## Milan Meloun, Josef Havel: COMPUTATION OF SOLUTION EQUILIBRIA, Part 2: Potentiometry, Folia Fac. Sci. Nat., Univ. Purkynianae Brunensis, Brno 1985. Click on picture

### INTRODUCTION

Great increase of computer application in natural science brought a new subdiscipline of chemistry called chemometrics representing the utilization of mathematical and statistical methods for handling, interpreting, and predicting chemical data. Powerful methodologies have opened new vistas for chemists and have provided easy solution of many advanced chemical problems. Solution equilibria calculation by regression analysis has proven to be one of the most potent techniques in chemometrics arsenal for years. The pioneering work of Sillén and the Scandinavian school established a first library of computer methods equilibrium constants determination by potentiometric data analysis.
In first part Spectrophotometry of our book Computation of Solution Equilibria, general aspects of solution equilibria, experimental techniques for spectrophotometric equilibria study, mathematical principles of numerical methods and an application of computer programs on various equilibria problems were introduced.
This second part Potentiometry brings, firstly a sufficient information of experimental technique recommended for potentiornetric (equilibria) titrimetry, secondly, an introduction to analytical aspects of potentiometric titrations, thirdly, an account of frequently used programs in potentiometric equilibria study. Finally, last chapter shows examples of chemical model building by analysis of potentiometric and spectrophotometric titration data as both instrumental techniques are often exploited commonly.
Illustrative problems contain task formulation:, experimental design, execution of computer data treatment and some comments of achieved parameter estimation with a reliability of their determination. Problems enable to reader to be more familiar with the input formulation and suitable program application of his own data.

### CONTENT

INTRODUCTION 9
CHAPTER 1 COMPLEX EQUILIBRIA AND EQUILIBRIUM CONSTANTS 10
1.1. Types of Complexes 10
1.2. Types of Equilibrium Constants 10
1.3. Thermodynamic, Stoichiometric and Conditional Stability Constants 12
1.4. Design of Estimated Equilibrium Model 13
1.5. References 13

CHAPTER 2 EXPERIMENTAL METHODS FOR STUDYING EQUILIBRIA 15
2.1. External Titration Outside the Cuvette l6
2.2. Internal Titration Inside the Cuvette 16
2.3. Rules of Absorbance Measurement 17
2.4. Methods Based on Various Concentration Variables 18
2.4.1. Mole Ratio Method 19
2.4.2. Method of Continuous Variations (Job's Method) 20
2.4.3. Method of Corresponding Solutions 21
2.4.4. Chelatometric Titration with Metallochromic Indicator 22
2.4.5. Combined pH-Spectrophotometric Titration 23
2.4.6. Absorbance Spectrum Measurements 23
2.5. Trends of Automation of Spectrophotometric Measurement 25
2.6. References 26

CHAPTER 3 GRAPHICAL AND CALCULATOR-ASSISTED TREATMENT OF DATA 27
3.1. Secondary Concentration Variables 27
3.2. Calculation of Stability Constants
3.2.1. Linear Extrapolation 29
3.2.2. Elimination Method. 30
3.3. Linear Least-Squares Method by Pocket Calculator 31
3.3.1. Dissociation Constant of Monoprotic Acid. 31
3.3.2. Equilibrium Constant of One Prevailing Complex 32
3.4. Illustrative Problems 35
Problem 3.1. Comparison of Linear Extrapolation and Elimination Methods 35
Problem 3.2. Dissociation Constant of Phenyl Red by Linear Regression 36
Problem 3.3. Determination of Equilibrium Model with the Use of linearized Transformations of A-pH Curve 38
3.5. References 39

CHAPTER 4 ELUCIDATION OF INDIVIDUAL PARTS OF EQUILIBRIUM PROGRAM 41
4.1. Structure of Spectrophotometric Equilibrium Program 41
4.2. INPUT, Random Errors and Systematic Errors 42
4.3. MINIMIZATION, Methods of Regression Analysis 44
4.3.1. Derivative-free Methods 45
(i) Direct Searching 46
(ii) Simplex Method 46
(iii) Pit Mapping Method 48
4.3.2. Derivative Methods, Gradient Methods 50
(i) Gauss-Newton and Newton-Raphson Methods 50
(ii) Steepest Descent Method 53
(iv) Davidon-Fletcher Powell Variable Metric Method 54
4.4. ERROR ANALYSIS, Correlation of Parameters, Confidence Intervals of Parameters 55
4.5. FITNESS TEST, Residuals Analysis, Hypothesis Testing 59
4.6. DATA SIMULATION 61
4.7. SPECIES NUMBER, Determining the Matrix Rank. 61
4.8. FREE CONCENTRATIONS, Calculation of Equilibrium Concentrations 65
4.8.1. Nonderivative Method in Subroutine COGS 65
4.8.2. Numerical Derivative Method in Subroutine COGSNR 66
4.8.3. Analytical Derivative Method 68
4.9. OUTPUT, Display of Results, Diagrams
4.10. Illustrative Problems 70
Problem 4.1. Heuristic Minimization by the Pit-Mapping Algorithm 70
Problem 4.2. Algorithmic Minimization by Various Algorithms 74
Problem 4.3. Determination of Ill-Conditioned Parameters 76
Problem 4.4. Various Derivative-free Least-Squares Algorithms 80
Problem 4.5. The Choice of Derivative Minimization Algorithms 83
Problem 4.6. Use of Random-Error Author in Simulated Data 83
Problem 4.7. Calculated Errors of Estimated Parameters 85
Problem 4.8. True Confidence Intervals of Estimated Parameters 87
Problem 4.9. Fitness Test by the Statistical Analysis of residuals 88
Problem 4.10.Regression Analysis of Spectra by the SQUAD Program 90
4.11. References 95

CHAPTER 5 AVAILABLE LEAST-SQUARES PROGRAMS 97
5.1. Treatment of Absorbance-Concentration Variable Curve 98
5.1.1. The MRLET and MRFIT Programs 99
5.1.2. The NCLET Program 101
5.1.3. The DCLET, DCMINUIT, DCSPONA and DCFIT Programs 103
5.1.4. The DHLET and DHMINUIT Programs 104
5.1.5. The EXLET Program 106
5.1.6. The SPLET Program. 106
5.1.7. The JOBCON Program 107
5.1.8. The CORSO Program 109
5.1.9. The SPEF-3Program 110
5.1.10.The LETAGROP-SPEFO Program I11
5.2. Treatment of Absorbance Spectra 112
5.2.2. The FA6OB+EY6OB Program 115
5.2.4. The DALSFEK Program. 118
5.3. Illustrative Problems 119
Problem 6.1. Analysis of Mole Ratio Curve by the MRFIT Program 119
Problem 5.2. Analysis of Chelatometric Titration Curve by the NCLET Program 121
Problem 5.3. Estimation of Two Near Dissociation Constants from A-pH Curve by the DCLET Program 125
Problem 5.4. Estimation of Thermodynamic Dissociation Constant from pKa-I Curve by the DHLET Program 126
Problem 6.5. Equilibria Constants and Molar Absorptivity from Spectrophotometric Extraction Data by the EXLET Program 126
Problem 5.6. Analysis of Absorbance-pH Curve by the SPLET Program 130
Problem 5.7. Curve Fitting of Continuous Variations Curve of a system Fe-SCN by the JOBCON Program 132
Problem 5.8. Stability Constants of two Consecutive Complexes by the CORSO Program 133
Problem 5.9. Analysis of Spectra by the FA6OB+EY6OB Program 134
Problem 5.10.Dissociation Constant from A-pH Curve by the LETAGROP-SPEFO and the SQUAD Programs 139
5.4. References 141

CHAPTER 6 DETERMINATION OF CHEMICAL MODEL 143
6.1. Absorption Spectra 144
6.2. Continuous Variation Method 144
Problem 6.1. Analysis of Continuous Variation Curves of Pr(III)-TAN-3, 6-S System by Nonlinear Regression 145
Problem 6.2. False Regression Approach to the Evaluation of Job's Curves 146
Problem 6.3. Analysis of Continuous Variation Curves by the Regression of Absorbance Matrix 151
6.3. Absorbance-pH or Absorbance-Concentration Curves. 155
Problem 6.4. Determination of Dissociation Constant from A-pH Curve. The Effect of a Number of Solutions and a Number of Wavelengths on Computed Dissociation Constant Value 156
Problem 6.5. The SPLET Program Evaluation of A-pH Curve at Single Wavelength. Determination of Equilibrium Model 157
6.4. Matrix Analysis of Spectra by General Least-Squares Programs 159
Problem 6.6. Dimerization of Proflavine Being Analyzed by the SQUAD Program 160
Problem 6.7. Chemical Model Determination from Spectra of Cadmium(II)-4-(2-Pyridylazo) Resorcine (PAR) System 165
Problem 6.8. Determination of Chemical Model for Reaction of Zinc(II) with 2-(2-Pyridylazo) Resorcine-1-Naphthol-4-Sulphonic Acid (lPAN-4S) 167
6.5. Concluding Remarks to the Problem of Chemical Model Determination 177
6.6 References 178

SUPPLEMENT 1 Basic Matrix Algebra 180
S 1.1. Matrix Definition 180
S 1.2. Special Types of Matrix 180
S 1.3. Basic Operations Involving Two Matrices 181
S 1.4. Determinant of a Matrix 182
S 1.5. lnverse of a Matrix 182
S 1.6. System of Linear Equations 182
S 1.7. Rank of a Matrix 183
S 1.8. Eigenvalues and Eigenvectors 183
S 1.9. References 184