4 = 10.34)

The diagram makes it possible to read the following pH values:

P 1 : aqueous solution of 0.02 mol L⁻¹ EDTE (H₄Y): $$ {c_{{{{{\rm H}}_3}{{{\rm O}}^{+}}}}}\approx {c_{{{{{\rm H}}_3}{{{\rm Y}}^{-}}}}} $$, pH = 1.89.

P 2 : aqueous solution of 0.02 mol L⁻¹ Na EDTE (NaH₃Y): $$ {c_{{{{{\rm H}}_4}{\rm Y}}}}\approx {c_{{{{{\rm H}}_2}{{{\rm Y}}^{2- }}}}} $$, pH = 2.41.

P 3 : aqueous solution of 0.02 mol L⁻¹ Na₂ EDTE (Na₂H₂Y): $$ {c_{{{{{\rm H}}_3}{{{\rm Y}}^{-}}}}}\approx {c_{{{\rm H}{{{\rm Y}}^{3- }}}}} $$, pH = 4.50.

P 4 : aqueous solution of 0.02 mol L⁻¹ Na₃ EDTE (Na₃HY): $$ {c_{{{{{\rm H}}_2}{{{\rm Y}}^{2- }}}}}\approx {c_{{{{{\rm Y}}^{4- }}}}} $$, pH = 8.29.

P 5 : aqueous solution of 0.02 mol L⁻¹ Na₄ EDTE (Na₄Y): $$ {c_{{{\rm O}{{{\rm H}}^{-}}}}}\approx {c_{{{\rm H}{{{\rm Y}}^{3- }}}}} $$, pH = 11.34.

The pH-logc i diagram is very useful to find the pH at which a solution should be buffered for a successful complexometric titration. This can be demonstrated as follows: given a solution containing 10⁻¹ mol L⁻¹ Ca²⁺ ions ($$ C_{{{\rm C}{{{\rm a}}^{2+}}}}^{ \circ } $$). The question is: at what pH this titration can be performed with an error not exceeding 1 %?

1 % of 10⁻¹ mol L⁻¹ $$ \hat{=}$$  10⁻³ mol L⁻¹

This is the maximum concentration of calcium $$ c_{{{\rm C}{{{\rm a}}^{2+}}}}^{{\rm eqp}} $$ which may be left uncomplexated at the equivalence point. Since complex formation proceeds in a 1:1 ratio, this is also the overall concentration of EDTA which may stay in the solution at the equivalence point. Figure 71 shows the pH-logc i diagram of EDTA for $$ \log C_{{\rm i}}^{ \circ } $$ = −3.

Fig. 71

pH-logc i diagram of EDTA ($$ \log C_{\mathrm{ i}}^{ \circ } $$ = −3, pK a1 = 2.07, pK a2 = 2.75, pK a3 = 6.24, pK a4 = 10.34)

From the stability constant of the Ca²⁺-EDTA complex (K stab = 1010.69 mol L⁻¹) one can calculate the allowable equilibrium concentration of the free ligand Y⁴⁻. For this, only the protonations of the ligand have to be taken into account, and possible side reactions of calcium ions may be disregarded. For the equilibrium concentration of the Ca²⁺-EDTA complex $$ {c_{{{\rm Ca}{{{\rm Y}}^{2- }}}}} $$ one can use the starting concentration of calcium ions $$ C_{{{\rm C}{{{\rm a}}^{2+}}}}^{ \circ } $$ as a good approximation, since the error should be smaller than 1 %:

(183)

(184)

This equilibrium concentration of Y⁴⁻ is in the solution at pH = 5.45 (cf. Fig. 71). This means that the pH should be kept at least at 5.45 (or higher), when the error should be kept equal to or smaller than 1 %.

Literature

1.

Narasaki H (1979) Talanta 26:605–607CrossRef

2.

Narasaki H (1980) Talanta 27:187–191CrossRef

3.

Narasaki H (1980) Talanta 27:193–199CrossRef

4.

Narasaki H (1980) Talanta 27:409–415CrossRef

5.

Henrion G, Schmidt R, Marquardt D (1980) Z Chem 20:433–436CrossRef

6.

Narasaki H (1987) Fresenius Z Anal Chem 328:633–638CrossRef

7.

Bliefert C (1978) pH-Wert-Berechnungen. Verlag Chemie, Weinheim

8.

Doğan A, Kılıç E (2007) Anal Biochem 365:7–13CrossRef

9.

Henrion G, Scholz F, Schmidt W, Zettler M (1982) Z Phys Chem (Leipzig) 263:634–636

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4_5© Springer-Verlag Berlin Heidelberg 2013

5. The Use of pH-logc i Diagrams for the Construction of Titration Diagrams

Heike Kahlert1  and Fritz Scholz¹

(1)

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Abstract

Diagrams displaying acid–base titrations should show the pH on the y-axis as a function of the advancement of the titration. The advancement of titration can be given by the added volume of titrator. However, using the volume of added titrator on the x-axis, each titration would need a different x-axis. To avoid this, one likes to use a normation, which is possible by calculating a so-called “degree of titration” τ (lower case Greek letter “tau”). The degree of titration can be defined as follows:

Diagrams displaying acid–base titrations show the pH on the y-axis as a function of the advancement of the titration. The advancement of titration can be given by the added volume of titrator. However, using the volume of added titrator on the x-axis, each titration would need a differently scaled x-axis. To avoid this, one likes to use a normation, which is possible by calculating a so-called “degree of titration” τ (lower case Greek letter “tau”). The degree of titration can be defined as follows:

Different definitions are given in the literature. However, since they are more difficult to grasp, the above definition is given here.

The degree of titration can assume values between zero and infinity. It is unity (1.00) at the equivalence point, that is, for an almost 100 % conversion of the titrated species, and exactly an addition of 100 % of the titrator with respect to the titrated species. To easily understand titration diagrams, it makes sense to plot them in the range of τ = 0 to τ = 2. The condition τ = 2 means a 100 % overtitration, which means that 200 % of the titrator has been added with respect to the amount of species to be titrated. One should understand here that the titration reaction takes place only up to τ = 1, and after τ = 1 the amount of reacting species tends to zero (although it is not zero because the equilibrium is still shifted). For small equilibrium constants of the titration reaction, the amount of species that are converted after τ = 1 may of course be more significant. Such titration reactions are certainly less useful for analytical purposes.

It is important to understand that the degree of titration is a normation giving the advancement of titration.

Comment: the lambda value measured with the help of lambda probes in cars is nothing but a degree of titration, here the degree of oxidation of the petrol by the oxygen of the air. The burning of petrol can be considered a redox titration!

When mixtures of acids or polybasic acids are titrated, there is the problem of how to define the “”: there is no common rule for this, but it makes sense to take here the overall amount of protons that can be titrated. This would mean that in the case of phosphoric acid, the first proton is titrated at τ = 1/3 (yielding a solution mainly containing $$ {{\mathrm{ H}}_2}\mathrm{ PO}_4^{-} $$), the second proton is titrated at τ = 2/3 (yielding a solution mainly containing $$ \mathrm{ HPO}_4^{2- } $$), and the third proton is titrated at τ = 3/3 = 1 (yielding a solution mainly containing $$ \mathrm{ PO}_4^{3- } $$). Of course one may define τ also in such way that it is 1 for the titration of the first, 2 for the titration of the second, and 3 for the titration of the third proton. In that case, one should plot the diagram in the range $$ 0\leq \tau \leq 4 $$ so as to reach a final concentration of NaOH that is equivalent to the starting concentration of phosphoric acid. Titrations of bases can be treated in an analogous way, as it will be shown in some examples.

Of course, titration curves can be also completely (and exactly) calculated, and software packages are available for this (e.g., at http:​/​/​www2.​iq.​usp.​br/​docente/​gutz/​Curtipot_​.​html). However, such calculations are beyond the scope of this book, which is focused on demonstrating how titration diagrams can be constructed in such a way that the most important features are displayed.

For constructing titration diagrams from pH-logc i diagrams, the latter have to be turned by 90° to the right, then the mirror image (with respect to the horizontal line at pH = 7) has to be produced (cf. Fig. 72a, b, c). The resulting pH-logc i diagram allows the parallel arrangement of the titration diagram (see Fig. 72d).

Fig. 72

(a) pH-logc i diagram. (b) pH-logc i following a turning by 90° to the right side (around the center). (c) pH-logc i diagram resulting in (b) and mirrored at the middle axis (pH = 7). (d) pH-logc i diagram together with the titration diagram, so that both pH scales are parallel aligned

For the construction of titration diagrams, the dilution which occurs during titration can be disregarded, as normally the volume of added titrator is smaller than the volume of titrand, and dilution affects the curves only marginally.

In constructing titration diagrams, the following idea is very helpful: a titration is a synthesis, i.e., by titrating hydrochloric acid with sodium hydroxide, one has synthesized a pure sodium chloride solution when τ = 1.0 ($$ {c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}={c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$) is reached. At τ = 0.5, one has performed only 50 % of the synthesis, and the solution contains equal concentrations of sodium chloride and hydrochloric acid. At τ = 1.5, the hydrochloric acid has been completely (100 %) converted to sodium chloride, and the solution contains an excess of 50 % of sodium hydroxide. At τ = 2.0, that excess amounts to 100 %.

5.1 Titration of Hydrochloric Acid of Various Concentrations with Sodium Hydroxide Solution

Figure 73 shows the titration diagram of hydrochloric acid ($$ \log C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$ = −1) with sodium hydroxide solution.

Fig. 73

Titration diagram of hydrochloric acid ($$ \log C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$ = −1) with sodium hydroxide solution

The following pH values can be taken from the pH-logc i diagram to construct the titration diagram (always disregarding the dilution):

P 1 : τ = 0.0; starting point of the titration (i.e., condition before titration), $$ C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ }={c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}={c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}} $$, pH = 1.

P 2 : τ = 0.9; the concentration of hydrochloric acid is 10 % of the starting concentration, i.e., 0.01 mol L⁻¹, pH = 2.

P 3 : τ = 0.99; the concentration of hydrochloric acid is 1 % of the starting concentration, i.e., 0.001 mol L⁻¹, pH = 3.

P 4 : τ = 1.0; the added amount of sodium hydroxide exactly equals the amount of hydrochloric acid at the start of the titration, i.e., there is only sodium chloride dissolved in the solution ($$ {c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}={c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$). The solution is neutral, i.e., pH = 7.

P 5 : τ = 1.01; this is a solution of sodium chloride with additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{100 }C_{\mathrm{ hydrochloricacid}}^{\circ }=0.001\ \mathrm{ mol} {{\mathrm{ L}}^{-1 }} $$ (the concentration of sodium ions is not relevant for the titration; however, it can be given as follows: $$ {c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}=\left( {\frac{1}{1}+\frac{1}{100 }} \right)C_{\mathrm{ hydrochloricacid}}^{\circ }=0.101\ \mathrm{ mol} {{\mathrm{ L}}^{-1 }} $$), pH = 11.

P 6 : τ = 1.1; this is a solution of sodium chloride with additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{10 }C_{\mathrm{ hydrochloric\ acid}}^{\circ }=0.01\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$ ($$ {c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}=\left( {\frac{1}{1}+\frac{1}{10 }} \right)C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ }=0.11\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$), pH = 12.

P 7 : τ = 2.0; this is a solution of sodium chloride with additional sodium hydroxide: $${c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ hydrochloric \ acid}}^{\circ }=0.1\ \mathrm{ mol} {{\mathrm{ L}}^{-1 }} $$ ($$ {c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}=\left( {\frac{1}{1}+\frac{1}{1}} \right)C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ }=0.2\ \mathrm{ mol} \ {{\mathrm{ L}}^{-1 }} $$), pH = 13.

Figure 74 displays a number of titration curves for different starting concentrations of hydrochloric acid. The important conclusions from this figure are: (i) the pH at the equivalence point is unaffected by the starting concentration, i.e., always 7.0, (ii) the pH jump around the equivalence point, defined as ΔpH between τ = 0.99 and τ = 1.01, decreases with decreasing initial concentration of hydrochloric acid. This means that $$ \frac{\mathrm{ d}\mathrm{ pH}}{{\mathrm{ d}\tau }} $$ decreases with decreasing initial concentration (this feature is not noticeable in the diagram because of the used scales). Decreasing $$ \frac{\mathrm{ d}\mathrm{ pH}}{{\mathrm{ d}\tau }} $$ leads to increasing random errors, which will be explained further down when errors are explained.

Fig. 74

Titration diagram of hydrochloric acid solutions of different concentrations with sodium hydroxide solution. (a) $$ \log C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$ = −3. (b) $$ \log C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$ = −2. (c) $$ \log C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$ = −1

5.2 Titration of Sodium Hydroxide Solution with Hydrochloric Acid

The following pH values can be taken from the pH-logc i diagram to construct the titration diagram (always disregarding the dilution) (Fig. 75):

Fig. 75

Titration diagram of sodium hydroxide with hydrochloric acid ($$ \log C_{\mathrm{ NaOH}}^{\circ } $$ = −1)

P 1 : τ = 0.0; starting point of the titration (i.e., condition before titration), $$ C_{\mathrm{ N}\mathrm{ aOH}}^{\circ }={c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}={c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$, pH = 13.

P 2 : τ = 0.9; the concentration of sodium hydroxide is 10 % of the starting concentration, i.e., 0.01 mol L⁻¹, pH = 12.

P 3 : τ = 0.99; the concentration of sodium hydroxide is 1 % of the starting concentration, i.e., 0.001 mol L⁻¹, pH = 11.

P 4 : τ = 1.0; the added amount of hydrochloric acid exactly equals the amount of sodium hydroxide at the start of the titration, i.e., there is only sodium chloride dissolved in the solution ($$ {c_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}}={c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ N}\mathrm{ aOH}}^{\circ } $$). The solution is neutral, i.e., pH = 7.

P 5 : τ = 1.01; this is a solution of sodium chloride with additional hydrochloric acid: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\frac{1}{100 }C_{\mathrm{ NaOH}}^{\circ }=0.001\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$ (the concentration of chloride ions is not relevant for the titration; however, it can be given as follows: $$ {c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}=\left( {\frac{1}{1}+\frac{1}{100 }} \right)C_{\mathrm{ NaOH}}^{\circ }=0.101\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$), pH = 3.

P 6 : τ = 1.1; this is a solution of sodium chloride with additional hydrochloric acid: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\frac{1}{10 }C_{\mathrm{ NaOH}}^{\circ }=0.01\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$ ($$ {c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}=\left( {\frac{1}{1}+\frac{1}{10 }} \right)C_{\mathrm{ NaOH}}^{\circ }=0.11\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$), pH = 2.

P 7 : τ = 2.0; this is a solution of sodium chloride with additional hydrochloric acid: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\frac{1}{1}C_{\mathrm{ NaOH}}^{\circ }=0.1\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$ ($$ {c_{{\mathrm{ C}{{\mathrm{ l}}^{-}}}}}=\left( {\frac{1}{1}+\frac{1}{1}} \right)C_{\mathrm{ NaOH}}^{\circ }=0.2\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$), pH = 1.

5.3 Titration of Different Concentrations of Acetic Acid with Sodium Hydroxide

The following pH values can be taken from the pH-logc i diagram to construct the titration diagram (always disregarding the dilution) (Fig. 76):

Fig. 76

Titration diagram of acetic acid ($$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −1; pK a = 4.75) with sodium hydroxide

P 1 : τ = 0.0; starting point of the titration (i.e., condition before titration), $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}\approx {c_{{\mathrm{ A}{{\mathrm{ c}}^{-}}}}} $$ (see Sect. 4.​1.​3.​1, Eq. (93)), pH = 2.88.

P 2 : τ = 0.5; acetic acid and acetate are present at equal concentrations. This is the so-called buffer point: $$ {c_{\mathrm{ HAc}}}={c_{{\mathrm{ A}{{\mathrm{ c}}^{-}}}}} $$, pH = pK a = 4.75.

P 3 : τ = 1.0; the added amount of sodium hydroxide exactly equals the amount of acetic acid at the start of the titration, i.e., there is only sodium acetate dissolved in the solution: $$ {c_{\mathrm{ H}\mathrm{ Ac}}}\approx {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$ (see Sect. 4.​1.​2.​3, Eq. (84)), pH = 8.875. The solution is basic at the equivalence point!

P 4 : τ = 1.01; this is a solution of sodium acetate with additional sodium hydroxide. From the presence of NaOH, the additional concentration of $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ ions is: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{100 }C_{\mathrm{ aceticacid}}^{\circ }=0.001\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$. The $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ concentration formed by protonation of acetate ions is smaller than 10⁻⁷ mol L⁻¹, and hence it can be disregarded: the pH is 11.

P 5 : τ = 1.1; this is a solution of sodium acetate with additional sodium hydroxide. From the presence of NaOH the additional concentration of $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ ions is: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{10 }C_{\mathrm{ aceticacid}}^{\circ }=0.01\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$. The $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ concentration formed by protonation of acetate ions is smaller than 10⁻⁸ mol L⁻¹, and hence it can be disregarded: the pH is 12.

P 6 : τ = 2.0; this is a solution of sodium acetate with additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ aceticacid}}^{\circ }=0.1\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$, pH = 13.

Figure 77 shows the titration diagrams of acetic acid of different concentrations.

Fig. 77

Titration diagrams of acetic acid of different concentrations (pK a = 4.75) with sodium hydroxide. (a) $$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −3. (b) $$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −2. (c) $$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −1

P 1 : Equivalence point for acetic acid with $$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −3, pH = 7.875.

P 2 : Equivalence point for acetic acid with $$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −2, pH = 8.375.

P 3 : Equivalence point for acetic acid with $$ \log C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = −1, pH = 8.875.

The important conclusions from this figure are: (i) the pH at the equivalence point decreases with decreasing concentration, (ii) the pH jump around the equivalence point, defined as ΔpH between τ = 0.99 and τ = 1.01, decreases with decreasing concentration. The latter also means that the slope $$ \frac{\mathrm{ d}\mathrm{ pH}}{{\mathrm{ d}\tau }} $$ decreases with concentration, and thus the random error will increase, since the random error is always related to a constant ΔpH typical for a certain indication method (see further down the discussion of random errors of titrations).

5.4 Titration of Moderately Strong Acids Having Different pK a Values with Sodium Hydroxide

Figures 78, 79, and 80 show titration diagrams of three moderately strong acids with different pK a values. In each diagram, only for one case are the distinct points of titration labeled. They are derived from the pH-logc i diagrams as in the case of acetic acid (see Sect. 5.3).

Fig. 78

Titration diagrams of three acids ($$ \log C_{\mathrm{ acid}}^{\circ } $$ = −1) with pK a = 3 (black line), pK a = 5 (orange line), and pK a = 7 (olive line) with sodium hydroxide solution. The distinctive points are given only for pK a = 3

Fig. 79

Titration diagrams of three acids ($$ \log C_{\mathrm{ acid}}^{\circ } $$ = −1) with pK a = 3 (black line); pK a = 5 (orange line) and pK a = 7 (olive line) with sodium hydroxide solution. The distinctive points are given only for pK a = 5

Fig. 80

Titration diagrams of three acids ($$ \log C_{\mathrm{ acid}}^{\circ } $$ = −1) with pK a = 3 (black line); pK a = 5 (orange line) and pK a = 7 (olive line) with sodium hydroxide solution. The distinctive points are given only for pK a = 7

The most important conclusions from these figures are: (i) the pH at the buffer point increases with increasing pK a values, (ii) the pH at the equivalence points increases with increasing pK a values, and (iii) the pH jump around the equivalence point, defined as ΔpH between τ = 0.99 and τ = 1.01, decreases with increasing pK a values. The latter means that the slope $$ \frac{\mathrm{ d}\mathrm{ pH}}{{\mathrm{ d}\tau }} $$ decreases also with increasing pK a values, which leads to increasing random errors, as any indication always relates to a specific constant ΔpH (see further down the discussion of random errors of titrations).

5.5 Titration of Sulfuric Acid

In Sect. 4.​2.​1.​1, it was explained how the pH of sulfuric acid solutions has to be calculated. For constructing the titration diagram, a number of the simplifications used there can be applied again (see Fig. 81).

Fig. 81

Titration diagram of sulfuric acid ($$ \log C_{\mathrm{ sulfuric}\ \mathrm{ acid}}^{\circ } $$ = −1), pK a1 = −3, pK a2 = 1.99 with sodium hydroxide

The following pH values can be taken from the pH-logc i diagram to construct the titration diagram (always disregarding the dilution):

P 1 : τ = 0.0; starting point of titration: it is a good simplification to write $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}={c_{{\mathrm{ H}\mathrm{ SO}_4^{-}}}} $$, and this gives pH = 1. This means disregarding the protolysis of $$ \mathrm{ HSO}_4^{-} $$, which should not (!) be done in calculating the pH (see Sect. 4.​2.​1.​1), as a calculation with Eq. (122) gives pH = 0.96.

P 2 : τ = 0.5; the added amount of sodium hydroxide is such that a solution containing 0.1 mol L⁻¹ sodium hydrogen sulfate results (see Sect. 4.​2.​2.​1), i.e., it holds that $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}\approx {c_{{{{\mathrm{ B}}^{2- }}}}} $$ (Eq. (139)): pH = 1.54.

P 3 : τ = 0.75; the solution contains equal concentrations of sulfate and hydrogen sulfate, i.e., it is hydrogen sulfate/sulfate buffer: $$ {c_{{\mathrm{ HSO}_4^{-}}}}={c_{{\mathrm{ SO}_4^{2- }}}} $$ pH = pK a2 = 1.99.

P 4 : τ = 1.0; the added amount of sodium hydroxide exactly equals the amount of titratable protons, it is twice the amount of sulfuric acid. It is a solution of sodium sulfate. Protolysis leads to $$ {c_{{\mathrm{ H}\mathrm{ SO}_4^{-}}}}={c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$: pH = 7.49.

P 5 : τ = 1.005; this is a solution of sodium sulfate with additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{100 }C_{\mathrm{ sulfuricacid}}^{\circ }=0.001\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$: pH = 11. The pH effect due to the protonation of $$ \mathrm{ SO}_4^{2- } $$ can be disregarded.

P 6 : τ = 1.05; this is a solution of sodium sulfate with additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{10 }C_{\mathrm{ sulfuricacid}}^{\circ }=0.01\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$, pH = 12. The pH effect due to the protonation of $$ \mathrm{ SO}_4^{2- } $$ can be disregarded.

P 7 : τ = 1.5; this is a solution of sodium sulfate with additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ sulfuricacid}}^{\circ }=0.1\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$, pH = 13. The pH effect due to the protonation of $$ \mathrm{ SO}_4^{2- } $$ can be disregarded.

5.6 Titration of Dibasic Amino Acids

Amino acids are commercially available as so-called hydrochlorides, i.e., in the form of the salts $$ {{\mathrm{ H}}_2}\mathrm{ A}{{\mathrm{ S}}^{+}}\mathrm{ C}{{\mathrm{ l}}^{-}} $$ (see Sects. 4.​2.​1.​2 and 4.​4.​8). This is the reason that the titration of $$ {{\mathrm{ H}}_2}\mathrm{ A}{{\mathrm{ S}}^{+}} $$ will be discussed here (see Fig. 82). In the pH-logc i diagram, the following three forms of the amino acid have been coded as follows:

Fig. 82

Titration diagram of alanine ($$ \log C_{\mathrm{ alanine}}^{\circ } $$ = −1, pK a1 = 2.3, pK a2 = 9.9) with sodium hydroxide

The following pH values can be taken from the pH-logc i diagram to construct the titration diagram (always disregarding the dilution):

P 1 : τ = 0.0; starting point of titration: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}\approx {c_{{\mathrm{ H}\mathrm{ A}{{\mathrm{ S}}^{+/- }}}}} $$.

P 2 : τ = 0.25; one half of the initial amount of H₂AS⁺ has been converted to HAS+/−, i.e., the solution contains a buffer: $$ {c_{{{{\mathrm{ H}}_2}\mathrm{ A}{{\mathrm{ S}}^{+}}}}}\approx {c_{{\mathrm{ H}\mathrm{ A}{{\mathrm{ S}}^{+/- }}}}} $$ and pH = pK a1.

P 3 : τ = 0.5; the added amount of sodium hydroxide exactly equals the initial amount of the hydrochloride. Hence the amino acid is present in its zwitterionic form HAS+/−: $$ {c_{{{{\mathrm{ H}}_2}\mathrm{ A}{{\mathrm{ S}}^{+}}}}}\approx {c_{{\mathrm{ A}{{\mathrm{ S}}^{-}}}}} $$ and Eq. (156) $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\sqrt{{{K_{\mathrm{ a}1}}{K_{\mathrm{ a}2}}}} $$ is valid so that $$ \mathrm{ pH}=\frac{{\mathrm{ p}{K_{\mathrm{ a}1}}+\mathrm{ p}{K_{\mathrm{ a}2}}}}{2} $$.

P 4 : τ = 0.75; one half of HAS+/− has been converted to AS–, i.e., another buffer results: $$ {c_{{\mathrm{ HA}{{\mathrm{ S}}^{+/- }}}}}={c_{{\mathrm{ A}{{\mathrm{ S}}^{-}}}}} $$ and pH = pK a2.

P 5 : τ = 1.0; the added amount of sodium hydroxide exactly equals the amount of titratable protons of the dibasic amino acid. This is a solution of Na⁺AS–. Protonation of AS– gives: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}\approx {c_{{\mathrm{ H}\mathrm{ A}{{\mathrm{ S}}^{+/- }}}}} $$.

P 6 : τ = 1.5; this is a solution of Na⁺AS– containing additional sodium hydroxide: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ alanine}}^{\circ }=0.1\ \mathrm{ mol}\ {{\mathrm{ L}}^{-1 }} $$. The pH effect due to the protonation of AS– can be disregarded.

5.7 Comparison of the Titrations of an Acid and its Corresponding Base: (a) Ammonium Ions with Sodium Hydroxide, (b) Ammonia with Hydrochloric Acid

This example is aimed at illustrating that titration curves may be significantly different for the case that an acid is titrated and the case where the corresponding base is titrated. Fig. 83 displays the titration curves for ammonium ions with sodium hydroxide and ammonia with hydrochloric acid.

Fig. 83

Titration diagram of ammonia with hydrochloric acid (orange line) and ammonium ions (e.g., ammonium chloride) with sodium hydroxide (black line) ($$ \log {C^{\circ }} $$ = −1, $$ \mathrm{ p}K_{\mathrm{ a}}^{{\mathrm{ NH}_4^{+}}} $$ = 9.25)

The titration curves are constructed as follows:

(a)

Titration of ammonium ions with NaOH:

P 1 : τ = 0.0; starting point of titration: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}\approx {c_{{\mathrm{ N}{{\mathrm{ H}}_3}}}} $$

P 2 : τ = 0.5; ammonium ions and ammonia are present at equal concentrations. This is the buffer point of the ammonium/ammonia buffer: $$ {c_{{\mathrm{ N}\mathrm{ H}_4^{+}}}}={c_{{\mathrm{ N}{{\mathrm{ H}}_3}}}} $$, pH = $$ \mathrm{ p}K_{\mathrm{ a}}^{{\mathrm{ NH}_4^{+}}} $$ = 9.25.

P 3 : τ = 1.0; the added amount of sodium hydroxide exactly equals the initial amount of ammonia. The solution contains ammonia and sodium chloride: $$ {c_{{\mathrm{ NH}_4^{+}}}}\approx {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$.

P 4 : τ = 1.1; this is a solution containing ammonia, sodium chloride and an additional concentration of sodium hydroxide. The $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ concentration due to NaOH is: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{10 }C_{\mathrm{ ammonium}}^{\circ }=0.01\ \mathrm{ mol} {{\mathrm{ L}}^{-1 }} $$. The $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ concentration due to protonation of ammonia is less than 10⁻⁴ mol L⁻¹ and can be disregarded. Hence the pH is 12.

P 5 : τ = 2.0; this is a solution containing ammonia, sodium chloride and an additional concentration of sodium hydroxide. The $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ concentration due to NaOH is: $$ {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}=\frac{1}{1}C_{\mathrm{ ammonium}}^{\circ }=0.1\ \mathrm{ mol} \ {{\mathrm{ L}}^{-1 }} $$, pH = 13. The $$ \mathrm{ O}{{\mathrm{ H}}^{-}} $$ concentration due to protonation of ammonia can be disregarded.

(b)

Titration of ammonia with hydrochloric acid:

P 1′ : τ = 0.0; starting point of titration: $$ {c_{{\mathrm{ NH}_4^{+}}}}\approx {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$

P 2′ : τ = 0.5; ammonium ions and ammonia are present at equal concentrations. This is the buffer point of the ammonium/ammonia buffer: $$ {c_{{\mathrm{ N}\mathrm{ H}_4^{+}}}}={c_{{\mathrm{ N}{{\mathrm{ H}}_3}}}} $$, pH = $$ \mathrm{ p}K_{\mathrm{ a}}^{{\mathrm{ NH}_4^{+}}} $$ = 9.25.

P 3′ : τ = 1.0; the added amount of hydrochloric acid exactly equals the initial amount of ammonia. This is a solution of ammonium chloride: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}\approx {c_{{\mathrm{ N}{{\mathrm{ H}}_3}}}} $$

P 4′ : τ = 1.01; this is a solution of ammonium chloride with an addition of hydrochloric acid. The $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ concentration due to the presence of hydrochloric acid is: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\frac{1}{100 }C_{\mathrm{ ammonia}}^{\circ }=0.001\ \mathrm{ mol} \ {{\mathrm{ L}}^{-1 }} $$. The $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ due to protolysis of ammonium ions is less than 10⁻⁷ mol L⁻¹, and it can be disregarded. The pH is 3.

P 5′ : τ = 1.1; this is a solution of ammonium chloride with an addition of hydrochloric acid. The $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ concentration due to the presence of hydrochloric acid is: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\frac{1}{10 }C_{\mathrm{ ammonia}}^{\circ }=0.01\ \mathrm{ mol} \ {{\mathrm{ L}}^{-1 }} $$. The $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ due to protolysis of ammonium ions is less than 10⁻⁸ mol L⁻¹, and it can be disregarded. The pH is 2.

P 6′ : τ = 2.0; this is a solution of ammonium chloride with an addition of hydrochloric acid. The $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ concentration due to the presence of hydrochloric acid is: $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}=\frac{1}{1}C_{\mathrm{ ammonia}}^{\circ }=0.1\ \mathrm{ mol} \ {{\mathrm{ L}}^{-1 }} $$. The $$ {{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$ due to protolysis of ammonium ions can be disregarded. The pH is 1.

Comparing the two titration curves one can recognize that the steepness of the titration curve at the equivalence point is much larger in case of the titration of ammonia with hydrochloric acid, than for the case of titration of ammonium ions with sodium hydroxide. This means that the random errors will be much smaller when ammonia is titrated with hydrochloric acid, and this titration is strongly to be preferred.

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4_6© Springer-Verlag Berlin Heidelberg 2013

6. Titration Errors

Heike Kahlert1  and Fritz Scholz¹

(1)

Institute of Biochemistry, University of Greifswald, Greifswald, Germany

Abstract

There are two types of titration errors, systematic and random errors. Systematic errors can be easily estimated with the help of pH-logc i diagrams. The diagrams are especially helpful for choosing the best color indicators. The diagrams are also useful to find, in the case of alternative titrations, the one with the smallest random error (see Sect. 5.​7). The following estimation of systematic titration errors is confined to cases where the autoprotolysis of water can be ignored, because a comprehensive and detailed treatment including the autoprotolysis would be completely beyond the scope of this book, and also only of importance in rather exotic cases of very dilute solutions.

There are two types of titration errors: systematic and random errors. Systematic errors can be easily estimated with the help of pH-logc i diagrams. The diagrams are especially helpful for choosing the best color indicators. The diagrams are also useful to find, in the case of alternative titrations, the one with the smallest random error (see Sect. 5.​7). The following estimation of systematic titration errors is confined to cases where the autoprotolysis of water can be ignored, because a comprehensive and detailed treatment including autoprotolysis would be completely beyond the scope of this book and only of importance in rather exotic cases of very dilute solutions.

6.1 Systematic Titration Errors

The reason for systematic titration errors is that the equivalence point is indicated too early or too late. This happens when the transition point of the indicator does not exactly match the pH of the equivalence point of the titration (systematic errors caused by wrongly calibrated pipettes or burettes will not be discussed here). The transition point of an indicator gives the experimental endpoint of the titration. Because the term endpoint can also be applied in the sense of “theoretical endpoint = equivalence point” we shall use here the term “transition point” to be clear. The same can happen in case of instrumental methods of indication when these methods do not identify the equivalence point correctly, but systematically deviate from it. Color indicators are themselves acid–base systems $$ \mathrm{ HI}/{{\mathrm{ I}}^{-}} $$ ($$ \mathrm{ HI}+{{\mathrm{ H}}_2}\mathrm{ O} \overset{{}}\ {\rightleftarrows}\ {{\mathrm{ I}}^{-}}+{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}} $$), the pK a value of which is usually denoted as the $$ \mathrm{ p}{K_{\mathrm{ I}}} $$ value, and it normally falls in the range of 2–12. There are bichromic and monochromic indicators. For example, a bichromic indicator may be red as an acid and blue as a base, and a monochromic may be colorless as an acid and violet as a base. In the case of bichromic indicators, the color changes when $$ {c_{\mathrm{ HI}}}={c_{{{{\mathrm{ I}}^{-}}}}} $$, that is at the buffer point of the indicator. Of course, the color change does not abruptly occur there, but it is smeared out in an interval (the so-called transition interval of an indicator), roughly in the range $$ \frac{1}{10}\leq {\displaystyle\frac{{{c_{\mathrm{ HI}}}}}{{{c_{{{{\mathrm{ I}}^{-}}}}}}}}\leq \frac{10 }{1} $$, in which a mixed color results. According to the buffer equation (see Chap.​ 2) this is equal to the pH range: $$ \Delta \left( {\mathrm{ pH}} \right) = \mathrm{ p}{K_{\mathrm{ I}}}\pm 1 $$. In Table 3, $$ \mathrm{ p}{K_{\mathrm{ I}}} $$ values and transition ranges of some indicators are given. In case of monochromic indicators one can detect the colored form already when it presents only 10 % of the overall indicator concentration (which additionally depends on the overall concentration). This leads to the phenomenon that the transition point of monochromic indicators appears to be systematically shifted from the $$ \mathrm{ p}{K_{\mathrm{ I}}} $$ value toward the colorless side.

Table 3

pK l values and transition intervals of some common acid–base indicators

Indicator

pK I

Transition interval

Color of the acid

Color of the base

Methyl orange

4.0

3.1–4.4

Red

Yellow–orange

Methyl red

5.8

4.2–6.3

Red

Yellow

Bromthymol blue

7.1

6.0–7.6

Yellow

Blue

Neutral red

7.0

6.8–8.0

Red

Yellow

Phenolphthalein

8.4

8.2–10.0

Colorless

Pink (fuchsia)

Thymolphthalein

10.0

9.3–10.6

Colorless

Blue

6.1.1 Systematic Titration Errors in Titrations of Acids with pK a Values Between 0 and 14 with Very Strong Bases (e.g., NaOH)

pH-logc i diagrams are very well suited to estimate systematic titration errors. At the equivalence point (eqp) of a titration, the amount of titrator added to the sample solution exactly equals the given amount of titrand (compound to be titrated). When the amount $$ n_{\mathrm{ HB}}^{\circ } $$ of an acid with a pK a value between 0 and 14 is titrated up to the equivalence point, the amount of added OH⁻ is $$ {{\left( {n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}^{\circ }} \right)}_{\mathrm{ eqp}}}=n_{\mathrm{ H}\mathrm{ B}}^{\circ } $$. When NaOH is used, this is equal to $$ {{\left( {n_{{\mathrm{ N}{{\mathrm{ a}}^{+}}}}^{\circ }} \right)}_{\mathrm{ eqp}}} $$. If one were to evaporate the solution at the equivalence point, pure NaB would be obtained. Hence, the solution has at the equivalence point exactly the pH of a NaB solution of the respective concentration (see Sects. 4.​1.​2.​3 and 4.​1.​3.​3): when the autoprotolysis of water is disregarded, the pH is given by the condition $$ {c_{\mathrm{ H}\mathrm{ B}}}\approx {c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}} $$ (P₃ in Fig.​ 76). In a real titration, the indicator will indicate the transition point (tp), which may deviate from the equivalence point. The absolute titration error $$ F_{\mathrm{ abs}}^{\rm{\prime}} $$, expressed as an amount in mol, is the difference between the amount of added titrator (OH⁻ ions) until reaching the transition point $$ {{\left( {n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}^{\circ }} \right)}_{\mathrm{ tp}}} $$ and the amount of acid to be titrated $$ n_{\mathrm{ HB}}^{\circ } $$:

(185)

The amount of OH⁻ ions added until the transition point $$ {{\left( {n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}^{\circ }} \right)}_{\mathrm{ tp}}} $$ has been used to form B⁻; however, not completely, as there is still present an amount $$ {{\left( {{n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$, which is understandable when looking at the reaction Eq. (15). Hence the following relation holds:

(186)

$$ {{\left( {{n_{{{{\mathrm{ B}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ is the amount of B⁻ at the equivalence point. For $$ n_{\mathrm{ HB}}^{\circ } $$ one can write:

(187)

and it follows that

(188)

Disregarding the dilution which always accompanies a titration, one can calculate the absolute error and give it in the unit (mol L⁻¹) as follows:

(189)

Relating the absolute error to the starting concentration of the acid $$ C_{\mathrm{ HB}}^{\circ } $$ yields the dimensionless relative error:

(190)

Figure 84 shows the example of a titration of acetic acid with sodium hydroxide to illustrate how the titration error can be derived from a pH-logc i diagram.

Fig. 84

Reading of the relevant concentration data from the pH-logc i diagram for estimation of the systematic titration errors in case of the titration of acetic acid ($$ C_{\mathrm{ acetic}\ \mathrm{ acid}}^{\circ } $$ = 0.1 mol L⁻¹) with sodium hydroxide. Peqp: equivalence point, P₁ to P₃ are transition points of the indicators methyl red, neutral red, and phenolphthalein

The concentrations $$ {{\left( {{c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ can be read at the logc i axis at P1a (methyl red), P2a (neutral red) and P3a (phenolphthalein). Similarly, the concentrations $$ {{\left( {{c_{\mathrm{ HB}}}} \right)}_{\mathrm{ tp}}} $$ can be read at P1b (methyl red), P2b (neutral red) and P3b (phenolphthalein). Table 4 gives these data, and also the systematic errors as calculated according to Eqs. (189) and (190).

Table 4

Systematic errors in case of the titration of acetic acid ($$ C_{{\rm acetic}\ { acid}}^{\circ } $$ = 0.1 mol L⁻¹) with sodium hydroxide when using different indicators. The concentration data are taken from Fig. 84

Indicator

$$ {{\left( {{c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ (mol L⁻¹)

(mol L⁻¹)

(mol L⁻¹)

$$ {F_{\mathrm{ rel}}} $$ (%)

Methyl red

5.8

7.2 × 10⁻⁹

7.4 × 10⁻³

−7.4 × 10⁻³

−7.4 × 10⁻²

−7.4

Neutral red

7.0

1.0 × 10⁻⁷

5.6 × 10⁻⁴

−5.6 × 10⁻⁴

−5.6 × 10⁻³

−0.56

Phenolphthalein

8.4

2.8 × 10⁻⁶

2.0 × 10⁻⁵

−1.72 × 10⁻⁵

−1.72 × 10⁻⁴

−0.0172 (≈ −0.02)

A negative systematic error means that the turning point is systematically showing up before the equivalence point, and a positive systematic error means that the turning point shows up after the equivalence point.

When the equivalence point is near to pH = 7, the autoprotolysis of water has to be considered, and the equations are more complicated [1–3]. However, when the equivalence point is very near to pH = 7, the titration can be treated as that of a very strong acid (see Sect. 6.1.3).

6.1.2 Systematic Titration Errors of Titrations of Bases with pK b Values Between 0 and 14 with a Very Strong Acid (e.g., Hydrochloric Acid)

This is completely analogous to the case of acids. When $$ n_{{{{\mathrm{ B}}^{-}}}}^{\circ } $$ denotes the amount (mol) of base to be titrated, and $$ {{\left( {n_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}^{\circ }} \right)}_{\mathrm{ tp}}} $$ is the amount of acid added until the transition point of the indicator, the absolute error is:

(191)

The absolute error in the unit (mol L⁻¹) follows as:

(192)

The relative error is then:

(193)

Figure 85 illustrates how the relevant data can be read from a pH-logc i diagram using the example of a titration of ammonia with hydrochloric acid.

Fig. 85

Reading of the relevant concentration data from the pH-logc i diagram for estimation of the systematic titration errors in case of the titration of ammonia ($$ C_{\mathrm{ ammonia}}^{\circ } $$ = 0.1 mol L⁻¹) with hydrochloric acid using different color indicators: Peqp: equivalence point, P₁ to P₃ are the transition points of the indicators phenolphthalein, neutral red, and methyl red

The concentrations $$ {{\left( {{c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}} \right)}_{\mathrm{ tp}}} $$ can be read at the logc i axis at P1a (phenolphthalein), P2a (neutral red) and P3a (methyl red). Similarly, the concentrations $$ {{\left( {{c_{{{{\mathrm{ B}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ can be read at P1b (phenolphthalein), P2b (neutral red) and P3b (methyl red). The data and the calculated systematic errors (using Eqs. (192) and (193)) are given in Table 5.

Table 5

Systematic errors in case of the titration of ammonia ($$ C_{{ammonia}}^{\circ } $$ = 0.1 mol L⁻¹) with hydrochloric acid when using different indicators. The concentration data are taken from Fig. 85

Indicator

$$ {{\left( {{c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}} \right)}_{\mathrm{ tp}}} $$ (mol L⁻¹)

$$ {{\left( {{c_{{{{\mathrm{ B}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ (mol L⁻¹)

(mol L⁻¹)

$$ {F_{\mathrm{ rel}}} $$ (%)

Phenolphthalein

8.4

4.2 × 10⁻⁹

1.4 × 10⁻²

−1.4 × 10⁻²

−1.4 × 10⁻¹

−14.0

Neutral red

7.0

1.0 × 10⁻⁷

6.0 × 10⁻⁴

−6.0 × 10⁻⁴

−6.0 × 10⁻³

−0.60

Methyl red

5.8

1.6 × 10⁻⁶

3.5 × 10⁻⁵

−3.34 × 10⁻⁵

−3.34 × 10⁻⁴

−0.0334 (≈ −0.03)

When the equivalence point is near to pH = 7, the autoprotolysis of water has to be considered, and the equations are more complicated [1–3]. However, when the equivalence point is very near to pH = 7, the titration can be treated as that of a very strong base (see Sect. 6.1.3).

6.1.3 Systematic Titration Errors of Titrations of Very Strong Acids with Very Strong Bases and Vice Versa

Examples for such titrations are the titration of hydrochloric acid with sodium hydroxide and vice versa. The systematic absolute error in (mol) is here:

(194)

$$ n_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}^{\circ } $$ is the amount (mol) of very strong acid HB to be titrated, $$ {{\left( {n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}^{\circ }} \right)}_{\mathrm{ tp}}} $$ is the amount (mol) of base (OH⁻) added until the transition point of the indicator. At the transition point, the amounts $$ {{\left( {{n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ and $$ {{\left( {{n_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}} \right)}_{\mathrm{ tp}}} $$ are present in the solution. The systematic absolute error in the unit (mol L⁻¹) is then:

(195)

Figure 86 illustrates how the relevant concentration data can be read from the pH-logc i diagram.

Fig. 86

Reading of the relevant concentration data from the pH-logc i diagram for estimation of the systematic titration errors in case of the titration of hydrochloric acid ($$ C_{\mathrm{ hydrochloric}\ \mathrm{ acid}}^{\circ } $$ = 0.1 mol L⁻¹) with sodium hydroxide. Peqp: equivalence point, P₁ to P₄ are the transition points of the color indicators methyl orange, methyl red, phenolphthalein, and thymolphthalein

The concentration $$ {{\left( {{c_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}}} \right)}_{\mathrm{ tp}}} $$ can be read on the logc i axis at P1a (methyl orange), P2a (methyl red), P3a (phenolphthalein) and P4a (thymolphthalein). Similarly, the concentration $$ {{\left( {{c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}}} \right)}_{\mathrm{ tp}}} $$ at P1b (methyl orange), P2b (methyl red), P3b (phenolphthalein) and P4b (thymolphthalein) can be read at the transition points of the indicators. Table 6 gives the relevant concentrations, and the systematic errors calculated according to Eqs. (194) and (195).

Table 6

Systematic errors in case of the titration of hydrochloric acid ($$ C_{{hydrochloric}\ {acid}}^{\circ } $$ = 0.1 mol L⁻¹) with sodium hydroxide when using different indicators. The concentration data are taken from Fig. 86

Indicator

(mol L⁻¹)

(mol L⁻¹)

(mol L⁻¹)

$$ {F_{\mathrm{ rel}}} $$ (%)

Methyl orange

4.0

10⁻¹⁰

10⁻⁴

−10⁻⁴

−10⁻³

−0.1

Methyl red

5.8

6.25 × 10⁻⁹

1.6 × 10⁻⁶

−1.6 × 10⁻⁶

−1.6 × 10⁻⁵

−1.6 × 10⁻³

Phenolphthalein

8.4

2.5 × 10⁻⁶

4.0 × 10⁻⁹

2.5 × 10⁻⁶

2.5 × 10⁻⁵

2.5 × 10⁻³

Thymolphthalein

10.0

10⁻⁴

10⁻¹⁰

10⁻⁴

10⁻³

0.1

When a very strong base is titrated with a very strong acid, the following equations are valid:

(196)

$$ n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}^{\circ } $$ is the amount of base B⁻ to be titrated. As it is a very strong base, this is equal to the amount $$ n_{{\mathrm{ O}{{\mathrm{ H}}^{-}}}}^{\circ } $$ present in the solution: the systematic absolute error is:

(197)

6.2 Random Titration Errors

Random titration errors have the origin in the fact that color indicators always have a transition range around the transition point. Also, instrumental methods of indication always “react” within a certain range of pH, never instantaneously at a certain pH point! Within that range, a randomized reading is made to determine the endpoint of titration. This means that in repeated titration this endpoints will be scattered within the transition range following the laws of statistic distribution (mostly indeed according to the Gaussian distribution). When the titration curve is very steep around the transition point (which hopefully will coincide with the equivalence point, so that no systematic error will arise) this will lead to a small range of τ values corresponding to a certain pH range. A small variation of τ values corresponds then also to a small range of volume readings at the indicated endpoints, i.e., a small standard deviation of results, i.e., a small random error. Figure 87 shows schematically how the steepness in the transition range effects the scattering of τ values and thus volume readings at the endpoints. .

Fig. 87

$$ \Delta \tau $$ range corresponding to a fixed pH range of the transition of an indicator (here bromthymol blue; transition range from pH 6.0 to 7.7): (a) for the case of a titration curve which is very steep in that pH range, and (b) for a titration curve which is less steep in that range

Figure 88 illustrates the fact that in case of the titration of ammonia with hydrochloric acid, the random error will be much smaller (see Δτ ₂) than in case of the titration of ammonium ions with sodium hydroxide (see Δτ ₁). Since τ is directly proportional to the volumes of titrator, a large Δτ means a large range Δv titrator, i.e., a wide range in which the results will be scattered randomly.

Fig. 88

pH-logc i diagram and diagram of the titrations of ammonia with hydrochloric acid (orange line with equivalence point Peqp2) and of ammonium ions with sodium hydroxide (black line with equivalence point Peqp1). The shadowed ranges indicate the transition ranges of the indicators alizarin yellow R and methyl red

Literature

1.

Buttler JN (1998) Ionic equilibrium. Solubility and pH calculations. Wiley, New York

2.

Seel F (1976) Grundlagen der Analytischen Chemie, 6th edn. Verlag Chemie, Weinheim

3.

Blumenthal G (1993) Grundlagen der quantitativen chemischen Analyse. In: Kolditz L (ed) Anorganikum, 13th edn. Barth, Leipzig

Heike Kahlert and Fritz ScholzAcid-Base Diagrams201310.1007/978-3-642-37902-4© Springer-Verlag Berlin Heidelberg 2013

Appendix A: Derivation of the Exact Functions and the Equations of the Asymptotes for Multibasic Acids

A1. Derivation of the Functions $$ \bf log {{\boldsymbol c}_{{\bf i}}}={\boldsymbol f}({\bf pH}) $$ for Dibasic Acids

Protolysis equilibria:

(A.1)

(A.2)

Laws of mass action:

(A.3)

(A.4)

Amount balance:

(A.5)

The following functions need to be derived:

1.

(A.6)

2.

(A.7)

3.

(A.8)

A1.1 $$ \bf log {{\boldsymbol c}_{{{{{\bf H}}_2}{\bf B}}}}={\boldsymbol f}({\bf pH}) $$

Equations need to be found for the concentrations $$ {c_{{\mathrm{ H}{{\mathrm{ B}}^{-}}}}} $$ and $$ {c_{{{{\mathrm{ B}}^{2- }}}}} $$ which have as unknown concentrations only $$ {c_{{{{\mathrm{ H}}_2}\mathrm{ B}}}} $$ and $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}} $$ . This can be done by rearranging Eqs. ( A.3 ) and ( A.4 ):

(A.9)

(A.10)

and insertion in Eq. ( A.5 ) gives:

(A.11)

Rearrangement gives:

(A.12)

With and follows:

(A.13)

A1.2 $$ \bf log {{\boldsymbol c}_{{{\bf H}{{{\bf B}}^{-}}}}}={\boldsymbol f}({\bf pH}) $$

Equations need to be found for the concentrations $$ {c_{{{{\mathrm{ H}}_2}\mathrm{ B}}}} $$ and $$ {c_{{{{\mathrm{ B}}^{2- }}}}} $$ which have as unknown concentrations only $$ {c_{{\mathrm{ H}{{\mathrm{ B}}^{\rm{-}}}}}} $$ and $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}} $$ . This can be done by rearranging Eqs. ( A.3 ) and ( A.4 ):

(A.14)

(A.15)

and insertion in Eq. ( A.5 ) gives:

(A.16)

Rearrangement gives:

(A.17)

With and follows:

(A.18)

A1.3 $$ \bf log {{\boldsymbol c}_{{{{{\bf B}}^{2- }}}}}={\boldsymbol f}({\bf pH}) $$

Equations need to be found for the concentrations $$ {c_{{{{\mathrm{ H}}_2}\mathrm{ B}}}} $$ and $$ {c_{{\mathrm{ H}{{\mathrm{ B}}^{-}}}}}, $$ which have as unknown concentrations only $$ {c_{{{{\mathrm{ B}}^{2- }}}}} $$ and $$ {c_{{{{\mathrm{ H}}_3}{{\mathrm{ O}}^{+}}}}} $$ . This can be done by rearranging Eqs. ( A.3 ) and ( A.4 ):

(A.19)

(A.20)

and insertion in Eq. ( A.5 ) gives:

(A.21)

Rearrangement gives:

(A.22)

With and follows:

(A.23)

A2. Derivation of the Functions $$ \bf log {{\boldsymbol c}_{{\bf i}}}={\boldsymbol f}({\bf pH}) $$ for a Tribasic Acid

Protolysis equilibria:

(A.24)

(A.25)

(A.26)

Laws of mass action:

(A.27)

(A.28)

(A.29)

Amount balance:

(A.30)

The following functions need to be derived:

1.

(A.31)

2.

(A.32)

3.

(A.33)

4.

(A.34)

A2.1 $$ \bf log {{\boldsymbol c}_{{{{{\bf H}}_3}{\bf B}}}}={\boldsymbol f}({\bf pH}) $$

For the concentrations $$ {c_{{{{\mathrm{ H}}_2}{{\mathrm{ B}}^{-}}}}} $$ , $$ {c_{{\mathrm{ H}{{\mathrm{ B}}^{2- }}}}} $$ and $$ {c_{{{{\mathrm{ B}}^{3- }}}}} $$ one can derive equations by rearranging Eqs. ( A.27 ), ( A.28 ) and ( A.29 ):

(A.35)

(A.36)

(A.37)

and insertion in Eq. ( A.30 ) gives:

(A.38)

(A.39)

With , and follows

(A.40)

A2.2 $$ \bf log {{\boldsymbol c}_{{{{{\bf H}}_2}{{{\bf B}}^{-}}}}}={\boldsymbol f}({\bf pH}) $$

Rearranging Eq. ( A.27 ), ( A.28 ), and ( A.29 ) gives:

(A.41)

(A.42)

(A.43)

and insertion in Eq. ( A.30 ) gives:

(A.44)

(A.45)

With , and follows:

(A.46)

A2.3 $$ \bf log {{\boldsymbol c}_{{{\bf H}{{{\bf B}}^{2- }}}}}={\boldsymbol f}({\bf pH}) $$

Rearranging Eqs. ( A.27 ), ( A.28 ), and ( A.29 ) gives:

(A.47)

(A.48)

(A.49)

and insertion in Eq. ( A.30 ) gives:

(A.50)

(A.51)

With , and follows:

(A.52)

A2.4 $$ \bf log {{\boldsymbol c}_{{{{{\bf B}}^{3- }}}}}={\boldsymbol f}({\bf pH}) $$

Rearranging Eqs. ( A.27 ), ( A.28 ), and ( A.29 ) gives:

(A.53)

(A.54)

(A.55)

and insertion in Eq. ( A.30 ) gives:

(A.56)

(A.57)

With , and follows:

(A.58)

A3. Derivation of the Functions $$ \bf log {{\boldsymbol c}_{{\bf i}}}={\boldsymbol f}({\bf pH}) $$ for Tetrabasic Acids

Protolysis equilibria:

(A.59)

(A.60)

(A.61)

(A.62)

Laws of mass action:

(A.63)

(A.64)

(A.65)

(A.66)

Amount balance:

(A.67)

The following functions need to be derived:

1.

(A.68)

2.

(A.69)

3.

(A.70)

4.

(A.71)

5.

(A.72)

A3.1 $$ \bf log {{\boldsymbol c}_{{{{{\bf H}}_4}{\bf B}}}}={\boldsymbol f}({\bf pH}) $$

Rearranging Eqs. ( A.63 ), ( A.64 ), ( A.65 ), and ( A.66 ) gives:

(A.73)

(A.74)

(A.75)

(A.76)

and insertion in Eq. ( A.67 ) gives:

(A.77)

(A.78)

With , , follows:

(A.79)

A3.2 $$ \bf log {{\boldsymbol c}_{{{{{\bf H}}_3}{{{\bf B}}^{-}}}}}={\boldsymbol f}({\bf pH}) $$

Rearranging Eqs. ( A.63 ), ( A.64 ), ( A.65 ), and ( A.66 ) gives:

(A.80)

(A.81)

(A.82)

(A.83)

and insertion in Eq. ( A.67 ) gives:

(A.84)

(A.85)

With and

(A.86)

A3.3 $$ \bf log {{\boldsymbol c}_{{{{{\bf H}}_2}{{{\bf B}}^{2- }}}}}={\boldsymbol f}({\bf pH}) $$

Rearranging Eqs. ( A.63 ), ( A.64 ), ( A.65 ), and ( A.66 ) gives:

(A.87)

(A.88)

(A.89)

(A.90)

and insertion in Eq. ( A.67 ) gives:

(A.91)

(A.92)

With , , , and follows:

(A.93)

A3.4 $$ \mathbf{log }\bf\boldsymbol {c_{{\mathrm{ H}{{\mathrm{ B}}^\mathbf{3- }}}}}=\bf\boldsymbol f(\mathrm{ pH}) $$

Rearranging Eqs. ( A.63 ), ( A.64 ), ( A.65 ), and ( A.66 ) gives:

(A.94)

(A.95)

(A.96)

(A.97)

and insertion in Eq. ( A.67 ) gives:

(A.98)

(A.99)

With , and follows:

(A.100)

A3.5 $$ \bf log {\boldsymbol c_{{{{{\boldsymbol B}}^\mathbf{4- }}}}}=\boldsymbol f({ pH}) $$

Rearranging Eqs. ( A.63 ), ( A.64 ), ( A.65 ), and ( A.66 ) gives:

(A.101)

(A.102)

(A.103)

(A.104)

and insertion in Eq. ( A.67 ) gives:

(A.105)

(A.106)

With and follows:

(A.107)

A4. Functions of the Asymptotes

A4.1 Dibasic acid H 2 B

$$ {{\mathrm{ \bf H}}_\mathbf{2}}\mathrm\mathbf{B} $$ :

$$ \mathrm{ \bf H}{{\mathrm{ \bf B}}^{-}} $$ :

$$ {{\mathrm\mathbf{B}}^\mathbf{2- }} $$ :

A4.2 Tribasic acids H 3 B

$$ {{\mathrm\mathbf{H}}_\mathbf{3}}\mathrm{ \bf B} $$ :

$$ {{\mathrm{ \bf H}}_\mathbf{2}}{{\mathrm{ \bf B}}^{-}} $$ :

$$ \mathrm\mathbf{H}{{\mathrm\mathbf{B}}^\mathbf{2- }} $$ :

$$ {{\mathrm\mathbf{B}}^\mathbf{3- }} $$ :

A4.3 Tetrabasic acids H 4 B

$$ {{\mathrm\mathbf{H}}_\mathbf{4}}\mathrm\mathbf{B} $$ :

$$ {{\mathrm\mathbf{H}}_\mathbf{3}}{{\mathrm\mathbf{B}}^\mathbf{-}} $$ :

$$ {{\mathrm\mathbf{H}}_\mathbf{2}}{{\mathrm\mathbf{B}}^\mathbf{2- }} $$ :

$$ \mathrm\mathbf{H}{{\mathrm\mathbf{B}}^\mathbf{3- }} $$ :

$$ {{\mathrm\mathbf{B}}^\mathbf{4- }} $$ :

Index

A

Acetate

Acetic acid

titration

Acetylsalicylic acid

Acid–base indicators, p K I values/transition intervals

Acid–base titrations

Acidity constant

Acids, dibasic

monobasic

p K a values

polybasic/multibasic

strong

titration

tetrabasic

tribasic

very strong

very weak

weak

Alanine

titration

Amino acids

dibasic, titration

Ammonia

titration, hydrochloric acid

Ammonium acetate

Ammonium chloride

Ammonium cyanide

Ammonium formate

Ammonium ions

titration, sodium chloride

Ammonium oxalate

Ampholytes

Ascorbic acid

Aspartic acid

Autoprotolysis

B

Bases, diacidic

strong

very strong

very weak

weak

Benzoic acid

Bjerrum, N.

Bromthymol blue

Brønsted (Brönsted)

Brønsted acids

C

Calcium benzoate

Concentration activities

Cyanide

E

Equilibrium concentrations

Ethanol

Ethylenediaminetetraacetic acid (EDTA)

F

Formate

Formic acid

G

Glycine

H

Hägg, G.

Hägg diagrams

Hydrochloric acid

pH-log c i diagram

titration

Hydrofluoric acid

Hydrogen cyanide

Hydrogen sulfide

I

Isoleucine

L

Leucine

Lowry

M

Manganese sulfide

Methionine

Methyl orange

Methyl red

Molar ratio activity

N

Neutral red

NH ₄ ⁺ /NH ₃

O

Oxalic acid

Oxalate ion

P

Phenolphthalein

Phenylalanine

pH-log c i diagrams

Phosphoric acid

Pipettes/burettes, calibration

Potassium benozoate

Proton acceptors

Proton donors (donators)

S

Salt solutions, protolyzing anions/cations

Sillén, L.G.

Sillén diagrams

Sodium acetate

Sodium benzoate

Sodium chloride

Sodium ethanolate

Sodium formate

Sodium hydrogen sulfate

Sodium hydroxide

titration

Sørensen, S.P.L.

Sulfide ion

Sulfuric acid

titration

T

Thymolphthalein

Titration, degree of

diagrams

errors

random

systematic

V

Valine

Vitamin C

W

Water, pH-log c i diagram

Z

Zwitterion