**Milan Meloun, Josef Havel: COMPUTATION OF SOLUTION EQUILIBRIA, Part 1: Spectrophotometry, Folia Fac. Sci. Nat., Univ. Purkynianae Brunensis, Brno 1984.**

### INTRODUCTION:

Obvious renaissance of solution equilibria in last years, unsubstitutional role of spectrophotometry and nowadays expansion of computing science made the situation that practicing chemist can hardly find the proper program in a jungle of computer-assisted methods published. Which program is the beat and most convenient one for a given problem? How to understand and how to interpret the program output, what experimental strategy to use to obtain reliable answers to the existing equilibria in solution? How many species and of which stoichiometry they are in equilibrium?

This book has been written to bring answers on all these questions in mind. Firstly, we tried to present up-to-date account of computer programs concerning a study of solutions equilibria with special respect to spectrophotometric indication. Secondly, to give brief mathematical information about the regression method and other statistical and calculus tools being applied in general equilibrium program and to elucidate principal parts of program structure and therefore enable a reader to formulate his own specific program. Thirdly, to recommend convenient experimental arrangement and sufficient experimental information to choose proper technique and strategy of experiment suitable for particular chemical system. Fourthly, to bring a survey of available computer programs up-to 1982 and to pay an attention to methods we have a good experience with. Finally, to demonstrate on many practical problems of solution equilibria study the procedure of "true" chemical model determination and reliability of its verification.

Throughout all the book problems and chemical examples always contain complete input data and all computational strategy which enables a reader to repeat the computation or to be more familiar with the formulation of input data of his own problem.

This first part of the book on Computation of Solution Equilibria relating with Spectrophotometry was delivered for print in February 1983.

### 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

(iii) Conjugate Gradients Method 54

(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

5.2.1. The SQUAD Program 113

5.2.2. The FA6OB+EY6OB Program 115

5.2.3. The PSEQUAD Program. 116

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