ChemicalKinetics andReactionDynamicsSantosh K. UpadhyaySpringer
PrefaceReactiondynamics is thepart of chemical kineticswhich is concerned withthemicroscopic-moleculardynamicbehaviorof reactingsystems.Molecularreaction dynamics is coming of age and much more refined state-to-stateinformation is becoming available on the fundamental reactions.Thecontributionofmolecularbeamexperimentsandlasertechniquestochemicaldynamics has become very useful in the study of isolated molecules andtheir mutual interactions not only in gas surface systems, but also in solute-solution systems.This book presents the important facts and theories relatingto the rateswith which chemical reactions occur and covers main points in a manner sothat the reader achieves a sound understanding of the principles of chemicalkinetics.A detailed stereochemical discussion of the reaction steps in eachmechanismand theirrelationshipwithkineticobservations hasbeenconsidered.I wouldliketotaketheopportunitytothankProfessorR.C.Srivastavaand Professor N.Sathyamurthywith whom Ihad the privilege of workingand who inspired my interest in the subject and contributed in one way oranotherto help completethis book.I express myheavy debt of gratitudetowards ProfessorM.C.Agrawal who was gracious enoughfor sparing timeoutof hisbusyscheduletogothrough themanuscript.Hisvaluablecommentsand suggestions, of course,enhanced the value and importanceof thisbook.I also express my gratitude to my colleagues, friends and research students,especially Dr. Neelu Kambo who took all the pains in helping me in preparing,typingand checkingthemanuscript.Finally, I thank my wife Mrs. Manju Upadhyay, daughter Neha and sonAnkur for their continuous inspiration during the preparation of thetext.SANTOSHK.UPADHYAY
Preface Reaction dynamics is the part of chemical kinetics which is concerned with the microscopic-molecular dynamic behavior of reacting systems. Molecular reaction dynamics is coming of age and much more refined state-to-state information is becoming available on the fundamental reactions. The contribution of molecular beam experiments and laser techniques to chemical dynamics has become very useful in the study of isolated molecules and their mutual interactions not only in gas surface systems, but also in solutesolution systems. This book presents the important facts and theories relating to the rates with which chemical reactions occur and covers main points in a manner so that the reader achieves a sound understanding of the principles of chemical kinetics. A detailed stereochemical discussion of the reaction steps in each mechanism and their relationship with kinetic observations has been considered. I would like to take the opportunity to thank Professor R.C. Srivastava and Professor N. Sathyamurthy with whom I had the privilege of working and who inspired my interest in the subject and contributed in one way or another to help complete this book. I express my heavy debt of gratitude towards Professor M.C. Agrawal who was gracious enough for sparing time out of his busy schedule to go through the manuscript. His valuable comments and suggestions, of course, enhanced the value and importance of this book. I also express my gratitude to my colleagues, friends and research students, especially Dr. Neelu Kambo who took all the pains in helping me in preparing, typing and checking the manuscript. Finally, I thank my wife Mrs. Manju Upadhyay, daughter Neha and son Ankur for their continuous inspiration during the preparation of the text. SANTOSH K. UPADHYAY
ContentsviiPreface11.Elementary11.1Rate of Reaction21.1.1 ExperimentalDeterminationofRate31.2RateConstant41.3Order and Molecularity61.4RateEquations1.4.1IntegralEquationsfornth OrderReactionof a Single6Reactant1.4.2Integral Equations for Reactions Involving More than7OneReactants81.5Half-life of a Reaction101.6Zero OrderReactions121.7FirstOrderReactions171.8RadioactiveDecayas a FirstOrderPhenomenon201.9Second Order Reactions281.10ThirdOrderReactions301.11DeterminationofOrderofReaction301.11.1Integration Method341.11.2Half-lifePeriodMethod341.11.3 Graphical Method351.11.4DifferentialMethod351.11.5OstwaldIsolationMethod391.12Experimental MethodsofChemicalKinetics391.12.1ConductometricMethod401.12.2PolarographicTechnique411.12.3PotentiometricMethod421.12.4Optical Methods421.12.5Refractometry431.12.6Spectrophotometry44Exercises462.TemperatureEffectonReactionRate462.1Derivationof ArrheniusEquation
Contents Preface vii 1. Elementary 1 1.1 Rate of Reaction 1 1.1.1 Experimental Determination of Rate 2 1.2 Rate Constant 3 1.3 Order and Molecularity 4 1.4 Rate Equations 6 1.4.1 Integral Equations for nth Order Reaction of a Single Reactant 6 1.4.2 Integral Equations for Reactions Involving More than One Reactants 7 1.5 Half-life of a Reaction 8 1.6 Zero Order Reactions 10 1.7 First Order Reactions 12 1.8 Radioactive Decay as a First Order Phenomenon 17 1.9 Second Order Reactions 20 1.10 Third Order Reactions 28 1.11 Determination of Order of Reaction 30 1.11.1 Integration Method 30 1.11.2 Half-life Period Method 34 1.11.3 Graphical Method 34 1.11.4 Differential Method 35 1.11.5 Ostwald Isolation Method 35 1.12 Experimental Methods of Chemical Kinetics 39 1.12.1 Conductometric Method 39 1.12.2 Polarographic Technique 40 1.12.3 Potentiometric Method 41 1.12.4 Optical Methods 42 1.12.5 Refractometry 42 1.12.6 Spectrophotometry 43 Exercises 44 2. Temperature Effect on Reaction Rate 46 2.1 Derivation of Arrhenius Equation 46
Contentsx2.2Experimental Determination of Energy of Activation and48Arrhenius Factor502.3Potential Energy Surface512.4Significance of Energy of Activation53Exercises553.ComplexReactions553.1Reversible Reactions3.1.1 ReversibleReactionWhen BoththeOpposing57Processes areSecondOrder593.2ParallelReactions593.2.1DeterminationofRateConstants633.3Consecutive Reactions643.3.1Concentration-TimeRelation663.4Steady-State Treatment673.5Chain Reactions683.5.1Rate Determination693.5.2ReactionbetweenHandBr2703.5.3Chain Length703.5.4Chain Transfer Reactions703.5.5BranchingChain Explosions713.5.6Kinetics of Branching ChainExplosion723.5.7FreeRadical Chains3.5.8ChainLength andActivationEnergyinChain75Reactions76Exercises794.Theories of Reaction Rate794.1Equilibrium andRateof Reaction4.2Partition Functions and Statistical Mechanics of80Chemical Equilibrium824.3PartitionFunctions andActivatedComplex834.4Collision Theory844.4.1CollisionFrequency864.4.2EnergyFactor874.4.3OrientationFactor874.4.4Rateof Reaction884.4.5Weakness of theCollisionTheory894.5Transition StateTheory914.5.1ThermodynamicApproach934.5.2PartitionFunctionApproach4.5.3Comparison withArrhenius Equation andCollision93Theory4.5.4ExplanationforStericFactorinTermsofPartition94Function
2.2 Experimental Determination of Energy of Activation and Arrhenius Factor 48 2.3 Potential Energy Surface 50 2.4 Significance of Energy of Activation 51 Exercises 53 3. Complex Reactions 55 3.1 Reversible Reactions 55 3.1.1 Reversible Reaction When Both the Opposing Processes are Second Order 57 3.2 Parallel Reactions 59 3.2.1 Determination of Rate Constants 59 3.3 Consecutive Reactions 63 3.3.1 Concentration-Time Relation 64 3.4 Steady-State Treatment 66 3.5 Chain Reactions 67 3.5.1 Rate Determination 68 3.5.2 Reaction between H2 and Br2 69 3.5.3 Chain Length 70 3.5.4 Chain Transfer Reactions 70 3.5.5 Branching Chain Explosions 70 3.5.6 Kinetics of Branching Chain Explosion 71 3.5.7 Free Radical Chains 72 3.5.8 Chain Length and Activation Energy in Chain Reactions 75 Exercises 76 4. Theories of Reaction Rate 79 4.1 Equilibrium and Rate of Reaction 79 4.2 Partition Functions and Statistical Mechanics of Chemical Equilibrium 80 4.3 Partition Functions and Activated Complex 82 4.4 Collision Theory 83 4.4.1 Collision Frequency 84 4.4.2 Energy Factor 86 4.4.3 Orientation Factor 87 4.4.4 Rate of Reaction 87 4.4.5 Weakness of the Collision Theory 88 4.5 Transition State Theory 89 4.5.1 Thermodynamic Approach 91 4.5.2 Partition Function Approach 93 4.5.3 Comparison with Arrhenius Equation and Collision Theory 93 4.5.4 Explanation for Steric Factor in Terms of Partition Function 94 x Contents
Contentsxi954.5.5ReactionbetweenPolyatomicMolecules1004.6UnimolecularReactionsandtheCollisionTheory1004.6.1Lindemann'sMechanism1034.6.2HinshelwoodTreatment1054.6.3Rice andRamsperger, andKassel (RRK)Treatment1064.6.4MarcusTreatment1074.6.5RRKMTheory4.7109Kinetic and Thermodynamic Control1104.8Hammond'sPostulate4.9111Probing of the Transition State113Exercises1155.Kineticsof Some Special Reactions1155.1Kinetics of Photochemical Reactions1155.1.1 Grotthuss-DraperLaw1155.1.2EinsteinLawofPhotochemicalEquivalence1165.1.3PrimaryProcessinPhotochemicalReactions1185.1.4 H2-Br2Reaction1195.1.5H2andCl2Reaction1205.2OscillatoryReactions1225.2.1Belousov-ZhabotinskiiReaction5.3124Kinetics of Polymerization1255.3.1StepGrowthPolymerization5.3.2PolycondensationReactions(inAbsenceof the125Catalyst)1265.3.3AcidCatalyzedPolycondensationReaction1275.3.4ChainGrowthPolymerization1275.3.5KineticsofFreeRadicalPolymerization1305.3.6Cationic Polymerization1315.3.7AnionicPolymerization1325.3.8Co-polymerization5.4135Kinetics of Solid StateReactions5.5139Electron TransferReactions1395.5.1OuterSphereMechanism1405.5.2InnerSphereMechanism141Exercises1426.Kinetics of Catalyzed Reactions1426.1Catalysis1426.1.1 Positive Catalysis1436.1.2Negative Catalysis1436.1.3Auto Catalysis1446.1.4InducedCatalysis1446.1.5Promoters
4.5.5 Reaction between Polyatomic Molecules 95 4.6 Unimolecular Reactions and the Collision Theory 100 4.6.1 Lindemann’s Mechanism 100 4.6.2 Hinshelwood Treatment 103 4.6.3 Rice and Ramsperger, and Kassel (RRK) Treatment 105 4.6.4 Marcus Treatment 106 4.6.5 RRKM Theory 107 4.7 Kinetic and Thermodynamic Control 109 4.8 Hammond’s Postulate 110 4.9 Probing of the Transition State 111 Exercises 113 5. Kinetics of Some Special Reactions 115 5.1 Kinetics of Photochemical Reactions 115 5.1.1 Grotthuss-Draper Law 115 5.1.2 Einstein Law of Photochemical Equivalence 115 5.1.3 Primary Process in Photochemical Reactions 116 5.1.4 H2-Br2 Reaction 118 5.1.5 H2 and Cl2 Reaction 119 5.2 Oscillatory Reactions 120 5.2.1 Belousov-Zhabotinskii Reaction 122 5.3 Kinetics of Polymerization 124 5.3.1 Step Growth Polymerization 125 5.3.2 Polycondensation Reactions (in Absence of the Catalyst) 125 5.3.3 Acid Catalyzed Polycondensation Reaction 126 5.3.4 Chain Growth Polymerization 127 5.3.5 Kinetics of Free Radical Polymerization 127 5.3.6 Cationic Polymerization 130 5.3.7 Anionic Polymerization 131 5.3.8 Co-polymerization 132 5.4 Kinetics of Solid State Reactions 135 5.5 Electron Transfer Reactions 139 5.5.1 Outer Sphere Mechanism 139 5.5.2 Inner Sphere Mechanism 140 Exercises 141 6. Kinetics of Catalyzed Reactions 142 6.1 Catalysis 142 6.1.1 Positive Catalysis 142 6.1.2 Negative Catalysis 143 6.1.3 Auto Catalysis 143 6.1.4 Induced Catalysis 144 6.1.5 Promoters 144 Contents xi
xiiContents1446.1.6 Poisons6.2145Theories of Catalysis1456.2.1IntermediateCompoundFormationTheory1456.2.2Adsorption Theory6.3146Characteristics of Catalytic Reactions1476.4Mechanism of Catalysis6.5149ActivationEnergiesofCatalyzedReactions1506.6Acid Base Catalysis1526.7Enzyme Catalysis1546.7.1Influence of pH6.8156Heterogeneous Catalysis1596.9Micellar Catalysis1616.9.1 Models for Micellar Catalysis1656.10Phase Transfer Catalysis1666.10.1GeneralMechanism6.10.2Differencebetween Micellar andPhaseTransfer-167CatalyzedReactions1686.11Kinetics of Inhibition1686.11.1 Chain Reactions1696.11.2EnzymeCatalyzedReactions1726.11.3Inhibition in Surface Reactions173Exercises1757.FastReactions1757.1Introduction7.2176FlowTechniques1777.2.1ContinuousFlowMethod1787.2.2AcceleratedFlowMethod1787.2.3StoppedFlowMethod1797.3Relaxation Method7.4181Shock Tubes7.5182Flash Photolysis1837.6ESRSpectroscopicTechnique7.7183NMRSpectroscopicTechniques184Exercises8.185Reactions in Solutions1858.1Introduction8.2185TheoryofAbsoluteReactionRate8.3187Influenceof InternalPressure8.4187Influence of Solvation8.5187Reactions between ons1898.6EntropyChange1908.7InfluenceoflonicStrength(SaltEffect)
6.1.6 Poisons 144 6.2 Theories of Catalysis 145 6.2.1 Intermediate Compound Formation Theory 145 6.2.2 Adsorption Theory 145 6.3 Characteristics of Catalytic Reactions 146 6.4 Mechanism of Catalysis 147 6.5 Activation Energies of Catalyzed Reactions 149 6.6 Acid Base Catalysis 150 6.7 Enzyme Catalysis 152 6.7.1 Influence of pH 154 6.8 Heterogeneous Catalysis 156 6.9 Micellar Catalysis 159 6.9.1 Models for Micellar Catalysis 161 6.10 Phase Transfer Catalysis 165 6.10.1 General Mechanism 166 6.10.2 Difference between Micellar and Phase TransferCatalyzed Reactions 167 6.11 Kinetics of Inhibition 168 6.11.1 Chain Reactions 168 6.11.2 Enzyme Catalyzed Reactions 169 6.11.3 Inhibition in Surface Reactions 172 Exercises 173 7. Fast Reactions 175 7.1 Introduction 175 7.2 Flow Techniques 176 7.2.1 Continuous Flow Method 177 7.2.2 Accelerated Flow Method 178 7.2.3 Stopped Flow Method 178 7.3 Relaxation Method 179 7.4 Shock Tubes 181 7.5 Flash Photolysis 182 7.6 ESR Spectroscopic Technique 183 7.7 NMR Spectroscopic Techniques 183 Exercises 184 8. Reactions in Solutions 185 8.1 Introduction 185 8.2 Theory of Absolute Reaction Rate 185 8.3 Influence of Internal Pressure 187 8.4 Influence of Solvation 187 8.5 Reactions between Ions 187 8.6 Entropy Change 189 8.7 Influence of Ionic Strength (Salt Effect) 190 xii Contents
Contentsxiii1928.8Secondary Salt Effect1938.9Reactions between theDipoles1958.10 Kinetic IsotopeEffect1978.11SolventIsotopeEffect1988.12HemmettEquation1998.13LinearFreeEnergyRelationship2008.14TheTaftEquation2018.15CompensationEffect202Exercises9.204Reaction Dynamics2049.1Molecular ReactionDynamics9.2205Microscopic-Macroscopic Relation9.3207ReactionRate and Rate Constant2099.4DistributionofVelocitiesofMolecules9.5Rateof Reaction for Collisions with aDistributionof209RelativeSpeeds9.6210Collision Cross Sections2109.6.1Cross SectionforHard SphereModel2119.6.2CollisionbetweenReactiveHard Spheres2139.7ActivationEnergy2169.8Potential Energy Surface2199.8.1 Features ofPotential EnergySurface2229.8.2 Ab initioCalculationof PotentialEnergySurface2259.8.3Fittingofab initioPotential EnergySurfaces2269.8.4Potential Energy Surfaces forTriatomic Systems2299.9Classical Trajectory Calculations2309.9.1 Initial State Properties2329.9.2Final StateProperties2329.9.3Calculationof ReactionCross Section2349.10Potential Energy Surface andClassicalDynamics2399.11Disposal of Excess Energy2409.12Influence of Rotational Energy2419.13ExperimentalChemicalDynamics2419.13.1Molecular Beam Technique2439.13.2StrippingandReboundMechanisms2449.13.3State-to-StateKinetics247Suggested Readings251Index
8.8 Secondary Salt Effect 192 8.9 Reactions between the Dipoles 193 8.10 Kinetic Isotope Effect 195 8.11 Solvent Isotope Effect 197 8.12 Hemmett Equation 198 8.13 Linear Free Energy Relationship 199 8.14 The Taft Equation 200 8.15 Compensation Effect 201 Exercises 202 9. Reaction Dynamics 204 9.1 Molecular Reaction Dynamics 204 9.2 Microscopic-Macroscopic Relation 205 9.3 Reaction Rate and Rate Constant 207 9.4 Distribution of Velocities of Molecules 209 9.5 Rate of Reaction for Collisions with a Distribution of Relative Speeds 209 9.6 Collision Cross Sections 210 9.6.1 Cross Section for Hard Sphere Model 210 9.6.2 Collision between Reactive Hard Spheres 211 9.7 Activation Energy 213 9.8 Potential Energy Surface 216 9.8.1 Features of Potential Energy Surface 219 9.8.2 Ab initio Calculation of Potential Energy Surface 222 9.8.3 Fitting of ab initio Potential Energy Surfaces 225 9.8.4 Potential Energy Surfaces for Triatomic Systems 226 9.9 Classical Trajectory Calculations 229 9.9.1 Initial State Properties 230 9.9.2 Final State Properties 232 9.9.3 Calculation of Reaction Cross Section 232 9.10 Potential Energy Surface and Classical Dynamics 234 9.11 Disposal of Excess Energy 239 9.12 Influence of Rotational Energy 240 9.13 Experimental Chemical Dynamics 241 9.13.1 Molecular Beam Technique 241 9.13.2 Stripping and Rebound Mechanisms 243 9.13.3 State-to-State Kinetics 244 Suggested Readings 247 Index 251 Contents xiii
1ElementaryChemical kineticsdeals with therates of chemical reactions,factors whichinfluence the rates and the explanation of the rates in terms of the reactionmechanisms of chemical processes.In chemical equilibria,the energyrelationsbetween thereactants and theproducts aregovernedbythermodynamics withoutconcerningthe intermediatestates or time.In chemical kinetics,the time variable is introduced and rateof change of concentrationof reactants or products with respect to time isfollowed. The chemical kinetics is thus, concerned with the quantitativedeterminationofrate of chemical reactions and of thefactors uponwhichtherates depend.With the knowledge of effect of various factors, such asconcentration,pressure,temperature,medium,effectofcatalystetc.,onreactionrate, one can consider an interpretation of the empirical laws in terms ofreaction mechanism. Let us first define the terms such as rate, rate constant,order,molecularity etc.beforegoingintodetail.1.1Rateof ReactionThe rate or velocity of a reaction maybe expressed in terms of any oneof thereactantsoranyoneoftheproductsofthereaction.The rate of reaction is defined as change in number of molecules ofreactantorproductperunittime,i.e.dnR_dnp(1.1)Rateofreaction==dtdtwherednganddn,arethechangesinnumberofmoleculesofreactantandproduct, respectively,for a small time interval dt.The reactant is beingconsumed, i.e.number of molecules of reactant decreaseswith time.Hence,minus sign is attached so thatrate will be positive numerically.For comparingtherates ofvarious reactions, the volume ofreaction systemmust be specifiedand rate of reaction is expressed per unit volume. If V, is the volume ofreactionmixture,then1 dnpIdnR(1.2)Rateofreaction=dVdt
1 Elementary Chemical kinetics deals with the rates of chemical reactions, factors which influence the rates and the explanation of the rates in terms of the reaction mechanisms of chemical processes. In chemical equilibria, the energy relations between the reactants and the products are governed by thermodynamics without concerning the intermediate states or time. In chemical kinetics, the time variable is introduced and rate of change of concentration of reactants or products with respect to time is followed. The chemical kinetics is thus, concerned with the quantitative determination of rate of chemical reactions and of the factors upon which the rates depend. With the knowledge of effect of various factors, such as concentration, pressure, temperature, medium, effect of catalyst etc., on reaction rate, one can consider an interpretation of the empirical laws in terms of reaction mechanism. Let us first define the terms such as rate, rate constant, order, molecularity etc. before going into detail. 1.1 Rate of Reaction The rate or velocity of a reaction may be expressed in terms of any one of the reactants or any one of the products of the reaction. The rate of reaction is defined as change in number of molecules of reactant or product per unit time, i.e. Rate of reaction = – = dnR p dt dn dt (1.1) where dnR and dnp are the changes in number of molecules of reactant and product, respectively, for a small time interval dt. The reactant is being consumed, i.e. number of molecules of reactant decreases with time. Hence, minus sign is attached so that rate will be positive numerically. For comparing the rates of various reactions, the volume of reaction system must be specified and rate of reaction is expressed per unit volume. If Vt is the volume of reaction mixture, then Rate of reaction = – 1 = 1 t R t p V dn dt V dn dt (1.2)
2Chemical Kinetics and Reaction DynamicsAt constant V.d(n/V)d(np/V)(1.3)RateofreactionddtAgainnp/Visthemolarconcentration of reactantand n,/Vthemolarconcentration of product.Therefore,in terms of molar concentrationsd[Reactant]d[Product](1.4)Rateofreaction=dtdtwhere[Reactant]and[Product|arethemolarconcentrations of reactantandproduct,respectively.Thisconventional wayof representing therateof reactionisvalid onlyat constantvolume.However,if thereisachange inthevolumed(nR/V)would yieldofthe reactants.dtdvd(nR/V)1dnR(np)(1.5)dtVdt(V)dtd[Reactant]1 dnRwillnot be equal toandcorrectionsand,therefore,dtV,d,need to be applied.At constant volume,therateof a general reaction,A+BC+D interms of molar concentration of reactant or product maybegiven asd[A] --d[B] - d[C] - d[D](1.6)Rateof reaction=dtdtdtdt[Decrease inmolarIncreaseinmolarRate of reactionconcentration of aconcentrationofareactantperunittimeproductperunittimeHowever, if reaction isnot of a simple stoichiometrybut involvesdifferentnumber of molesofreactants orproducts,therateshouldbedivided bycorresponding stoichiometric coefficient in thebalanced chemical equationfor normalizing it and making it comparable.For example,fora generalreactionaA+bB→cC+dD1d[A]d[B]1d[C]1 d[D]Rate of reaction =(1.7)ddtadtbdtdtc1.1.1Experimental Determination of RateFor thedetermination ofrate of reaction at constant volumethe concentrationof a chosen reactant orproduct is determined at various timeintervals.Thechange in concentration AC,for a given time interval Ar(t2-t)is obtained.Anaveragerateofreactionisthenobtainedbycalculating△C/t.ThesmallerthevalueofAt.thecloserthevalueoftheratewill betothereal rateattime(t)+t2)/2 because
2 Chemical Kinetics and Reaction Dynamics At constant V, Rate of reaction = – ( /) = ( /) dn V R p dt dn V dt (1.3) Again nR/V is the molar concentration of reactant and np /V the molar concentration of product. Therefore, in terms of molar concentrations Rate of reaction = – [Reactant] = d [Product] dt d dt (1.4) where [Reactant] and [Product] are the molar concentrations of reactant and product, respectively. This conventional way of representing the rate of reaction is valid only at constant volume. However, if there is a change in the volume of the reactants, – dn V dt ( /) R t would yield – dn V dt V dn dt n V dV dt ( /) = 1 + ( ) ( ) R t t R R t 2 ⎛ t ⎝ ⎞ ⎠ (1.5) and, therefore, – d[Reactant] dt will not be equal to – 1 t R V t dn d and corrections need to be applied. At constant volume, the rate of a general reaction, A + B → C + D in terms of molar concentration of reactant or product may be given as Rate of reaction = – [A] = – [B] = [C] = d [D] dt d dt d dt d dt (1.6) Rate of reaction = Decrease in molar concentration of a reactant per unit time = Increase in molar concentration of a product per unit time ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ However, if reaction is not of a simple stoichiometry but involves different number of moles of reactants or products, the rate should be divided by corresponding stoichiometric coefficient in the balanced chemical equation for normalizing it and making it comparable. For example, for a general reaction aA + bB → cC + dD Rate of reaction = – 1 a [A] = – 1 b [B] = 1 c [C] = 1 d d [D] dt d dt d dt d dt (1.7) 1.1.1 Experimental Determination of Rate For the determination of rate of reaction at constant volume the concentration of a chosen reactant or product is determined at various time intervals. The change in concentration ∆C, for a given time interval ∆t(t2 – t1) is obtained. An average rate of reaction is then obtained by calculating ∆C/∆t. The smaller the value of ∆t, the closer the value of the rate will be to the real rate at time (t1 + t2)/2 because
3ElementaryACdclim(1.8)-0dtTherate of reaction can also beobtained byplotting concentration ofreactant orproductagainst time and measuring the slope of the curve (dcldt)at the required time.The rate of reaction obtained from such method isknown as instantaneous rate.The concentration of thereactant or productvaries exponentially or linearly with time as shown in Fig.1.1.d[P][[pod]Slope =dtd[R]Slope dtTimeTimeFig. 1.1Concentration variation of thereactant/productwith timeFordeterminationoftheinstantaneous rateatanypointa,theslopeof thecurve is determined.It may also be notedfromFig.1.Ithat if the concentrationvaries linearly withtime,the slope of the curve orrate of the reaction willremain same throughout the course of reaction.However, if concentration ofthereactant orproduct varies exponentiallywith time the slope of the curveortherateof reaction will bedifferent atdifferenttimeintervals.Thus,itisnotnecessarythatrateof reaction mayalways remain samethroughoutthecourse of reaction. The reaction may proceed with a different rate in theinitial stageand mayhave different rate in the middle or near the end of thereaction.Inplaceof concentrationof reactant or product anyphysical property,whichisdirectlyrelatedwithconcentration,suchasviscosity,surfacetension,refractive index, absorbance etc.can be measured for the determination ofthe rate of reaction.1.2RateConstantForageneralreactionaA+bB→cC+dDthe rate is proportional to [Aja × [B]b, i.e.Rate = k [A]" [B]b(1.9)whereproportionality constantk,relating rate with concentration terms,isknown as rateconstantor velocityconstantat agiventemperature.When the reactants are present at their unit concentrations,Rate= k
Elementary 3 lim ∆ 0 ∆ t ∆ C t dC → dt → (1.8) The rate of reaction can also be obtained by plotting concentration of reactant or product against time and measuring the slope of the curve (dc/dt) at the required time. The rate of reaction obtained from such method is known as instantaneous rate. The concentration of the reactant or product varies exponentially or linearly with time as shown in Fig. 1.1. Slope = d R[ ] dt Slope = d P[ ] dt Time [Reactant] [Product] Time a Fig. 1.1 Concentration variation of the reactant/product with time. For determination of the instantaneous rate at any point a, the slope of the curve is determined. It may also be noted from Fig. 1.1 that if the concentration varies linearly with time, the slope of the curve or rate of the reaction will remain same throughout the course of reaction. However, if concentration of the reactant or product varies exponentially with time the slope of the curve or the rate of reaction will be different at different time intervals. Thus, it is not necessary that rate of reaction may always remain same throughout the course of reaction. The reaction may proceed with a different rate in the initial stage and may have different rate in the middle or near the end of the reaction. In place of concentration of reactant or product any physical property, which is directly related with concentration, such as viscosity, surface tension, refractive index, absorbance etc. can be measured for the determination of the rate of reaction. 1.2 Rate Constant For a general reaction aA + bB → cC + dD the rate is proportional to [A]a × [B]b , i.e. Rate = k [A]a [B]b (1.9) where proportionality constant k, relating rate with concentration terms, is known as rate constant or velocity constant at a given temperature. When the reactants are present at their unit concentrations, Rate = k