Chemical Reaction Kinetics

Chemical Reaction Kinetics

Concepts, Methods and Case Studies

Jorge AncheytaInstituto Mexicano del PetróleoMexico City, Mexico

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Names: Ancheyta, Jorge.Title: Chemical reaction kinetics : concepts, methods and case studies /Prof. Jorge Ancheyta.

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Contents

About the Author xiPreface xiii

1 Fundamentals of Chemical Reaction Kinetics 11.1 Concepts of Stoichiometry 11.1.1 Stoichiometric Number and Coefficient 11.1.2 Molecularity 21.1.3 Reaction Extent 31.1.4 Molar Conversion 41.1.5 Types of Feed Composition in a Chemical Reaction 51.1.6 Limiting Reactant 61.1.7 Molar Balance in a Chemical Reaction 71.1.8 Relationship between Conversion and Physical Properties of the

Reacting System 81.2 Reacting Systems 111.2.1 Mole Fraction, Weight Fraction and Molar Concentration 111.2.2 Partial Pressure 131.2.3 Isothermal Systems at Constant Density 131.2.3.1 Relationship between Partial Pressure (pA) and

Conversion (xA) 161.2.3.2 Relationship between Partial Pressure (pA) and Total

Pressure (P) 161.2.3.3 Relationship between Molar Concentration (CA) and Total

Pressure (P) 161.2.4 Isothermal Systems at Variable Density 181.2.5 General Case of Reacting Systems 221.2.6 Kinetic Point of View of the Chemical Equilibrium 221.3 Concepts of Chemical Kinetics 241.3.1 Rate of Homogeneous Reactions 241.3.2 Power Law 261.3.2.1 Relationship between kp and kc 27

v

1.3.2.2 Units of kc and kp 271.3.3 Elemental and Non-elemental Reactions 291.3.4 Comments on the Concepts of Molecularity and Reaction

Order 301.3.5 Dependency of k with Temperature 301.3.5.1 Arrhenius Equation 301.3.5.2 Frequency Factor and Activation Energy 321.3.5.3 Evaluation of the Parameters of the Arrhenius Equation 321.3.5.4 Modified Arrhenius Equation 421.4 Description of Ideal Reactors 431.4.1 Batch Reactors 431.4.1.1 Modes of Operation 441.4.1.2 Data Collection 461.4.1.3 Mass Balance 481.4.2 Continuous Reactors 491.4.2.1 Space–Time and Space–Velocity 501.4.2.2 Plug Flow Reactor 501.4.2.3 Continuous Stirred Tank Reactor 52

2 Irreversible Reactions of One Component 552.1 Integral Method 562.1.1 Reactions of Zero Order 582.1.2 Reactions of the First Order 592.1.3 Reaction of the Second Order 612.1.4 Reactions of the nth Order 642.2 Differential Method 692.2.1 Numerical Differentiation 712.2.1.1 Method of Approaching the Derivatives (−dCA/dt) to (ΔCA/Δt)

or (dxA/dt) to (ΔxA/Δt) 712.2.1.2 Method of Finite Differences 722.2.1.3 Method of a Polynomial of the nth Order 742.2.2 Graphical Differentiation 742.2.2.1 Method of Area Compensation 742.2.2.2 Method of Approaching the Derivative (−dCA/dt) to

(ΔCA/Δt) 762.2.2.3 Method of Finite Differences 772.2.2.4 Method of a Polynomial of the nth Order 782.2.2.5 Method of Area Compensation 802.2.2.6 Summary of Results 822.3 Method of Total Pressure 832.3.1 Reactions of Zero Order 842.3.2 Reactions of the First Order 852.3.3 Reactions of the Second Order 852.3.4 Reactions of the nth Order 86

Contentsvi

2.3.5 Differential Method with Data of Total Pressure 882.4 Method of the Half-Life Time 912.4.1 Reactions of Zero Order 922.4.2 Reactions of the First Order 922.4.3 Reaction of the Second Order 932.4.4 Reaction of the nth Order 932.4.5 Direct Method to Calculate k and n with Data of t1/2 952.4.6 Extension of the Method of Half-Life Time (t1/2) to Any

Fractional Life Time (t1/m) 972.4.7 Calculation of Activation Energy with Data of Half-Life

Time 972.4.8 Some Observations of the Method of Half-Life Time 992.4.8.1 Calculation of n with Two Data of t1/2 Measured with

Different CAo 992.4.8.2 Generalization of the Method of Half-Life Time for Any

Reaction Order 101

3 Irreversible Reactions with Two or Three Components 1033.1 Irreversible Reactions with Two Components 1033.1.1 Integral Method 1033.1.1.1 Method of Stoichiometric Feed Composition 1043.1.1.2 Method of Non-stoichiometric Feed Composition 1093.1.1.3 Method of a Reactant in Excess 1173.1.2 Differential Method 1203.1.2.1 Stoichiometric Feed Composition 1203.1.2.2 Feed Composition with a Reactant in Excess 1203.1.2.3 Non-stoichiometric Feed Compositions 1213.1.3 Method of Initial Reaction Rates 1233.2 Irreversible Reactions between Three Components 1273.2.1 Case 1: Stoichiometric Feed Composition 1273.2.2 Case 2: Non-stoichiometric Feed Composition 1293.2.3 Case 3: Feed Composition with One Reactant in Excess 1303.2.4 Case 4: Feed Composition with Two Reactants in Excess 131

4 Reversible Reactions 1354.1 Reversible Reactions of First Order 1354.2 Reversible Reactions of Second Order 1394.3 Reversible Reactions with Combined Orders 146

5 Complex Reactions 1535.1 Yield and Selectivity 1535.2 Simultaneous or Parallel Irreversible Reactions 1555.2.1 Simultaneous Reactions with the Same Order 155

Contents vii

5.2.1.1 Case 1: Reactions with Only One Reactant 1555.2.1.2 Case 2: Reactions with Two Reactants 1615.2.2 Simultaneous Reactions with Combined Orders 1635.2.2.1 Integral Method 1655.2.2.2 Differential Method 1665.3 Consecutive or In-Series Irreversible Reactions 1675.3.1 Consecutive Reactions with the Same Order 1675.3.1.1 Calculation of CR

max and t∗ 1715.3.1.2 Calculation of CR

max and t∗ for k1= k2 1725.3.2 Consecutive Reactions with Combined Orders 174

6 Special Topics in Kinetic Modelling 1796.1 Data Reconciliation 1806.1.1 Data Reconciliation Method 1816.1.2 Results and Discussion 1826.1.2.1 Source of Data 1826.1.2.2 Global Mass Balances 1856.1.2.3 Outlier Determination 1876.1.2.4 Data Reconciliation 1876.1.2.5 Analysis of Results 1896.1.3 Conclusions 1956.2 Methodology for Sensitivity Analysis of Parameters 1966.2.1 Description of the Method 1986.2.1.1 Initialization of Parameters 1996.2.1.2 Non-linear Parameter Estimation 2016.2.1.3 Sensitivity Analysis 2016.2.1.4 Residual Analysis 2026.2.2 Results and Discussion 2026.2.2.1 Experimental Data and the Reaction Rate Model from the

Literature 2026.2.2.2 Initialization of Parameters 2046.2.2.3 Results of Non-linear Estimation 2066.2.2.4 Sensitivity Analysis 2076.2.2.5 Analysis of Residuals 2106.2.3 Conclusions 2106.3 Methods for Determining Rate Coefficients in Enzymatic

Catalysed Reactions 2116.3.1 The Michaelis–Menten Model 2136.3.1.1 Origin 2136.3.1.2 Development of the Model 2136.3.1.3 Importance of Vmax and Km 2146.3.2 Methods to Determine the Rate Coefficients of the

Michaelis–Menten Equation 214

Contentsviii

6.3.2.1 Linear Regression 2146.3.2.2 Graphic Method 2156.3.2.3 Integral Method 2156.3.2.4 Non-linear Regression 2166.3.3 Application of the Methods 2176.3.3.1 Experimental Data 2176.3.3.2 Calculation of Kinetic Parameters 2206.3.4 Discussion of Results 2226.3.5 Conclusions 2256.4 A Simple Method for Estimating Gasoline, Gas and Coke Yields

in FCC Processes 2266.4.1 Introduction 2266.4.2 Methodology 2276.4.2.1 Choosing the Kinetic Models 2276.4.2.2 Reaction Kinetics 2286.4.2.3 Estimation of Kinetic Parameters 2296.4.2.4 Evaluation of Products Yields 2306.4.2.5 Advantages and Limitations of the Methodology 2306.4.3 Results and Discussion 2316.4.4 Conclusions 2346.5 Estimation of Activation Energies during Hydrodesulphurization

of Middle Distillates 2346.5.1 Introduction 2346.5.2 Experiments 2356.5.3 Results and Discussion 2366.5.3.1 Experimental Results 2366.5.3.2 Estimation of Kinetic Parameters 2376.5.3.3 Effect of Feed Properties on Kinetic Parameters 2406.5.4 Conclusions 241

Problems 243Nomenclature 273References 277Index 283

Contents ix

About the Author

Jorge Ancheyta, PhD, graduated with a bachelor’s degree in PetrochemicalEngineering (1989), master’s degree in Chemical Engineering (1993) andmaster’s degree in Administration, Planning and Economics of Hydrocar-bons (1997) from the National Polytechnic Institute (IPN) of Mexico. Hesplits his PhD between the Metropolitan Autonomous University (UAM)of Mexico and the Imperial College London, UK (1998), and was awardeda postdoctoral fellowship in the Laboratory of Catalytic Process Engineer-ing of the CPE-CNRS in Lyon, France (1999). He has also been visitingprofessor at the Laboratoire de Catalyse et Spectrochimie (LCS), Univer-sité de Caen, France (2008, 2009 and 2010), Imperial College London, UK(2009), and Mining University at Saint Petersburg, Russia (2016).Prof. Ancheyta has worked for the Mexican Institute of Petroleum

(IMP) since 1989, and his present position is Manager of Products forthe Transformation of Crude Oil. He has also worked as professor atthe undergraduate and postgraduate levels for the School of ChemicalEngineering and Extractive Industries at the National Polytechnic Insti-tute of Mexico (ESIQIE-IPN) since 1992 and for the IMP postgrade since2003. He has been supervisor of more than 100 BSc, MSc and PhD theses.Prof. Ancheyta has also been supervisor of a number of postdoctoral andsabbatical year professors.Prof. Ancheyta has been working in the development and application of

petroleum refining catalysts, kinetic and reactor models, and processtechnologies, mainly in catalytic cracking, catalytic reforming, middle dis-tillate hydrotreating and heavy oils upgrading. He is author and co-authorof a number of patents and books and about 200 scientific papers; he hasbeen awarded the highest distinction (Level III) as National Researcher bythe Mexican government and is a member of the Mexican Academy ofScience. He has also been guest editor of various international journals,for example Catalysis Today, Petroleum Science and Technology, Indus-trial Engineering Chemistry Research, Chemical Engineering Communica-tions and Fuel. Prof. Ancheyta has also chaired numerous internationalconferences.

xi

Preface

Reaction kinetics is mainly focused on studying the rate at which chemicalreactions proceed. It is also used to analyse the factors that affect the reac-tion rates and the mechanisms by means of which they take place.The study of the chemical kinetics of a reaction is a fundamental tool to

perform in the design of chemical reactors, to predict the reactor’s perfor-mance and to develop new processes. In fact, the first step for designing achemical reactor is always the generation of experimental data wherebythe reaction rate expressions are determined.Chemical Reaction Kinetics: Concepts, Methods and Case Studies is

devoted to describing the fundamentals of reaction kinetics, with partic-ular emphasis on the mathematical treatment of the experimental data.The book is organized in six chapters, each one having detailed deduc-tions of the kinetic models with examples.Chapter 1 deals with the definitions of the main concepts of stoichiom-

etry, reacting systems, chemical kinetics and ideal reactors.Chapter 2 gives details about the mathematical methods to determine

the reaction order and the reaction rate coefficient for irreversible reac-tions with one component. The methods described here include the inte-gral method, differential method, total pressure method and half-life timemethod.Chapter 3 reports the mathematical methods for evaluating the kinetics

of irreversible reactions with two or three components by employing theintegral method, differential method and initial reaction rate method. Allof the mathematical treatments are performed according to the type offeed composition: stoichiometric, non-stoichiometric and with a reactantin excess.Chapter 4 describes the reversible reactions of first order, second order

and combined orders.Chapter 5 presents themathematical treatment of complex reactions, that

is, simultaneous or parallel irreversible reactions and consecutive or in-series irreversible reactions, with the same order or with combined orders.

xiii

Chapter 6 is devoted to special topics in kinetic modelling, whichinclude reconciliation of data generated during experiments to minimizethe inconsistencies of mass balances due to experimental errors, a methodfor sensitivity analysis to assure that kinetic parameters are properly esti-mated and the convergence of the objective function to the global mini-mum is achieved, estimation of kinetic parameters of enzymatic reactionsby means of different approaches, estimation of kinetic parameters ofcatalytic cracking reaction using a lumping approach and estimation ofkinetic parameters of hydrodesulphurization of petroleum distillates.Each chapter illustrates the application of the different methods with

detailed examples by using experimental information reported in the lit-erature. Step-by-step solutions are provided so that the methods can beeasily followed and applied for other situations. Some exercises are pro-vided at the end to allow the reader to apply all of the methods developedin the previous chapters.Chemical Reaction Kinetics: Concepts, Methods and Case Studies is

oriented to cover the contents of undergraduate and postgraduate courseson reaction kinetics of chemical engineering and similar careers. It isanticipated that Chemical Reaction Kinetics: Concepts, Methods and CaseStudies will become an outstanding and distinctive textbook because itemphasizes detailed description of fundamentals, mathematical treat-ments and examples of chemical reaction kinetics, which are notdescribed with such details in previous textbooks related to the topic.The particular manner in which the kinetic models are developed will helpthe readers adapt to their own reaction studied and experimental data.I would like to acknowledge Prof. Miguel A. Valenzuela from the School

of Chemical Engineering and Extractive Industries at the National Poly-technic Institute of Mexico, who contributed some ideas during the prep-aration of the Spanish version of this book, and also to hundreds ofstudents who during more than 20 years of delivering lectures encouragedme to write this book.

Jorge Ancheyta

Prefacexiv

1

Fundamentals of Chemical Reaction Kinetics

In homogeneous reacting systems, all the reactants and products are inthe same phase. If the reaction involves a catalyst, it is also in the samephase. To determine the rates of reaction, experimental information isneeded, which is generated by using properly designed small-scale reac-tors and experiments. These reaction rates cannot be directly measured,but they are obtained by means of experimental data such as the variationof time with respect to concentration of reactants or products, partialpressures and total pressure, among others.To obtain the kinetic expression that represents the studied reaction,

there are various approaches that correlate the experimental data withthe variables that affect them.When a reaction proceeds, one ormore reactants can take part. It can be

carried out in either liquid or gas phase, the reaction extent is measured bymeans of variations of reactants or product properties, or simply the reac-tion mechanisms are unknown. In any case, it is necessary before startingwith the mathematical treatment of the experimental data to know thefundamentals of stoichiometry, thermodynamics and kinetics that willbe further used for elucidating the specific mathematical expression foreach type of reaction. This chapter is then devoted to introducing thereaders to these topics.

1.1 Concepts of Stoichiometry

1.1.1 Stoichiometric Number and Coefficient

A chemical reaction can be represented as follows:

aA+ bB+… rR+ sS +… 1 1

where A, B, R and S are the chemical species, and a, b, r and s are theircorresponding stoichiometric coefficients, which are the positive num-bers before the chemical formula that balance the reaction.

1

Chemical Reaction Kinetics: Concepts, Methods and Case Studies, First Edition.Jorge Ancheyta.© 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

Eq. (1.1) can be transformed as follows (Chopey, 1994):

−aA−bB−…+ rR+ sS +…= 0 1 2

or with positive values:

υ1A1 + υ2A2 + υ3A3 +…+ υn−1An−1 + υnAn = 0 1 3

which can be generalized as:

n

i=1

υiAi = 0 1 4

where Ai is the chemical formula and υi is the corresponding stoichiomet-ric numbers.Stoichiometric numbers (υi) are numerically equal to stoichiometric

coefficients (a, b, r and s), but they have a negative sign for reactantsand positive sign for products.

Example 1.1 Determine the stoichiometric coefficients and numbersfor the following reaction for synthesis of ammonia:

N2 + 3H2 2NH3 aA + bB rR

Solution

According to stoichiometry, the stoichiometric coefficients and num-bers are:

Stoichiometric coefficient Stoichiometric number

a = 1 νN2 = −1

b = 3 νH2 = −3

r = 2 νNH3 = 2

1.1.2 Molecularity

Molecularity is defined as the number of molecules of reactants that takepart in a chemical reaction. Most of the reactions exhibit a molecularity ofone or two, and in rare cases it reaches the value of three (Hill, 1977).Molecularity is an appropriate concept for a process in which a simple

or elemental step is occurring. Reactions in which one or several reactantsproduce one or several products in a simple path are scarce. For complexreactions, it is necessary to know the molecularity of each individual stepof the reaction.

Chemical Reaction Kinetics2

Based on this concept, the chemical reactions can be classified mainly asmono-molecular, bi-molecular and tri-molecular. A mono-molecularreaction involves one molecule of reactant. In a bi-molecular reaction,two molecules of reactants (either the same or different) are combinedto form one or more products. Tri-molecular reactions are rare since theyneed the simultaneous collision of three molecules to produce one or sev-eral products. Examples of the different types or reactions according totheir molecularity are shown in Table 1.1.

1.1.3 Reaction Extent

To follow the performance of a chemical reaction, it is necessary to definea parameter which properly represents the conversion of the reactants. In1920, De Donder (1920) introduced the concept of reaction extent (ξ), byconsidering that the change in the number of moles of the chemical spe-cies is directly related to the stoichiometric number as follows:

Δn2Δn1

=υ2υ1

orΔn3Δn1

=υ3υ1

or, in differential form:

dn2dn1

=υ2υ1

ordn2υ2

=dn1υ1

dn3dn1

=υ3υ1

ordn3υ3

=dn1υ1

Table 1.1 Chemical reactions with different molecularity.

Molecularity Examples

1 A R n−C4H10 i−C4H10

A R + S SO2Cl2 SO2 + Cl2

A Products (CH3)2O CH4 + H2 + CO

2 A + B R + S O3 + NO O2 + NO2

A + B 2R H2 + Br2 2HBr

A + B R C2 H4 + HI C2H5I

2A Products 2N2O 2N2 + O2

3 A + B + C Products C2H5NO2 + C5H5N + I2C2H4INO2 + C5H5NH

+ + I−

2A + B Products 2NO + O2 2NO2

Fundamentals of Chemical Reaction Kinetics 3

For all chemical species, these equations can be generalized in the follow-ing manner:

dn1υ1

=dn2υ2

=dn3υ3

= … =dniυi

1 5

Defining the parameter ξ, as the reaction extent:

dniυi

= dξ 1 6

The integration of Eq. (1.6) gives:

ni

nio

dni = υi

ξ

0

dξ

ξi =ni−nioυi

1 7

It is then observed that if amoles ofA1 react with bmoles ofA2 to producer moles of An−1 and s moles of An, the reaction extent ξ is equal to 1.Therefore, in general, it can be stated that ξa moles of A1 react with ξbmoles of A2 to produce ξr moles of An−1 and ξs moles of An.

1.1.4 Molar Conversion

The molar fractional conversion (xi) is an intensive normalized parameterreferred preferably to the limiting reactant; it is defined as the fraction ofsuch a reactant that is transformed into products (Froment et al., 2010):

xi =Moles of reactant “i” transformed

Initial moles of reactant “i”=nio−ninio

1 8

where 0 ≤ xi ≤ 1.Subindex “o” refers to the number of moles at zero time (i.e. the begin-

ning of the reaction). Conversion can be correlated with reaction extent bymeans of Eqs. (1.7) and (1.8):

ni = nio + υiξi 1 9

ni = nio−nioxi 1 10

where:

ξi = −nioυi

xi 1 11

Themaximum reaction extent (ξimax) can be calculated from Eq. (1.11) for

the maximum conversion value ximax = 1 :

ξmaxi = −

nioυi

1 12

Chemical Reaction Kinetics4

which implies that the minimum and maximum values of ξi are in the fol-lowing range:

0 ≤ ξi ≤ −nioυi

1.1.5 Types of Feed Composition in a Chemical Reaction

When a chemical reaction involves more than one reactant, the feed com-position is different depending on the relative initial concentrations of thechemical species:

• Stoichiometric feed composition: This occurs when the ratio between thestoichiometric coefficients of the reactants is equal to the ratio betweenthe amount of moles or the molar initial concentrations of reactants.

• Non-stoichiometric feed composition: This is when the ratio between thestoichiometric coefficients of the reactants is different from the ratiobetween the amount of moles or the molar initial concentrations ofreactants.

• Equimolar feed composition: This is when the same amount of reactantsare used at the beginning of the reaction to keep the ratio between theamount of moles or the molar initial concentration equal to unity nomatter the stoichiometric coefficients of the reaction.

• Reactant in excess: This is when the ratio between the amount of molesor the molar initial concentrations of the reactants with respect to thelimiting reactant is much higher than the ratio between the stoichio-metric coefficients.

Some feed compositions can be considered close to the stoichiometricfeed composition, and this happens when the ratio between the amountof moles or themolar initial concentrations of reactants is more or less thesame as the ratio between the stoichiometric coefficients.If at the beginning of the reaction there are inert components, although

they are not reacting, they must be considered to define the type of feedcomposition.

Example 1.2 Define the different feed compositions for the followingreaction of formation of nitrogen dioxide:

2NO + O2 2NO2 2A + B 2R

Solution

If a feed consists of 4 moles of NO and 2 moles of O2, the ratio of molesbetween them is nO2/nNO = 2/4 = 0.5, and the ratio between stoichiomet-ric coefficients is b/a = 1/2 = 0.5. Since nO2/nNO = b/a, the feed composi-tion is stoichiometric.

Fundamentals of Chemical Reaction Kinetics 5

If the feed consists of 3moles ofNO and 2moles ofO2, the ratio ofmolesbetween them is nO2/nNO = 2/3 = 0.66, and the ratio between stoichiomet-ric coefficients is b/a = 1/2 = 0.5. Since nO2/nNO b/a, the feed composi-tion is non-stoichiometric.If the feed consists of 4 moles of NO and 4 moles of O2, the ratio of

moles between them is nO2/nNO = 4/4 = 1, and therefore the feed compo-sition is equimolar. This feed composition is also non-stoichiometricsince nO2/nNO b/a.If the feed consists of 1 mol of NO and 20 moles of O2, the ratio of

moles between them is nO2/nNO = 20/1 = 20, and the ratio betweenstoichiometric coefficients is b/a = 1/2 = 0.5. Since nO2/nNO >> b/a, it isconsidered that reactant B (O2) is in excess.If the feed consists of 4 moles of NO and 1.8 moles of O2, the ratio of

moles between them is nO2/nNO = 1.8/4 = 0.45, and the ratio between stoi-chiometric coefficients is b/a = 1/2 = 0.5. Since nO2/nNO≈ b/a, the feedcomposition is assumed to be close to stoichiometric.An equimolar feed composition would also be 4moles ofNO, 4 moles of

O2 and 4 inert moles.

1.1.6 Limiting Reactant

The limiting reactant is the chemical species that in a chemical reaction isconsumed before all of the other reactants (Himmelblau, 1970). If thereaction is carried out with only one reactant, the limiting reactant con-cept does not have meaning since it is obvious that it is the limiting one.For reactions between two or more components with stoichiometric

feed composition, any of the reactants can be the limiting one since theyare consumed at the same rate. For other feed compositions, the definitionof limiting reactant will depend on such a composition and on the reac-tion stoichiometry.To know the limiting reactant in certain reactions, the concept of reac-

tion extent can be used according to the following definition: “the limitingreactant is the chemical species that has the lowest value of maximumreaction extent (ξi

max).”

Example 1.3 Determine the limiting reactant if, in the following reaction,5moles of ethylene bromide (A) and 2moles of potassium iodide (B) are fed:

C2H4Br2 + 3KI C2H4 + 2KBr + KI3 A + 3B R + 2S + T

Solution

The number of moles of each reactant in the feed and the correspondingstoichiometric numbers are:

nAo = 5moles,nBo = 2moles,υA = −1 and υB = −3

Chemical Reaction Kinetics6

According to Eq. (1.12):

ξmaxA = −

nAoυA

= −5−1

= 5

ξmaxB = −

nBoυB

= −2−3

= 23

Since ξmaxB < ξmax

A , thus the limiting reactant is B.The results of this example can be confirmed by analysing the stoichi-

ometry of the reaction, in which for each mole of A, 3 moles of B arerequired. For the case of this example, if 5 moles of A are used, then15moles of Bwill be required to complete the reaction, and if only 2molesof B are present, then B is consumed first and A is in excess.

1.1.7 Molar Balance in a Chemical Reaction

If, in the reactionaA + bB rR + sS,A is assumed tobe the limiting reactant,andnAo,nBo,nRo andnSo are thenumberofmolesofA,B,R andS, respectively,at the beginning of the reaction, then from Eq. (1.7) for reactant A:

nA = nAo + υAξA

Substituting Eq. (1.11) in this equation:

nA = nAo + υA −nAoxAυA

= nAo−nAoxA = nAo 1−xA 1 13

For reactant B:

nB = nBo + υBξA

nB = nBo + υB −nAoxAυA

Since υB = −b and υA = −a, then:

nB = nBo + −b −nAoxA−a

= nBo−banAoxA 1 14

which can also be written as follows to introduce the ratio nBo/nAo:

nB = nAonBonAo

−baxA

Fundamentals of Chemical Reaction Kinetics 7

Defining the following feed molar ratio of B with respect to A:

MBA =nBo

nAo

nB = nAo MBA−baxA

1 15

Following the same procedure for R and S:

nR = nRo +ranAoxA 1 16

nR = nAo MRA +raxA 1 17

nS = nSo +sanAoxA 1 18

nS = nAo MSA +saxA 1 19

where:

MRA =nRo

nAo and MSA =nSo

nAo

1.1.8 Relationship between Conversion and Physical Propertiesof the Reacting System

When it is not possible to generate experimental information in terms ofcommon properties (concentration, total pressure, partial pressure, etc.),it is necessary to measure the reaction extent as a function of any physicalproperty of the system, such as absorbance, electric conductivity, refrac-tive index and the like, since they are additive functions of the contribu-tions of all chemical species and in general they vary linearly with theconcentration (Levenspiel, 1972).For any physical property (λ), the contribution of all the chemical spe-

cies can be represented by:

λ=n

i=1

yiλi 1 20

The relationship between λ and Ci can be written as:

λi Ci or λi = kλiCi 1 21

Dividing Eq. (1.7) between the volume to obtain the volumetric reactionextent (ξi ):

ξi =ξiV

=ni−nioVυi

=

niV

−nioV

υi=Ci−Cio

υi

Ci =Cio + υiξi

Chemical Reaction Kinetics8

And, substituting in Eq. (1.21):

λi = kλi Cio + υiξi = kλiCio + kλiυiξi

Substituting λi in Eq. (1.20) and considering a constant value of ξi for aspecific chemical species:

λ=n

i= 1

yi kλiCio + kλiυiξi =n

i= 1

yikλiCio + ξi

n

i= 1

yikλiυi 1 22

Since kλi and υi are constant, the following equation can be derived:

n

i= 1

yikλiυi = kλiυin

i=1

yi = kλiυi =Kλ

Moreover, at zero time, Eq. (1.20) is:

λo =n

i=1

yiλio =n

i=1

yikλiCio

Substituting Kλ and λo in Eq. (1.22):

λ= λo +Kλξi or λ−λo =Kλξi 1 23

Applying Eq. (1.23) at the maximum point of reaction extent:

λ∞ −λo =Kλξmaxi 1 24

Dividing Eq. (1.23) by Eq. (1.24):

λ−λoλ∞ −λo

=Kλ ξi

Kλ ξmaxi

=ξi

ξ maxi

=ξi

V

ξmaxi V

=ξiξmaxi

1 25

Since the maximum reaction extent (ξimax) is:

ξmaxi = −

nioυi

the ratio (ξi/ξimax) is:

ξiξmaxi

=−nio υi

xi

−nio υi

= xi 1 26

And, finally, Eq. (1.25) is:

xi =λ−λoλ∞ −λo

1 27

Fundamentals of Chemical Reaction Kinetics 9

where:

λ: Physical property at time t;λo: Physical property at time zero (t = 0);λ∞: Physical property that does not change with time; andxi: Conversion.

Example 1.4 ReactantA is preparedunder refrigerationand is introducedin a small capillary that acts as a reaction vessel, in which the decompositionreaction A R + S is carried out. The vessel is rapidly introduced in a bathcontaining water at the boiling point. During handling, there is no reaction.During the experiments, several data of the capillary length occupied by thereactingmixture (L)were collected (Levenspiel, 1979). Evaluate the values ofconversion for the capillary length at different times indicated in Table 1.2.

Solution

In this case, Eq. (1.27) can be written as:

xA =L−LoL∞ −Lo

where:

L: Capillary length at time t;Lo: Capillary length at time zero (t = 0);L∞: Capillary length that does not change with time; andxA: Conversion of reactant A.

In this equation, the initial capillary length (Lo) is unknown. However,from the analysis of the stoichiometry, it is deduced that the reaction is

Table 1.2 Data and results of Example 1.4.

Time (min) Capilar length (cm) xA

0.5 6.1 0.2979

1.0 6.8 0.4468

1.5 7.2 0.5319

2.0 7.5 0.5957

3.0 7.85 0.6702

4.0 8.1 0.7234

6.0 8.4 0.7872

10.0 8.7 0.8511

∞ 9.4 1.0000

Chemical Reaction Kinetics10

irreversible; hence, at t =∞, the conversion is 100% (xA = 1.0), that is, allthe reactant A has been transformed into R and S. In other words, 1 moleof reactant has been transformed into 2 moles of products. This indicatesthat at t =∞, the number of moles is duplicated, as well as the volume andthe capillary length, so that:

L∞ = 2Lo

Lo = L∞ 2= 9 4 2 = 4 7cm

The application of the previous equation, at t = 1 min, gives:

xA =L−LoL∞ −Lo

=6 8−4 79 4−4 7

= 0 2979

The results of conversion for all the capillary lengths are reported inTable 1.2.

1.2 Reacting Systems

1.2.1 Mole Fraction, Weight Fraction and Molar Concentration

If the total number of moles and weight of all the chemical species presentin the reacting mixture are nt and wt, respectively, and if ni moles and wi

weight units of component i are present, the mole or molar fraction (yi)and the weight fraction (ywi) of species i in the system are defined as:

yi =nint

mole fraction 1 28

ywi =wi

wtweight fraction 1 29

By definition, the sum of fractions of all the components must be equalto unity:

n

i= 1

yi = y1 + y2 +…+ yn =n1nt

+n2nt

+…+nnnt

=n1 + n2 +…+ nn

nt=ntnt

= 1

n

i= 1

ywi = yw1 + yw2 +…+ ywn =w1

wt+w2

wt+…+

wn

wt=w1 +w2 +…+wn

wt=wt

wt= 1

To convert a mole fraction in a weight fraction or vice versa, the followingrelationship is used, which is obtained by using the definition of number ofmoles (n = w/MW):

yi =nint

=wi

MWi

wtMWt

=wi

wt

MWt

MWi= ywi

MWt

MWi1 30

Fundamentals of Chemical Reaction Kinetics 11

where the molecular weight of the mixture (MWt) is:

MWt =n

i= 1

yiMWi 1 31

The molar concentration is defined as the ratio between the number ofmoles of a chemical species (ni) by unit of the system volume (V) andis related with density (ρi) as follows:

Ci =niV

=wi

PMi

V=

wi

V PMi=

ρiPMi

1 32

Example 1.5 Evaluate the initial mole and weight fractions of the reac-tants if the following reaction of nitrogen dioxide formation starts with3 moles of NO and 2 moles of O2.

2NO + O2 2NO2 2A + B 2R

Solution

Mole fractions. Using Eq. (1.28):

yNOo =nNOonto

=nNOo

nNOo + nO2o=

33 + 2

= 0 6

yO2o =nO2onto

=nO2o

nNOo + nO2o=

23 + 2

= 0 4

Weight fractions. Using Eq. (1.29):

wNOo = nNOo MWNO = 3 30 = 90 g

wO2o = nO2o MWO2 = 2 32 = 64 g

wto =wNOo +wO2o = 90 + 64 = 154 g

ywNOo =wNOo

wto=

90154

= 0 584

ywO2o =wO2o

wto=

64154

= 0 416

Using Eq. (1.30):

MWt = yNOMWNO + yO2MWO2 = 0 6 30 + 0 4 32 = 30 8 g gmol

ywNOo = yNOoMWNO

MWt= 0 6

3030 8

= 0 584

ywO2o = yO2oMWO2

MWt= 0 4

3230 8

= 0 416

Chemical Reaction Kinetics12

1.2.2 Partial Pressure

Partial pressure is defined as the pressure that a gas in a mixture of gaseswould exert if it alone occupied the whole volume occupied by the mix-ture at the same temperature. Therefore, partial pressure pi of gas i in amixture of gases is calculated by multiplying its mole fraction (yi) bythe total pressure of the system (P):

pi = yiP

This is the so-called Dalton law, which also states that the total pressureexerted of a mixture of gases is equal to the sum of partial pressures of allthe gases of the mixture (Smith et al., 1980):

n

i= 1

pi =n

i= 1

yiP =Pn

i=1

yi =P

Partial pressure of a gas in a mixture of gases is related to its molar con-centration by means of the ideal gas law:

piV = niRT

pi = yiP =niV

RT =CiRT1 33

where R is the universal gas constant, the common values of which are:

R= 1 987Cal

gmol K= 1 986

BTUlbmol R

=82 057atm cm3

gmol K= 8 314

Jgmol K

= 0 08205atm ltgmol K

= 10 73psia ft3

gmol K0 7302

atm ft3

lbmol R=62 361

mmHg ltgmol K

= 1 315atm ft3

lbmol K=8 31

KPa ltgmol K

5 83 × 10−4 KWhlbmol R

= 7 82 × 10−4 hp hlbmol R

1.2.3 Isothermal Systems at Constant Density

When a system operates at constant density, the corresponding volumerefers to the reacting mixture and not to the volume of the reactor. Tothis type of system belongs those reactions conducted in liquid phaseor gas phase that either do not experience change in the number of molesor are carried out in hermetic vessels.For an homogeneous reaction, which is carried out in the hermetic ves-

sel shown in Figure 1.1, in gas phase, isothermally, with change in thenumber of moles and consequently with an increase or decrease in the

Fundamentals of Chemical Reaction Kinetics 13

pressure by expansion or compression due to the reaction, the molar bal-ance at any time for the reaction aA + bB rR + sS is the following:

nA = nAo−nAoxA

nB = nBo−banAoxA

nR = nRo +ranAoxA

nS = nSo +sanAoxA

Assuming that the system follows the ideal gas law (PV = nRT) for the sys-tem at constant density, then at the beginning of the reaction (Figure 1.1):

PoVo = nToRTo 1 34

nTo = nAo + nBo + nRo + nSo 1 35

and at time t:

PVo = nTRTo 1 36

nT = nA + nB + nR + nS 1 37

Substituting the equations of the molar balance (Eqs. 1.13, 1.15, 1.17 and1.19) in Eq. (1.37):

nT = nAo + nBo + nRo + nSo +ranAoxA +

sanAoxA−nAoxA−

banAoxA

nT = nTo +nAoa

r + s−a−b xA = nTo +nAoΔn

axA

1 38

Time

Initial conditions

t = 0ToVoPonTo

Final conditions

t > 0ToVoPnT

Figure 1.1 Reacting system at constant density.

Chemical Reaction Kinetics14