Hydrolysis of Metal Ions

By Paul L. Brown and Christian Ekberg

Filling the need for a comprehensive treatment that covers the theory, methods and the different types of metal ion complexes with water (hydrolysis), this handbook and ready reference is authored by a nuclear chemist from academia and an industrial geochemist.
The book includes both cation and anion complexes, and approaches the topic of metal ion hydrolysis by first covering the background, before proceeding with an overview of the dissociation of water and then all different metal-water hydrolysis complexes and compounds.

A must-have for scientists in academia and industry working on this interdisciplinary topic.

• Language: English
• Publisher: Wiley
• Release date: Feb 23, 2016
• ISBN: 9783527656202

Paul L. Brown

Paul Brown is a Chief Geochemist in the Growth and Innovation group of Rio Tinto in Australia. He previously was a Chief Executive Officer of the Australian Sustainability and Industry Research Centre and a Principal Research Scientist at the Australian Nuclear Science and Technology Organisation. He received both his undergraduate and graduate degrees from the University of Wollongong in Australia.

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Preface

If gold has been prized because it is the most inert element, changeless and incorruptible, water is prized for the opposite reason – its fluidity, mobility, changeability make it a necessity and a metaphor for life itself. To value gold over water is to value economy over ecology, that which can be locked up over that which connects all things.

Rebecca Solnit – Storming the Gates of Paradise, Landscapes for Politics

Water is a weak acid. The acidity of water molecules in the hydration sphere of a metal ion is much larger than that of water itself. This enhancement of the acidity may be interpreted qualitatively as the result of repulsion of the protons of the water molecules by the positive charge of the metal ion. The acidity will increase as the metal ion size decreases and its charge increases.

Werner Stumm and James Morgan – Aquatic Chemistry

Numbers are the product of counting. Quantities are the product of measurement. This means that numbers can conceivably be accurate because there is a discontinuity between each integer and the next. Between two and three there is a jump. In the case of quantity, there is no such jump; and because jump is missing in the world of quantity, it is impossible for any quantity to be exact. You can have exactly three tomatoes. You can never have exactly three gallons of water. Always quantity is approximate.

Gregory Bateson – Number is Different from Quantity

Water and metals are ever present. Water has the power to erode any material that might contain the metals, thereby releasing them into the water. The liberated metals can hydrolyse the water, releasing protons, the hydroxide binding to the metal. All metals can undergo this process. Nature has conspired to make many of these processes essential for life itself, utilising the unique features of some metal ions within essential mechanisms in plants and animals. Conversely, nature also conspires to make just as many of these processes able to destroy life. Moreover, the processes are important in many other scientific and industrial fields. Consequently, knowledge of hydrolytic reactions and their magnitude is seen as being essential.

The study of hydrolysis reactions was pioneered within two preeminent institutions. The first was at the Royal Institute of Technology in Stockholm, Sweden, founded by the work of Lars Gunnar Sillén. His group entitled their work ‘Studies on the Hydrolysis of Metal Ions’. The second group hailed from the Oak Ridge National Laboratory (ORNL) in the United States. The driving force of this work was by Charles Baes and Bob Mesmer who wrote the seminal book ‘The Hydrolysis of Cations’. This book has been an invaluable tool for aqueous chemists over several scientific generations. Although hydrolysis work at ORNL preceded Baes and Mesmer, it was their work that cemented the institution as preeminent in the field. Subsequently, their work was continued by Don Palmer and others. One of the present authors had the pleasure to spend part of their PhD dissertation work at ORNL under the guidance of Baes, Mesmer and Palmer. The title for this book ‘Hydrolysis of Metal Ions’ has been adopted in honour of the work of these two groups.

About 20 years ago, the authors started their collaboration on the hydrolysis of tetravalent metal ions and since then have continued to work together in several areas of aqueous chemistry. Some 10 years ago, we realised that there had been significant progress in the art of hydrolysis constant determination and that many new studies were available. Thus, we decided to try to compile the progress in the area. In our attempt to create a new ‘Baes and Mesmer’, we have tried to approach hydrolysis from a thermodynamic standpoint, that is, by first trying to go through the concept of activities instead of concentrations and discuss the different models developed for assessing activities and inevitably the method of obtaining activity coefficients. Using these activity coefficients, the obtained stability constants can be calculated back to a standard state for comparison and general utilisation, that is, infinite dilution at 25 °C and 10⁵ Pa pressure.

A comprehensive collection of data as presented in this book cannot be undertaken without the help of many people. We would like to express our gratitude to the personnel at the library of Chalmers University of Technology for their ability to trace papers we never thought were available, Natallia Torapava and Artem Matyskin for their help with translation of very important papers written in Russian and Teodora Retegan for her help with finding papers as well as helping with translations from Romanian and Italian. We would also like to sincerely thank our families, Teodora, Susan, Tara and Liam, for their understanding and patience at the very many long nights collating, interpreting and writing about stability and solubility constant data. For those who helped and have not been mentioned, you are still not forgotten.

We hope that this book will provide a good companion to scientific work that describes the hydrolysis of metal ions in the same way as Baes and Mesmer's version was helpful to us as we did our hydrolysis work over the last three decades.

January 2016

Paul L. Brown

Kiama

Christian Ekberg

Göteborg

1

Introduction

Water constitutes approximately 70% of the world's surface and 60% of the human body, where aqueous chemical reactions dominate the foundations of life. Even life itself is assumed to have originated in water. For many scientists, life and what has evolved around it are closely coupled to chemistry. For these reasons, the understanding of aqueous chemical systems is of great importance. However, no attempt has been made herein to try and grasp the whole subject of aqueous chemical reactions, but, instead, to focus on reactions associated with the self-ionisation of water and the reactions of water with cations. Such reactions are termed hydrolysis.

Water can self-ionise according to the reaction

(1.1)

for which the stability (protolysis) constant is given by

(1.2)

Water can also hydrate ions present in an aqueous solution where, for example, a metal ion, Mz+, will be present as the hydrated ion M(H2O)nz+, where z is the charge of the metal ion and n is its coordination number in the aqueous solution. The water bound to the metal ion can also ionise (hydrolyse) to produce a proton and a metal hydroxide complex. The reaction can be written as

(1.3)

The conditions (i.e. pH) under which the metal ion will hydrolyse is dependent on its physical properties, in particular its ionic charge z and its ionic radius r. Typically, the larger the charge and the smaller the radius, the lower the pH at which the metal will hydrolyse.

Multiple hydrolysis steps are possible as the pH increases, with the species containing the largest number of hydroxide groups likely to be anionic. The reaction for the formation of these species can be expressed by

(1.4)

A reasonably unique feature of water bound to metal ions is its ability to ionise to produce polymeric hydrolysis species. A relatively large number of polymeric species stoichiometries are possible, and those which can form for a particular metal ion are again related to the physical properties of the ion (size, charge, coordination number). These polymeric species are formed according to the reaction

(1.5)

The parameters p and q are the stoichiometric coefficients of the hydrolysis species that has been formed. In the formation of the polymeric species, some water molecules are lost from the reacting metal ion without being hydrolysed as shown in reaction (1.5).

Not only can soluble species be produced from the reaction between metal and hydroxide ions. At a given temperature and under appropriate pH and metal ion concentrations, the hydrated water molecules on the metal ion will hydrolyse, resulting in the formation of a solid hydroxide and/or oxide phase. The formation of such phases can be represented by reaction (1.6):

(1.6)

For odd charged metal ions, the value of x can be fractional having values of 0.5, 1.5 and 2.5 for metal charges of 1, 3 and 5, respectively. In these cases, no hydroxide ions would be in the phase formed, and the phase would be MOx(s) (x = 0.5, 1.5 or 2.5), which is equivalent to M2O2x(s) (e.g. AgO½(s) is equivalent to Ag2O(s)).

Highly charged metal ions (+4 to +6) hydrolyse water so strongly that in aqueous solution most cannot occur as the free metal ion, but appear as oxo-metal ions where one or two oxide ions are bound to the metal ion. Metal ions that fall into this group include titanium(IV), vanadium(IV), technetium(IV) and polonium(IV) (where the ion MO²+ is formed); vanadium(V), uranium(V), neptunium(V), plutonium(V) and americium(V) (where the ion MO2+ is formed), protactinium(V) (where the ion MO³+ is formed); and uranium(VI), neptunium(VI), plutonium(VI) and americium(VI) (where the ion MO2²+ is formed). There are no pentavalent or hexavalent metal ions that exist as the free metal ion and only a few tetravalent ions (e.g. zirconium(IV), hafnium(IV), thorium(IV) and cerium(IV) and possibly tin(IV) and lead(IV)).

The ubiquity of water and metals that are ever present in the water means that hydrolysis reactions are essential aspects of many areas of science, industry and nature. Hydrolysis is an important component in water purification and treatment. In water treatment, many metals are removed from solution by precipitation as solid hydroxide phases which, in turn, may remove other metals and metalloids via adsorption onto the formed solids. In many countries, arsenic is removed from groundwater recovered for drinking via adsorption onto solid iron hydroxide, and knowledge of the precipitation behaviour of magnesium hydroxide is important in thermal desalination processes.

Gibbsite (aluminium hydroxide) is purified from bauxite (impure aluminium oxide) via the Bayer process, where the bauxite is reacted with sodium hydroxide to form the aluminium hydrolysis species, Al(OH)4−, from which gibbsite can then be recovered. Corundum (aluminium oxide) can then be obtained from the gibbsite via calcination. Pyrolusite (manganese dioxide) is often utilised in the recovery of uranium, which occurs in ores principally in the form of uraninite (uranium dioxide). The manganese dioxide oxidises ferrous to ferric iron which, in turn, oxidises the uraninite to soluble uranium that is then separated from the ore tailings. The operation of some batteries relies on hydrolytic reactions. The oxidative process in a nickel hydride battery occurs via the interaction of nickel(II) hydroxide with hydroxide ions to produce nickel(III) oxyhydroxide. Conversely, in lithium batteries conditions are formulated such that the hydrolysis of the lithium does not occur. The non-reactivity of chromium alloys and stainless steels results because of the formation of oxide coatings on the surfaces of the alloys or steels.

There are a number of biological processes that rely on hydrolytic reactions. The regulation of iron in the body is carried out by the protein ferritin. This protein stores the iron in the form of a ferric oxyhydroxide phase, removing iron from the body when it is in excess and releasing it when the body is deficient in iron. Manganese has an essential role in photosynthesis. Manganese is a highly redox active metal, and it forms the strongest trivalent hydrolysis species that aids in the production of oxygen from water. Vanadium is an important metal for some marine organisms. These organisms convert vanadium(V) to either vanadium(III) or vanadium(IV), and hydrolysis is an essential component of this conversion process.

There are many different experimental techniques that can be utilised to obtain a hydrolysis constant. Each has advantages and disadvantages and no clear preference can be given to a particular technique, but use of a combination of techniques is generally desired since each can enable the derivation of the stability constants of different species for the single metal ion. However, few studies exist where multiple techniques have been employed successfully, where the relative strengths of each technique have been utilised under specific conditions, enabling a whole set of hydrolysis species (and their associated stability constants) to be derived, including all monomeric species that can be formed by the metal ion as well as all polymeric hydrolysis species.

There are two formulations by which stability or solubility constants can be expressed (see Section 2.1). The first is as a stoichiometric constant where the concentrations of the reacting and produced species are utilised. The second and more important formulation is as a thermodynamic constant where the activities of the species are used. An activity is the product of the concentration and the corresponding activity coefficient, and it is essential to understand models developed for determining activities as well as methods for deriving activity coefficients. Using derived activity coefficients, stoichiometric stability or solubility constants can be used to calculate the thermodynamic constants at the standard state, that is, at infinite dilution and a temperature of 25 °C and pressure of 10⁵ Pa. The majority of critical reviews of the stability or solubility constants of metal hydroxide species and phases have derived thermodynamic stability constants.

There have been a number of reviews of hydrolysis reactions that have appeared in the literature. Baes and Mesmer (1976) produced a seminal review of the hydrolysis of metal ions. This was a fundamental study and compilation of hydrolysis reactions and associated stability constants and derivation of thermodynamic stability and solubility constants. More recently, critical reviews have been undertaken on the thermochemistry of individual metals. The Nuclear Energy Agency of the Organisation for Economic Co-operation and Development began a series of reviews of the thermochemistry of elements related to nuclear waste management in 1992 on uranium (Grenthe et al., 1992). Metals which were the focus of subsequent reviews have included americium, technetium, neptunium, plutonium, nickel, zirconium, thorium, tin and iron (Silva et al., 1995; Rard et al., 1999; Lemire et al., 2001, 2013; Guillaumont et al., 2003; Gamsjäger et al., 2005, 2012; Brown, Curti and Grambow, 2005; Rand et al., 2007). IUPAC also sponsored a series of reviews on environmentally relevant metal ions (Powell et al., 2005, 2007, 2009, 2011, 2013) that included the metal ions mercury, copper, lead, cadmium and zinc. Other reviews have also been undertaken and thermochemical data for hydrolysis species derived. However, the focus of virtually all of these reviews has been on the selection of thermochemical data for the standard state (i.e. zero ionic strength, 25 °C and 10⁵ Pa). There is a dearth of selected information relating to stability and solubility constants relating to the wider range of temperatures over which liquid water exists (i.e. 0–375 °C).

The purpose of the current review was to critically evaluate the hydrolytic reactions of metal ions (cations) and select stability and solubility constants, where available, across the temperature range of 0–375 °C. In carrying out the review, an attempt has been made to be as thorough as possible and utilise as much data as are available in the literature as possible. However, it is not feasible to review all data that have been published on a given metal ion, and, undoubtedly, some studies have been missed. This certainly has not been by intention, and it is possible that the inclusion of some of these missing data may have led to different conclusions being reached for metals where little data exists. For those metals where substantial information is available, or for the protolysis constant of water, the exclusion of some data would unlikely affect the thermochemical data derived or the conclusions reached regarding the speciation of the metal ion.

References

Baes, C.F. and Mesmer, R.E. (1976) The Hydrolysis of Cations, John Wiley & Sons, Inc., New York.

Brown, P.L., Curti, E., and Grambow, B. (2005) Chemical Thermodynamics of Zirconium, vol. 8, Elsevier, Amsterdam, 512 pp.

Gamsjäger, H., Bugajski, J., Gajda, T., Lemire, R.J., and Preis, W. (2005) Chemical Thermodynamics of Nickel, vol. 6, Elsevier, 617 pp.

Gamsjäger, H., Gajda, T., Sangster, J., Saxena, S.K., and Voigt, W. (2012) Chemical Thermodynamics of Tin, vol. 12, OECD Publishing, Paris, 609 pp.

Grenthe, I., Fuger, J., Konings, R.J.M., Lemire, R.J., Muller, A.B., Nguyen-Trung, C., and Wanner, H. (1992) Chemical Thermodynamicsof Uranium, vol. 1, North-Holland, 715 pp.

Guillaumont, R., Fanghänel, T., Neck, V., Fuger, J., Palmer, D.A., Grenthe, I., and Rand, M.H. (2003) Update on the Chemical Thermodynamics of Uranium, Neptunium, Plutonium, Americium and Technetium, vol. 5, Elsevier, Amsterdam, 919 pp.

Lemire, R.J., Berner, U., Musikas, C., Palmer, D.A., Taylor, P., and Tochiyama, O. (2013) Chemical Thermodynamics of Iron. Part 1, vol. 13a, OECD Publishing, Paris, 1082 pp.

Lemire, R.J., Fuger, J., Spahiu, K., Sullivan, J.C., Nitsche, H., Ullman, W.J., Potter, P., Vitorge, P., Rand, M.H., Wanner, H., and Rydberg, J. (2001) Chemical Thermodynamics of Neptunium and Plutonium, vol. 4, Elsevier, Amsterdam, 845 pp.

Powell, K.J., Brown, P.L., Byrne, R.H., Gajda, T., Hefter, G., Leuz, A.-K., Sjöberg, S., and Wanner, H. (2009) Chemical speciation of environmentally significant metals with inorganic metals. Part 3: The Pb²+ + OH−, Cl−,, and systems. Pure Appl. Chem., 81, 2425–2476.

Powell, K.J., Brown, P.L., Byrne, R.H., Gajda, T., Hefter, G., Leuz, A.-K., Sjöberg, S., and Wanner, H. (2011) Chemical speciation of environmentally significant metals with inorganic ligands. Part 4: The Cd²+ + OH−, Cl−,, and systems. Pure Appl. Chem., 83, 1163–1214.

Powell, K.J., Brown, P.L., Byrne, R.H., Gajda, T., Hefter, G., Sjöberg, S., and Wanner, H. (2005) Chemical speciation of environmentally significant heavy metals with inorganic ligands. Part 1: the Hg²+–Cl−, OH−,, and aqueous systems. Pure Appl. Chem., 77, 739–800.

Powell, K.J., Brown, P.L., Byrne, R.H., Gajda, T., Hefter, G., Leuz, A.-K., Sjöberg, S., and Wanner, H. (2013) Chemical speciation of environmentally significant metals with inorganic ligands. Part 5. The Zn²+−OH−, Cl−,, and systems. Pure Appl. Chem., 85, 2249–2311.

Powell, K.J., Brown, P.L., Byrne, R.H., Gajda, T., Hefter, G., Sjöberg, S., and Wanner, H. (2007) Chemical speciation of environmentally significant metals with inorganic ligands. Part 2. The Cu²+−OH−, Cl−,, and systems. Pure Appl. Chem., 79, 895–950.

Rand, M., Fuger, J., Grenthe, I., Neck, V., and Rai, D. (2007) Chemical Thermodynamics of Thorium, vol. 11, OECD Publishing, Paris, 942 pp.

Rard, J.A., Rand, M., Anderegg, G., and Wanner, H. (1999) Chemical Thermodynamics of Technetium, vol. 3, Elsevier, Amsterdam, 544 pp.

Silva, R.J., Bidoglio, G., Rand, M.H., Robouch, P.B., Wanner, H., and Puigdomenech, I. (1995) Chemical Thermodynamics of Americium, vol. 2, Elsevier, Amsterdam, 374 pp.

2

Theory

2.1 Hydrolysis Reactions and Stability/Solubility Constants

The general hydrolysis reaction (2.1) that also involves another ligand can be written as

(2.1)

where M is a metal cation of charge z, L is an anionic ligand of charge n, and p, q and r are the stoichiometric coefficients relating to the metal M, water and the ligand L, respectively. The stability constant for reaction (2.1) is defined by

(2.2)

that is termed the stoichiometric stability constant. A thermodynamic stability constant can also be defined for reaction (2.1):

(2.3)

where each {X} represents the activity of the species X and the activity of the species is equivalent to the product of the concentration of the species and the relevant activity coefficient γX. The stoichiometric stability constant is most commonly determined from measurements conducted in a medium of fixed ionic strength such that the activity coefficients remain constant.

The stoichiometric and thermodynamic stability constants can be related through Eq. (2.4):

(2.4)

where γ(pqr) is the activity coefficient of the species MpLr(OH)q(pz−rn−q) and a(H2O) is the activity of water.

Hydrolysis reactions are a simplified version of reaction (2.1) since r is zero. The general hydrolysis reaction can be written as

(2.5)

and the stoichiometric stability constant for this reaction is

(2.6)

The formation of the generic hydrolysis species can also be described via the reaction of the metal with the hydroxide ion, as shown by reaction (2.7):

(2.7)

The stoichiometric stability constant for reaction (2.7) is

(2.8)

The stoichiometric stability constants, as defined by Eqs. (2.6) and (2.8), are related through the protolysis constant of water relevant to the experimental conditions (ionic strength and temperature) used to study the stoichiometric reaction. Stability constants that include an asterisk denote conditions where water is used as the reactant and no asterisk is used to denote where hydroxide ion is the reactant (i.e. see Eqs. (2.6) and (2.8)). Equation (2.5) can also be used to denote the reaction for the protolysis of water. In this circumstance, p = 0 and q = 1 in the equation, with the right-hand product of the reaction being the hydroxide ion.

Stepwise hydrolysis reactions for monomeric species can be written as

(2.9)

and the stoichiometric stability constant for this reaction is

(2.10)

Note that the stability constants of monomeric species (and their stepwise constants) are often referred to without the preceding 1 (i.e. as *βq rather than *β1q). The formation of the stepwise hydrolysis species M(OH)q(z−q) can also be described via the reaction of the metal with the hydroxide ion, as shown by reaction (2.11):

(2.11)

The stoichiometric stability constant for reaction (2.11) is

(2.12)

The generic reaction for the solubility of oxide, hydroxide or mixed oxide/hydroxide phases is described by reaction (2.13):

(2.13)

where again z is the charge of the metal cation and x can range from 0 to z/2. For metals with non-even valencies, for oxide phases, Eq. (2.13) produces MOz/2 for the chemical formula of the phase; this is correct and is equivalent to ½M2Oz.

The stoichiometric solubility constant for reaction (2.13) is

(2.14)

The solubility reaction can also be written with respect to the formation of hydroxide ions, and the solubility constant for such a reaction is denoted Ks10 (or often Ksp (which is defined as the solubility product); again, *Ks10 and Ks10 are related through the protolysis constant of water), as defined in reaction (2.15):

(2.15)

The solubility of the generic phase given in reaction (2.13) can be such that rather than the free metal ion being the reaction product, other hydrolysis species may form depending on the reaction pH. The generic formula for such reactions can be derived by combination of reactions (2.5) and (2.13). The combined reaction is

(2.16)

The solubility constant for this reaction is

(2.17)

Again, the solubility of the generic phase to form the hydrolysis species Mp(OH)q(pz−q) can be expressed in terms of the hydroxide ion, as defined by reaction (2.18):

(2.18)

2.2 Debye–Hückel Theory

The Debye–Hückel equation derives from a combination of the Poisson equation and a statistical-mechanical distribution formula (Debye and Hückel, 1923). The Poisson equation is a general expression of the Coulomb law of force between charged bodies and can be written as

(2.19)

where ψ is the potential at a point where the charge density is ρ and ε is the dielectric constant of the medium which contains the charges. In the special case of symmetry about the origin, ψ depends only on the distance r at which a point is removed from the origin. Here, the partial differential operator reduces to a total differential operator, namely,

(2.20)

which by substitution of Eq. (2.19) into Eq. (2.20) gives

(2.21)

Debye and Hückel assumed the Boltzmann distribution law which states that since the electrical potential energy of a particular ion is zeψ, the average local concentration n′ of those ions at a point is

(2.22)

where n is the total number of the particular ions. Since each of these ions carries a charge equal to ze, the net charge density, when summing for all ionic species, is given by Eq. (2.23):

(2.23)

Using the Taylor series to expand the exponential term, the expression for the charge density becomes

(2.24)

In Eq. (2.24), the first term equals zero as a result of electrical neutrality, and, if , only the linear term in ψ is appreciable; therefore Eq. (2.24) reduces to

(2.25)

Substituting Eq. (2.25) into Eq. (2.21), the Poisson equation, leads to

(2.26)

where

(2.27)

which is a function of concentration, ionic charge, temperature and the dielectric constant of the solvent. In the equation, I is the ionic strength expressed in moles per litre and N is Avogadro's number. Substitution of u = ψr into Eq. (2.26), the derived expression for the Poisson equation, leads to the simplified differential equation, given by Eq. (2.28):

(2.28)

Equation (2.28) has the general solution

(2.29)

or, alternatively, by substituting u = ψr into the equation

(2.30)

where A and B are constants of integration dependent on the physical conditions of the problem. Since the potential must remain finite at large values of r, it is necessary that B = 0. By substitution of Eq. (2.30) into Eq. (2.25), the expression for the charge density becomes (note that an expression containing κ² from Eq. (2.27), following rearrangement, is included in Eq. (2.31))

(2.31)

The average charge density ρ at a point depends on the probabilities of an element of volume, at that point, being occupied by various kinds of ions. If a cation is chosen as the ion occupying the origin, then the condition of neutrality requires that the net charge outside this point equals −ze. The charge at any given point can therefore be derived from

(2.32)

where a represents the distance within which no other ion can approach the central ion (i.e. the distance of closest approach). Combination of Eqs. (2.31) and (2.32) leads to

(2.33)

On integration of Eq. (2.33) by parts, Eq. (2.34) is obtained:

(2.34)

Substitution of Eq. (2.34) into Eq. (2.30), remembering that B = 0, leads to the following equation for the potential:

(2.35)

This last equation is the fundamental expression derived by Debye and Hückel for the time-average potential at a point of distance r from an ion of valency z in the absence of external forces. However, for an isolated central ion of valency z, in a medium of dielectric constant ε, the potential at a distance r is given by

(2.36)

Taking into account Eqs. (2.35) and (2.36), the potential due to all other ions is

(2.37)

which, at the distance of closest approach, that is, when r = a, is equal to

(2.38)

The electrical energy of the central ion is reduced by the product of its charge ze and the potential, due to its interaction with its neighbours. However, if this argument was applied to all ions, each would be counted twice, once as the central ion and once as part of the surroundings of other ions. The change ΔG in the electrical energy of a given ion due to ionic interactions is therefore

(2.39)

The corresponding quantity for 1 mol of such ions is

(2.40)

where again N is Avogadro's number. In the absence of interionic forces, a solution would exhibit ideal behaviour (i.e. G(ideal) = RT ln a), and it is then possible to conclude that

(2.41)

where a is the activity and γ the activity coefficient of the ion. Therefore, the activity coefficient can be calculated by considering Eqs. (2.40) and (2.41):

(2.42)

If κ is replaced by its definition given in Eq. (2.27), then the expression for the activity coefficient takes the form

(2.43)

where A and B are given by (when corrected to units of moles per kilogram rather than moles per litre)

(2.44)

(2.45)

where ρ is the density of the medium and aj is the effective diameter of the hydrated ion j. At 25 °C, Baj has generally been assigned a value of 1.5. Table 2.1 gives values of A and B at various temperatures. The numerator in Eq. (2.43) shows the effect of long-range Coulomb forces, and the denominator represents how these forces can be modified by short-range ionic interactions.

Table 2.1 Calculated values of A and B in Eqs. (2.44) and (2.45)

2.3 Osmotic Coefficient

As shown by Eq. (2.41), the activity of an ion is related to its Gibbs energy. Similarly, the activity of a solvent (as) can be defined by

(2.46)

where γ± is the activity coefficient and m the concentration of the solvent. Additionally, the osmotic coefficient Φ is defined by

(2.47)

where Ws is the molecular weight of the solvent and v is the number of dissociating ions. The chemical potential is the partial molal derivative of the Gibbs energy, and, as such, the Gibbs–Duhem equation applies, namely,

(2.48)

where n is the number of moles of a particular species, the summation covering both solvent and solute species. For the case where both the temperature and pressure are constant

(2.49)

When a solution only contains one solute, Eq. (2.49) becomes

(2.50)

Multiplying both sides by (1000/W1n1) leads to

(2.51)

Combination of Eq. (2.51) with the definition of ln(as) (Eq. (2.47)) for an aqueous solution (i.e. where water is the solvent) gives

(2.52)

From Eq. (2.52) and Eq. (2.47) for the osmotic coefficient, it can be shown that

(2.53)

that can be converted into a form from which the osmotic coefficient can be derived, namely,

(2.54)

By inclusion of the expression for the activity coefficient (log γ) given in Eq. (2.43) and then by making the substitution u = (m½/(1 + Bajm½), it can be shown that m = u²/(1 − Baju)² from which, in turn, leads to the following expression from which the osmotic coefficient can be calculated

(2.55)

2.4 Specific Ion Interaction Theory

Equation (2.43) describes the effect of long-range forces and how they can be modified by short-range interactions between ions. In a solution, however, short-range interactions between ions and solvent molecules need to be considered, and it has been found that such reactions have an approximate variation which is proportional to the concentration of the ionic medium. Therefore, the expression for the activity coefficient can be extended to

(2.56)

According to the specific ion interaction theory, the activity coefficient of an ion j of charge z in a solution of ionic strength I can be described by Eq. (2.56), when expressed in the form of Eq. (2.57):

(2.57)

where ε(j,k) is the ion interaction constant between the ion j and an ion k of the ionic medium of concentration mk.

An individual activity coefficient cannot be measured directly by experimental methods. Therefore, a mean activity coefficient γ± of an electrolyte NX of concentration m dissociating into v1 cations of valence z1 and v2 anions of valence z2 is given by

(2.58)

Since mX/v2 = mN/v1 = mNX, Eq. (2.58) becomes

(2.59)

Thus, a plot of log γ± + |z1z2|D versus mNX should be linear, pass through the origin and have a slope of 2v1v2ε(N,X)/(v1+v2), where D is given by

(2.60)

With respect to the osmotic coefficient, described by Eq. (2.55), incorporation of the expression for log γ± as given in Eq. (2.58) into the integral used to derive Eq. (2.55) leads to an expanded equation for the osmotic coefficient, namely,

(2.61)

If the first term on the right-hand side of the above equation is called Q, then a plot of (1 − φ − Q) versus mNX should be linear, pass through the origin and have a slope of ln(10)v1v2ε(N,X)/(v1 + v2).

Equations (2.59) and (2.61), depending on the ionic medium, describe reasonably well activity and osmotic coefficient data to moderate ionic strength. At high temperatures (typically above 100 °C), they only describe such data to relatively low ionic strength. Ciavatta (1980) demonstrated that an extension to the specific ion interaction theory could be used to describe data to much higher ionic strength. In this extension, the ion interaction parameter, ε(N,X), is modified to the following form:

(2.62)

Moreover, the use of Eq. (2.62) to describe ion interaction coefficients is also excellent at describing activity and osmotic coefficient (as well as stability and solubility constant) data at high temperatures.

With the inclusion of Eq. (2.62) into Eqs. (2.59) and (2.61), they become

(2.63)

(2.64)

where

(2.65)

According to the specific ion interaction theory, for hydrolysis reactions, ionic strength (activity) corrections can be illustrated in relation to the case of the formation reaction of the general hydrolysis complex described by reaction (2.5). The stability constant for this reaction, *βpq (Eq. (2.6)), determined in an ionic medium of ionic strength, Im, is related to the corresponding value at zero ionic strength, βpq°, by Eq. (2.66) (this is the specific form of Eq. (2.4) describing hydrolysis stability constants):

(2.66)

where γM is the activity coefficient of the metal ion, γpq is that of the hydrolysis complex formed (Mp(OH)q(pz−q)), γH is that of the hydrogen ion, and a(H2O) is the activity of water. Equation (2.63) can be used to describe each of the activity coefficients, and therefore, Eq. (2.66) becomes

(2.67)

where D is given in Eq. (2.60) and Δz² and Δεn (Δε1 or Δε2) are given by Eqs. (2.68) and (2.69), respectively:

(2.68)

(2.69)

In Eq. (2.68), z is the charge of the reacting metal ion. In Eq. (2.69), εn(H,X) is the ion interaction parameter between hydrogen ions and the anion of the ionic medium, εn(M,X) is that between the metal ion and the anion of the ionic medium, and εn(p,q,(N or X)) is that between the hydrolysis species formed and that of either the cation or anion of the ionic medium (depending on whether the species formed is cationic or anionic).

Hydrolysis reactions, as described by Eq. (2.5), involve H2O as a reactant and consequently require a correction for the activity of water, a(H2O), when deriving the zero ionic strength (thermodynamic) stability constant as shown in Eq. (2.67). The activity of water in an electrolyte mixture can be calculated from

(2.70)

Thus, the activity of water is a function of the osmotic coefficient of the solution, and the summation extends over all ions k with molality mk present in solution. In the presence of an ionic medium NX in dominant concentration, the equation can be simplified by neglecting the contributions of all minor species, that is, the reacting ions. For a 1 : 1 electrolyte of ionic strength, therefore, I ≈ mNX and Eq. (2.70) becomes

(2.71)

2.5 Determination of Temperature-Dependent Parameters

Determination of stability constants at temperatures other than 25 °C requires an expression to predict the values of the Debye–Hückel constants, A and B (given in Eqs. (2.44) and (2.45), respectively), as well as the ion size parameter, aj (Eq. (2.43)). The values of A and B can be determined using multiple regression with an equation of the form

(2.72)

and the values calculated using such an equation are compared with those values given previously (Table 2.1) in Table 2.2.

Table 2.2 Calculated values of the Debye–Hückel parameters A and B using a multiple regression equation

a) Literature data for A and B given in, for example, Brown, Curti and Grambow (2005).

Calculated values of the ion size parameter, aj, are in accord with Oelkers and Helgeson (1990) and are listed in Table 2.3. The variation of aj is given by

(2.73)

where g(T,p) is a temperature- and pressure-dependent function and is approximately zero at temperatures below 175 °C and is expressed (at the saturation pressure of water) by

(2.74)

Table 2.3 Comparison of calculated and literature values of the ion size parameter at the saturation pressure of water

a) Data for g(T,p) from Shock et al. (1992).

From Eq. (2.67), aj(298.15, 1 bar) = 4.5676 Å.

2.6 Determination of Ion Interaction Parameters from Activity and Osmotic Coefficient Data

Literature data (Robinson and Stokes, 1959) for the activity and osmotic coefficients of potassium hydroxide and sodium hydroxide solutions, at 25 °C, are listed in Table 2.4. The activity and osmotic coefficients of each of these alkali hydroxides have been fitted simultaneously using Eqs. (2.63) and (2.64), respectively (to keep the uncertainty in each point similar, a weight of 1/x² has been used where x is the measured value). The fits obtained are illustrated in Figures 2.1 and 2.2. From these fits, the values obtained for the ion interaction parameters are

Table 2.4 Osmotic and activity coefficient data (Robinson and Stokes, 1959) for potassium and sodium hydroxide solutions at 25 °C

Graphs of ion interaction parameter derivation ɛn (K+, OH−) from activity (top) and osmotic coefficients (bottom) of KOH solutions, both displaying ascending plots of solid and dotted lines.

Figure 2.1 Derivation of ion interaction parameters εn(K+, OH−) from (a) activity and (b) osmotic coefficients of KOH solutions. The solid line is the line of best fit, and the dotted lines are the 95% uncertainties projected out from I = 0–10 mol kg−1.

Graphs of ion interaction parameter derivation ɛn (K+, OH−) from activity (top) and osmotic coefficients (bottom) of NaOH solutions, both displaying ascending plots of solid and dotted lines.

Figure 2.2 Derivation of ion interaction parameters εn(Na+, OH−) from (a) activity and (b) osmotic coefficients of NaOH solutions. The solid line is the line of best fit, and the dotted lines are the 95% uncertainties projected out from I = 0–10 mol kg−1.

The differences (ε1(Na+, OH−) − ε1(K+, OH−)) and (ε2(Na+, OH−) − ε2(K+, OH−)) are equal to −(0.050 ± 0.003) kg mol−1 and 0.021 ± 0.003 kg mol−1, respectively. These can be compared with the same differences obtained from ion interaction parameter values determined from the ionic strength dependence of the protolysis constant of water in NaCl and KCl media (see Table 5.5). These latter values are (ε1(Na+, OH−) − ε1(K+, OH−)) = (0.103 ± 0.012 − 0.152 ± 0.011) = − (0.049 ± 0.016) kg mol−1 and (ε2(Na+, OH−) − ε2(K+, OH−)) = (0.062 ± 0.019 − 0.015 ± 0.020) = 0.047 ± 0.028 kg mol−1. The 95% confidence intervals of the latter values overlap the former values, and consequently, the hypothesis that they are similar cannot be rejected. Therefore, the values for ε1(K+, OH−) and ε2(K+, OH−) determined from the activity and osmotic coefficient data of Robinson and Stokes (1959) will be used as the starting point for the calculation of the ion interaction parameters.

It can be noted from Figures 2.1 and 2.2 that the values given above fit the activity coefficient data very well. However, for both sets of data, there is a significant underprediction of the osmotic coefficients. It is not clear why this is the case, but given the agreement between the interaction coefficients obtained from analysis of the protolysis of water data with those from the activity and osmotic coefficient data, the obtained ion interaction data for ε1(K+, OH−) and ε2(K+, OH−) are retained.

2.7 Determination of Ion Interaction Parameters for KOH at Temperatures Other than 25 °C

As indicated in the previous section, the ion interaction parameters ε1(K+, OH−) and ε2(K+, OH−) have been used, at 25 °C, as the starting point for the calculation of other ion interaction parameters. The same strategy will also be used for other temperatures. Activity coefficient data for other temperatures have been provided by Li and Pitzer (1996) and are reproduced here in Table 2.5 but in the moles per kilogram scale rather than mole fraction.

Table 2.5 Activity coefficient data for KOH over the temperature range of 0–300 °C and 0–12 mol kg−1

Values for the ion interaction coefficients were obtained at each of the temperatures listed in Table 2.5 by fitting Eq. (2.63). The values obtained are listed in Table 2.6 together with the values calculated at 25 °C that were determined in the previous section.

Table 2.6 Ion interaction parameters (εm(K+, OH−)) at various temperatures

The data in Table 2.6 were then fitted to the following equation to determine the temperature-dependent ion interaction parameters for KOH:

(2.75)

and the values determined for the parameters given in Eq. (2.75) are listed in Table 2.7. These values will be subsequently utilised to derive other temperature-dependent interaction parameters.

Table 2.7 Ion interaction parameters (εmn(K+, OH−))

2.8 Activity of Water

As indicated earlier, the stability constant of a hydrolysis species is dependent on the activity of water as shown in Eq. (2.67). The activity of water is related to the osmotic coefficient through Eq. (2.70). Thus, for an electrolyte NaXb of molality m, the activity of water can be described by

(2.76)

At a given temperature, it is possible to describe the activity of water using

(2.77)

where a1 and a2 are constants that depend on temperature and the electrolyte under consideration. Typically, Eq. (2.77) describes literature water activity data with a coefficient of determination higher than 0.999.

The coefficients a1 and a2 given in Eq. (2.77) can be described by the temperature-dependent equations:

(2.78)

where b3n−3, b3n−2 and b3n−1 are constants. Again, Eq. (2.78) fits the derived a1 and a2 data with very high coefficients of determination.

Table 2.8 provides values of a1 and a2 for the electrolytes that have been utilised to determine the stability constants cited in this study. Each reference listing the osmotic coefficient data utilised to derive the activity of water values is also provided. Values for the constants b3n−3, b3n−2 and b3n−1 are provided in Table 2.9 for those electrolytes where hydrolysis reactions have been studied at temperatures other than 25 °C. Data for a1 and a2 for these electrolytes and various temperatures have been given in Table 2.8.

Table 2.8 Activity of water parameters, a1 and a2, for various electrolytes

Table 2.9 Activity of water parameters, b3n−3, b3n−2 and b3n−1, for various electrolytes

2.9 Enthalpy and Entropy

The standard relationship between Gibbs energy, enthalpy and entropy is

(2.79)

Further, the Gibbs energy is related to the stability constant of a reaction, log K, via Eq. (2.80):

(2.80)

Combining Eqs. (2.79) and (2.80) gives

(2.81)

If the change in heat capacity, ΔCP, of a reaction is zero, the enthalpy and entropy of a reaction do not change with temperature and are constants. Thus, plotting log K values at different temperatures against the reciprocal of absolute temperature, 1/T, can be used to determine the values of ΔH° and ΔS°.

When the change in heat capacity is a non-zero constant, the enthalpy and entropy of a reaction at a particular temperature can be related to that at 25 °C (298.15 K) through the following equations:

(2.82)

(2.83)

Substituting these last two expressions into Eq. (2.79) gives

(2.84)

where

(2.85)

(2.86)

(2.87)

A variety of Eq. (2.84) can also be used with the stability constant by combining the equation with Eq. (2.80):

(2.88)

In this latter equation, the values of the constants A, B and C are given by

(2.89)

(2.90)

(2.91)

Thus, Eq. (2.88) can be utilised with regression analysis to determine the values of the enthalpy and entropy of reaction at 25 °C (298.15 K) and the change in heat capacity of the reaction. In this work, either Eq. (2.81) or (2.88) has been used to determine the enthalpy and entropy of reaction.

2.10 Estimation of Stability and Solubility Constants

The reaction of a metal ion, M, with a hydroxide ion to produce the species Mp(OH)q(pz−q) has been described by reaction (2.5). The stability constant for this species is shown in Eq. (2.6). Brown, Sylva and Ellis (1985) demonstrated that the stability constant, βpq, could be predicted from chemical and physical properties of the reacting metal ion. The stability constant could be described by Eq. (2.92):

(2.92)

where Int1 and Slp1 and Int2 and Slp2 are, respectively, the intercept values and the linear regression slope for the MOH(z−1) species and the polymeric Mp(OH)q(pz−q) species and r is the ionic radius of the reacting metal ion. For the hydroxide ion, in Eq. (2.92), g1 and g2 are defined by Brown, Sylva and Ellis (1985)

(2.93)

(2.94)

In these equations, S depends on the presence (S = 1) or absence (S = 0) of s-electrons in the outermost shell of the ion (i.e. those metal ions exhibiting the inert pair effect), D depends on the availability (D = 1) or not (D = 0) of d-orbitals for bonding, z is the charge of the metal ion, g(n) is a Slater function that is dependent on the principal quantum number n (when n is unity, g(n) = 0; otherwise it equals unity), and d is the number of d-electrons in the outermost shell of the ion. In Eq. (2.92), log Upq is given by

(2.95)

where k is a proportionality that relates two consecutive stability constants.

Solid-state measurements have shown that the ionic radius of oxo-cations such as dioxouranium(VI) is a function of both the nature and number of ligands bonded to the central metal ion (Zachariasen, 1954). Subsequently, it was shown that a similar dependence occurs in aqueous solution (Brown and Sylva, 1987). This latter study demonstrated that the value of the function g1(z/r² + g2) was dependent on the nature of the binding ligand, specifically its dissociation constant, where r in the function could be replaced by rapp (i.e. the function g1(z/rapp² + g2) could be utilised). For the hydroxide ion, the value of rapp can be derived from

(2.96)

where rapp is the apparent ionic radius and r0 is the maximum allowable ionic radius in the case of a ligand bound in the primary hydration sphere of the metal ion (Brown and Wanner, 1987).

Equation (2.92) was found to calculate stability constants that were in very good agreement with literature stability constants relevant to 25 °C and zero ionic strength. However, it has been correctly stated that the ability of the equation to provide reasonable stability constants decreased as the molecularity increased (i.e. as the number of ligands bound to the central metal ion increased) (Moriyama et al., 2005). Thus, the poorest agreement between stability constants calculated using Eq. (2.92) and measured values was found for monomeric species with large numbers of bound ligands.

Baes and Mesmer (1976) examined the relationship between the solubility of a hydroxide or oxide phase and the formation of the first monomeric hydrolysis species. For the reaction

(2.97)

the solubility constant is defined by

(2.98)

Baes and Mesmer found that the latter expression could be approximated by a constant, that is,

(2.99)

Subsequently, Brown and Sylva (1987) re-expressed Eq. (2.99) where the left-hand side of the expression was not constant:

(2.100)

Rearrangement of Eq. (2.100) and taking logarithms give

(2.101)

It was found by Brown and Sylva that the constant βp could be expressed by a similar expression as given for βpq in Eq. (2.92), that is,

(2.102)

where N is the number of water molecules lost in the formation of the solid phase. Equation (2.102) can then be combined with Eq. (2.92) (with respect to the formation of MOH(z−1)) to provide a method for estimating the solubility constant *Ks10.

References

Baes, C.F. and Mesmer, R.E. (1976) The Hydrolysis of Cations, John Wiley & Sons, Inc., New York.

Brown, P.L., Curti, E., and Grambow, B. (2005) Chemical Thermodynamics of Zirconium, vol. 8, Elsevier, Amsterdam, 512 pp.

Brown, P.L. and Sylva, R.N. (1987) Unified theory of metal ion complex formation constants. J. Chem. Res., (S) 4-5, (M) 0110–0181.

Brown, P.L., Sylva, R.N., and Ellis, J. (1985) An equation for predicting the formation constants of hydroxo-metal complexes. J. Chem. Soc., Dalton Trans., 723–730.

Brown, P.L. and Wanner, H. (1987) Predicted Formation Constants Using the Unified Theory of Metal Ion Complexation, Nuclear Energy Agency. Organisation for Economic Co-operation and Development, Paris, 102 pp.

Ciavatta, L. (1980) The specific interaction theory in evaluating ionic equilibria. Ann. Chim. (Rome), 551–567.

Debye, P. and Hückel, E. (1923) De la théorie des électrolytes. I. Abaissement du point de congélation et phénomènes associés. Phys. Z., 24, 185–206.

Goldberg, R.N. and Nuttall, R.L. (1978) Evaluated activity and osmotic coefficients for aqueous solutions: the alkaline earth metal halides. J. Phys. Chem. Ref. Data, 7, 263–310.

Hamer, W.J. and Wu, Y.-C. (1972) Osmotic coefficients and mean activity coefficients of uni-univalent electrolytes in water at 25 °C. J. Phys. Chem. Ref. Data, 1, 1047–1099.

Li, Z. and Pitzer, K.S. (1996) Thermodynamics of aqueous KOH over the full range of saturation and to 573 K. J. Solution Chem., 25, 813–823.

Lindenbaum, S. and Boyd, G.E. (1964) Osmotic and activity coefficients for the symmetrical tetraalkyl ammonium halides in aqueous solution at 25°. J. Phys. Chem., 68, 911–917.

Liu, C. and Lindsay, W.T. (1972) Thermodynamics of sodium chloride solutions at high temperatures. J. Solution Chem., 1, 45–69.

Lobo, V.M.M. (1989) Handbook of Electrolyte Solutions. Part A and B, Physical Sciences Data, vol. 41, Elsevier, Amsterdam, 2360 pp.

Moriyama, H., Sasaki, T., Kobayashi, T., and Takagi, I. (2005) Systematics of hydrolysis constants of tetravalent actinide ions. J. Nucl. Sci. Technol., 42, 626–635.

Oelkers, E.H. and Helgeson, H.C. (1990) Triple-ion anions and polynuclear complexing in supercritical electrolyte solutions. Geochim. Cosmochim. Acta, 54, 727–738.

Rard, J.A., Palmer, D.A., and Albright, J.G. (2003) Isopiestic determination of the osmotic and activity coefficients of aqueous sodium trifluoromethanesulfonate at 298.15 K and 323.15 K, and representation with an extended ion-interaction (Pitzer) model. J. Chem. Eng. Data, 48, 158–166.

Rard, J.A., Wijesinghe, A.M., and Wolery, T.J. (2004) Review of the thermodynamic properties of Mg(NO3)2(aq) and their representation with the standard and extended ion-interaction (Pitzer) models at 298.15 K. J. Chem. Eng. Data, 49, 1127–1140.

Robinson, R.A. and Stokes, R.H. (1959) Electrolyte Solutions, Butterworth and Company, London, 574 pp.

Shock, E.L., Oelkers, E.H., Johnson, J.W., Sverjensky, D.A., and Helgeson, H.C. (1992) Calculation of the thermodynamic properties of aqueous species at high pressures and temperatures. Effective electrostatic radii, dissociation constants and standard partial molal properties to 1000 °C and 5 kbar. J. Chem. Soc., Faraday Trans., 88, 803–826.

Zachariasen, W.H. (1954) Crystal chemical studies of the 5f-series of elements. XXIII. On the crystal chemistry of uranyl compounds and of related compounds of transuranic elements. Acta Crystallogr., 7, 795–799.

3

Methodologies for Determining Stability/Solubility Constants

3.1 Introduction

Many techniques are available for the measurement of metal–ligand interactions and the determination of their stability or solubility constants. These techniques include the following (the list is not exhaustive – abbreviations typically used for the techniques are given in parentheses):

Anion and cation exchange may often be referred to as just ion exchange (ix), and more specific information may often be given when referring to potentiometry including the type of potentiometric electrode used in carrying out the investigation (glass electrode (gl), quinhydrone electrode (qh), redox electrode (red), amalgam electrode (MHg)).

The most common techniques utilised for the determination of the stability or solubility constants of metal hydroxide species and phases include ion exchange (solid–liquid extraction), distribution (liquid–liquid extraction), potentiometry, solubility and spectrophotometry. A brief outline of each of these techniques will be discussed.

3.2 Potentiometry

The main principle behind potentiometric titrations is the measurement of the activity of one or several aqueous species using ion-selective electrodes versus a selected standard (reference) electrode. In the case of the determination of stability and solubility constants using potentiometry, the hydrogen ion activity is typically determined. The activity of the hydrogen ion can be measured using various types of electrodes, the most common being the glass electrode, but also the hydrogen and quinhydrone electrodes have been utilised.

Potentiometric titrations are used for the determination of stoichiometric and thermodynamic properties of elements in solution and are usually based on the Nernst equation (Nernst, 1889):

(3.1)

where E is the potential between two vessels containing aqueous solutions, R is the molar gas constant, T is the temperature in kelvin, F is the Faraday constant, ν is the charge of the ion, and ai the activity of the determined ion in the two vessels. Equation (3.1) was later generalised to a pair of oxidation states in the same solution and was formulated according to Eq. (3.2):

(3.2)

where E° is the potential at some standard state. In this equation, ai represents the activity of two different valency states of an element in the solution. In the following text, Eq. (3.2) is used and denoted as the Nernst equation.

There are two methods that are considered fundamental in the use of potentiometric titrations to determine the properties of a solution. The first was developed by Bodländer and Fittig (1902). The main feature of this method was to obtain a description of the stoichiometric constant in a reaction between a metal (M) and a ligand (L). However, it is usually desirable to also obtain a value for the stability constant itself since only how the expression looks for the reaction is not sufficient. A method for doing this using potentiometric titrations was introduced by Bjerrum (1941). In this method, Bjerrum used the average ligand number defined by

(3.3)

where [L−]bound is the concentration of the ligand bound to the metal and [M]T is the total metal concentration. When Eq. (3.3) is expressed as a function of the ligand (hydroxide) concentration, it is called the formation function. For the case of a ligand releasing one proton in the complexing action and the assumption that the deprotonisation of the ligand (hydroxide) is negligible compared to the other contributors of protons, the derivation of the formation function will be as follows:

(3.4)

where [H+] is the measured hydrogen ion concentration (which is related to the hydrogen ion activity), [H+]0 is the initial hydrogen ion