An Introduction to Aqueous Electrolyte Solutions

Preface xix

Preliminary Chapter Guidance to Student xxiii

List of symbols xxv

1 Concepts and Ideas: Setting the Stage 1

1.1 Electrolyte solutions – what are they? 2

1.2 Ions – simple charged particles or not? 4

1.3 The solvent: structureless or not? 7

1.4 The medium: its structure and the effect of ions on this structure 8

1.5 How can these ideas help in understanding what might happen when an ion is put into a solvent? 9

1.6 Electrostriction 11

1.7 Ideal and non-ideal solutions – what are they? 11

1.8 The ideal electrolyte solution 14

1.9 The non-ideal electrolyte solution 14

1.10 Macroscopic manifestation of non-ideality 15

1.11 Species present in solution 17

1.12 Formation of ion pairs from free ions 17

1.13 Complexes from free ions 21

1.14 Complexes from ions and uncharged ligands 21

1.15 Chelates from free ions 22

1.16 Micelle formation from free ions 22

1.17 Measuring the equilibrium constant: general considerations 23

1.18 Base-lines for theoretical predictions about the behaviour expected for a solution consisting of free ions only, Debye-Hu¨ckel and Fuoss-Onsager theories and the use of Beer’s Law 24

1.19 Ultrasonics 26

1.20 Possibility that specific experimental methods could distinguish between the various types of associated species 29

1.21 Some examples of how chemists could go about inferring the nature of the species present 29

2 The Concept of Chemical Equilibrium: An Introduction 33

2.1 Irreversible and reversible reactions 34

2.2 Composition of equilibrium mixtures, and the approach to equilibrium 34

2.3 Meaning of the term ‘position of equilibrium’ and formulation of the equilibrium constant 35

2.4 Equilibrium and the direction of reaction 39

2.5 A searching problem 44

2.6 The position of equilibrium 45

2.7 Other generalisations about equilibrium 46

2.8 K and pK 46

2.9 Qualitative experimental observations on the effect of temperature on the equilibrium constant, K 47

2.10 Qualitative experimental observations on the effect of pressure on the equilibrium constant, K 49

2.11 Stoichiometric relations 49

2.12 A further relation essential to the description of electrolyte solutions – electrical neutrality 50

3 Acids and Bases: A First Approach 53

3.1 A qualitative description of acid–base equilibria 54

3.2 The self ionisation of water 56

3.3 Strong and weak acids and bases 56

3.4 A more detailed description of acid–base behaviour 57

3.5 Ampholytes 60

3.6 Other situations where acid/base behaviour appears 62

3.7 Formulation of equilibrium constants in acid–base equilibria 66

3.8 Magnitudes of equilibrium constants 67

3.9 The self ionisation of water 67

3.10 Relations between Ka and Kb: expressions for an acid and its conjugate base and for a base and its conjugate acid 68

3.11 Stoichiometric arguments in equilibria calculations 70

3.12 Procedure for calculations on equilibria 71

4 Equilibrium Calculations for Acids and Bases 73

4.1 Calculations on equilibria: weak acids 74

4.2 Some worked examples 80

4.3 Calculations on equilibria: weak bases 85

4.4 Some illustrative problems 90

4.5 Fraction ionised and fraction not ionised for a weak acid; fraction protonated and fraction not protonated for a weak base 97

4.6 Dependence of the fraction ionised on pKa and pH 98

4.7. The effect of dilution on the fraction ionised for weak acids lying roughly in the range: pKa ¼ 4.0 to 10.0 101

4.8 Reassessment of the two approximations: a rigorous expression for a weak acid 103

4.9 Conjugate acids of weak bases 104

4.10 Weak bases 105

4.11 Effect of non-ideality 105

5 Equilibrium Calculations for Salts and Buffers 107

5.1 Aqueous solutions of salts 108

5.2 Salts of strong acids/strong bases 108

5.3 Salts of weak acids/strong bases 108

5.4 Salts of weak bases/strong acids 109

5.5 Salts of weak acids/weak bases 117

5.6 Buffer solutions 119

6 Neutralisation and pH Titration Curves 139

6.1 Neutralisation 140

6.2 pH titration curves 141

6.3 Interpretation of pH titration curves 149

6.4 Polybasic acids 153

6.5 pH titrations of dibasic acids: the calculations 161

6.6 Tribasic acids 166

6.7 Ampholytes 168

7 Ion Pairing, Complex Formation and Solubilities 177

7.1 Ion pair formation 178

7.2 Complex formation 184

7.3 Solubilities of sparingly soluble salts 195

8 Practical Applications of Thermodynamics for Electrolyte Solutions 215

8.1 The first law of thermodynamics 216

8.2 The enthalpy, H 217

8.3 The reversible process 217

8.4 The second law of thermodynamics 217

8.5 Relations between q, w and thermodynamic quantities 218

8.6 Some other definitions of important thermodynamic functions 218

8.7 A very important equation which can now be derived 218

8.8 Relation of emfs to thermodynamic quantities 219

8.9 The thermodynamic criterion of equilibrium 220

8.10 Some further definitions: standard states and standard values 221

8.11 The chemical potential of a substance 221

8.12 Criterion of equilibrium in terms of chemical potentials 222

8.13 Chemical potentials for solids, liquids, gases and solutes 223

8.14 Use of the thermodynamic criterion of equilibrium in the derivation of the algebraic form of the equilibrium constant 224

8.15 The temperature dependence of DHu 230

8.16 The dependence of the equilibrium constant, K, on temperature 231

8.17 The microscopic statistical interpretation of entropy 236

8.18 Dependence of K on pressure 237

8.19 Dependence of DGu on temperature 242

8.20 Dependence of DSu on temperature 242

8.21 The non-ideal case 244

8.22 Chemical potentials and mean activity coefficients 247

8.23 A generalisation 251

8.24 Corrections for non-ideality for experimental equilibrium constants 258

8.25 Some specific examples of the dependence of the equilibrium constant on ionic strength 263

8.26 Graphical corrections for non-ideality 270

8.27 Comparison of non-graphical and graphical methods of correcting for non-ideality 270

8.28 Dependence of fraction ionised and fractiion protonated on ionic strength 271

8.29 Thermodynamic quantities and the effect of non-ideality 271

9 Electrochemical Cells and EMFs 273

9.1 Chemical aspects of the passage of an electric current through a conducting medium 274

9.2 Electrolysis 275

9.3 Electrochemical cells 280

9.4 Some examples of electrodes used in electrochemical cells 285

9.5 Combination of electrodes to make an electrochemical cell 292

9.6 Conventions for writing down the electrochemical cell 293

9.7 One very important point: cells corresponding to a ‘net chemical reaction’ 298

9.8 Liquid junctions in electrochemical cells 298

9.9 Experimental determination of the direction of flow of the electrons, and measurement of the potential difference 305

9.10 Electrode potentials 305

9.11 Standard electrode potentials 306

9.12 Potential difference, electrical work done and DG for the cell reaction 308

9.13 DG for the cell process: the Nernst equation 312

9.14 Methods of expressing concentration 315

9.15 Calculation of standard emfs values for cells and DGu values for reactions 317

9.16 Determination of pH 320

9.17 Determination of equilibrium constants for reactions where K is either very large or very small 322

9.18 Use of concentration cells 324

9.19 ‘Concealed’ concentration cells and similar cells 326

9.20 Determination of equilibrium constants and pK values for reactions which are not directly that for the cell reaction 328

9.21 Use of concentration cells with and without liquid junctions in the determination of transport numbers 343

10 Concepts and Theory of Non-ideality 349

10.1 Evidence for non-ideality in electrolyte solutions 350

10.2 The problem theoretically 351

10.3 Features of the simple Debye-Hu¨ckel model 351

10.4 Aspects of electrostatics which are necessary for an understanding of the procedures used in the Debye-Hückel theory and conductance theory 353

10.5 The ionic atmosphere in more detail 360

10.6 Derivation of the Debye-Hückel theory from the simple Debye-Hückel model 363

10.7 The Debye-Hückel limiting law 380

10.8 Shortcomings of the Debye-Hückel model 382

10.9 Shortcomings in the mathematical derivation of the theory 384

10.10 Modifications and further developments of the theory 385

10.11 Evidence for ion association from Debye-Hückel plots 391

10.12 The Bjerrum theory of ion association 393

10.13 Extensions to higher concentrations 401

10.14 Modern developments in electrolyte theory 402

10.15 Computer simulations 402

10.16 Further developments to the Debye-Hückel theory 404

10.17 Statistical mechanics and distribution functions 409

10.18 Application of distribution functions to the determination of activity coefficients due to Kirkwood; Yvon; Born and Green; and Bogolyubov 414

10.19 A few examples of results from distribution functions 417

10.20 ‘Born-Oppenheimer level’ models 419

10.21 Lattice calculations for concentrated solutions 419

11 Conductance: The Ideal Case 421

11.1 Aspects of physics relevant to the experimental study of conductance in solution 422

11.2 Experimental measurement of the conductivity of a solution 425

11.3 Corrections to the observed conductivity to account for the self ionisation of water 427

11.4 Conductivities and molar conductivities: the ideal case 428

11.5 The physical significance of the molar conductivity, L 431

11.6 Dependence of molar conductivity on concentration for a strong electrolyte: the ideal case 432

11.7 Dependence of molar conductivity on concentration for a weak electrolyte: the ideal case 433

11.8 Determination of L0 436

11.9 Simultaneous determination of K and L0 438

11.10 Problems when an acid or base is so weak that it is never 100% ionised, even in very, very dilute solution 441

11.11 Contributions to the conductivity of an electrolyte solution from the cation and the anion of the electrolyte 441

11.12 Contributions to the molar conductivity from the individual ions 442

11.13 Kohlrausch’s law of independent ionic mobilities 443

11.14 Analysis of the use of conductance measurements for determination of pKas for very weak acids and pKbs for very weak bases: the basic quantities involved 447

11.15 Use of conductance measurements in determining solubility products for sparingly soluble salts 451

11.16 Transport numbers 453

11.17 Ionic mobilities 457

11.18 Abnormal mobility and ionic molar conductivity of H3Oþ(aq) 463

11.19 Measurement of transport numbers 464

12 Theories of Conductance: The Non-ideal Case for Symmetrical Electrolytes 475

12.1 The relaxation effect 476

12.2 The electrophoretic effect 480

12.3 Conductance equations for strong electrolytes taking non-ideality into consideration: early conductance theory 480

12.4 A simple treatment of the derivation of the Debye-Hu¨ckel-Onsager equation 1927 for symmetrical electrolytes 483

12.5 The Fuoss-Onsager equation 1932 488

12.6 Use of the Debye-Hu¨ckel-Onsager equation for symmetrical strong electrolytes which are fully dissociated 488

12.7 Electrolytes showing ion pairing and weak electrolytes which are not fully dissociated 490

12.8 Empirical extensions to the Debye-Hu¨ckel-Onsager 1927 equation 492

12.9 Modern conductance theories for symmetrical electrolytes – post 1950 493

12.10 Fuoss-Onsager 1957: Conductance equation for symmetrical electrolytes 493

12.11 A simple illustration of the effects of ion association on experimental conductance curves 500

12.12 The Fuoss-Onsager equation for associated electrolytes 500

12.13 Range of applicability of Fuoss-Onsager 1957 conductance equation for symmetrical electrolytes 503

12.14 Limitations of the treatment given by the 1957 Fuoss-Onsager conductance equation for symmetrical electrolytes 504

12.15 Manipulation of the 1957 Fuoss-Onsager equation, and later modifications by Fuoss and other workers 505

12.16 Conductance studies over a range of relative permittivities 506

12.17 Fuoss et al. 1978 and later 506

Appendix 1 512

Appendex 2 515

13 Solvation 517

13.1 Classification of solutes: a resume´ 518

13.2 Classification of solvents 518

13.3 Solvent structure 519

13.4 The experimental study of the structure of water 522

13.5 Diffraction studies 522

13.6 The theoretical approach to the radial distribution function for a liquid 526

13.7 Aqueous solutions of electrolytes 526

13.8 Terms used in describing hydration 528

13.9 Traditional methods for measuring solvation numbers 530

13.10 Modern techniques for studying hydration: NMR 533

13.11. Modern techniques of studying hydration: neutron and X-ray diffraction 538

13.12 Modern techniques of studying solvation: AXD diffraction and EXAFS 541

13.13 Modern techniques of studying solvation: computer simulations 542

13.14 Cautionary remarks on the significance of the numerical values of solvation numbers 543

13.15 Sizes of ions 544

13.16 A first model of solvation – the three region model for aqueous electrolyte solutions 544

13.17 Volume changes on solvation 551

13.18 Viscosity data 552

13.19 Concluding comment 552

13.20 Determination of DGu hydration 552

13 21 Determination of DHu hydration 553

13.22 Compilation of entropies of hydration from DGu hydration and DHu hydration 554

13.23 Thermodynamic transfer functions 554

13.24 Solvation of non-polar and apolar molecules – hydrophobic effects 554

13.25 Experimental techniques for studying hydrophobic hydration 556

13.26 Hydrophobic hydration for large charged ions 559

13.27 Hydrophobic interaction 560

13.28 Computer simulations of the hydrophobic effect 560

Subject Matter of Worked Problems 561

Index 563