Thermodynamics for Drug Product Design

SI Defining Constants xv Foreword xix Preface xxi Acknowledgments xxv 1 Data Reporting and Analysis 1 1.1 Introduction 1 1.1.1 Fundamental/Base and Derived Physical Quantities 1 1.1.2 Basic Mathematics and Statistics 2 1.2 Physical Quantity Dimensions and Dimensional Analysis 3 1.2.1 Application of Dimensional Analysis 3 1.2.2 Limitations of Dimensional Analysis 4 1.2.3 Dimensionless Numbers 4 1.2.4 Dimensional Nomenclature 4 1.2.5 Dimensional Quantity Algebra or Dimensional Algebra 4 1.3 Data Reporting Using Decimal, Scientific, Normalized Scientific, and E Scientific Numerical Notation 7 1.4 Accuracy 8 1.5 Precision as Measurement Variability and Relative Uncertainty 8 1.6 Valid Measurements 8 1.7 Uncertainty 9 1.7.1 Uncertainty Associated with a Single Measurement 9 1.7.2 Uncertainty Associated with Replicate Measurements 10 1.7.3 Propagation of Error of Combined Measurements and Calculations 10 1.7.4 Author’s Anecdote: A Priori Estimation of Dose Variation Using Propagation of Error Equations 10 1.8 Significant Figures (Digits) for Measured Values and Calculations 12 1.8.1 Rules for Determining Significant Figures for a Measured or Reported Value 12 1.8.2 Determination of Significant Figures for Addition and Subtraction Calculations 13 1.8.3 Determination of Significant Figures for Multiplication and Division Calculations 13 1.8.4 Logarithm Significant Figures 14 1.8.5 Antilogarithm Significant Figures 14 1.9 Rounding Numbers 15 1.10 How Many Digits and Decimals to Use in Research Reports, Tables, and Presentations 15 1.11 Exponents 16 1.11.1 Exponent Properties and Operations 16 1.11.2 Examples of Scientific Exponent Phenomena 17 1.12 Logarithms 18 1.12.1 Logarithmic Properties and Operations 18 1.12.2 Scientific Examples of Base-10 Logarithms (Log10 or Log) 19 1.12.3 Scientific Examples of Base-e Logarithms (Loge or Ln) 20 1.13 Differential and Partial Differential Equations 22 1.13.1 Differential Order of Ordinary Differential Equations 22 1.13.2 First-Order Ordinary Differential Equations 23 1.13.3 First-Order Partial Derivatives 24 1.14 Integral Equations 25 1.15 Basic Descriptive Statistics 26 1.15.1 Central Tendency of Single Values of Sample Data 27 1.15.2 Dispersion of Single Values of Sample Data 28 1.16 Application: Linear Least Squares and Coefficient of Determination 29 References 31 Chapter 1 Problem Set 32 2 Underlying Thermodynamic Physical Property Relationships and Definitions 35 2.1 Introduction 35 2.2 Temperature 35 2.3 Energy, Heat, and Work 36 2.4 System, Surroundings, Boundary, and Universe 36 2.4.1 Isolated System 37 2.4.2 Closed System 37 2.4.3 Open System 37 2.5 Macroscopic Properties and Intensive–Extensive Variables 38 2.6 Thermodynamic State Variables 38 2.6.1 Ideal Gas Law 38 2.6.2 Dalton’s Law 39 2.6.3 Pharmaceutical Application of the Ideal Gas Law and Dalton’s Law: Sterile Vial and Prefilled Syringe Manufacturing 39 2.6.4 Pharmaceutical Application of Dalton’s Law: Determination of Total Gas Pressure of Pressurized Metered Dose Inhalers Formulated with a Mixture of Propellant Gases 41 2.6.5 Pharmaceutical Application of Ideal Gas Law: Determination of a Pressurized Metered-Dose Inhaler when Stored at −20.0 Fahrenheit (−28.9 C) in a Car Glove Compartment in the Middle of Winter 42 2.7 Isothermal, Adiabatic, Isovolumetric, and Isobaric Thermodynamic State Processes 42 2.7.1 Pharmaceutical Application: Using an Adiabatic Calorimeter to Determine the Enthalpy of Solution of a Drug or Material of Interest 43 2.7.2 Pharmaceutical Application: Freeze-Drying as an Isochoric or Isovolumetric Process 45 2.8 Thermodynamic State Functions 46 2.9 Spontaneous Processes and Nonspontaneous Processes 47 2.9.1 Author’s Anecdote - Unexpected Polymorphic Conversion in Preparation of Toxicology Supplies 49 2.9.2 Nonspontaneous Processes 49 2.10 Reversible Processes 49 2.10.1 Reversible Isothermal Expansion 50 2.11 Irreversible Processes 51 References 51 Chapter 2 Problem Set 52 3 The Laws of Thermodynamics 53 3.1 Introduction 53 3.2 The Zeroth Law of Thermodynamics: Thermal Equilibrium and Temperature 54 3.3 The First Law of Thermodynamics: Conservation of Energy and Internal Energy as a State Function 55 3.3.1 Example of Change in Internal Energy for a Liquid Undergoing Vaporization 57 3.3.2 Enthalpy: A Thermodynamic State Function 59 3.3.3 Enthalpy and Hess’s Law 61 3.3.4 Pharmaceutical Application: Thermochemistry Using Differential Scanning Calorimetry 62 3.3.5 Heat Capacity: A Thermodynamic State Function 68 3.3.6 Heat Capacity at Constant Volume 68 3.3.7 Heat Capacity at Constant Pressure 69 3.3.8 Pharmaceutical Application: Thermochemistry Using Heat Capacity and Modulated Temperature Differential Scanning Calorimetry 70 3.3.9 Other Pharmaceutical Applications Using Heat Capacity 73 3.4 Second Law of Thermodynamics: Entropy Is Unavailable to Do Useful Work and Does Not Decrease – Entropy Is a State Function 74 3.4.1 What is Entropy? 74 3.4.2 Entropy: A Statistical and Molecular Interpretation 74 3.4.3 Entropy: State Function That Explains Spontaneous Reactions 76 3.4.4 Entropy: The Effect of Volume and Temperature Changes 77 3.4.5 Entropy of Mixing Ideal Gases 78 3.4.6 Determination of Entropy Changes for Phase Transitions 80 3.4.7 Other Reversible Processes 81 3.4.8 Factors Affecting Entropy 81 3.4.9 Determination and Prediction of ΔS Sign and Relative Value 82 3.4.10 Entropy’s Shortcomings for Prediction of Spontaneous Processes 82 3.5 Third Law of Thermodynamics: Entropy of a System Approaches a Constant Value as Its Temperature Approaches Absolute Zero 83 References 83 Chapter 3 Problem Set 85 4 Gibbs Free Energy 87 4.1 Introduction 87 4.2 Relationship Between Gibbs Free Energy and Internal Energy 88 4.3 Gibbs Free Energy to Predict Spontaneous Processes 89 4.4 Temperature Influence on Gibbs Free Energy 93 4.5 Pressure Influence on Gibbs Free Energy 94 4.6 Standard Gibbs Free Energy of Reaction 95 4.7 Gibbs Free Energy, Chemical Equilibrium, and Extent of Reaction 97 4.8 Gibbs Free Energy and Chemical Potential: Introduction to the Fundamental Equations for Open Systems 99 4.8.1 Chemical Potential Dependence on Temperature and Pressure 101 4.8.2 Gibbs-Duhem Equation for a One-Phase Two- Component System 101 4.9 Molar Gibbs Free Energy and Chemical Potential for an Ideal Gas 104 4.9.1 Chemical Potential for a Nonideal System 105 4.10 Gibbs Free Energy and Non-Pressure-Volume Work 105 4.11 Gibbs Free Energy and Phase Changes 106 4.11.1 Determination of the Transition Entropy 107 4.12 Application of Gibbs Free Energy and Tertiary Protein Structure and Stability 107 4.12.1 Enthalpic Energy Favoring the Native Folded State 109 4.12.2 Entropic “Hydrophobic Effect” Favoring the Native Folded State 110 4.12.3 Entropic Conformational Freedom Favoring Denaturation and Unfolding 110 4.12.4 Application: Overall Thermodynamic Effects on an Insulin-like Proteins Folded Conformation 110 References 111 Chapter 4 Problem Set 112 5 Equilibria 115 5.1 Introduction 115 5.1.1 Brief Review of Chemical Equilibrium 115 5.1.1.1 Law of Mass Balance 115 5.1.1.2 Law of Mass Action 116 5.1.1.3 Equilibrium State 116 5.1.1.4 Equilibrium State and Gibbs Free Energy 116 5.2 Le Châtelier’s Principle 118 5.2.1 Effect of Concentration Change Stress on Equilibrium 118 5.2.2 Effect of Pressure Change Stress on Equilibrium 121 5.2.3 Effect of Temperature Change Stress on Equilibrium 122 5.2.4 Author’s Anecdote – Solving an Expensive Production Efficiency Improvement Misadventure by Applying Heat of Solution, Le Châtelier’s Principle, and the Nernst and Brunner Equation 123 5.3 Phase Equilibrium of a Pure Substance 125 5.4 Gibbs Phase Rule 126 5.5 Temperature-Composition Phase Equilibrium Diagrams of a Two-Component System 127 5.5.1 Classic Two-Component Eutectic Phase Diagram 128 5.5.2 Application: Commercialized Eutectic Product and Its Eutectic Phase Diagram 129 5.5.3 Application: Complex Two-Component Equilibrium Phase Diagram – NaCl and Water 130 5.6 Phase Equilibrium of a Three-Component System 133 5.7 Phase Transitions of Pure Substances 134 5.7.1 The Clapeyron Equation: Vapor Equilibrium Boundary 135 5.7.2 Derivation of the Clausius-Clapeyron Equation: Vapor Equilibrium Boundary 136 5.7.3 Application of the Clausius-Clapeyron Equation 137 5.7.4 Ehrenfest Classification of Phase Transition Order 138 5.8 Pharmaceutical Application of Chemical Equilibrium 139 5.9 Ionic Equilibrium 142 5.9.1 Ionization Equilibrium of Weak Acids and Bases 142 5.9.2 pH and pKa 144 5.9.3 Henderson-Hasselbalch Equation 144 5.9.4 Le Châtelier’s Principle and Common Ion Effect on pH 145 5.10 Effect of Temperature on the Equilibrium Constant: The van’t Hoff Equation 147 5.10.1 The Effect of Temperature on the Ionic Product of Water (Kw) 149 References 149 Chapter 5 Problem Set 150 6 Drug Solubility Equilibrium 153 6.1 Introduction 153 6.1.1 Author’s Anecdote – Too Often Solubility Studies Emphasize Analyzing the Concentration of Solute in Solution While Neglecting the Critical Importance of the Identity of the Undissolved Solid Phase Whose Chemical Potential is Responsible for the Observed Solubility 154 6.2 Brief Review of Solubility Concentration Scales Used in this Chapter 155 6.3 Solubility-Related Intermolecular Interactions 156 6.3.1 Solvent and Solute Interactions 156 6.3.2 Types of Intermolecular Attraction Forces 156 6.3.3 Solvent Classification 158 6.4 Nonelectrolyte Enthalpy of Solution and Hess’s Law 159 6.5 Nonelectrolyte Entropy of Solution 161 6.5.1 Hydrophobic Effect: The Self-Association of Water in the Presence of Nonpolar Solutes and the Subsequent Self-Association of the Nonpolar Solutes 161 6.6 Nonelectrolyte Gibbs Free Energy of Solution 162 6.6.1 Application: Using Gibbs Free Energy of Solution to Calculate the Solubility Advantage of Amorphous Compounds 162 6.7 Nonelectrolyte: Determining the Solution Enthalpy and Entropy Using the van’t Hoff Equation 166 6.8 Nonelectrolyte Ideal Solution – Effect of Enthalpy of Fusion and Melting Temperature 166 6.9 Nonelectrolyte Ideal Solution Properties and Henry’s Law 168 6.9.1 Ideal Solution Formation of a Nonelectrolyte: Entropy Driven Process 168 6.9.2 Determining the Enthalpy of Fusion and Melting Point of a Crystalline Solute Using the van’t Hoff Equation 169 6.9.3 Henry’s Law 169 6.10 Nonideal Nonelectrolyte Solutions 170 6.10.1 Reference and Standard States 172 6.10.2 Activity Conventions (Adapted from Connors and Mecozzi) 173 6.10.3 Activity Coefficients 173 6.11 Electrolyte Solutions 177 6.12 Solubility of Slightly Soluble Strong Electrolyte Salts 178 6.13 Author’s Anecdote – Common Ion Effect and the Ability of Thermodynamically Unstable Systems to be Manufactured and Used for Years Without Failure, Until One Day, Catastrophe 181 6.14 Solubility of Weak Acids and Bases 183 6.14.1 Solubility of a Weak Base as Function of the Drug’s pKa, the System pH, and Solubility of the Unionized Base 183 6.14.2 When pH Equals pKa: The Total Weak Base Solubility is 2 Sunionized 185 6.14.3 A Weak Base is Nearly 100% Ionized at 2 pH Units Below its pKa 185 6.14.4 A Weak Base is Nearly 100% Unionized at 2 pH Units Above its pKa 186 6.14.5 Similar Equations can be Developed for Weak Acids 186 6.14.6 Solubility-pH Profile for a Weak Acid and a Weak Base Drug 186 6.14.7 Slope of the Ionized Portion of the Solubility Profile is Negative-One for a Weak Base and Positive-One for a Weak Acid 187 6.14.8 The Solubility Product of a Weak Acid or Weak Base Counterion Limits the Total pH-Solubility of These Ionized Weak Electrolytes 188 6.14.9 Weak Acid or Base Salt Total Solubility as a Function of pH, pKa, and Ksp 188 6.14.10 Consolidated Solubility Profile for a Weak Base 189 6.15 Weak Acid and Base Salt Disproportionation 191 6.16 Application: Effect of Colloidal Sized Particles on Solubility 191 References 192 Chapter 6 Problem Set 192 7 Surface Thermodynamics 197 7.1 Introduction to Interfacial Surface Region, Interfacial Tension, and Interfacial Energy 197 7.1.1 What Is a Surface or Interfacial Surface? 197 7.1.2 Interfacial Tension 198 7.1.3 Interfacial Energy 199 7.2 Surface Tension of Liquid-Vapor Interfaces 200 7.2.1 Capillary Method 200 7.2.2 Du Noüy Ring Method 203 7.2.3 Wilhelmy Plate Method 203 7.2.4 Maximum Bubble Pressure Method 204 7.2.5 Surface Tension of Selected Pharmaceutical Liquids 206 7.3 Effect of Temperature on Surface Tension 206 7.4 Surface Tension as Force Per Unit Length and as Energy per Unit Area 207 7.5 Surface Tension as a Measure of Intermolecular Cohesive Force and Work of Cohesion 209 7.6 Interfacial Tension and Work of Adhesion 211 7.6.1 Interfacial Tension of Pharmaceutical Liquids 211 7.7 Liquid-Liquid Cohesive and Adhesive Forces: Spreading of Two Immiscible Liquids 211 7.8 Wetting and Spreading of Liquids on Solids, Work of Solid-Liquid Adhesion, and Solid-Liquid Interfacial Tension 213 7.8.1 Spreading of a Liquid on a Solid 215 7.8.2 Work of Solid-Liquid Adhesion 217 7.9 Solid Surface Tension/Energy 218 7.9.1 Underlying Solid Surface Tension/Energy Theory: Combining Berthelot’s Rule, Dupré Work of Cohesion, and Anges Pockles’ Unifying Young-Dupré Equations 218 7.9.2 Underlying Solid Surface Tension/Energy Theory: Fowkes Concept 220 7.9.3 Owen, Wendt, Rabel, and Kaelble (OWRK) Surface Tension/Energy Equation 221 7.9.4 Wu Surface Tension/Energy Equation Using Harmonic Mean for Adhesive Work 222 7.9.5 Van Oss-Chaudhury-Good (vOCG) Surface Tension/Energy Equation Using Acid-Base Adhesion Theory 223 7.9.6 Solid Surface Tension/Energy Analytical Methods 223 7.9.7 Pharmaceutical Application: Prediction of Dry Powder Inhaler Formulation Performance Using Surface Tension/Energy Measurements 223 7.10 Relationship Between Surface Curvature, Surface Tension/Energy, and the Pressure Difference Across a Bubble’s Curved Surface 224 7.11 The Effect of Curvature and Surface Tension on Vapor Pressure 225 7.12 The Effect of Curvature on Solubility 228 7.12.1 Pharmaceutical Application of the Kelvin-Ostwald-Freundlich Equation: Increasing Drug Solubility and Dissolution Rate by Decreasing Drug Particle Size to One Micron and Smaller 228 7.12.2 Author’s Anecdote – Albuterol Sulfate Pressurized Suspension Inhaler: Catastrophic Crystallization in Chlorofluorocarbon Propellant 229 References 229 Chapter 7 Problem Set 231 8 Adsorption Phenomena 235 8.1 Introduction 235 8.1.1 Physisorption and Chemisorption 236 8.1.2 Adsorption and Absorption 236 8.1.3 Adsorption and Desorption Profiles 237 8.1.4 Porous Absorbents 237 8.2 Adsorption Data 237 8.2.1 Data Reporting 237 8.2.2 Reversible Type I Isotherms 237 8.2.3 Reversible Type II Isotherm 238 8.2.4 Reversible Type III Isotherm 238 8.2.5 Reversible Type IV Isotherm 239 8.2.6 Reversible Type V Isotherm 239 8.2.7 Reversible Type VI Isotherm 239 8.2.8 General Observations 239 8.3 Freundlich, Langmuir, and Brunauer, Emmett, and Teller Adsorption Isotherm Equations 240 8.3.1 Freundlich Adsorption Isotherm Equation 241 8.3.2 Langmuir Adsorption Isotherm Equation 242 8.3.2.1 Langmuir Equation for Vapor/Gas Adsorption onto an Adsorbent 242 8.3.2.2 Langmuir-Like Equation for Adsorptive-Solute Solution Adsorption onto an Adsorbent 245 8.3.3 Brunauer, Emmett, and Teller (BET) Adsorption Isotherm Equation 247 8.4 Other Adsorption Isotherm Equations 250 8.5 Factors Influencing Adsorption 255 8.5.1 Adsorbent Nature 255 8.5.2 Adsorptive Nature 255 8.5.3 Adsorptive Pressure, Concentration, and Solubility 256 8.5.4 Temperature 256 8.5.5 pH 256 8.6 Adsorption–Desorption Hysteresis 256 8.7 Solid-Water Vapor Adsorption Systems 262 8.7.1 Author’s Anecdote- Failure of a Commercial Controlled-Release Product Caused by Sodium Chloride Deliquescence 265 8.8 Solid-Solution Adsorption Systems 266 References 267 Chapter 8 Problem Set 268 Solved Problem Set 273 Index 309