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Lecture notes on Physical Chemistry

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PHYSICAL CHEMISTRY IN BRIEF Prof. Ing. Anatol Malijevsk´ y, CSc., et al. (September 30, 2005) Institute of Chemical Technology, Prague Faculty of Chemical EngineeringAnnotation The Physical Chemistry In Brief offers a digest of all major formulas, terms and definitions needed for an understanding of the subject. They are illustrated by schematic figures, simple worked-out examples, and a short accompanying text. The concept of the book makes it different from common university or physical chemistry textbooks. In terms of contents, the Physical Chemistry In Brief embraces the fundamental course in physical chemistry as taught at the Institute of Chemical Technology, Prague, i.e. the state behaviour of gases, liquids, solid substances and their mixtures, the fundamentals of chemical thermodynamics, phase equilibrium, chemical equilibrium, the fundamentals of electrochemistry, chemical kinetics and thekineticsoftransportprocesses,colloidchemistry,andpartlyalsothestructureofsubstances andspectra. Thereaderisassumedtohaveareasonableknowledgeofmathematicsatthelevel of secondary school, and of the fundamentals of mathematics as taught at the university level.3 Authors Prof. Ing. Josef P. Nov´ak, CSc. Prof. Ing. Stanislav Lab´ık, CSc. Ing. Ivona Malijevska, ´ CSc.4 Introduction Dear students, Physical Chemistry is generally considered to be a difficult subject. We thought long and hard about ways to make its study easier, and this text is the result of our endeavors. The book provides accurate definitions of terms, definitions of major quantities, and a number of relations including specification of the conditions under which they are valid. It also contains a number of schematic figures and examples that clarify the accompanying text. The reader will not find any derivations in this book, although frequent references are made to the initial formulas from which the respective relations are obtained. Intermsofcontents,wefollowedthesyllabiof“PhysicalChemistryI”and“PhysicalChem- istry II” as taught at the Institute of Chemical Technology (ICT), Prague up to 2005. However the extent of this work is a little broader as our objective was to cover all the major fields of Physical Chemistry. This publication is not intended to substitute for any textbooks or books of examples. Yet we believe that it will prove useful during revision lessons leading up to an exam in Physical Chemistry or prior to the final (state) examination, as well as during postgraduate studies. Even experts in Physical Chemistry and related fields may find this work to be useful as a reference. Physical Chemistry In Brief has two predecessors, “Breviary of Physical Chemistry I” and “Breviary of Physical Chemistry II”. Since the first issue in 1993, the texts have been revised and re-published many times, always selling out. Over the course of time we have thus striven to eliminate both factual and formal errors, as well as to review and rewrite the less accessible passages pointed out to us by both students and colleagues in the Department of Physical Chemistry. Finally,asthenumberofforeignstudentscomingtostudyatourinstitutecontinues togrow, wedecidedtogivethemaproventoolwrittenintheEnglishlanguage. Thistextisthe result of these efforts. A number of changes have been made to the text and the contents have been partially extended. We will be grateful to any reader able to detect and inform us of any errors in our work. Finally, the authors would like to express their thanks to Mrs. Flemrov´a for her substantial investment in translating this text.CONTENTS CONTENTS 5 Contents 1 Basic terms 24 1.1 Thermodynamic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.1.1 Isolated system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.1.2 Closed system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.3 Open system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.4 Phase, homogeneous and heterogeneous systems . . . . . . . . . . . . . . 25 1.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.1 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.1 Intensive and extensive thermodynamic quantities . . . . . . . . . . . . . 28 1.4 The state of a system and its changes . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.1 The state of thermodynamic equilibrium . . . . . . . . . . . . . . . . . . 29 1.4.2 System’s transition to the state of equilibrium . . . . . . . . . . . . . . . 30 1.4.3 Thermodynamic process . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.4 Reversible and irreversible processes. . . . . . . . . . . . . . . . . . . . . 31 1.4.5 Processes at a constant quantity . . . . . . . . . . . . . . . . . . . . . . . 31 1.4.6 Cyclic process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.5 Some basic and derived quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.1 Mass m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.2 Amount of substance n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.3 Molar mass M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34CONTENTS CONTENTS 6 1.5.4 Absolute temperature T . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.5 Pressure p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5.6 Volume V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.6 Pure substance and mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 th 1.6.1 Mole fraction of the i component x . . . . . . . . . . . . . . . . . . . . 36 i 1.6.2 Mass fraction w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 i 1.6.3 Volume fraction φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 i 1.6.4 Amount-of-substance concentration c . . . . . . . . . . . . . . . . . . . . 40 i 1.6.5 Molality m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 i 2 State behaviour 42 2.1 Major terms, quantities and symbols . . . . . . . . . . . . . . . . . . . . . . . . 43 2.1.1 Molar volume V and amount-of-substance (or amount) density c . . . . 43 m 2.1.2 Specific volume v and density ρ . . . . . . . . . . . . . . . . . . . . . . . 43 2.1.3 Compressibility factor z . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1.4 Critical point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1.5 Reduced quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.6 Coefficient of thermal expansion α . . . . . . . . . . . . . . . . . . . . . 45 p 2.1.7 Coefficient of isothermal compressibility β . . . . . . . . . . . . . . . . . 47 T 2.1.8 Partial pressure p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 i 2.2 Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.1 Concept of the equation of state . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.2 Equation of state of an ideal gas . . . . . . . . . . . . . . . . . . . . . . 48 2.2.3 Virial expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.4 Boyle temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.5 Pressure virial expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.6 Van der Waals equation of state . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.7 Redlich-Kwong equation of state . . . . . . . . . . . . . . . . . . . . . . 52 2.2.8 Benedict, Webb and Rubin equation of state . . . . . . . . . . . . . . . . 53 2.2.9 Theorem of corresponding states . . . . . . . . . . . . . . . . . . . . . . 53 2.2.10 Application of equations of state . . . . . . . . . . . . . . . . . . . . . . 54 2.3 State behaviour of liquids and solids . . . . . . . . . . . . . . . . . . . . . . . . 56CONTENTS CONTENTS 7 2.3.1 Descriptionofstatebehaviourusingthecoefficientsofthermalexpansion α and isothermal compressibility β . . . . . . . . . . . . . . . . . . . . . 56 p T 2.3.2 Rackett equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.3 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4 State behaviour of mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.1 Dalton’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.4.2 Amagat’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4.3 Ideal mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.4 Pseudocritical quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4.5 Equations of state for mixtures . . . . . . . . . . . . . . . . . . . . . . . 61 2.4.6 Liquid and solid mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3 Fundamentals of thermodynamics 63 3.1 Basic postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.1 The zeroth law of thermodynamics . . . . . . . . . . . . . . . . . . . . . 63 3.1.2 The first law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . 64 3.1.3 Second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 65 3.1.4 The third law of thermodynamics . . . . . . . . . . . . . . . . . . . . . . 66 3.1.4.1 Impossibility to attain a temperature of 0K . . . . . . . . . . . 67 3.2 Definition of fundamental thermodynamic quantities . . . . . . . . . . . . . . . 68 3.2.1 Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.2 Helmholtz energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2.3 Gibbs energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2.4 Heat capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.5 Molar thermodynamic functions . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.6 Fugacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.7 Fugacity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.8 Absolute and relative thermodynamic quantities . . . . . . . . . . . . . 75 3.3 Some properties of the total differential . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.1 Total differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.2 Total differential and state functions . . . . . . . . . . . . . . . . . . . . 79 3.3.3 Total differential of the product and ratio of two functions . . . . . . . . 81 3.3.4 Integration of the total differential . . . . . . . . . . . . . . . . . . . . . 81CONTENTS CONTENTS 8 3.4 Combined formulations of the first and second laws of thermodynamics . . . . . 83 3.4.1 Gibbs equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.4.2 Derivatives of U, H, F, and G with respect to natural variables . . . . . 83 3.4.3 Maxwell relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.4 Total differential of entropy as a function of T, V and T, p . . . . . . . . 85 3.4.5 Conversion from natural variables to variables T, V or T, p . . . . . . . . 85 3.4.6 Conditions of thermodynamic equilibrium . . . . . . . . . . . . . . . . . 87 3.5 Changes of thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.1 Heat capacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.1.1 Temperature dependence. . . . . . . . . . . . . . . . . . . . . . 90 3.5.1.2 C dependence on pressure . . . . . . . . . . . . . . . . . . . . 91 p 3.5.1.3 C dependence on volume . . . . . . . . . . . . . . . . . . . . . 91 V 3.5.1.4 Relations between heat capacities . . . . . . . . . . . . . . . . . 91 3.5.1.5 Ideal gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.5.2 Internal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5.2.1 Temperature and volume dependence for a homogeneous system 92 3.5.2.2 Ideal gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.5.2.3 Changes at phase transitions . . . . . . . . . . . . . . . . . . . 93 3.5.3 Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.5.3.1 Temperature and pressure dependence for a homogeneous system 94 3.5.3.2 Ideal gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.5.3.3 Changes at phase transitions . . . . . . . . . . . . . . . . . . . 95 3.5.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.5.4.1 Temperature and volume dependence for a homogeneous system 96 3.5.4.2 Temperature and pressure dependence for a homogeneous system 97 3.5.4.3 Ideal gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.5.4.4 Changes at phase transitions . . . . . . . . . . . . . . . . . . . 98 3.5.5 Absolute entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.5.6 Helmholtz energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.5.6.1 Dependence on temperature and volume . . . . . . . . . . . . . 101 3.5.6.2 Changes at phase transitions . . . . . . . . . . . . . . . . . . . 102 3.5.7 Gibbs energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.5.7.1 Temperature and pressure dependence . . . . . . . . . . . . . . 103CONTENTS CONTENTS 9 3.5.7.2 Changes at phase transitions . . . . . . . . . . . . . . . . . . . 103 3.5.8 Fugacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.5.8.1 Ideal gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.5.8.2 Changes at phase transitions . . . . . . . . . . . . . . . . . . . 104 3.5.9 Changes of thermodynamic quantities during irreversible processes . . . . 104 4 Application of thermodynamics 107 4.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1.1 Reversible volume work . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1.2 Irreversible volume work . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.1.3 Other kinds of work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.1.4 Shaft work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2.1 Adiabatic process—Poisson’s equations . . . . . . . . . . . . . . . . . . . 112 4.2.2 Irreversible adiabatic process . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.3 Heat engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3.1 The Carnot heat engine . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3.2 Cooling engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3.3 Heat engine with steady flow of substance . . . . . . . . . . . . . . . . . 120 4.3.4 The Joule-Thomson effect . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.5 The Joule-Thomson coefficient . . . . . . . . . . . . . . . . . . . . . . . . 123 4.3.6 Inversion temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5 Thermochemistry 127 5.1 Heat of reaction and thermodynamic quantities of reaction . . . . . . . . . . . . 128 5.1.1 Linear combination of chemical reactions . . . . . . . . . . . . . . . . . . 129 5.1.2 Hess’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 ◦ 5.2 Standard reaction enthalpy Δ H . . . . . . . . . . . . . . . . . . . . . . . . . . 131 r ◦ 5.2.1 Standard enthalpy of formation Δ H . . . . . . . . . . . . . . . . . . . . 131 f ◦ 5.2.2 Standard enthalpy of combustion Δ H . . . . . . . . . . . . . . . . . . . 132 c 5.3 Kirchhoff’s law—dependence of the reaction enthalpy on temperature . . . . . . 134 5.4 Enthalpy balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.4.1 Adiabatic temperature of reaction . . . . . . . . . . . . . . . . . . . . . . 137CONTENTS CONTENTS 10 6 Thermodynamics of homogeneous mixtures 139 6.1 Ideal mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.1.1 General ideal mixture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.1.2 Ideal mixture of ideal gases . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.2 Integral quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2.1 Mixing quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2.2 Excess quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2.3 Heat of solution (integral) . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2.3.1 Relations between the heat of solution and the enthalpy of mix- ing for a binary mixture . . . . . . . . . . . . . . . . . . . . . . 146 6.3 Differential quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3.1 Partial molar quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.3.2 Properties of partial molar quantities . . . . . . . . . . . . . . . . . . . . 148 6.3.2.1 Relations between system and partial molar quantities . . . . . 148 6.3.2.2 Relations between partial molar quantities . . . . . . . . . . . . 149 6.3.2.3 Partial molar quantities of an ideal mixture . . . . . . . . . . . 149 6.3.3 Determination of partial molar quantities . . . . . . . . . . . . . . . . . . 150 6.3.4 Excess partial molar quantities . . . . . . . . . . . . . . . . . . . . . . . 152 6.3.5 Differential heat of solution and dilution . . . . . . . . . . . . . . . . . . 153 6.4 Thermodynamics of an open system and the chemical potential . . . . . . . . . 155 6.4.1 Thermodynamic quantities in an open system . . . . . . . . . . . . . . . 155 6.4.2 Chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.5 Fugacity and activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.5.1 Fugacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.5.2 Fugacity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.5.3 Standard states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.5.4 Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.5.5 Activity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 x 6.5.5.1 Relation between γ and γ . . . . . . . . . . . . . . . . . . . . 168 i i 6.5.5.2 Relation between the activity coefficient and the osmotic coef- ficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.5.6 Dependence of the excess Gibbs energy and of the activity coefficients on composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169CONTENTS CONTENTS 11 6.5.6.1 Wilson equation . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.5.6.2 Regular solution . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7 Phase equilibria 171 7.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1.1 Phase equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1.2 Coexisting phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7.1.3 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.1.4 Boiling point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.1.5 Normal boiling point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.1.6 Dew point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.1.7 Saturated vapour pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.1.8 Melting point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.1.9 Normal melting point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.1.10 Freezing point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.1.11 Triple point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.2 Thermodynamic conditions of equilibrium in multiphase systems . . . . . . . . . 177 7.2.0.1 Extensive and intensive criteria of phase equilibrium . . . . . . 177 7.2.1 Phase transitions of the first and second order . . . . . . . . . . . . . . . 178 7.3 Gibbs phase rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3.1 Independent and dependent variables . . . . . . . . . . . . . . . . . . . . 179 7.3.2 Intensive independent variables . . . . . . . . . . . . . . . . . . . . . . . 179 7.3.3 Degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3.4 Gibbs phase rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.4 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.4.1 General terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.4.2 Phase diagram of a one-component system . . . . . . . . . . . . . . . . . 182 7.4.3 Phase diagrams of two-component (binary) mixtures . . . . . . . . . . . 184 7.4.4 Phase diagrams of three-component (ternary) mixtures . . . . . . . . . . 186 7.4.5 Material balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.4.5.1 Lever rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.5 Phase equilibria of pure substances . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.5.1 Clapeyron equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190CONTENTS CONTENTS 12 7.5.2 Clausius-Clapeyron equation . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.5.3 Liquid-vapour equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7.5.4 Solid-vapour equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.5.5 Solid-liquid equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.5.6 Solid-solid equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 7.5.7 Equilibrium between three phases . . . . . . . . . . . . . . . . . . . . . . 194 7.6 Liquid-vapour equilibrium in mixtures . . . . . . . . . . . . . . . . . . . . . . . 195 7.6.1 The concept of liquid-vapour equilibrium . . . . . . . . . . . . . . . . . . 195 7.6.2 Raoult’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.6.3 Liquid-vapour equilibrium with an ideal vapour and a real liquid phase . 196 7.6.4 General solution of liquid-vapour equilibrium . . . . . . . . . . . . . . . . 198 7.6.5 Phase diagrams of two-component systems . . . . . . . . . . . . . . . . . 198 7.6.6 Azeotropic point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.6.7 Effect of the non-volatile substance content on the boiling pressure and temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.6.8 High-pressure liquid-vapour equilibrium. . . . . . . . . . . . . . . . . . . 204 7.7 Liquid-gas equilibrium in mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.7.1 Basic concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.7.2 Henry’s law for a binary system . . . . . . . . . . . . . . . . . . . . . . . 205 7.7.3 Estimates of Henry’s constant . . . . . . . . . . . . . . . . . . . . . . . . 207 7.7.4 Effect of temperature and pressure on gas solubility . . . . . . . . . . . . 208 7.7.4.1 Effect of pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.7.5 Other ways to express gas solubility . . . . . . . . . . . . . . . . . . . . . 208 7.7.6 Liquid-gas equilibrium in more complex systems . . . . . . . . . . . . . . 210 7.8 Liquid-liquid equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.8.1 Conditions of equilibrium at constant temperature and pressure . . . . . 212 7.8.2 Two-component system containing two liquid phases . . . . . . . . . . . 212 7.8.3 Two-componentsystemcontainingtwoliquidphasesandonegaseousphase212 7.8.4 Three-component system containing two liquid phases . . . . . . . . . . . 213 7.9 Liquid-solid equilibrium in mixtures . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.9.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.9.2 General condition of equilibrium . . . . . . . . . . . . . . . . . . . . . . . 216CONTENTS CONTENTS 13 7.9.3 Two-component systems with totally immiscible components in the solid phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.9.4 Two-component systems with completely miscible components in both the liquid and solid phases . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.9.5 Two-componentsystemswithpartiallymisciblecomponentsineitherthe liquid or the solid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 7.9.6 Formation of a compound in the solid phase . . . . . . . . . . . . . . . . 221 7.9.7 Three-component systems . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.10 Gas-solid equilibrium in mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.10.1 General condition of equilibrium . . . . . . . . . . . . . . . . . . . . . . 223 7.10.2 Isobaric equilibrium in a two-component system . . . . . . . . . . . . . . 223 7.10.3 Isothermal equilibrium in a two-component system . . . . . . . . . . . . 223 7.11 Osmotic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 8 Chemical equilibrium 226 8.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 8.2 Systems with one chemical reaction . . . . . . . . . . . . . . . . . . . . . . . . . 228 8.2.1 General record of a chemical reaction . . . . . . . . . . . . . . . . . . . . 228 8.2.2 Material balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 8.2.3 Gibbs energy of a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.2.4 Condition of chemical equilibrium . . . . . . . . . . . . . . . . . . . . . . 234 8.2.5 Overview of standard states . . . . . . . . . . . . . . . . . . . . . . . . . 236 8.2.6 Equilibrium constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 8.2.7 Reactions in the gaseous and liquid phases . . . . . . . . . . . . . . . . . 238 8.2.8 Reactions in the solid phase . . . . . . . . . . . . . . . . . . . . . . . . . 243 8.2.9 Heterogeneous reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 8.3 Dependence of the equilibrium constant on state variables . . . . . . . . . . . . 246 8.3.1 Dependence on temperature . . . . . . . . . . . . . . . . . . . . . . . . . 246 8.3.1.1 Integrated form . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 8.3.2 Dependence on pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.3.2.1 Integrated form . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 8.4 Calculation of the equilibrium constant . . . . . . . . . . . . . . . . . . . . . . . 249 8.4.1 Calculation from the equilibrium composition . . . . . . . . . . . . . . . 249CONTENTS CONTENTS 14 8.4.2 Calculation from tabulated data . . . . . . . . . . . . . . . . . . . . . . . 249 8.4.3 Calculation from the equilibrium constants of other reactions . . . . . . . 252 8.4.4 Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 8.5 Le Chatelier’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.5.1 Effect of initial composition on the equilibrium extent of reaction . . . . 255 8.5.2 Effect of pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.5.2.1 Reactions in condensed systems . . . . . . . . . . . . . . . . . . 256 8.5.3 Effect of temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 8.5.4 Effect of inert component . . . . . . . . . . . . . . . . . . . . . . . . . . 257 8.6 Simultaneous reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.6.1 Material balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.6.2 Chemical equilibrium of a complex system . . . . . . . . . . . . . . . . . 260 9 Chemical kinetics 262 9.1 Basic terms and relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 9.1.1 Rate of chemical reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 262 9.1.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.1.3 Simple reactions, order of reaction, rate constant . . . . . . . . . . . . . 265 9.1.4 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.1.5 Material balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.1.6 Methods of solving kinetic equations . . . . . . . . . . . . . . . . . . . . 269 9.2 Simple reactions systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.1 Zero-order reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.1.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.1.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.1.3 Integrated form of the kinetic equation . . . . . . . . . . . . . . 271 9.2.1.4 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 271 9.2.2 First-order reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.2.2.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.2.2.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.2.2.3 Integrated form of the kinetic equation . . . . . . . . . . . . . . 273 9.2.2.4 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 273 9.2.3 Second-order reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274CONTENTS CONTENTS 15 9.2.3.1 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.2.3.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.2.3.3 Integrated forms of the kinetic equation . . . . . . . . . . . . . 275 9.2.3.4 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.2.3.5 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 9.2.3.6 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.2.3.7 Integrated forms of the kinetic equation . . . . . . . . . . . . . 277 9.2.3.8 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.2.3.9 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 9.2.3.10 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 278 9.2.3.11 Pseudofirst-order reactions . . . . . . . . . . . . . . . . . . . . . 278 9.2.4 Third-order reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 9.2.4.1 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.2.4.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.2.4.3 Integrated forms of the kinetic equation . . . . . . . . . . . . . 280 9.2.4.4 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.2.4.5 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.2.4.6 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.2.4.7 Integrated forms of the kinetic equation . . . . . . . . . . . . . 281 9.2.4.8 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.2.4.9 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.2.4.10 Integrated forms of the kinetic equation . . . . . . . . . . . . . 281 9.2.4.11 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.2.4.12 Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.2.4.13 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.2.4.14 Integrated forms of the kinetic equation . . . . . . . . . . . . . 283 th 9.2.5 n -order reactions with one reactant . . . . . . . . . . . . . . . . . . . . 283 9.2.5.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 283 9.2.5.2 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 283 9.2.5.3 Integrated forms of the kinetic equation . . . . . . . . . . . . . 283 9.2.5.4 Reaction half-life . . . . . . . . . . . . . . . . . . . . . . . . . . 284 th 9.2.6 n -order reactions with two and more reactants . . . . . . . . . . . . . . 284 9.2.6.1 Kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 284CONTENTS CONTENTS 16 9.2.7 Summary of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 9.3 Methods to determine reaction orders and rate constants . . . . . . . . . . . . . 287 9.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 9.3.2 Integral method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 9.3.3 Differential method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 9.3.4 Method of half-lives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 9.3.5 Generalized integral method . . . . . . . . . . . . . . . . . . . . . . . . . 291 9.3.6 Ostwald’s isolation method . . . . . . . . . . . . . . . . . . . . . . . . . . 292 9.4 Simultaneous chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 9.4.1 Types of simultaneous reactions . . . . . . . . . . . . . . . . . . . . . . . 293 9.4.2 Rate of formation of a substance in simultaneous reactions . . . . . . . . 294 9.4.3 Material balance in simultaneous reactions . . . . . . . . . . . . . . . . . 295 9.4.4 First-order parallel reactions . . . . . . . . . . . . . . . . . . . . . . . . . 296 9.4.4.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 296 9.4.4.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 296 9.4.4.3 Integrated forms of the kinetic equations . . . . . . . . . . . . . 297 9.4.4.4 Wegscheider’s principle . . . . . . . . . . . . . . . . . . . . . . . 297 9.4.5 Second-order parallel reactions . . . . . . . . . . . . . . . . . . . . . . . . 297 9.4.5.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 297 9.4.5.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.4.5.3 Integrated forms of the kinetic equations . . . . . . . . . . . . . 298 9.4.6 First- and second-order parallel reactions . . . . . . . . . . . . . . . . . . 298 9.4.6.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.4.6.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 299 9.4.6.3 Integrated forms of the kinetic equations . . . . . . . . . . . . . 299 9.4.7 First-order reversible reactions . . . . . . . . . . . . . . . . . . . . . . . . 300 9.4.7.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 300 9.4.7.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 300 9.4.7.3 Integrated forms of the kinetic equations . . . . . . . . . . . . . 300 9.4.8 Reversible reactions and chemical equilibrium . . . . . . . . . . . . . . . 300 9.4.9 First-order consecutive reactions . . . . . . . . . . . . . . . . . . . . . . . 301 9.4.9.1 Type of reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 301 9.4.9.2 Kinetic equations . . . . . . . . . . . . . . . . . . . . . . . . . . 301CONTENTS CONTENTS 17 9.4.9.3 Integrated forms of the kinetic equations . . . . . . . . . . . . . 302 9.4.9.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 9.5 Mechanisms of chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.5.1 Elementary reactions, molecularity, reaction mechanism . . . . . . . . . 304 9.5.2 Kinetic equations for elementary reactions . . . . . . . . . . . . . . . . . 305 9.5.3 Solution of reaction mechanisms . . . . . . . . . . . . . . . . . . . . . . 305 9.5.4 Rate-determining process . . . . . . . . . . . . . . . . . . . . . . . . . . 307 9.5.5 Bodenstein’s steady-state principle . . . . . . . . . . . . . . . . . . . . . 307 9.5.6 Lindemann mechanism of first-order reactions . . . . . . . . . . . . . . . 308 9.5.7 Pre-equilibrium principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 9.5.8 Mechanism of some third-order reactions . . . . . . . . . . . . . . . . . . 310 9.5.9 Chain reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.5.10 Radical polymerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.5.11 Photochemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.5.11.1 Energy of a photon  . . . . . . . . . . . . . . . . . . . . . . . . 313 9.5.11.2 Quantum yield of reaction . . . . . . . . . . . . . . . . . . . . . 314 9.5.11.3 Rate of a photochemical reaction . . . . . . . . . . . . . . . . . 314 9.6 Temperature dependence of the rate of a chemical reaction . . . . . . . . . . . . 315 9.6.1 Van’t Hoff rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 9.6.2 Arrhenius equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 9.6.3 Collision theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9.6.4 Theory of absolute reaction rates . . . . . . . . . . . . . . . . . . . . . . 317 9.6.5 General relation for temperature dependence of the rate constant . . . . 319 9.7 Chemical reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.7.1 Types of reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.7.2 Batch reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.7.3 Flow reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 9.8 Catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 9.8.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 9.8.2 Homogeneous catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 9.8.3 Heterogeneous catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 9.8.3.1 Transport of reactants . . . . . . . . . . . . . . . . . . . . . . . 327 9.8.3.2 Adsorption and desorption . . . . . . . . . . . . . . . . . . . . . 328CONTENTS CONTENTS 18 9.8.3.3 Chemical reaction . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.8.4 Enzyme catalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 10 Transport processes 330 10.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 10.1.1 Transport process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 10.1.2 Flux and driving force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 10.1.3 Basic equations of transport processes . . . . . . . . . . . . . . . . . . . 332 10.2 Heat flow—thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.2.1 Ways of heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.2.2 Fourier’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.2.3 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 10.2.3.1 Dependence on state variables . . . . . . . . . . . . . . . . . . . 334 10.2.4 Fourier-Kirchhoff law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 10.3 Flow of momentum—viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 10.3.1 Newton’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 10.3.2 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 10.3.2.1 Dependence on state variables . . . . . . . . . . . . . . . . . . . 337 10.3.3 Poiseuille’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 10.4 Flow of matter—diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 10.4.1 Fick’s first law of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 340 10.4.2 Diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 10.4.2.1 Dependence on state variables . . . . . . . . . . . . . . . . . . . 340 10.4.3 Fick’s second law of diffusion . . . . . . . . . . . . . . . . . . . . . . . . 341 10.4.4 Self-diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 10.4.5 Thermal diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 10.5 Kinetic theory of transport processes in dilute gases . . . . . . . . . . . . . . . . 343 10.5.1 Molecular interpretation of transport processes . . . . . . . . . . . . . . . 343 10.5.2 Molecular models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 10.5.3 Basic terms of kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . 344 10.5.4 Transport quantities for the hard spheres model . . . . . . . . . . . . . . 345 10.5.5 Knudsen region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346CONTENTS CONTENTS 19 11 Electrochemistry 347 11.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 11.1.1 Electric current conductors . . . . . . . . . . . . . . . . . . . . . . . . . . 347 11.1.2 Electrolytes and ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 11.1.3 Ion charge number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 11.1.4 Condition of electroneutrality . . . . . . . . . . . . . . . . . . . . . . . . 349 11.1.5 Degree of dissociation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 11.1.6 Infinitely diluted electrolyte solution . . . . . . . . . . . . . . . . . . . . 351 11.1.7 Electrochemical system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 11.2 Electrolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 11.2.1 Reactions occurring during electrolysis . . . . . . . . . . . . . . . . . . . 353 11.2.2 Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 11.2.3 Coulometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 11.2.4 Transport numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 11.2.5 Concentration changes during electrolysis . . . . . . . . . . . . . . . . . . 358 11.2.6 Hittorf method of determining transport numbers . . . . . . . . . . . . . 359 11.3 Electric conductivity of electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . 361 11.3.1 Resistivity and conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 361 11.3.2 Conductivity cell constant . . . . . . . . . . . . . . . . . . . . . . . . . . 362 11.3.3 Molar electric conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 362 11.3.4 Kohlrausch’s law of independent migration of ions . . . . . . . . . . . . . 363 11.3.5 Molar conductivity and the degree of dissociation . . . . . . . . . . . . . 364 11.3.6 Molar conductivity and transport numbers . . . . . . . . . . . . . . . . . 364 11.3.7 Concentration dependence of molar conductivity . . . . . . . . . . . . . . 365 11.4 Chemical potential, activity and activity coefficient in electrolyte solutions . . . 366 11.4.1 Standard states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 11.4.1.1 Solvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 11.4.1.2 Undissociated electrolyte . . . . . . . . . . . . . . . . . . . . . . 367 11.4.1.3 Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 11.4.2 Mean molality, concentration, activity and activity coefficient . . . . . . 368 11.4.3 Ionic strength of a solution . . . . . . . . . . . . . . . . . . . . . . . . . . 369 11.4.4 Debye-Huc ¨ kel limiting law . . . . . . . . . . . . . . . . . . . . . . . . . . 370 11.4.5 Activity coefficients at higher concentrations . . . . . . . . . . . . . . . . 372CONTENTS CONTENTS 20 11.5 Dissociation in solutions of weak electrolytes . . . . . . . . . . . . . . . . . . . . 373 11.5.1 Some general notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 11.5.2 Ionic product of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 11.5.3 Dissociation of a week monobasic acid . . . . . . . . . . . . . . . . . . . 375 11.5.4 Dissociation of a weak monoacidic base . . . . . . . . . . . . . . . . . . . 377 11.5.5 Dissociation of weak polybasic acids and polyacidic bases . . . . . . . . . 377 11.5.6 Dissociation of strong polybasic acids and polyacidic bases . . . . . . . . 378 11.5.7 Hydrolysis of salts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 11.5.8 Hydrolysis of the salt of a weak acid and a strong base . . . . . . . . . . 379 11.5.9 Hydrolysis of the salt of a weak base and a strong acid . . . . . . . . . . 380 11.5.10Hydrolysis of the salt of a weak acid and a weak base . . . . . . . . . . . 381 11.6 Calculation of pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 11.6.1 Definition of pH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 11.6.2 pH of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 11.6.3 pH of a neutral solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 11.6.4 pH of a strong monobasic acid . . . . . . . . . . . . . . . . . . . . . . . . 384 11.6.5 pH of a strong monoacidic base . . . . . . . . . . . . . . . . . . . . . . . 385 11.6.6 pH of a strong dibasic acid and a strong diacidic base . . . . . . . . . . . 385 11.6.7 pH of a weak monobasic acid . . . . . . . . . . . . . . . . . . . . . . . . 386 11.6.8 pH of a weak monoacidic base . . . . . . . . . . . . . . . . . . . . . . . . 388 11.6.9 pH of weak polybasic acids and polyacidic bases . . . . . . . . . . . . . . 388 11.6.10pH of the salt of a weak acid and a strong base . . . . . . . . . . . . . . 389 11.6.11pH of the salt of a strong acid and a weak base . . . . . . . . . . . . . . 390 11.6.12pH of the salt of a weak acid and a weak base . . . . . . . . . . . . . . . 390 11.6.13Buffer solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 11.7 Solubility of sparingly soluble salts . . . . . . . . . . . . . . . . . . . . . . . . . 393 11.8 Thermodynamics of galvanic cells . . . . . . . . . . . . . . . . . . . . . . . . . . 396 11.8.1 Basic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 11.8.2 Symbols used for recording galvanic cells . . . . . . . . . . . . . . . . . . 397 11.8.3 Electrical work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.8.4 Nernst equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 11.8.5 Electromotive force and thermodynamic quantities . . . . . . . . . . . . 400 11.8.6 Standard hydrogen electrode . . . . . . . . . . . . . . . . . . . . . . . . . 401