Thermodynamics for Drug Product Design

William Craig Stagner, RPh, MS, PhD is Professor Emeritus and formerly Director of the Campbell University Pharmaceutical Sciences Institute and Director of the Center for Analysis of Pharmaceutical Biomaterials at Campbell University College of Pharmacy & Health Sciences at Buies Creek, NC. Prior to joining the faculty, he founded the Pharmaceutics Department at the Glaxo Research Institute in Research Triangle Park, NC. Among his publications are co-authoring both editions of the Wiley title Integrated Pharmaceutics: Applied Preformulation, Product Design, and Regulatory Science.

• SI Defining Constants xvForeword xixPreface xxiAcknowledgments xxv1 Data Reporting and Analysis 11.1 Introduction 11.1.1 Fundamental/Base and Derived Physical Quantities 11.1.2 Basic Mathematics and Statistics 21.2 Physical Quantity Dimensions and Dimensional Analysis 31.2.1 Application of Dimensional Analysis 31.2.2 Limitations of Dimensional Analysis 41.2.3 Dimensionless Numbers 41.2.4 Dimensional Nomenclature 41.2.5 Dimensional Quantity Algebra or Dimensional Algebra 41.3 Data Reporting Using Decimal, Scientific, Normalized Scientific, and E Scientific Numerical Notation 71.4 Accuracy 81.5 Precision as Measurement Variability and Relative Uncertainty 81.6 Valid Measurements 81.7 Uncertainty 91.7.1 Uncertainty Associated with a Single Measurement 91.7.2 Uncertainty Associated with Replicate Measurements 101.7.3 Propagation of Error of Combined Measurements and Calculations 101.7.4 Author’s Anecdote: A Priori Estimation of Dose Variation Using Propagation of Error Equations 101.8 Significant Figures (Digits) for Measured Values and Calculations 121.8.1 Rules for Determining Significant Figures for a Measured or Reported Value 121.8.2 Determination of Significant Figures for Addition and Subtraction Calculations 131.8.3 Determination of Significant Figures for Multiplication and Division Calculations 131.8.4 Logarithm Significant Figures 141.8.5 Antilogarithm Significant Figures 141.9 Rounding Numbers 151.10 How Many Digits and Decimals to Use in Research Reports, Tables, and Presentations 151.11 Exponents 161.11.1 Exponent Properties and Operations 161.11.2 Examples of Scientific Exponent Phenomena 171.12 Logarithms 181.12.1 Logarithmic Properties and Operations 181.12.2 Scientific Examples of Base-10 Logarithms (Log10 or Log) 191.12.3 Scientific Examples of Base-e Logarithms (Loge or Ln) 201.13 Differential and Partial Differential Equations 221.13.1 Differential Order of Ordinary Differential Equations 221.13.2 First-Order Ordinary Differential Equations 231.13.3 First-Order Partial Derivatives 241.14 Integral Equations 251.15 Basic Descriptive Statistics 261.15.1 Central Tendency of Single Values of Sample Data 271.15.2 Dispersion of Single Values of Sample Data 281.16 Application: Linear Least Squares and Coefficient of Determination 29References 31Chapter 1 Problem Set 322 Underlying Thermodynamic Physical Property Relationships and Definitions 352.1 Introduction 352.2 Temperature 352.3 Energy, Heat, and Work 362.4 System, Surroundings, Boundary, and Universe 362.4.1 Isolated System 372.4.2 Closed System 372.4.3 Open System 372.5 Macroscopic Properties and Intensive–Extensive Variables 382.6 Thermodynamic State Variables 382.6.1 Ideal Gas Law 382.6.2 Dalton’s Law 392.6.3 Pharmaceutical Application of the Ideal Gas Law and Dalton’s Law: Sterile Vial and Prefilled Syringe Manufacturing 392.6.4 Pharmaceutical Application of Dalton’s Law: Determination of Total Gas Pressure of Pressurized Metered Dose Inhalers Formulated with a Mixture of Propellant Gases 412.6.5 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 422.7 Isothermal, Adiabatic, Isovolumetric, and Isobaric Thermodynamic State Processes 422.7.1 Pharmaceutical Application: Using an Adiabatic Calorimeter to Determine the Enthalpy of Solution of a Drug or Material of Interest 432.7.2 Pharmaceutical Application: Freeze-Drying as an Isochoric or Isovolumetric Process 452.8 Thermodynamic State Functions 462.9 Spontaneous Processes and Nonspontaneous Processes 472.9.1 Author’s Anecdote - Unexpected Polymorphic Conversion in Preparation of Toxicology Supplies 492.9.2 Nonspontaneous Processes 492.10 Reversible Processes 492.10.1 Reversible Isothermal Expansion 502.11 Irreversible Processes 51References 51Chapter 2 Problem Set 523 The Laws of Thermodynamics 533.1 Introduction 533.2 The Zeroth Law of Thermodynamics: Thermal Equilibrium and Temperature 543.3 The First Law of Thermodynamics: Conservation of Energy and Internal Energy as a State Function 553.3.1 Example of Change in Internal Energy for a Liquid Undergoing Vaporization 573.3.2 Enthalpy: A Thermodynamic State Function 593.3.3 Enthalpy and Hess’s Law 613.3.4 Pharmaceutical Application: Thermochemistry Using Differential Scanning Calorimetry 623.3.5 Heat Capacity: A Thermodynamic State Function 683.3.6 Heat Capacity at Constant Volume 683.3.7 Heat Capacity at Constant Pressure 693.3.8 Pharmaceutical Application: Thermochemistry Using Heat Capacity and Modulated Temperature Differential Scanning Calorimetry 703.3.9 Other Pharmaceutical Applications Using Heat Capacity 733.4 Second Law of Thermodynamics: Entropy Is Unavailable to Do Useful Work and Does Not Decrease – Entropy Is a State Function 743.4.1 What is Entropy? 743.4.2 Entropy: A Statistical and Molecular Interpretation 743.4.3 Entropy: State Function That Explains Spontaneous Reactions 763.4.4 Entropy: The Effect of Volume and Temperature Changes 773.4.5 Entropy of Mixing Ideal Gases 783.4.6 Determination of Entropy Changes for Phase Transitions 803.4.7 Other Reversible Processes 813.4.8 Factors Affecting Entropy 813.4.9 Determination and Prediction of ΔS Sign and Relative Value 823.4.10 Entropy’s Shortcomings for Prediction of Spontaneous Processes 823.5 Third Law of Thermodynamics: Entropy of a System Approaches a Constant Value as ItsTemperature Approaches Absolute Zero 83References 83Chapter 3 Problem Set 854 Gibbs Free Energy 874.1 Introduction 874.2 Relationship Between Gibbs Free Energy and Internal Energy 884.3 Gibbs Free Energy to Predict Spontaneous Processes 894.4 Temperature Influence on Gibbs Free Energy 934.5 Pressure Influence on Gibbs Free Energy 944.6 Standard Gibbs Free Energy of Reaction 954.7 Gibbs Free Energy, Chemical Equilibrium, and Extent of Reaction 974.8 Gibbs Free Energy and Chemical Potential: Introduction to the Fundamental Equations for Open Systems 994.8.1 Chemical Potential Dependence on Temperature and Pressure 1014.8.2 Gibbs-Duhem Equation for a One-Phase Two- Component System 1014.9 Molar Gibbs Free Energy and Chemical Potential for an Ideal Gas 1044.9.1 Chemical Potential for a Nonideal System 1054.10 Gibbs Free Energy and Non-Pressure-Volume Work 1054.11 Gibbs Free Energy and Phase Changes 1064.11.1 Determination of the Transition Entropy 1074.12 Application of Gibbs Free Energy and Tertiary Protein Structure and Stability 1074.12.1 Enthalpic Energy Favoring the Native Folded State 1094.12.2 Entropic “Hydrophobic Effect” Favoring the Native Folded State 1104.12.3 Entropic Conformational Freedom Favoring Denaturation and Unfolding 1104.12.4 Application: Overall Thermodynamic Effects on an Insulin-like Proteins Folded Conformation 110References 111Chapter 4 Problem Set 1125 Equilibria 1155.1 Introduction 1155.1.1 Brief Review of Chemical Equilibrium 1155.1.1.1 Law of Mass Balance 1155.1.1.2 Law of Mass Action 1165.1.1.3 Equilibrium State 1165.1.1.4 Equilibrium State and Gibbs Free Energy 1165.2 Le Châtelier’s Principle 1185.2.1 Effect of Concentration Change Stress on Equilibrium 1185.2.2 Effect of Pressure Change Stress on Equilibrium 1215.2.3 Effect of Temperature Change Stress on Equilibrium 1225.2.4 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 1235.3 Phase Equilibrium of a Pure Substance 1255.4 Gibbs Phase Rule 1265.5 Temperature-Composition Phase Equilibrium Diagrams of a Two-Component System 1275.5.1 Classic Two-Component Eutectic Phase Diagram 1285.5.2 Application: Commercialized Eutectic Product and Its Eutectic Phase Diagram 1295.5.3 Application: Complex Two-Component Equilibrium Phase Diagram – NaCl and Water 1305.6 Phase Equilibrium of a Three-Component System 1335.7 Phase Transitions of Pure Substances 1345.7.1 The Clapeyron Equation: Vapor Equilibrium Boundary 1355.7.2 Derivation of the Clausius-Clapeyron Equation: Vapor Equilibrium Boundary 1365.7.3 Application of the Clausius-Clapeyron Equation 1375.7.4 Ehrenfest Classification of Phase Transition Order 1385.8 Pharmaceutical Application of Chemical Equilibrium 1395.9 Ionic Equilibrium 1425.9.1 Ionization Equilibrium of Weak Acids and Bases 1425.9.2 pH and pKa 1445.9.3 Henderson-Hasselbalch Equation 1445.9.4 Le Châtelier’s Principle and Common Ion Effect on pH 1455.10 Effect of Temperature on the Equilibrium Constant: The van’t Hoff Equation 1475.10.1 The Effect of Temperature on the Ionic Product of Water (Kw) 149References 149Chapter 5 Problem Set 1506 Drug Solubility Equilibrium 1536.1 Introduction 1536.1.1 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 1546.2 Brief Review of Solubility Concentration Scales Used in this Chapter 1556.3 Solubility-Related Intermolecular Interactions 1566.3.1 Solvent and Solute Interactions 1566.3.2 Types of Intermolecular Attraction Forces 1566.3.3 Solvent Classification 1586.4 Nonelectrolyte Enthalpy of Solution and Hess’s Law 1596.5 Nonelectrolyte Entropy of Solution 1616.5.1 Hydrophobic Effect: The Self-Association of Water in the Presence of Nonpolar Solutes and the Subsequent Self-Association of the Nonpolar Solutes 1616.6 Nonelectrolyte Gibbs Free Energy of Solution 1626.6.1 Application: Using Gibbs Free Energy of Solution to Calculate the Solubility Advantage of Amorphous Compounds 1626.7 Nonelectrolyte: Determining the Solution Enthalpy and Entropy Using the van’t Hoff Equation 1666.8 Nonelectrolyte Ideal Solution – Effect of Enthalpy of Fusion and Melting Temperature 1666.9 Nonelectrolyte Ideal Solution Properties and Henry’s Law 1686.9.1 Ideal Solution Formation of a Nonelectrolyte: Entropy Driven Process 1686.9.2 Determining the Enthalpy of Fusion and Melting Point of a Crystalline Solute Using the van’t Hoff Equation 1696.9.3 Henry’s Law 1696.10 Nonideal Nonelectrolyte Solutions 1706.10.1 Reference and Standard States 1726.10.2 Activity Conventions (Adapted from Connors and Mecozzi) 1736.10.3 Activity Coefficients 1736.11 Electrolyte Solutions 1776.12 Solubility of Slightly Soluble Strong Electrolyte Salts 1786.13 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 1816.14 Solubility of Weak Acids and Bases 1836.14.1 Solubility of a Weak Base as Function of the Drug’s pKa, the System pH, and Solubility of the Unionized Base 1836.14.2 When pH Equals pKa: The Total Weak Base Solubility is 2 Sunionized 1856.14.3 A Weak Base is Nearly 100% Ionized at 2 pH Units Below its pKa 1856.14.4 A Weak Base is Nearly 100% Unionized at 2 pH Units Above its pKa 1866.14.5 Similar Equations can be Developed for Weak Acids 1866.14.6 Solubility-pH Profile for a Weak Acid and a Weak Base Drug 1866.14.7 Slope of the Ionized Portion of the Solubility Profile is Negative-One for a Weak Base and Positive-One for a Weak Acid 1876.14.8 The Solubility Product of a Weak Acid or Weak Base Counterion Limits the Total pH-Solubility of These Ionized Weak Electrolytes 1886.14.9 Weak Acid or Base Salt Total Solubility as a Function of pH, pKa, and Ksp 1886.14.10 Consolidated Solubility Profile for a Weak Base 1896.15 Weak Acid and Base Salt Disproportionation 1916.16 Application: Effect of Colloidal Sized Particles on Solubility 191References 192Chapter 6 Problem Set 1927 Surface Thermodynamics 1977.1 Introduction to Interfacial Surface Region, Interfacial Tension, and Interfacial Energy 1977.1.1 What Is a Surface or Interfacial Surface? 1977.1.2 Interfacial Tension 1987.1.3 Interfacial Energy 1997.2 Surface Tension of Liquid-Vapor Interfaces 2007.2.1 Capillary Method 2007.2.2 Du Noüy Ring Method 2037.2.3 Wilhelmy Plate Method 2037.2.4 Maximum Bubble Pressure Method 2047.2.5 Surface Tension of Selected Pharmaceutical Liquids 2067.3 Effect of Temperature on Surface Tension 2067.4 Surface Tension as Force Per Unit Length and as Energy per Unit Area 2077.5 Surface Tension as a Measure of Intermolecular Cohesive Force and Work of Cohesion 2097.6 Interfacial Tension and Work of Adhesion 2117.6.1 Interfacial Tension of Pharmaceutical Liquids 2117.7 Liquid-Liquid Cohesive and Adhesive Forces: Spreading of Two Immiscible Liquids 2117.8 Wetting and Spreading of Liquids on Solids, Work of Solid-Liquid Adhesion, and Solid-Liquid Interfacial Tension 2137.8.1 Spreading of a Liquid on a Solid 2157.8.2 Work of Solid-Liquid Adhesion 2177.9 Solid Surface Tension/Energy 2187.9.1 Underlying Solid Surface Tension/Energy Theory: Combining Berthelot’s Rule, Dupré Work of Cohesion, and Anges Pockles’ Unifying Young-Dupré Equations 2187.9.2 Underlying Solid Surface Tension/Energy Theory: Fowkes Concept 2207.9.3 Owen, Wendt, Rabel, and Kaelble (OWRK) Surface Tension/Energy Equation 2217.9.4 Wu Surface Tension/Energy Equation Using Harmonic Mean for Adhesive Work 2227.9.5 Van Oss-Chaudhury-Good (vOCG) Surface Tension/Energy Equation Using Acid-Base Adhesion Theory 2237.9.6 Solid Surface Tension/Energy Analytical Methods 2237.9.7 Pharmaceutical Application: Prediction of Dry Powder Inhaler Formulation Performance Using Surface Tension/Energy Measurements 2237.10 Relationship Between Surface Curvature, Surface Tension/Energy, and the Pressure Difference Across a Bubble’s Curved Surface 2247.11 The Effect of Curvature and Surface Tension on Vapor Pressure 2257.12 The Effect of Curvature on Solubility 2287.12.1 Pharmaceutical Application of the Kelvin-Ostwald-Freundlich Equation: Increasing Drug Solubility and Dissolution Rate by Decreasing Drug Particle Size to One Micron and Smaller 2287.12.2 Author’s Anecdote – Albuterol Sulfate Pressurized Suspension Inhaler: Catastrophic Crystallization in Chlorofluorocarbon Propellant 229References 229Chapter 7 Problem Set 2318 Adsorption Phenomena 2358.1 Introduction 2358.1.1 Physisorption and Chemisorption 2368.1.2 Adsorption and Absorption 2368.1.3 Adsorption and Desorption Profiles 2378.1.4 Porous Absorbents 2378.2 Adsorption Data 2378.2.1 Data Reporting 2378.2.2 Reversible Type I Isotherms 2378.2.3 Reversible Type II Isotherm 2388.2.4 Reversible Type III Isotherm 2388.2.5 Reversible Type IV Isotherm 2398.2.6 Reversible Type V Isotherm 2398.2.7 Reversible Type VI Isotherm 2398.2.8 General Observations 2398.3 Freundlich, Langmuir, and Brunauer, Emmett, and Teller Adsorption Isotherm Equations 2408.3.1 Freundlich Adsorption Isotherm Equation 2418.3.2 Langmuir Adsorption Isotherm Equation 2428.3.2.1 Langmuir Equation for Vapor/Gas Adsorption onto an Adsorbent 2428.3.2.2 Langmuir-Like Equation for Adsorptive-Solute Solution Adsorption onto an Adsorbent 2458.3.3 Brunauer, Emmett, and Teller (BET) Adsorption Isotherm Equation 2478.4 Other Adsorption Isotherm Equations 2508.5 Factors Influencing Adsorption 2558.5.1 Adsorbent Nature 2558.5.2 Adsorptive Nature 2558.5.3 Adsorptive Pressure, Concentration, and Solubility 2568.5.4 Temperature 2568.5.5 pH 2568.6 Adsorption–Desorption Hysteresis 2568.7 Solid-Water Vapor Adsorption Systems 2628.7.1 Author’s Anecdote- Failure of a Commercial Controlled-Release Product Caused by Sodium Chloride Deliquescence 2658.8 Solid-Solution Adsorption Systems 266References 267Chapter 8 Problem Set 268Solved Problem Set 273Index 309