Catalytic Kinetics

Catalytic Kinetics by Dmitry Murzin, Tapio Salmi

•

ISBN: 0444516050

•

Pub. Date: September 2005

•

Format: Hardcover, 492pp

•

Publisher: Elsevier Science & Technology Books

Preface Chemistry and chemical technology have been at the heart of the revolutionary developments of the 20th century. The chemical industry has a long history of combining theory (science) and practice (engineering) to create new and useful products. Worldwide, the process industry (which includes chemicals, petrochemicals, petroleum refining, and pharmaceuticals) is a huge, complex, and interconnected global business with an annual production value exceeding $4 trillion dollars. The performance of a majority of chemical reactors (and hence the processes) is significantly influenced by the performance of the catalysts. Catalyst research has been devoted to increase the catalyst activity and selectivity to improve process economics and reduce environmental impact through better feedstock utilization. Catalysis-based chemical synthesis accounts for 60% of today's chemical products and 90% of current chemical processes. Catalysis development and understanding thus is essential to the majority of chemical synthesis advances. Because the topic of chemical synthesis is so broad and catalysis is so crucial to chemical synthesis, catalysis should be specifically addressed. Although in industry special focus is in heterogeneous catalysis; homogeneous, enzymatic, photochemical and electrochemical catalysis should not be overlooked, as the major aim is to produce certain chemicals in the best possible way, applying those types of catalysis, which suit a particular process in the most optimal way. For instance bioprocesses have become widely used in several fields of commercial biotechnology, such as production of enzymes (used, tbr example, in tbod processing and waste management) and antibiotics. As techniques and instrumentation are refined, bioprocesses may have applications in other areas where chemical processes are now used. Advantages of bioprocesses over conventional chemical methods of production are lower temperature, pressure, and pH and application of renewable resources as raw materials with less energy consumption. Catalyst development in industry is inseparable from understanding of catalysis on microscopic (elementary reactions) and macroscopic levels (transport phenomena). This book presents an attempt to unify the main sub disciplines forming the cornerstone of practical catalysis. Catalysis according to the very definition of it deals with enhancement of reaction rates, i.e. with catalytic kinetics. Diversity of catalysts, e.g. catalysis by acids, organometallic complexes, solid inorganic materials, enzymes resulted in the fact, that these topics are usually treated separately in textbooks, despite the fact, that there are very many analogues in the kinetic treatment of homogeneous, heterogeneous and enzymatic catalysis. Catalytic engineering includes as an essential part also macroscopic considerations, more specifically transport phenomena. Such an integrated approach to kinetics and transport phenomena in catalysis, still recognizing the fundamental differences between different types of catalysis, could be seldom found in the literature, where quite often artificial borders are build, preventing free exchange of useful ideas and concepts. Cross-disciplinary approach can be only beneficial for the advancement of catalytic reaction engineering. it should be mentioned, that it is not the aim of the authors to provide exhaustive bibliography. Contrary, as we are trying to cover a variety of topics, we would like to limit ourselves to the main monographs, review articles and key references. The hope of the authors is that the book could be also used as a textbook in catalytic kinetics and catalytic reaction engineering.

vi This book is partially based on several courses, which the authors have taught at Abo Akademi University over the recent years, namely "Heterogeneous Catalysis", "Chemical Kinetics", Chemical Reaction Engineering", "Chemical Reactors", "Chemical Technology", "Bioreaction Engineering", where topics covered in the present textbook were touched in one way or another. Chapters 1-8, 9.4, 9.6-9.11, 10.1-10.2, 10.7-10.9 were written by D.Yu. Murzin, material for chapters 9.1-9.3, 9.5. and 10.3-10.6 was prepared by T. Salmi. The authors are very grateful to many colleagues from academia and industry who shared their knowledge and expertise in kinetics and mass transfer. In particular the late Professor M.I. Temkin introduced one of the authors into the field of heterogeneous catalysis and chemical reaction engineering in the broader context of physical chemistry and practical industrial needs and was a role model as a scientist and a person. Special thanks go to Dr. Nikolai DeMartini, who carethlly read the manuscript and corrected the language, also giving several advices regarding the presentation of material. Finally help ofElena Murzina in making the corrections is appreciated, as well as her patience during the many weekends and evenings when I was working on the book. The authors understand that it is very difficult to cover the whole field in one book, therefore the selection of topics and examples and especially allocated space to particular topics might be considered biased. We will be delighted to receive critics and comments, which will help to improve the text.

Dmitry Murzin June, 2005, Turku/Abo

Table of Contents

Ch. 1

Setting the scene

1

Ch. 2

Catalysis

27

Ch. 3

Elementary reactions

73

Ch. 4

Complex reactions

111

Ch. 5

Homogeneous catalytic kinetics

149

Ch. 6

Enzymatic kinetics

189

Ch. 7

Heterogeneous catalytic kinetics

225

Ch. 8

Dynamic catalysis

285

Ch. 9

Mass transfer and catalytic reactions

341

Ch. 10

Kinetic modelling

419

Chapter 1. Setting the scene 1.1 History All processes occur over a time ranging from femtosecond to billions of years. The same holds for chemical and biochemical transformations. Kinetics (derived from the Greek word KtvrlxtZo ¢ meaning dissolution) is a science which investigates fine rates of processes. Chemical kinetics is the study of reaction rates. However complex a process is, it can be in principle divided into a number of elementary processes which can be studied separately. Chemical kinetics emerged as a branch of physical chemistry in the 1880-s with seminal works of Harcourt and Esson demonstrating the dependence of reaction rates on the concentrations of reactants. It was a German scientist K. Wenzel who stated that the affinity of solid materials towards a solvent is inversely proportional to dissolution time and 100 years before Guldberg and Waage (Norway) formulated a law, which was later coined the "law of mass action," meaning that the reaction "forces" are proportional to the product of the concentrations of the reactants. When the rate of a certain process is measured, especially if it is of practical importance, a curious mind is always eager to know if it is possible to accelerate its velocity. Moreover, one could even imagine a situation that for a system demonstrating complete inertness introduction of a foreign substance could enhance the rate dramatically. Conversion of startch to sugars in the presence of acids, combustion of hydrogen over platinum, decomposition of hydrogen peroxide in alkaline and water solutions in the presence of metals, etc. were critically summarized by a Swedish scientist J. J. Berzelius in 1836, who proposed the existance of a certain body, which "effectiing the (chemical) changes does not take part in the reaction and remains unaltered through the reaction". He called this unknown tbrce, catalytic force, and defined catalysis as decomposition of bodies by this force.

J6ns Jakob Berzelius

Wilhelm Ostwald and Svante Arrhenius

This new concept was immediately critized by Liebig, as this notion was putting catalysis somewhat outside other chemical disciplines. A catalyst was later defined by Ostwald as a compound, which increases the rate of a chemical reaction, but which is not consumed by the reaction. This definition allows for the possibility that small amounts of the catalyst are lost in the reaction or that the catalytic activity is slowly lost.

1.2. Catalysis Already from these definitions it is clear that there is a direct link between chemical kinetics and catalysis, as according to the very definition of catalysis it is a kinetic process. There are different views, however, on the interrelation between kinetics and catalysis. While some authors state that catalysis is a part of kinetics, others treat kinetics as a part of a broader phenomenon of catalysis. Despite the fact that catalysis is a kinetic phenomenon, there are quite many issues in catalysis which are not related to kinetics. Mechanisms of catalytic reactions, elementary reactions, surface reactivity, adsorption of reactants on the solid surfaces, synthesis and structure of solid materials, enzymes, or organometallic complexes, not to mention engineering aspects of catalysis are obviously outside the scope of chemical kinetics. Some discrepancy exists whether chemical kinetics includes also the mechanisms of reactions. In fact if reaction mechanisms are included in the definition of catalytic kinetics it will be an unnecessary generalization, as catalysis should cover mechanisms. Catalysis is of crucial importance for the chemical industry, the number of catalysts applied in industry is very large and catalysts come in many different forms, from heterogeneous catalysts in the form of porous solids to homogeneous catalysts dissolved in the liquid reaction mixture to biological catalysts in the form of enzymes. Catalysis is a multidisciplinary field requiring efforts of specialists in different fields of chemistry, physics and biology to work together to achive the goals set by the mankind. Knowledge of inorganic, organometallic, organic chemistry, materials and surface science, solid state physics, spectroscopy, reaction engineering, and enzymology is required for the advancements of the discipline of catalysis. Despite the fundamental differences between elementary steps in catalytic process on surfaces, with enzymes or homogeneous organometalics there are stricking similarities also in terms of chemical kinetics. Although superficially it is difficult to find something in common between the reaction of nitrogen and hydrogen forming ammonia on a surface of iron, Dfructose 6-phosphate with ATP involving an enzyme phosphofructokinase, or ozone decomposition in the atmosphere in the presence of NOx, all these trasnformations require that bonds are formed with the reacting molecules. Such a complex then reacts to products leaving the catalyst unaltered and ready for taking part in a next catalytic cycle.

iiiiiiiiiii ..................

iiiiiiii~

Figure 1.1. Catalytic cycle

iiiiiiiiiiiiiiii

:~iiiiiiii

Figure 1.1 is an example of a catalytic reaction between two molecules A and B with the involvment of a catalyst. In order to understand how a catalyst can accelerate a reaction a potential energy diagram should be considered. x~

0

P÷Q R e . o n ¢oerdinate

Figure 1.2. Potential energy diagram

Figure 1.2 represents a concept for a non-catalytic reaction of An'henius, who suggested that reactions should overcome a certain barrier before a reaction can proceed. X* "1

I

\

/F~', If

G

~/

\

~

(the reduction in AG~ bythe catalyst)

Catalyzed

A+B A+B .

" P+Q

~

Reactioncoordinate Figure 1.3. Potential energy diagram for catalytic reactions

The change in the Gibbs free energy between the reactants and the products AG does not change in case of a catalytic reaction, however the catalyst provides an alternative path for the reaction (Figure 1.3). In general reaction rates increase with increasing temperature. Kooij and van't Hoff (1893) proposed an equation for the temperature dependence of reaction rates k = AT"

e -E~T

(1.1)

where A is pre-exponential factor and activation energy, Ea, is related to the potential energy barrier. This equation, which could be derived on the basis of transition sate theory, in a slightly simplified tbrm

4

k = ko e K G

(1.2)

was applied by Arrhenius and is reffered to as the Arrhenius law. It is immediately clear from equation (1.2) that a decrease in activation energy will lead to an increase of the rate constant and thus the reaction rate (a discussion on the relationship between the rate and rate constant will be given below). At the same time the catalyst (heterogeneous, homogeneous or enzymatic) affects only the rate of the reaction, it changes neither the thermodynamics of the reaction (Gibbs energy) nor the equilibrium composition. An important conclusion is thus that a catalyst can change kinetics but not thermodynamics of a reaction and if a process is thermodynamically unfavorable, there is no need to apply any modern and fancy methods (high throughput screening and alike) to find such a catalyst. Concentration

Time

Figure 1.4. Concentration vs time dependences for a reversible reaction The dashed line in Figure 1.4 demonstrates the equlibrium that cannot be ovecome for a given set of parameters. Furthermore the ratio of rate constants in the forward and reverse direction for catalytic and noncatalytic reactions is the same. _

[Pl,,,

kc,,/ [AL,

= x

(1.3)

It also implies that if a catalyst is active in enhancing a rate of the forward reaction, it will do the same with a reverse reaction. Figure 1.3 is somewhat simplified as it does not take into account possible bonding of the catalyst and reactant. In order for a catalyst to be effective, the energy barrier between the catalyst -substrate and activated complex must be less than between substrate and activated complex in the uncatalyzed reaction. The binding of substrate to an enzyme lowers the free energy of the catalyst substrate complex relative to the substrate (Figure 1.5). This is a general feature of catalysis and is relevant for heterogeneous, homogeneous and enzymatic catalysis. If the energy is lowered too much, without a greater lowering of the activation energy then catalysis would not take place, meaning that bonding between a catalyst and a reactant should not be too strong. Alternatively if it is too weak, then the catalytic cycle could not proceed.

0

bmulb~g

reactlott

sq~aration

read:ion coordinate

Figure 1.5. Potential energy diagram of a heterogeneous catalytic reaction (1. Chorkendorfl, J.W. Niemantsverdriet, Concepts of Modern Catalysis and Kinetics, Wiley, 2003). Chemical kinetics as a discipline adresses how the reaction rates depend on reactant concentration, temperature, nature of catalysts, pH, solvent, to name a few- reaction parameters. Chemical kinetics together with other means of studying catalytic reactions, like spectroscopy of catalysts and catalyst models, quantum-chemical calculations for reactants, intermediates and products, calculation of the thermodynamics of reactants, intermediates and products from measured spectra and quantum-chemical calculations form the modern basis for understanding catalysis. Kinetic investigations are one of the ways to reveal reaction mechanisms. The following problems can be solved using the kinetic model: • choosing the catalyst and comparing the selectivity and activity of catalysts and their performance under optimum conditions for each catalyst; • the determination of the optimum sizes and structure of catalyst grains and the necessary amount of the catalyst to achieve the specified values of the selectivity of the process and conversion of the starting products; • the determination of the composition of all byproducts formed during the process; • the determination of the stability of steady states and parametric sensitivity; that is, the influence of deviations of all parameters on the steady-state regime and the behavior of the reactor under unsteady state conditions; • the study of the dynamics of the process and deciding if the process should be carried out under unsteady-state conditions; • the study of the influence of mass and heat transfer processes on the chemical reaction rate and the determination of the kinetic region of the process; • choosing the type of a reactor and structure of the contact unit that provide the best approximations to the optimum conditions. Very often the rates of chemical transformations are affected by the rates of other processes, such as heat and mass transfer. The process should be treated as a part of kinetics. The gas/liquid mass transfer in multiphase heterogeneous and homogeneous catalytic reactions could be treated in a similar way. The mathematical framework for modelling diffusion inside solid catalyst particles of supported metal catalysts or immolisided enzymes does not differ that much, but proper care should be taken of the reaction kinetics.

The immense importance o f catalysis in chemical industry is manisfested by the tact that roughly 85-90% o f all chemical products have seen a catalyst during the course o f production.

1997

Chemical

2003

Chemical

................... olymer

Polymer

Refinerl

Refiner

Environmental

Billion US$ 7.4

Environmental

9.0

* toll manufacturing fees only The Catalyst Group: The Intelligence Report: Global Shifts in the Catalyst Industt2¢

Figure 1.6. Worldwide catalyst market Figure 1.6 demonstrates applications o f catalysis in industry. In the last years there is an increase o f catalytic applications also for non-chemical industries: treatment o f exhaust gases from cars and other mobile sources, as well as power plants (Figure 1.7).

Figure 1.7. Catalytic treatment of NOx in a) mobile b) stationary sources A comparison between homogeneous and heterogeneous catalysts from the viewpoint o f a homogeneous catalysis expert is presented below

Activity Selectivity Conditions of reaction Life time of catalyst Sensitivity to deactivation Problems due to diffusion Recycling of catalyst Steric and electronic properties Mechanism

Homogeneous

Heterogeneous

high high mild variable low none usually difficult easily changed realistic models exist

variable variable harsh long high difficult to solve can easily be done no vm'iation possible not obvious

The topics adressed above will be dicussed in more detail in the subsequent chapters. A great variety of homogeneous catalysts are known: metal complexes and ions, Bronsted and Lewis acid, enzymes. Homogeneous transition metals are used in several industrial processes, a few of them are given below Process Acetaldehyde Acetic acid Oxo-alcohols Dimethyl terephthalate Terephthalic acid

World capacity (million t/a) 2.5 4.0 7 3.3 9.4

Catalyst Pd/Cu Rh Co or Rh Co Co

Temperature (K) 375-405 425-475 335-470 415-445 450-505

Pressure (bar) 3-8 30-60 200/30 4-8 15-30

Metal complexes can have a very sophisticated structure with a variety of ligands. An example of such ligands for Rh catalysed hydroformylation is given below (Fig.l.8) along with some images of heterogeneous catalysts (Fig. 1.9)

Figure 1.8. A ligand for Rh catalysedhydroformylation

Figure 1.9. Images of heterogeneous catalysts Enzymes represent a special type of homogeneous catalyst. They are large proteins (Figure 1.10) capable of increaing the reaction rates by a factor of 106 to 106 at mild reaction conditions and displaying very high specificity and capability of regulation.

Figure I. 10. A schematic view on an enzyme structure Specificity (Figure 1.11) is controlled by the enzyme structure, more precisely a unique fit of substrate with the enzyme controls the selectivity for the substrate and the product yield.

Figure 1.11. Specificityof enzyme catalysis Superficially there is not that much in common between a large protein and a Pt/A1203 heterogeneous catalyst. At the same time the chemical reactions which occur with both types of catalysts involve certain active sites, e.g. regions where catalysis occurs. Whatever the specific reaction, these active sites can be represented by Figure 1.5, which is a schematic representation of a catalytic reaction. This in turn means that the kinetics of either heterogeneous or homogeneous catalytic reactions can be very similar and in fact they are.

1.3. F o r m a l k i n e t i c s

Chemical kinetics as a dispipline concerns the rates (the velocities) of chemical reactions and deals with experimental measurements of the velocities in batch, semibatch or continuous reactors. Interpretation of the experimental data is currently done using the laws of physical chemistry. One of the fathers of chemical kinetics, Louis Jacques Th6nard, discovered hydrogen peroxide and measured its decomposition rates. He demonstrated for the first time, that rates of chemical reactions varied with the concentrations of the reactants. In later study Ludwig Ferdinand Wilhelmy investigated the inversion of cane sugar in the presence of acids and

9 developed a rate equation, which was the first attempt to interpret the temperature dependence of the rate constant. Unfortunately this work remained in oblivion until 1884. In 1865 rate laws combined with mass balances for a batch reactor were proposed by Augustus George Vernon Harcourt and William Esson, giving a mathematical expression for concentration vs t for first order, second order and consecutive reactions, representing a major breakthrough for modern chemical kinetics. Following the footsteps of the great scientists of the 19 th century, let us try to consider reaction rates for a chemical reaction described by the following equation aA + bB = cC+dD

(1.4)

where A and B are reactants, C and D products, and a,b,c, and d are stoichiometric coefficients. An equation for a chemical reaction is written in such a way that all the molecules particpating in the reaction are balanced. Very often in chemical reaction egineering the stoichiometric coefficient v, is defined as the amount of product produced after one run of the reaction. It implies that the stoichiometric coefficient is positive for a product and negative for a reactant. Thus for the reaction A+B~ C

(1.5)

The following stoichiometric coefficients hold: VA = -1, V~ = -1, VC= +2

(1.6)

An extensive quantity describing the progress of a chemical reaction equal to the number of chemical transformations (the total number of reaction runs) divided by the Avogadro number (it is essentially the amount of chemical transibrmations) is called the extent of reaction. The change in the extent of reaction is given by d~ = dn]v~, where v~ is the stoichiometric number of any reaction entity i (reactant or product) and ni is the corresponding amount in moles. Thus d ~/dt is an extensive property, which is measured in moles and cannot be considered a reaction rate, as it is proportional to the size of the reactor. In general, for a homogeneous reaction for which the reaction rate changes with time and also it is not unitbrm over a volume of a reactor the reaction rate is

t" -

~ t dV

(1.7)

where V stands for the reactor volume. If the reactor volume is constant then the reaction rate is simply -

1 dC,

v, dt

(1.8)

where i is the reactant or product with corresponding stoichiometric coefficient vi, and Ci is the concentration of component i. For a reaction

10 A+B~ C

(1.9)

the rate of consumption of reactant A is then

rA .

l dn A . . . v A Vdt

.

dC A

d[A]

dt

dt

(1.10)

where nA is the number of moles of A in the reactor and [A] is the concentration of A. Similarly for the reaction 2NOC1 (g) ~ 2 NO(g) + 1 C12 (g)

(1.11)

2 moles of NOC1 disappear for every 1 mole C12 formed so the rate is defined as rate

-

1 d[NOCl]_ 2

dt

+

1 d[NO] 2

- ~

dt

d[Cl2]

(1.12)

dt

For a heterogeneous reaction occurring on the surface S of a catalyst the following expression holds,

r -

324

(1.13)

c3t~

which can be further simplified, if the rate is uniform across the surface, to r -

ld~

(1.14)

Sc~t

Rate laws express how the rate depends on concentration. If a reaction follows eq. (1.4), the law of mass action could be applied leading to a following equation r = r+ - r_ = k + [ A ] ~ [ B ] h - k _ [ C ] C [ D ] d

(1.15)

where k+ and k. are reaction rate constants and stoichiometric coefficients appear as the powers (reaction orders towards particular components). In reallity the chemical equation (1.4) does not tell us how reactants become products - it is a summary of the overall process. In fact it is molecularity, e.g. the number of species that must collide to produce the reaction which determines the form of a rate equation. Reactions whose rate law can be written from its molecularity are called elementary. The kinetics of the elementary step depends only on the number of reactant molecules in that step. For the reaction 2NO2 (g) + Cl2 (g) ~ 2 NO2C1 (g)

(1.16)

the rate expression based on the formal kinetics is r = k [NO2] 2 [C12]

(1.17)

11 with the overall order defined as the sum of orders to each reactant being equal to 3. However the reaction mechanism is more complicated and consists of several elementary steps. a) NO2 (g) + C12 (g) ~ NO2C1 (g) + C1 (g) b) NO2 (g) + C1 (g) =:> NO2C1 (g)

(1.18)

fthe rate of the overall process is determined by the first step a, then the rate is defined as F = k [ N O 2 ] [C121

(1.19)

and the overall order is just two. For elementary reactions the reaction orders have orders that are integers which are usually equal to one or two (Figure 1.12), and occasionally three for trimolecular reactions.

I

r

a

t

e

/

/

/

S

~

r

First order ~ A ] or 0 th order

Figure 1.12. Representationof reaction kinetics of differentorders. In practice, reaction orders can be fractional, indicating a complex reaction mechanism. The majority of this book is devoted to such cases, as catalytic reaction mechanisms, which follow from the general considerations above, are typical examples of complex reactions. Reaction orders for a reaction A ~ P described by a following equation for the rate dc A -

rA = -kc

t~

(1.20)

A

dt

are determined using logarithmic plots In(- ff~tA) = In k + n in c,~ with the reaction order corresponding to the slope (Figure 1.13)

Ink ~ ~

~ Inc

Figure l. 13. Determinationof reaction orders

(1.21)

12 1.4. Acquisition of kinetic data Kinetic data for a chemical reaction is gathered in different type of reactors and we will briefly mention some requirements for chemical reactors from the viewpoint of kinetic analysis. A high precision of the data is needed as large deviations in the values of the experimentally measured rates will be a serious obstacle for quantitative considerations. Reproducibility of rate measurements over a broad range of parameters is also of importance. Another necessary feature is the possibility to reach a goal of obtaining the maximum amount of kinetic information in minimum time. Analysis of products as well as reactor lay-out should preferbably be as easy as possible. Essential features for catalytic reactions is the readiness in reduction/activation of heterogeneous catalysts and a possibility to utilize them in the needed geometrical form. Despite the strict definition of catalysis, which states, that the catalyst does not change during the catalytic reactions, some activity deterioration takes place and therefore measurements of catalytic kinetics should always monitor the catalyst activity. Different types of reactors are applied in practice (Figure 1.14). Stirred tank reactors (STR), very often applied for homogeneous, enzymatic and nmltiphase heterogeneous catalytic reactions, can be operated batchwise (batch reactor, BR), semi-batchwise (semibatch reactor, SBR) or continuously (continuous strirred tank reactor, CSTR)

formly dxed

Uniformly mixed

Product

Figure 1.14. Different types of sth'red tank reactors. Alternatively, tubular reactors with plug flow (piston flow) (PFR) are used and operated in continuous mode (Figure 1.15). /

F~

W

Pr~uct

Figure 1.15. Tubular reactors

1.4.1. Batch reactors

The batch mode of operation (Figure 1.16) brings several advantages as it allows us to monitor the progress of the reaction over time and thus to acquire the whole kinetic curve in one experiment.

13

.

.

.~

[~

e ~

.

kaa

Hs~e ~ae £6e e rg~/pot'~ ~S

Figure 1.16. Approach to kinetic analysis in batch reactors The high precision and wide range of parameters afforded by this operation mode made batch reactors very popular for kinetic studies especially in the field of fine and pharmaceutical chemicals• Another advantage is the possibility to utilize heterogeneous catalysts of different geometrical shape• Such reactors can be made either of glass or stainless steel to sustain high pressures (Figure 1.17) and can be applied in a parallel mode.

Figure 1.17. Batch reactors : a) glass, b) high pressure, c) in parallel mode. At the same time, for heterogeneous catalytic reactions, activity control presents a challenge and will be discussed further in Chapter 8. Moreover, catalyst pretreatment (reduction) and regeneration are not straightforward• Quantitative treatment is not easy and will be briefly discussed below• j

V

flow

flow

no A Figure 1.18. A volume element. For an infinitesimal volume element AV in Figure 1.18 the mass balance could be written in a form 1N + G E N E R A T I O N

=OUT + ACCUMULATION

leading to a ~bllowing equation in terms of moles

14

dn A

boa + rlArAAV = hA + - -

all

(1.22)

where h0A and h A are the mole fluxes, q,4 is the catalyst effectiviness factor (chapter 9) taking into account mass transfer. For a batch reactor it holds that IN =0, OUT = 0 , therefore r&GV-

(1.23)

dnA dt

If the volume is constant one gets dnA -- d ( c A V ) - V dcA dt

dt

dt

(1.24)

From equations (1.23) and (1.24) dcA dt

= rlArA

(1.25)

making use of the relationship between concentration and conversion (~ cA = cA,, (1 - a )

(1.26)

assuming that the catalyst effectiviness factor r/A is equal to 1 and taking into account boundary conditions (t-0, c~-0) we arrive at t -- C % ~j -d- a o -rA

(1.27)

In fact treatment of heterogeneous, homogeneous and enzymatic reactions is basically the same with the only difference in the expressions of reaction rates, which reflect different reaction mechanisms. Some specific cases will be discussed in Chapters 5-7. Here we present few examples. Inserting the expression of reaction rate in eq. (1.25) for a reaction A ~ P , which occurs over a catalyst d C A - q k C n, dt

(1.28)

and assuming that n-1 and r/A - 1 we arrive at dCA - kC4, (1 - cz) all

(1.29)

15 which after integration gives an expression for reaction time

1 ~ da o"1-

1 ln(1- c~)

(1.30)

-

For the first order reaction k has units of S-1 (the rates are given in mol 11 s b . For a second order reaction reaction with qA -1 instead of (1.30) one arrives

1

t

1

1

cA)

(1.31)

with k in 1/(mol s), and for a zero order reaction units o f k are mol/1 s and 1

0

t = ~-(C x - Cx)

(1.32)

Equations (1.30-1.32) can be applied for batch reactors independent of the which type of catalysis is operative, if reactions could be described by zero, first or second order. More complicated cases for Langmuir kinetics or Michaelis-Menten kinetics will be considered further. 1.4.2. CSTR

Examples of continuous stirred tank reactors are presented in Figure 1.14. Such a system can be applied for both homogeneous and heterogeneous systems. Figure 1.19 illustrates the differences between batch and CSTR reactors. batch stirred tank reactor

c o n t i n u o u s stirred tank r e a c t o r

subsll

Immobilized catalyst

Immobilized catalyst

Figure 1.19. Stirred tank reactors in a) batch, b) continuous mode.

For a perfectly mixed CSTR at steady state it holds that there is no accumulation dn• = 0 dt

therefore

(1.33)

16

boa - h A + 7 / j s V = 0

(1.34)

Substituting concentration and volumetric flows for molar flows c0AV0 - c a r +

= 0

(1.35)

and assuming that volumetric flows in and out are equal (e.g. density is constant)

l)0 = / )

(1.36)

and introducing residence time as the ratio o f reactor volume to volumetric flow-rate V / l ) o = z-

(1.37)

we arrive at CoA - c A + ~71~r~lr= 0

(1.38)

which can be further transformed to Z - -cA- -- c°A -

(1.39)

r/ArA

In the case o f a first order reaction, the expression for the residence time becomes CA -- C°A rl A k C A

1.4. 3 . P l u g J l o w r e a c t o r s

An example o f a tubular reactor is presented in Fig. 1.20.

Figure 1.20. A tubular reactor.

(1.40)

17 For heterogeneous catalysis, the catalyst is packed in such reactors, which are easy to design and control, as the gases or liquids pass through the reactor and are analyzed. Such reactors are efficient for catalyst screening, especially when they are arranged in a parallel mode (Figure 1.21).

Figure 1.21. Multitubular reactor. The apparent drawback is that one experiment leads to only one data point. On the other hand catalyst deactivation with time on stream could be easily seen (Figure 1.22). 40-

35: 30 2520 15-" lO 5-

activity

•

,

20

•

,

•

40

,

60

•

,

80

•

,

1O0

•

,

120

•

,

140

•

r

160

•

,

•

180

time on stream

Figure 1.22. Catalyst deactivation with time-on-stream at different conditions Quantitative treatment of plug flow- reactors is somewhat cumbersome, therefore several assumptions are usually made. The fluid composition is considered to be unform along the reactor cross section (i.e. there is no radial dispersion). This is valid only when

d - - 2H20 which can be formally written as a two-step sequence H2 +NO2 2NO + 02

: H20 + N O ~ 2NO2

The steps could be even more complicated H2 + NO2 H'+ "NO 2 "OH + H2

J" HNO2 + H ° NO +'OH ~ H20 + H ° ~

02 + 1~IO 1~103+ I'~lO

• 1~O3 •" 21{102

The mechanism of this reaction can be written in the following form

I

For gas-phase catalytic reactions often the catalyst should have a radical nature with a relatively low value of activation energy for formation of a catalytic complex. If the catalyst is a simple molecule the negative effect of the entropy factor is diminished.

28 2.2. A c i d - b a s e c a t a l y s i s

General Bronsted acid catalysis begins with the addition of a proton whilst general Bronsted base catalysis begins with the removal of a proton: X+HA ~ XH++A YH+B ~ Y-+BH + Specific acid catalysis starts with the substrate activation by H30 + (H +) species and the reaction rate is given by r = kii+ [S] [H+]

(2.1)

An example is the hydration of unsaturated aldehydes O

O

II

II ./H

R -C -OR' +Ha0+

R -C -O \

÷

H20

R'

O

O

II +/. R-C-O\

...... + 2H20 R'

II

"

R-C-OH +H30 +

+R'OH

Specific base catalysis starts with the substrate (S) activation by OH- species (hydrolysis of esters, aldol condensation) O

O

II

II

R -C -OR' +

H OH

R - C - O H +OH +R'OH

OH"

and the reaction rate is then expressed (2.2)

r = koH-[S] [OH-]

For a reaction S ~ P which is catalyzed by both acid and bases S + H+ ~

P+ H ÷ • S + O H - ~

P+OH-

the rate follows the expressions r = ko[S] + k H+[S] [H+]+ koH-[S] [ON-] = k' [g]

(2.3)

where for water solutiions k'

ko + k H+[H+]+ k oH-Kw/[H +]

(2.4)

29

as [OH-] ~ Kw/[H +] with Kw= l0 -14 mol2dm-6 at 25°C) It follows from eq. (2.4) that at high acid concentrations, catalysis by hydroxide ions is minor, while at high base concentrations, catalysis by hydrogen ions is minor.

1

lgklgk°

pH

lgk'-lgkoH_+lgKw+PH Figure 2.1. Dependence of the rate constant on pH.

Acid catalysed reactions involve formation of rc complexes

and carbonium ions

which react further and the proton is fully recovered

zc I~+x-

/

+ H-R

I -~c~x "

z c I® \+

/C

H + R+

/

I~

+ "'\H

" ±-~o / I/ ®I

/C~O~H

2.3. Catalysis by transition metals Catalysis by organometallic compounds is based on activation of the substrates by coordinating it to the metal, which lowers the activation energy of the reaction between substrates. As in other types of catalysis the use of a homogeneous catalyst in a reaction provides a new pathway, because the reactants interact with the metallic complex first. Homogeneous transition metal catalysts are increasingly being applied in industrial processes to obtain bulk chemicals, fine chemicals and polymers. Examples of metal complex catalysts are: RhCI(PPh3)3 for olefins hydrogenation, C02(CO)s for carbonylation and metallocenes for polymerization.

30 Industrial applications are toluene and xylene oxidation to acids, oxidation of ethene to aldehyde, carbonylation of methanol and methyl acetate, polymerization over metallocenes (Figure 2.2), hydroformylation of alkenes, etc.

M(CH3)2

Figure 2.2. Metallocenecatalysts A facinating application of homogeneous catalysis is asymmetric catalysis. The 2001 Nobel Prize in Chemistry was given for research in the field of chiral transition metal catalysts for stereoselective hydrogenations and oxidations.

Ryoii Noyori

William S. Knowles

K. Barry Sharpless

Many of the compounds associated with living organisms are chiral (not superposable on its mirror image), for example DNA, enzymes, antibodies and hormones.

Figure 2.3. Chiral enantiomers Therefore enantiomers (Figure 2.3) of compounds (e.g. pairs of optical isomers) may have distinctly different biological activity. For many drugs, only one of these enantiomers has a beneficial effect, and the other enantiomer can be inactive or even toxic. In 1968 Knowles at Monsanto Company, St. Louis showed that a chiral transition metal based catalyst could transfer chirality to a nonchiral substrate resulting in a chiral product with one of the enantiomers in excess. Knowles's aim was to develop an industrial synthesis process for the rare amino acid LDOPA which had proved useful in the treatment of Parkinson's

31 disease. Knowles and co-workers at Monsanto discovered that a cationic rhodium complex containing DiPAMP (Figure 2.4), a chelating diphosphine with two chiral phophorus atoms, catalyzes highly enantioselective hydrogenations of enamides (Figure 2.5). H

H2N--C 1

C\ OH

d ,00

HC" / C ' ~ c H

II

I

HO"-~C~C "-..0 H

I OH

DiPAMP

Figure 2.4. a) L-DOPA, b) DiPAMP

The pathway to L-DOPA including asymmetric hydrogenation is depicted in Figure 2.5. Rh(DiPAMP}

C

D(97.5%1

L- DOPA (S- DOPA)

Me = ella Ac = CHACO

Figure 2.5. Synthesis of L-DOPA.

Noyori discovered a chiral diphosphine complex, BINAP. Rh(I) complexes of the enantiomers of BINAP are remarkably effective in various kinds of hydrogenation reactions. Ph

(SFBNAP

i

mirrorplane

Ph

Fh

IR)-BNAP

H2 99.5%

Figure 2.6. Structure and application of BINAP complexes.

Using titanium(IV) tetraisopropoxide, tert-butyl hydroperoxide, and an enantiomerically pure dialkyl tartrate, the Sharpless reaction (Figure 2.7) accomplishes the epoxidation of allylic alcohols with excellent stereoselectivity. Organometallic catalysts also include specific ligands besides the atom or group of metal atoms. They can be easily modified by ligand exchange. A very large number of different types of ligands can coordinate to transition metal ions. Once the ligands are coordinated, the reactivity of the metals may change dramatically. The rate and selectivity of a given process can be optimized to the desired level by controlling the ligand enviro~nent.

32

(S~'SJ- d i eth yl ta rt rat e (-)-DET

1 l

R" R'" R'~.-~o H

Re (R/R) diethyl tartrate

(+)-DE-

Figure 2.7. Sharplessepoxidation. Transition metals have partially occupied d-orbitals, the symmetry of which is suitable for formation of chemical bonds with neutral molecules. These metals also have several stable oxidation states and can have different coordination numbers as a result of the changes in the number of d-electrons (Figure 2.8). d,c,, d×,, dyz J

d6

d8

d lo

P d ( I V ) ~ Pd(lll) ~ Pd(ll) ~ Pd(I) ~ Pd(O) d2sp 3 dsp 2 spa m

dxy, dx=, dyz

-.x z yZ~ - z z

A ~80

250kJ/mole

Figure 2.8. Formationof chemicalbonds duringcatalysis. Different types of elementary steps are possible with organometallic catalysts (Figure 2.9)

Substitution

Isomerization

/.-7 Reductiveelimination

insertion

X~~BB A > ~ A + B x ~'A-X Oxidative addition

X~-~

+ A-B = = = ~ ~ B

13_elimination

A

H r M_CH2/CH-R

Figure 2.9. Elementarysteps in organometalliccatalysis.

.

M ~H + CH2=CHR

33 Catalytic cylcles in homogeneous catalysis involve changes in the state of the central ion during the reaction. At the same time the initial state of the central ion could be the same or different from the final state. Examples for the mechanisms of double bond migration or hydrogenations reactions which occur in the systems forming catalytic cycles are given in Figures 2.10 and 2.11 1 ),

1

C I 4 P d -2 +

"

Cl3Pd

3 ...i- C I

1 2 Cl3Pd

-

Cl2Pd'~2 -

~

Cl2Pd

+

Cl

1 Cl3Pd~3 2

1

Cl3Pd~3 2 -

2

1

Cl4Pd-2 + ~3 2

Figure 2.10. Mechanism for double bond migration over an organometallic catalyst.

cu L\ /H L/M "CH 2

L\ /H L.-'M"-.H

02H4

Figure 2.11. Mechanism of a hydrogenation reaction over an organometallic catalyst.

which could be presented in a general form for a reaction A ~ P (Figure 2.12)

MC: Figure 2.12. A general form of a catalytic cycle for a reaction A ~ P

In some other cases, like hydrolysis of ethylene the catalytic cycle is not closed

34

M'

P

and requires furter transformations of palladium in the zero oxidation state to Pd 2+. " PdCl3(C2H4)- + Cl-

PdCl4:-2+ C2H4

PdCI3(C2H4)- + H20

"

PdCI2(C2H4)(H20) PdCI2(C2H4)(OH)

PdCI2(C2H4)(H20) + Cl-

" PdCl2(C2H4)(OH)- + H + " PdCI2(CH2CH2OH)-

CI2Pd-CH2CH2OH C2H4 + P d C ~ + H20

~" P@+ 2Cl- + CH3CHO+ H ÷

"

C H 3 C H O + P{f'+ 2HCl

This can be done, for example, by coupling ethylene hydrolysis with palldium reduction. C2H4+PdC12 +H20 ~ CH3CHO+2HCI+Pd Pd+2CuC12 ~ PdC12+2CuC1 2Cu + +1/2 O2+2H + ~ 2Cu2++H20 finally leading to a production of acetaldehyde from ethylene C2H4+ 1/202~CH3CHO and closing the catalytic cycle

............. m ~ - J

"-----~p

In industrial practice the process is organized in such a way that reaction and re-oxidation of Cu in performed in one reactor. Polymerization reactions can be described by similar types of catalytic cycles ~

/M H

CHz=CH2

Figure 2.13. Catalytic cycles for polymerization reactions. M = Ti, Zr, Cr, V; L = PR3.

35 Due to the fact, that organometallic complexes are highly soluble in organic solvents their behavior throughout the catalytic reaction can be studied even in-situ using various spectroscopic techniques, like NMR, IR, Raman. These measurements may provide information about the structure of complexes. Kinetic studies are much rarer in the study of homogeneous catalysis by transition complexes then for heterogeneous catalysis. One of the reasons could be that the generally adopted reaction schemes sometimes look too complicated for non specialists in kinetics, as analytical expressions could be very cumbersome to derive. Recent attempts were devoted to heterogenization of metal complexes to inorganic supports.

+

- - O \ ~ ~.. .% - - O ~ ~'' " / O

~

"0

]


>d l n y dx

(2.121)

dx

Taking 0 = N m / L the chemical potential can be obtained /~ = - k T ( # l n Q / # N ) N , L , T

(2.122)

The chemical potential of a gas molecule (Pgi) is expressed with the lowest energy state assigned as energy zero: (2.123)

/~gi = - kT In ( qg i/c)

Concentration c in the gas phase is defined by: (Pid/T) PiNATst/TPstVst (st- standard conditions). As the equilibrium condition is the equality of chemical potentials in the gas phase and in the adsorbed state, an adsorption isotherm follows naturally =

e

(2.124)

qm - O.n(O. )

The adsorption isotherm for an ideal adsorbed layer for molecules occupying more than one site on the adsorbent surface is then given by the following equation: a P = 0/(q~n - n(O))

Variation of the screening parameter with coverage is approximately linear

(2.125)

69 n = v - ~bO

(2.126)

where v is the screening parameter at zero coverage. Values for several systems are given in Table 2.2.

Table 2.2. Values of parameters for different adsorption with shielding. System

33

m

v

¢

1. Two-centred symmetrical species on a chain

1

2

3

1

2.Two-centred symmetrical species on a square lattice

2

2

7

3

3.Four-centred symmetrical species on a square lattice

1

4

9

5

4.Seven centred molecules (benzene) on a lattice

1

7

19

12

of equilateral triangles Let us consider an interacting adsorbed layer which consists o f three distinct species: activated complexes o f arbitrary configuration (total number o f different kind o f complexes M0; multi-centred adsorbed species (total number o f different types Mm); uni-centred adsorbed species (total number = Mh). The partition function for the adsorbed layer is given by Ma

A,/t

A,Ih

Q. = [-[(f, 'c,;)I--Iff,,,c..:)l-I(f;,'ch,) i=l

j=l

(2.127)

h= l

The number o f ways in which multi-centred molecules can be distributed is formulated

C

N,3tx i

k=l

~-~---Z-Pn,,:,(O:,) - k - L1 p = l l:~p

nlitJ

(Op )

(2.128)

where p refers to species which are already adsorbed on the surface. For instance, for the species o f third type, the middle term in eq. (2.128) is given by

o,

,71.. k2, eq. (3.36) is simplified to k lk 2C~ C B k 2Cz = KIk2CAC B - k 2Cz kl

r-

(3.38)

with the equilibrium constant for the first step K1 kl/kO ,'(

04 • S

~!

:i(

8888

'

",~=0

8i2

'

8i4

'

2-"---J8

,8

Coverage Figure .3.14. Dependence o£ reaction rate on coverage for non-ideal surfaces.

In the case of ideal surfaces using a similar approach as for adsorption (3.73) also for desorption r =2"0=2"

at)

(3.92)

l+ap

the ratio of rate is then r /r+ - 2" ap 2"+ P

(3.93)

At equilibrium the gas pressure is equal to the fugacity p P and the rate in the reverse direction is equal to the rate in the forward direction r_=r+ giving the ratio of rate constants on each site Z

-

a

2"+ and consequently

(3.94)

99

r = r+ p

(3.95)

This equation is valid for the contributions of each site of a nonuniform surface to adsorption and desorption rates and hence for total r ,r+ values on a nonuniform surfaces. From (3.86) it follows with n 1-m r = r + Pp = k + 7 P7 -- " K±p 1-,,, = k ± p n

(3.96)

Replacing the fugacity by coverage p = (1 / a o )e re we arrive at the Langmuir desorption equation, which shows the dependence of the desorption rate on coverage

r_ = k± ao-'~e ~r°

(3.97)

where n=l-m=l-(z=[3 and 0

k_+ =

~ Y 2"+ sin(m~) e ~ - 1 a o'

(3.98)

As m+n=l, sin(m•) = sin(nr 0, 2`o / 2o = ao and k+ =

~r 2" Z °a0' sin(nit) e g - 1

(3.99)

the desorption rate for evenly nonuniform surfaces is

r -

~r Z sin(,6'K) f

0

e m° = k e #°

(3.100)

Often in the literature the adsorption rate is expressed by r+ = S ( 0 ) P

(1 - O )

(3.101)

assuming that the sticking coefficient is coverage dependent (Figure 3.15). In the treatment presented above a special model of nonuniformity was presented which takes into account sophisticated surface structure and the existance of different crystallographical planes with different reactivity. An interesting and industrially relevant situation is when nm size metal crystallites on various supports act as catalytically active material. Metal nanoparticles supported on inorganic and organic matrices have shown promising features, like higher catalytic activity and/or selectivity than conventional catalysts in many catalytic reactions. The origin for

100 these effects is the size quantization of most electronic properties. The metal nanoparticles exhibit unique properties that differ from the bulk substances, e.g. different heat capacity, vapor pressure and melting point. Moreover, as indicated above, when decreasing the metal particle size sufficiently enough, there occurs the transition of the electronic state from metallic to a nonmetallic one. Additionally metal nanoparticles exhibit large surface-to-volume ratio and increased number of edges, corners and faces leading to altered catalytic activity and selectivity. Sticking probability 1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

Figure 3.15. Illustration of the possible dependence of the sticking coefficientsof coverage.

Figure 3.16. Schematic illustration of the metal crystallite with different crystallographical planes. In such a case the kinetics will be an interplay of kinetics on different facets of a catalyst particle and it is possbile, that the activity of a catalyst particle may be higher than that calculated by assuming that the facets operate independently. There are then essential grounds to expect that future kinetic modeling on catalytic reactions over nm- size metal particles, which we can coin as nanokinetics, will include surface heterogeneity (e.g. apriori due to different activities of distinct catalyst sites or/and induced due to lateral interactions, which will be considered below). It should also be mentioned, that crystallite shape transformations due to adsorbed reactants may affect the steady-state kinetics of catalytic reactions. 3.6. 2. Lateral interactions

Similar to adsorption on real interactions and more specifically widely used lattice gas model the coverage is complex and cannot be

surfaces, kinetic models based on involvement of lateral lattice gas models have been developed in literature. In the relationships between the rate of an elementary reaction and written in a closed form when this model is used.

101 In the model each adsorbate is assumed to be localized on a two-dimensional array of surface sites and each site is assumed to be either vacant or occupied by a single adsorbate. Quasi chemical approximation of the lattice gas model assumes that the adsorbate maintains an equilibrium distribution on the surface. The lattice gas model with this approximation was used for the description of the reaction of gases on metal surfaces. Among the studied reactions was the steady state oxidation of CO over Ir, hydrogen over Pt, the kinetics of the CO - NO and CO-O2 reactions over Rh and Pt, showing a complex dynamic behavior. Another example is the reaction of NO+H2 on Pt(110) which shows quite complex behaviour, i.e. the multiplicity of steady states, the reaction oscillations were observed (Figure 3.17)

g g i' o

"I~AE (~)

TIUIE ~s)

Fig. 3.17. Rate of N2 desorption as a function of time during the N O H 2 reaction on Pt(l 0 0) at PNO 3×10 9 bar and T-460 K: (a) period-I oscillations at PNo/PH2--1;(b) period-2, (c) period-4, and (d) and aperiodic oscillations at PNo/PII2=I.4. (V. P. Zhdanov, impact of surface science on the understanding of kinetics of heterogeneous catalytic reactions. Smjace Scence, 500 (2002) 966).

Theoretical investigations of this model (A. G. Makeev, B. E. Nieuwenhuys, Mathematical modeling of the NO + H2/Pt(100) reaction: "Surface explosion," kinetic oscillations, and chaos, Journal of Chemical Physics, 108 (1998) 3740-3749) with 11 reversible and irreversible elementary steps included lateral interactions for only two steps in the forward direction and two steps in the reverse direction, leading to the following rate expressions r, = k f l , 1 a

1 = 0, +

0, exp(G, / RT)

(3.102)

where 0, is the coverage of vacant sites, m is the number of nearest -neighbour sites, and e is the energy of lateral interactions, which were determined by a fitting procedure to provide the best description of all experimental data. This model was able to reproduce many kinds of non-linear behaviour including kinetic oscillations and the transition to chaos. Unfortunately the cumbersome character of the lattice gas

102 model reduces its application to the kinetics of concrete catalytic reactions, especially when the reaction is a complex one. In distinction from the more refined, and thus much more complicated lattice-gas model, the form of the model of the surface electronic gas provides possibilities for its application to chemisorption of gas mixtures and thus to modelling of kinetics of complex reactions. Derivation of multicomponent chemisorption isotherms based on thermodynamic approach was presented in the previous chapter. Within the framework of this model the following generalized elementary reaction A + Z I + Z ~ S is considered. This reaction is written as a three-body collision, which is highly improbable, but is presented here only for illustrative purposes of how to express the reaction rate r=k

f ~ PA 00 0

(3.103)

where 0 is coverage and d 2

J; =exp(~,r/i

* O,C* /T) I-Iexp(co,~7~r//0/C/T)

(3.104)

.l=j I~j.~i ,f

./,'~ =exp o; #q,o,C /r)

1--Iexp(coj q, q

OjC/T)

(3.105)

.j-/[,/~i

here f ~ a n d f are the activity coefficients of the transition state and the substrate 1 in the adsorbed condition, q is the effective charge acquired by an adsorbed particle, rl~ is the effective charge of the transition state, proportional to rl via the Polanyi relationship (the bridge between kinetics and thermodynamics), ooi and co, can be either +1 (repulsive interactions) or -1 (attractive). Constant C* is expressed by C* = h 2/4N~m*, where h is the Plank constant, m* is the effective electron mass and k is the Boltzmann constant. Eq. (3.104) and (3.105) take into account all the possible lateral interactions between all the surface adsorbed species present on the surface and thus are given in a generalized form for a gas mixture, which contains components of i andj type. We do not present in this book examples of data fitting and demonstrating the superiority of the models of non uniform surfaces, as this has been done many times in the past for adsorption and reaction kinetics. A series of processes such as hydrogenation of organic compounds, configuration isomerization, oxidation of methane, ethylene, CO and alcohols, oxidative ammoxidation, oxidative dehydrogenation, oxidative chlorination, methanol synthesis and isotopic exchange, were mentioned in the literature when the expressions corresponding to a definite scheme on non uniform surfaces more adequately described the experimental data than classical kinetic equations, based on the models of ideal adsorbed layers. In these examples the reaction mechanisms were treated as if they occur through a two-step sequence on biographical non uniform surfaces. This sequence will be discussed in more detail in Chapter 7, where it will be

103 demonstrated that for the two-step sequence in the region of medium coverages the kinetic equations for biographical and induced non uniform surfaces have the same form. The treatment based on the two-step sequence for nonideal surfaces originates from the complexity of deriving the explicit form of rate equations for other reaction mechanisms on biographical non uniform surfaces. At the same time models based on lateral interactions have no restrictions from this point of view as the implicit form can be and is used for the data fitting. In some particular cases when the reaction occurs either at low or at high surface coverage the kinetics is insensitive to surface non uniformity. Thus the description has to be exactly the same when applying either concepts of uniform or non uniform surfaces, as generally speaking the uniform surface can be treated just as a special case of the more general model of non uniform surfaces. Using models assuming lateral interactions one arrives at the description based on the assumption of uniform surface simply setting the effective charges of adsorbed species equal to zero in the surface electronic gas model or setting the energy of interactions to zero in more refined lattice gas models. Therefore, within the framework of these models if the parameter estimation is statistically correct the kinetic models of uniibrm surfaces can never be superior to non uniform surfaces. That seems to be another advantage of utilizing adsorbate-adsorbate lateral interaction models in distinction from the models of biographical non uniformity with certain distribution of adsorption energies. For example, if a comparison is made of models for ideal and non-ideal surfaces using the Hougen-Watson approach (only one elementary step is rate controlling) and applying the expressions for adsorption isotherms for non uniform surfaces in the region of medium coverages, it could lead to a result which looks rather puzzling from the first glance, namely that in several cases the fits based on the models of non uniform surfaces were extremely bad in comparison with uniform surfaces. The reason for these statistically unacceptable descriptions is that the assumptions of the medium coverage did not hold.

3.6. 3. Limited mobility of adsorbed species in chemically reactive adsorbed overlayers, the probabilities of arrangements of adsorbed particles and accordingly the reaction rate are defined by the interplay between adsorption, reaction, and adsorbate diffusion. The activation energy for surface diffusion is often relatively low-, therefore the adsorption overlayer is close to equilibrium giving a framework for analysis as presented above. When diffusion of some of the reactants is slow compared to other steps the arrangements of adsorbed particles is often far from equilibrium. In particular, immobile reactants may form islands (Figure 3.18). Hi{iiiiiiiHiiiiiiiiiiiiiiiiiii

Figure 3.18. Formationof islands on the surfaces.

104 If the sites interact attractively, the occupied sites will form more or less round islands. The sites at the edge of the island have fewer occupied neighbors and are thus less stable than the sites in the interior of the island. intuitively it is clear that if there is a reaction of adsobed atoms of type A forming an island and another adsorbed atom B, then A atoms adsorbed in the interior of the islands are inactive, and only those on the island perimeters are reactive. Then the rate is proportional to the fraction of the total area which is reactive. 3.7. Deterministic and stochastic models

Mathematical models of catalytic systems in the general form are rather sophisticated. Often, they consist of nonlinear systems of differential equations containing both conventional equations and equations with partial derivatives of parabolic, hyperbolic, and other forms. Efficient simulation is only possible if a well developed qualitative theory of differential equations (mainly, equations with partial derivatives) and high performance programs for computational experiments exist. In the modeling of catalytic reactions at the molecular level, the stochastic approach is also fruitful along with the simulation based on equations (the deterministic approach). Stochastic simulations (the dynamic Monte Carlo method) makes it possible to penetrate into the microlevel and monitor detailed changes in the adsorption layer, and explain the observed phenomena. For the stochastic methods the equation of motion is artificial and tries to generate configurations with certain statistical properties giving a chance to follow the evolution of the adsorbed layer. In general, Monte Carlo methods refer to any procedures which involve sampling from random numbers. These methods are used in simulation of natural phenomena, simulation of experimental apparatus and numerical analysis. An important feature is the simple structure of the computational algorithm. The method was developed by yon Neuman, Ulam and Metroplois, during World War II to study the diffusion of neutrons in fissionable materials (e.g., atomic bomb design). Let us consider atom diffusion and demonstrate the principle of the Monte Carlo method. A two-dimensional square grid (Figure 3.19) represents interstitial sites in a solid. O O O O Figure 3.19. Two-dimentional grid. The diffusion of the interstitial atoms can be simulated as a 'hop' of an atom from an occupied site to an open site. h°l© 0 O C} Q Figure 3.20. Illustration of diffusion.

105 A convenient way to choose which atom should hop and in which direction is to use random numbers. For example, using a random number to choose a particular lattice site, than another random number to decide which direction the interstitial atom should hop. If the neighbor site is open, the hop is effective. In the opposte case another site is picked. The application of Monte-Carlo simulations to non-equilibrium reaction systems in heterogeneous catalysis started by Ziff, Gulari and Barshad on the lattice-gas version of a simple Langmuir-Hinshelwood model of CO oxidation on a transition metal surface. The ZGB-model is a lattice-gas version of the Langmuir-Hinshelwood-model of CO oxidation. The reactions are: 1.CO (gas)+*~--~CO(ads), 2.O2(gas)+2*+-~20(ads), 3. CO(ads)+O(ads)--~CO2(ads)+*, 4.COffads) --~COffgas)+*. The probabilities of steps 1 and 2 are between 0 and 1, while probabilities of other steps are P(3) = 1, P(4) = 1, P(-1)= O; P(-2) = 0, P(4)=0. The ZGB-model shows the effect of heterogeneity in the adlayer; because of the infinitely fast formation of CO2, there is a segregation of the reactants in CO and oxygen islands. The original model has later been extended and modified by numerous people to include desorption of the reactants, diffusion, an Eley Rideal mechanism for the oxidation step, physisorption of the reactants, lateral interactions, an oxidation step with a finite rate constant, surface reconstruction and additional poisoning adsorbates. The unit of time is the Monte-Carlo step which corresponds to one trial per site. The relation between a Monte-Carlo step and real time is not always made explicit, but usually one MonteCarlo step is 1/R on average, where R is either the sum of the rate constants, or the maximum rate constant. In the simulations a label is attached to each grid point, specifying the adsorbate at the site. Due to reactions, this implies that the labels will change during a simulation, which is in fact a change of the labels according to reactions, and the determination of times when the reactions take place. The specification of a reaction consists of a set of grid points, labels attached to them corresponding to the adsorbates before the reaction has taken place (reactants), labels corresponding to the adsorbates after the reaction has taken place (products), and some rate constant. The set of grid points should be regarded as a representation for all sets of grid points where the reaction may occur. All these sets are related via translational symmetry and possibly (combinations with) rotations and reflections. The evolution of the adlayer and the substrate is described by the Master equation alP, = ~_, (W~,~ps~ _ W~,pPp ) dt

(3.106)

where o~ and f3 refer to the configuration of the adlayer, the P's are the probabilities of the configurations, t is time, and the Vf's are transition probabilities per unit time, calculated within the framework of transition state theory. These transition probabilities give the rates with which reactions change the occupations of the sites. They are very similar to reaction rate constants. W ~ corresponds to the reaction that changes [3 into (~. For desorption A(ads) ~*+A(gas), where A is the particle that desorbs, and * is a vacant site, the coverage of A is 04 = 1~-~ P~A~ •

S

a

(3.107)

106 with S the number o f sites in the system and A~ is the number o f A's o f configuration c~. The desorption rate is then proportional to coverage. Monte-Carlo methods are able to simulate rather complicated nonlinear phenomena, like periodic oscillations and formation o f waves on the catalyst surfaces, mimicking experimental observations (Figuresw 3.21-3-23).

Figure 3.21. FEM images of the oscillating CO oxidation over the (1 0 0)-oriented Pt-tip. (E.I. Latkin, V. I. EIokhin, V.V. Gorodetskii, Chemical Engineering Journal 91 (2003) 123-131).

/Vv'4V"~4Vwuv'm~uVV'~"v"/V~%W~t ~0 • O

2000

4000

J

6000

,

~

8000

,

m

-

10000

Time, MCS

Figure 3.22. Dynamics of the specific rate of CO2 formation (E.I. Latkin, V. I. Elokhin, V.V. Gorodetskii, Chemical Engineering Journal 91 (2003) 123-131).

Figure 3.23. Snapshots showing the initial stages of the spiral wave formation. (E.I. Latkin, V. 1. Elokhin, V.V. Gorodetskii, Chemical Engineering Journal 91 (2003) 123-131).

107

3.8. Microkinetic modelling Microkinetic modeling is a framework for assembling the microscopic information provided by atomistic simulations and electronic structure calculations to obtain macroscopic predictions of physical and chemical phenomena in systems involving chemical transformations. In such an approach the particular catalytic reaction mechanism is expressed in terms of its most elementary steps. In contrast to the Langmuir-Hinshelwood-Hougen-Watson (LHHW) formulations, no rate-determining mechanistic step (RDS) is assumed.

Olaf Hougen The LHHW approach started to be popular in the 40-s, when powerful computers and corresponding software were not at hand to perform rigorous kinetic analysis of complex systems. Interestingly already in the 50-60s there was an understanding that the formulation of the rate expressions based on the original theory of Langmuir adopted by Hinshelwood and widely applied in this form for technical process development is a crude approximation. That the theory of complex reactions kinetics went beyond LHHW treatment is due in major part to Horiuti and Temkin. Up to now it serves as a basis for mathematical modeling of catalytic processes and reactors at stationary conditions. The reaction and the process are considered to be stationary, if the concentration of all reactants and products in any element of the reactor space, including the active catalyst surthce, do not change in time. At stationary conditions, concentrations of the intermediates are time independent as the rates of their generation in elementary steps are equal to the rates of consumption in other elementary reactions. Kinetic treatment based on the theory of complex reactions introduced the necessity to calculate quite many parameters (pre-exponential factors, activation energies of elementary reactions, etc.). Therefore a need to estimate independently the rates and surface coverage called for the application of theoretical approaches, based on thermodynamics and transition state theory, as well as other tools (ultra-high vacuum studies, spectroscopy) to get necessary data and reduce the number of parameters in statistical data fitting. This approach started to be developed in the 70s, when more powerful computers became available and an input in the computer program required all the elementary reactions. Then such programs constructed matrices of mass balance equations, comprising all of the components in the reactions (in the case of heterogeneous catalysis, this included the surface and the adsorbed species) and solved these equations by Newton-Raphson iteration. Later on this approach was refined and coined micro-kinetic modeling. Microkinetic models are much more widely applicable than LHHW traditional models which assume an RDS, since the RDS can change with reaction conditions. Because all postulated elementary steps are included explicitly, accurate rate parameters for all of the forward and

108 reverse reactions are needed to solve the equations comprising the model. This requirement greatly increases the amount of information needed to create a microkinetic model, but this is also the power of the technique. Often, the information needed to create a microkinetic model cannot be determined from one set of experiments but requires the compilation of information from many different experiments and theoretical investigations. The wider utilization of microkinetic models is somewhat retarded by the vast amount of information needed about interactions of chemical intermediates with complex, heterogeneous catalysts. The microkinetic approach has been applied to numerous diverse chemistries including cracking, hydrogenation, hydrogenolyis, hydrogenation, oxidation reactions and ammonia synthesis to name a few-. Microkinetic modeling assembles molecular-level information obtained from quantum chemical calculations, atomistic simulations and experiments to quantify the kinetic behavior at given reaction conditions on a particular catalyst sur~hce. In a postulated reaction mechanism the rate parameters are specified for each elementary reaction. For instance adsorption preexponential terms, which are in units of c m 3 mo1-1 s-~, have been typically assigned the values of the standard collision n u m b e r (1013 c m 3 mol l s-l). The pre-exponential term (cm -2 mol sl ) of the bimolecular surface reaction in case of immobile or moble transition state is 1021 . The same number holds for the bimolecular surface reaction between one mobile and one immobile adsorbate producing an immobile transition state. However, often parameters must still be fitted to experimental data, and this limits the predictive capability that microkinetic modeling inherently offers. A detailed account of microkinetic modelling is provided by P. Stoltze, Progress in Surface Science, 65 (2000) 65-150.

3.9. Compensation effect The activation energy for a reaction is sometimes measured under different reaction conditions. An example might be a measurement using a very active catalyst at a moderate temperature and a measurement using a less active catalysts at higher temperatures. The increase in temperature partially compensates for the lower activity. Contrary to expectations of equal activation energy, however, different activation energies and different preexponential factors are found. A large value for the activation energy is correlated with a large prefactor and all lines in the Arrhenius plot intersect in a single point, the isokinetic point. The correlation between activation energy and preexponential factor is known as the compensation effect (Figure 3.24). Although many attempts have been made to explain the compensation effect within the framework of TST, the effect itself does not follow from the conventional TST, (e.g. any connection between activation entropy and activation enthalpy). The majority of examples of compensation in heterogeneous catalysis are founded in the use of apparent kinetic parameters. Hence a complex reaction mechanism is involved and there is no point in trying to establish a relatrionship between the pre-exponential factor and activation energy based on for instance transition state theory. For a catalytic reaction where the surface coverage on different catalysts is not the same, either a zero or first oder dependence is possible. Therefore, at a low surface coverage one value for the activation energy will be obtained, while at a high surface coverage another one.

109 In(r)

,,

'\\ \

\

\ m 1

Figure 3.24. Illustration of compensation effect. It should thus be kept in mind, that the value of Eexp will, in general, be a function of reactant concentration, it cannot be emphasized too strongly that the value of E~xp has no fundamental significance, except perhaps in the case of zero-order reactions, and for it to have any meaning at all, it is essential for the conditions used for its measurement to be precisely stated. It was realized many years ago that compensation effects can be considered to be based on either analysis of multistep complex reactions or elementary reactions. The supposition that if the compensation effect exists it cannot be attributed to an elementary process was introduced at least 40 years ago by Kiperman and was pointed once more by Bond (G.C.Bond, M A. Keane, H. Kral, J. A. Lercher, Compensation phenomena in heterogeneous ctalysis: General principles and a possible explanation, CatalysisReviews, 42 (2000) 323) that very frequently observed correlation in the literature between activation energy and pre-exponential factor arises from the use of apparent rather than true activation energies, with the most common explanation for that being either the surface heterogeneity or the occurrence of two or more concurrent reactions.

MCKC04.fm Page 111 Friday, August 12, 2005 2:40 PM

112 Fractional stoichiometric numbers are allowable, for example it can be written for S02 oxidation. 2 Z + 0 2 2ZO ZO+SO2_Z+SOa.

1 2

2Z +02 2ZO ZO+SO2Z+SO~

0.5 1

1//2 0 2 + S 0 2 = S 0 3

02 +2S02=2803

The overall equation of the last scheme is obtained from the overall equation of the previous one when it is multiplied by 1/2. Such an operation is senseless for the equations of simple stages, if the reaction Z +1/202 ZO does not occur. The reaction 1/2 02 +SO2=SO3 describes only stoichiometry, but not the reaction mechanism. A set of stoichiometric numbers of the stages producing an overall reaction equation is called after Horiuti a "reaction route". Routes must be essentially different and it is impossible to obtain one route from another through multiplication by a number, although their respective overall equations can be identical. N (1) N (2) 1. O2+2Z---~2 ZO 2. H2+2ZZC + Z'

0

1

0

0

3.ZB + Z, H2 4. ZC + Z, Hz

k~ >ZD+ Z' k4 ) ZD+ Z'

0 0

0 0

1 0

0 1

I.Z+A---ZA II. Z + B = ZB III. Z+C---ZC IV.Z+D--ZD V. Z'+H 2 =Z'H

2

N (2) N (3) 1 0 0 1 -1 0 0 -1 1 1 0 0

N °) : A + H 2 ~ B; N (2) • A + H 2 ~ C;

N (3) " g +

N (4) 0 0 1 -1 1 0

H 2 ~ D;

(4.129)

N (4) " C +

U2 ~ D

where ZA represents adsorbed cinnamaldehyde, ZB adsorbed cinnamylalcohol etc. In the above mechanism, Z is a surface site o f the first type and Z' o f the second type, where hydrogen is molecularly adsorbed. On the right hand side, stoichiometric numbers for the different routes (N 0~, N (2) , etc.) are given. Deriving kinetic equations from the mechanism above one arrives at following equations:

=

kt KA KH~cA P a ' o ~ ~ (1 + KAc A + KBc ~ + K c c c + Kncj))(1 + K n 2 p , 2) k 2K

/'2

(4.130)

AKn,, c Apn2

(4.131)

(I+KAc4 + K , c , + K c c c + K , c , ) ( I + K H ~ p n ~ ) k3KBKH2 cB PH,

=

(4.132)

-

(1 + K ~cA + Knc B + Kcc c + K~)cj))(1 + KH2p,2) r4 =

k4KcKH2ccpH~ ° (1 + KAC4 + K , c , + K c c c + Kj>cI>)(1 + K n . p , 2 )

,, (4.1.~.~)

where kl is rate constant o f reaction 1, KA is adsorption constant and CA concentration o f component (A) etc., PH2 is hydrogen partial pressure. Equations (4.130-4.133) should be further combined to the mass balances o f the components 1 dc A -

pdt

r~-r2;

1 dc B ---rl-r;; pdt

1 dc c ---rz-r4; pdt

1 dc, ---r;+r pdt

4

(4.134)

140 where p is the mass of catalyst-to-liquid volume ratio. Although numerical parameter estimation shows a relatively good agreement between the experimentally obtained and the predicted concentrations with a high value for the degree of explanation (Figure 4.20) and it superficually seems that the model predicts reaction behavior, it is important to verify whether selectivity is correctly predicted. 120 433 K, 37 bar

'~&

80

°=

[]

[]

[]

[] []

40 D

0 0

50

100

150

200

Time [rain]

Figure 4.20. Reactants concentration as a function of time in cinnamaldehyde hydrogenation- mechanism (4.129) ( J. Hfijek, J. W~irnfi, D.Yu. Murzin, Liquid-phase hydrogenation of cimlamaldehyde over Ru-Sn sol-gel catalyst. Part II Kinetic modeling, industrial & Engineering Chemistry Research, 43 (2004) 2039).

Example plots in the Figure 4.21 indicate that the proposed kinetic model cannot fully explain the development of the selectivities with hydrogen pressure. The intrinsic discrepancy between the calculated and experimental selectivity is apparently clear from analysis of the ratio of initial selectivities to (B) and (C) at low conversions, which is defined as r~

kl KA cA PH:

kl

r(

k 2 K Ac,4 P it-:

k2

(4.135)

0.5 •[] 0.4

g~ 0.3

[] O

0.2 0.1

1 [

O

°% []

433 K

/

0

0.25

0.5

0.75

Convm~ion [%]

Figure 4.21. Selectivity vs conversion at different hydrogen partial pressures in cinnamaldehyde hydrogenationmechanism (4.129) (J. Hfijek, J. Wfirnfi, D.Yu. Murzin, Industrial & Engineering Chemistry Research, 43 (2004) 2039).

141 It is clearly seen, that the ratio in eq. 4.135 does not depend either on conversion or hydrogen pressure. Note that assumption o f competitive hydrogen adsorption does not change the conclusion above, as this assumption will inevitable lead to equation (4.135) as well. The model (4.129) thus has to be revised, as dependence o f selectivity on hydrogen pressure was observed experimentally. The modified mechanism is N (1)

N (2)

N (3)

N (4)

1. ZA + H 2 ¢:> ZAH 2

0

1

0

0

2. ZAH 2 ~ ZC

0

1

0

0

3. ZB + H 2 ¢:>

0

0

1

0

4. ZBH 2 ~ ZD

0

0

1

0

5. ZC + 82 4::5ZCH 2

0

0

0

1

6. ZCH 2 ~ ZD 7. Z + A = Z A 8. Z + B = Z B 9. z + c = z c 10. Z+D---ZD 11. Z * + A - - Z * A 12. Z * A + H 2 4; 2=> 1; 3=>2; 4=>3 and denominator should include S=2,

(5.79)

0)10)20)3;0)-40)2 0)3;0)-40)-I 0)3;0)-40)-I 0)-I

Continuing this proceduer if now counting starts from step 3, the subscripts in eq. (5.78) are replaced following the new sequence 1=>3; 2=>4; 3 =>1; 4=>2 s-3,

(5.80)

0)40)10)2;0)_30)1 0.)2;0)_30)_4 0)2;0)_30)_4 O.)_1

Finally if step 4 is taken as the starting point in counting 1=>2; 2 ~ 3 ; 3=>2; 4=>1 the following terms are added to denominator in (5.76) s-4,

(5.81)

0)3(-O40)1;0)-2034 0)1;O)-20)-3 0)1;(-O-20)-3 0)-4

For irreversible reactions equations (5.76-5.77) are simplified to 0)] 0)20)30)4

r =

(5.82)

O)20)30/4 q- 0)10)20)3 -}- O)40)10)2 -1-0)30)40)1

which in the particular case of the ~bur step positional isomerization reaction will give (the first step in the catalytic cycle is the reaction of ML2 with the olefin)

Q,

F=

k CCH6; 2=>1; 3=>2; 4=>3; 5=>4; 6=>5; meaning that step numbers in eq. (5.85) should be consequently replaced (e.g. instead of say 0)2 we should use 0)1; 0)-~is replaced by o0_6, etc.). Finally the denominator in eq. (5.84) should contain 36 terms. The overall equation will look pretty complicated, however the reduced form (all steps are irreversible) has a rather simple form F = O-)10)20)30)40950)6

Ccot

(5.86)

D'

where D ' = 0)/o20)30)40) 5 + O.)1g02003 0)4 (2)6 + (/)10.)20)3 (/)5 0.)6 -}-

(5.87)

(O1(O20)4(O5(O 6 -]- O)10)3 0.)4 0)5 (O6 -t- (02(03(04050) 6

In order to get the explicit expression for the reaction rate, the step frequencies should be replaced by rate constants and reactant concentrations. For instance, instead of the frequency of step 1 0)1 in the carbonylation mechanism we have k l C H x , etc. An even more complicated case is the situation of rhodium catalyzed hydroformylation, which can be represented by the following mechanism with 8 intermediates in the catalytic cycle (Figure 5.18) CH 3 CH=CH 2 +CO+H 2

C H 3 (CH2) 2 C(O)H

'

H

Ph3PJ,,.,,,.RIh__CO / ~ . ,

j/

Ph3P~B'~CO I

CO R Ph3P° OC~.Rh~ . . . . . . ~H PPh3 ~-. ..~

\

Pho;.!h.~ph3

R

Ph3Pto,,, ~ ,~H OC

!L

H phap/R!h%HPh3P/""'i'"xC(O)CH2CH2R CO H2 PhsP~,~.Rh£%,~C(O)CH2CH2R ~ Ph3P CO

PPh 3

.CH2CH2R Ph3D~°""lh--CO" Ph3P~'" ] CO

Figure 5.18. Hy&'oformylation o f olefms.

Ph3Pr,....... ~xCH2CH2R OC PPh3 j'

~ ~'~x CO

171 The general form of the rate equation is similar to the 6 step cycle ~' ~ 0 ) 1 ( ' 0 2 0 ) 3 0 ) 4 0 ) 5 0 ) 6 0 ) 7 0 ) 8

- - 0 ) - 10) _ 2 ~9 _ 3 (`0 _ 4 (`0 _.50) _ 60) _ 70) _ 8 ~;(~cat

\~I(qRR]

D"

but now the denominator in eq. (5.88) in its more general fourm should contain 64 terms. Simplification for the case when all the steps are irreversible gives r = (2)10)20)30)4(2)50)60)70)8

C~a;

(5.89)

D"

D"=

(O1(2)2(2)30)40)5(O60) 7 -]- (O10)20)30)40)50)6(O 8 Jr- ~010)2 0)3 0)4 0)5 0)7 0 ) 8 -]- O)10)20).30)40)60)70) 8

(5.90)

O,)10)20)30050)60)70)8 -1- 001002004(2)50060)7008 -1- (O1(O_tO)40)5(O6(O7(O8 Jr- 0)20)_3(04(050)6(070) 8

In a particular case of the hydroformylation reaction, if the first step is the reaction of a catalytic complex with an olefin the general form of equations (5.89-5.90) could be written as F ~

kiCol4;.k2k3P(ok4ksPH 2 k 6 k 7 P~70 k 8 D"

D"=

klClq/ink2k3]gcok4k5PH2k6k7~,o

~

(5.91)

•cat

.-}- klClqfmk2k3~cok4k5~2k6k8

+

+ k, Co;4;.k2k~P(:ok4&P. kTPco & + k,C..;4;.k2&P(:ok4k6kTPco & +

(5.92)

+ k, Cos~:~.k2k3P(ok5PH2k6k7Pcok8 + k, Co;~¢,.k2k4ksPmk6kTP~oks + +

k, Co;¢l;,k3Pcok4k5gs2k~,k7Pcoks

+

k2k3Pcok4k~gsk6kTPcok s

+

which could be simplified to

r=

tCo,,<J?of'.

, . 2 . . . . . . . . . 2 +kwp(2op.. C ..... k ColR1

mo

@+s K4L RI+ S Figure 6.28. Binding orS to Ro

the equilibrium constant is defined (once again in a "bio"logical but not a chemical way) K R = 4[e 0][s] [R1]

(6.62)

The 4 is included in the numerator because there are 4 free binding sites on Ro. In other words, the concentration of fi'ee binding sites is 4[R0]. The next step (Figure 6.29) is defined as S+R~> 1 at c < l ) similar considerations as above lead to (1 + a)" >> L(1 + ca)"

(6.77)

210

giving

an expression for Y since R ~ 1 (Figure 6.32)

a

(6.78)

[S]

Y

O~

Figure 6.32 Dependence of the fraction of occupied sites as a function of a (high affinity binding) Numerical analysis of eq. 6.71 reveals (Figure 6.33), that the cooperativity is more marked when the constant L = [TO]/[R 0] i s large, e.g. when the equilibrium between R 0 and TO is shifted in favour of tense sites. The cooperativity is also favoured at lower values of the dissociation constant c (Figure 6.33).

Y

e

.................... iii ..............

/ 0.5

..........

;

, :

0.0 ~~'~ 0

;:.~:5

5

a

Figure 6.33. Theoretical dependence of the saturation function (eq. 6.71 with n=4).

The model of Monod was later extended to reversible one- and two-substrate reactions. The results are however more complicated than justified by the experimental evidence in enzymes kinetics. For a two site model an expression for Y can be obtained from a general case taking the form l+cL y =

1 + c2L K~S°

(I+L) , (I+c2L) K ~

2(1+cL) 1+c2 L KI~So + S O

(6.79)

211 where the reaction rate is for the enzyme at rapid equilibrium conditions r = F/~,,,~x

(6.80)

and c is defined according to eq. 6.69. This model always gives positive cooperativity and with L - 0 or infinity there is no cooperativity. The sequential hypothesis (Koshland, Nemethy and Filmer) accepts the possibility that mixed enzymes, containing both types may occur. Once again an equilibrium would occur in solution in which the complete R and T structures simply represent the extremes (Figure 6.34).

iiiiiiiiiiiiiiiiiiiii!! ii q

iii ii!! iiiii

iiiiiiii

~i!i!iiiii~i i!i!~¸'"¸~i!i!i!iiiii! ....i.iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiG Figure 6.34. Equilibria according to the sequential hypothesis. The sequential hypothesis assumes that the substrate has a more direct influence on the shape of the enzyme. In the absence of substrate the enzyme would exist more or less entirely in the T-form which has a very poor fit for the substrate. As the substrate enters the active site, as usual by purely random collision, the enzyme protein moulds itself around the substrate to produce a good fit. This is known as i n d u c e d fit. The process of induced fit has resulted in the subunit to which the substrate has bound being converted to the R-conformation of the enzyme. For the model with two enzyme forms (R and T) with Kv=[R]/[T], the substrate only binds to the R form Ks=[RS]/[R][S]. At the same time the free site is mostly in the form of T (KT ES ~

> E+P

IES

Figure 6.46. Schemeof uncompetitiveinhibition. Kinetically speaking this case is very similar to competitive inhibition

r= s

or

ZCHOH+H2

3.ZCHOHH2+ZCO 4.ZCOZ+CO CH4+ H20-CO+3H2

(7.18)

The general form of this equation is given by eq. (5.76) which can be adopted for the scheme (7.18)

r=

klPcH k2PH~ok~k4 - k 1PH k 2PH k BPHk 4P(o -

-

D

'- . . . . .

Q,,~

(7.19)

with

D=k2PH2ok3k4

+k

~PH k3k4 + k

,PH k

2PH2k4 +k ~PH k 2PH k 3PH~

+ klP(,H k2Pv~ok3 + k 4P( ok2PH2ok3 + k 4Pcok 1PH k3 + k 4P( ok 1PH k 2PH~ + k4ktP{.H~k2PH~o + k_3PH~klP(,H~k2PH~o + k_3PH~k_4P(,ok2PH9 + k_3PH~k_4P(,okZN2

(7.97)

2.ZN_2+3He =Z+2 NH3 N2+3H2=2NH3 Here ZN2 is an adsorbed intermediate in the form of dinitrogen, step 1 is reversible and step 2 is in equilibrium. Based on the experimental data which covered the pressure range from below 1 atm up to 500 atm, it was proposed that near equilibrium, the reaction rate is described by the following equation, which is often referred to in the literature as the Temkin-Pyzhev equation

(7.98) NH 3

H 2

where m is a constant (0 < m < 1). Under equilibrium, the reaction rate equals 0; therefore, k+/k_=K, where K is the equilibrium constant. Hence, only one of the constants, either k+ or k_, together with m should be determined from the experimental data. Equation 7.98 was supported by numerous studies on various types of catalysts• This equation was based on the supposition that nitrogen chemisorption on a energetically nonuniform surface determines the rate of the overall reaction• The second step is, in fact, an equilibrium between the adsorbed nitrogen and the gas-phase concentrations between hydrogen and ammonia. The adsorption rate is determined by equation (3•86) which takes the form r+ -

k+P% tl5

'

PN2 while desorption is given by eq. (3.96)

(7.99)

243 1 m

n

r = G PN~ = k±Px~

(7.100)

where Pu~ is dinitrogen pressure in the gas phase and Px, is the fugacity of adsorbed nitrogen (i.e. the presure of dinitrogen in the state of ideal gas that would correspond to adsorption equilibrium with the surface at the present coverage). Fugacity is determined by the equilibrium of the second step

(7.101)

Taking into account eq. (7.101) the adsorption and desorption rates are 3

r+ = A:+t'N2I ~ )

(7.102

r = k ( PpN~H ~)

(7.103

I m

with k+ = k+_K" ; k = k+_K-~'

(7.104

The reaction rate is then given as r =t+ -r_ leading to eq. (7.98). At high pressures, eq. 7.98 should be modified to include the deviations from the laws of ideal gases and to incorporate the effect of pressure on the reaction rate depending on the volume change at activation. Therefore, eq 7.98 at high pressures contains not partial pressures, but fugacities, and additionally, the right-hand side of it includes a factor exp[-(V>~ - m V z ~ . ) P / R T ], where V~,~is the partial molar volume of the activated complex of nitrogen adsorption, VzN2is the partial molar volume of the adsorbed nitrogen and P is the total pressure. Although in the original derivation it was supposed that nitrogen is adsorbed in the molecular form, an assumption on the dissociation nitrogen adsorption also leads to the same equation. In the region far from equilibrium, it was suggested that the reaction rate is determined by two slow irreversible steps. The first step is nitrogen chemisorption, and the second one is the addition of hydrogen to molecular adsorbed nitrogen 1 .N2+Z¢=~ ZN2 2.ZN2+ H2 ¢v~Z N2H2 3. ZN2H_2 +2H2 =Z+ 2NH3_ N2+3H2=2NH3

The reaction rate is expressed as

(7.105)

244

k p 1-'( 1 r =

I (g, +

P£2'H~ -

p2

NH~ KPu= P~4,~

Fu=

(7.106) + 1)' "

with l = k . / k + 2 . This more general equation proposed by Temkin, Morozov and Shapatina was successfully tested over a wide range of operating conditions. Eq (7.106) can also be derived with the supposition of dissociative nitrogen adsorption. At relatively high ammonia pressures, eq. (7.106) is transformed into eq. (7.98). Similarly to the concept of the biographical nonuniform surfaces mechanism (7.97) can be used to derive a rate expression in supposition of lateral interactions. Assuming that surface species other than chemisorbed nitrogen are present on the surface in inferior quantities, which is backed by experimental evidences showing that nitrogen adsorption on iron catalysts proceeds at a rate approximately equal to that of ammonia synthesis, the equilibrium constant of step 2 in eq. (7.97) can be expressed, following the general treatment, as 2

(P;< ~ O° e ,?o~"~

(7.174)

r4 = k4Oce °-~)~''~

(7.175)

r 1 = k 104e0-~0'~AW

(7.176)

The steady-state conditions require that r~ - r~ = r2, thus Oo = (k2PBO AeO ~2),1,1~-+ k ,OAe 0 which are linked to the H chemisorption site. Thus a patch of Z first neighbor potential free sites is an active site for hydrocarbon chemisorption. For the simplification we assume equilibrium adsorption of organic molecule and hydrogen. Quasi-equilibria for the adsorption steps of hydrocarbon and hydrogen give K 1 = OiqC /~PH(,O Z

(7.186)

K 2 = Ok/PH2Os~

(7.187)

where OHdenotes the coverage of hydrogen, OHc the coverage of hydrocarbon, 0s the fraction of vacant sites and Puc and Pu2 are respectively the partial pressures of hydrocarbon and hydrogen. The rate of step 3 is expressed by

257 (7.188)

r3 = k30;~cO;~

Combining equation (7.188) with equations (7.186) and (7.187) leads to k U K°SP /~

=

'~3~1~2

P°-stoz+l

(7.189)

~ H ( '~ H 2 ~ S

and the site balance equation is given by 0n~ + 0, + 0~' = 1. We then have K °2 S P~ H° 52 0~ 5 ' + K1P;~(O z + 0~,• = 1

(7.190)

The system of equations (7.189) and (7.190) cannot be solved analytically (except for z=l). The estimation of reaction rates and comparison with experimental data should be done by minimization of the sum of residual squares, while the value of surface coverage from balance equations should be solved numerically using, for instance, the Newton-Raphson procedure. A similar but slightly more complicated example is hydrogenation of ethylbenzene H2(g) +j *¢::>jH*2/j E(g) + S ~ ES

KH KA

1. ES +jH*2/j ¢:>EH2S+j* 2. EH2S +jH*2/j EH4S+j* 3. EH4S +j H*2/j ~ EH6S+j* 4. EH_6S ~ EtCH + S(fast) E+3H2~EtCH

(7.191)

Where * is the active site for hydrogen (H), S is the active site for the aromatic compound (E), EtCH is ethylcyclohexane, and j is equal to unity for nondissociative adsorption or 2 for dissociative. For competitive adsorption, S is equal to X, where X is the number of * sites, where ethylbenzene is adsorbed. Taking into account steady-stated conditions r = r1 = r2 = ~

(7.192)

w-here the rates are expressed as r~ = k,O~ O~ - k ,OLH20J

(7.193) (7.194)

Z+Z'+RHz R + H2 = RH2

(7.222)

Symbols Z and Z' denote two types of sites. For the sake of simplicity, an irreversible reaction is considered. One can see that for scheme (7.222) there are 5 "intermediates" in the spirit of Temldn considerations (Z, Z', ZH', Z'H + and ZRH" ), three steps, one route and thus according to the Horiuti-Temkin rule (eq. 4.3, 4.4) there should be three balance equations. As there are two types of sites in (7.222) then these balance equations take the form (7.223)

[Z'I+[z'H+]=I

(7.224)

Besides these two equations there should be another one which corresponds to the overall electroneutrality of the catalyst (7.225) which then leads to [Z]= [Z']

(7.226)

and coverage of surface protons is

[Z'H+]=I-[ZI

(7.227)

At steady state conditions the reaction rates of steps divided by their stoichiometric numbers are equal to each other. Equalities q = r2 and ~ = r3 subsequently give

k,PH2[Z][Z']=k2[ZIt ]PR

(7.228)

264

kiPH2[Z ][Z']

=

k3

[ZRH-][Z' H + ]

(7.229)

where P ZN2 2.ZN2+H2 ZN2H2 3. ZN2H2 +Z=2ZNH 4. ZNH+ H~=Z+NH~ NR+3H2=2NH3

1 1 1 2

(7.243)

The transfer of lablled atoms in the presence of deuterim can be represented by two independent routes of isotope transfer (steps without transfer of isotopes are not considered) I* 1. Z N2+HD ZN2HD 1 2.ZN2HD+Z ¢:>ZNH+ZND 1 3. ZNH +HD=Z--NH2D 0 4. ZND+ Hp -Z+NHzD 1 HD---~NH2D

II* 0 0 1 0

(7.244)

There are two independent intermediates ZN2HD and ZND, which results in two independent routes. Combining scheme (7.243) and (7.244) leads to I II III 1 1 1 1 .N2+Z ZN2 1 0 1 2.ZN2..H2 ZN2H2 0 1 0 3.ZN2+HD ZN2HD 4. ZN2H2 + Z - 2 Z N H 1 0 1 5.ZN2HD+Z ¢=>ZNH+ZND 0 1 0 2 1 1 6.ZNH-- H2 -Z--NH3 7.ZNH +HD-Z+NH2D 0 0 1 8.ZND-- H2-Z--NH2.D 0 1 0 NI:N2+3H2=2NH3, N 1I, N m : N2,,2H2+HD=NH3+NH2D

(7.245)

There are 5 independent intermediates in the framework of Horiuti considerations (ZN2, Z N2H2, ZN2HD, ZNH, ZND) which for 8 steps gives 3 routes. From this basis of the routes one can come to another basis of routes I' II' III' 1 .N2+Z ZN2 1 0 0 2.ZN2+H2 ZN2H2 1 -1 0 3.ZN2+HD ZN2HD 0 1 0 4. ZN2H2 "Z=2ZNH 1 -1 0 5.ZN2HD+Z ZNH+ZND 0 1 0 6.ZNH--H2 -Z+NH3 2 -1 -1 7.ZNH +HD=Z+NH2D 0 0 1 8.ZND+Hp_-Z+NHzD 0 1 0 NC:N2+3Hz=2NH3; N", N IIl: NH3+HD=H2+NH2D where

(7.246)

267 VI' J . H' 11 I . Ill' z VIII 1 s =Vs,Vs' =Vs --Vs,Vs s' - - V s

(7.247)

The rate o f the transfer o f labelled atom from HD to NH2D is p = r H + r H1 = r rt' + r Hr'

(7.248)

The aim o f the following treatment is to relate the rate o f the transfer o f a labelled atom with the rate o f the overall chemical reaction. For the sake o f clarity we will consider a one-stage reversible reaction between the molecules A and A' with the formation o f molecules B and B'. It is supposed that the atoms are substituted by their isotopes. The rate o f transfer o f an isotope is defined as the number o f labelled atoms transfered from the molecules o f one species to the molecules o f another species per unit time per unit reaction space. The fraction o f labelled atoms in the molecules A and B will be denoted by CAand ~B, the number o f equivalent atoms that could be labelled and are transfered from the substance A to the substance B in one elementary act o f the reaction is denoted by ~t. Thus the rate o f transfer o f the label by the forward reaction is p+ = p r + ¢ A

(7.249)

where r+ is the rate o f forward reaction. The transfer in the reverse direction is p

= ¢trf#

(7.250)

It is the net rate o f transfer p = p+ - p

(7.251)

that is directly observable and it could be expressed by p = ¢t(r+¢ A - ~¢,)

(7.252)

Besides the isotopes transfer rate also the reaction rate r = r+ - r_ can be measured

r+-

p-rz¢z

(7.253)

/~(¢A - ¢ ~ ) r_ -

/ 9 - r/~- A ~(¢A - ¢ ~ )

(7.254)

In the general case, the transfer stages must be supplied with two numbers, o f which the first s is the number o f the corresponding stage o f the reaction while the second cr is the number o f the variant o f transfer. As an example in hydrogenation o f ethylene for the step C2HsZ +HZ =C2H6 + 2 Z , there are the following possibilities for the isotope transfer C2HsZ +DZ -C2HsD+2Z and C2H4DZ +HZ -C2HsD+2Z. Then the number o f steps, which constitute the transfer routes, should increase. The reaction rates o f transfer are then defined as

268 p,,,o =/a,,or,~-,,,~

(7.255)

p_,,~ =/a,,~c,g_,,~

(7.256)

where g.,.,~ and g_,,~ are the fractions of labelled atoms in the labelled reactant and the labelled product of the transfer stage

s , cr respectively.

s, cr for the transfer path is denoted by 2 ''°

The stoichiometric number of the transfer stage Since the motion of only one labelled atom is

always traced, Z can only be 1 or 0. The stage steady state conditions in relation to the label transfer, take the form N*

IT*

~-~2,~p

I1"

=p,,~-p

.~,~

(7.257)

n*=l

Let us now consider the case of a reversible reaction with one basic route and one basic path for the transfer of the label from the reactant molecule A to the product molecule B. If i,t is the number of equivalent atoms that could be labelled and are transfered from A to B in one turnover of the reaction under consideration and one turnover of the whole reaction corresponds to v, turnovers of the step s we get l, =/,t,v,

(7.258)

The reaction rate of the label transfer can be described by an equation equivalent to 4.69 and is presented here for 2 =1 for all steps s* (s* could be smaller than s, as the label transfer could occur not in all reaction steps) s*

s*

p =

(7.259) P+2 ""P+s* + P-lP+3 ""P+s* + "" + P - l P - 2 ""Ps*-I

Eq. (7.259) can be combined with the eq. (7.257) and (7.258) giving

p =/a

(7.260) 1211+2...F+,~, -1- Fl122F+3...F+s.,

-}- F 1F 2...V ,

It follows from (7.260) that the rate of the isotope transfer can be calculated from the rates of the steps in a similar fashion as the rate of the overall reaction. If the reactants are completely labelled ( gA = gU = 1 ) then p =/ar

(7.261)

As an example of the application of eq. (7.260) to isotope exchange we will discuss the heteromolecular exchange of oxygen

269 1. ZO+O2*ZO 02* 2.ZO 02# 2 H + + 2 e-

and a cathode reaction

1/202 + 2 H + + 2 e - - >

H20

Polymer electrolyte membrane (PEM) fuel cells use a solid polymer as an electrolyte and porous carbon electrodes containing a platinum catalyst. They need only hydrogen, oxygen from the air, and water to operate and do not require corrosive fluids like some fuel cells. They are typically fueled with pure hydrogen supplied from storage tanks or onboard reformers. Such reformers could use different type of fuels, for instance methanol (Figure 7.9).

iiiiiiiiiiiiiiW~Niiii~i~ikliiiiiiiiiiiii~ Figure 7.9. Application of a fuel cell in combination of a catalytic reformer. Polymer electrolyte membrane fuel cells operate at relatively low temperatures, around 80°C. Low- temperature operation allows them to start quickly (less warm-up time) and results in better durability. However, it requires that a noble-metal catalyst (typically platinum) be used to separate the hydrogen electrons and protons. The platinum catalyst is also extremely

272 sensitive to CO poisoning, making it necessary to employ an additional reactor to reduce CO in the fuel gas if the hydrogen is derived from an alcohol or hydrocarbon fuel. Platinum/ruthenium catalysts are currently exploited as they are more resistant to CO. The membrane in a PEM cell is made from a sulfinate polymer Nafion, which only lets protons through because there are sulfinate (SO4) molecules in the polymer, which contain oxygen atoms that are slightly negatively charged. The positively charged protons can weakly bind to them, which allows protons to permeate the membrane, and jump from one sulfinate molecule to another across the membrane with help from thermal fluctuations and the electric field created across the membrane by the electron flow (Figure 7.10).

PEM FUEL CELL Ele~4c~l C u ~ n t ~ter an :::::::::::::::::::::::::: Heat O*Jt

Fuel

C~

Fuel ~ln

Anoa~

Elect,ofyte

~ca~ode

Figure 7.10. Polymer electrolyte membrane fuel cell.

Another application of electrochemistry to heterogeneous catalysis is cyclic voltammetry, which is an important electroanalytical technique. Cyclic voltammograms (CV) trace the transfer of electrons during an oxidation-reduction (redox) reaction (Figure 7.11). electron flow

Figure 7.11. Schematic picture of electron transfer in a electrochemical cell.

The reaction begins at a certain potential (voltage). As the potential changes, it controls the point at which the redox reaction will take place. Electrodes are placed in an electrolyte solution. The electrolyte contains analyte that will undergo the redox reaction. In CV, the current in the cell is measured as a function of potential. The potential of an electrode in

273 solution is linearly cycled from a starting potential to the final potential and back to the starting potential. This process, in turn, cycles the redox reaction. Multiple cycles can take place. A plot of potential versus current is then produced. The system starts off with an initial potential at which no redox can take place. At a critical potential during the forward scan, the electroactive species will begin to be reduced. After reversal of the potential scan direction and depletion of the oxidized species, the reverse reaction, oxidation, takes place. The most important electrode in CV is the working electrode. Another electrode is the auxiliary electrode also known as the counter electrode. A third electrode is used to conduct electricity from the signal source into the solution, maintaining the correct current. This reference electrode is usually made from silver/silver chloride (Ag/AgC1) or saturated calomel (SCE) and its potential is known and constant. The potential that is cycled is the potential difference between the working electrode and the reference electrode• Examples for CV for different platinum surfaces are given in Figure 7.12. ~'too ?

tl~[,

R ( ~ = 6(~11)x(Ilt) (1'11)x(1tl) = (110)s~ps

;° ~t00 0~1 02 o~3 oA o~ o.o &7 o~ o~ to Potential/V Pd/H

~I 0.2 ~ 0.4 0.5 0.6 0.7 0.8 11.91.0 PotentiaW Pd/H

'7

lS0 Pt(tlo)

I-

R0 t,tJ) = 6(100)x (11"~)

.1-0~x(111)stem

j(loo)m
R(s)

(7.271 )

the current flowing in either the reductive or oxidative steps can be predicted using the following expressions

i, =-FAkox[R]o

(7.272)

i~ = -FAk,.~t[O]o

(7.273)

For the reduction reaction the current i~ is related to the electrode area (A), the surface concentration of the reactant [O]o, the rate constant for the electron transfer (kRed or ko~,.) and Faraday's constant (F-96500 coulombs/mole). A similar expression is valid for the oxidation i,, dependent on the surface concentration of species R. By definition, the reductive current is negative and the oxidative positive, the difference in sign tells us that current flows in opposite directions across the interface depending upon whether oxidation or reduction is under consideration. Application of transition state theory to electrochemical reactions gives (compare with eq.3.20) -A

k,~,d = k ' e x p ( ~ )

~

(7.274)

ko~ = k' e x p ( - AG~;, ) RT

(7.275)

where the Gibbs activation energy of oxidation and reduction is illustarted in Figure 7.13.

~W

O+e Fleacti~ Coordinate

Figure 7.13. Illustration of activation processes for oxidation- reduction reactions

For a single applied voltage the free energy profiles appear qualitatively to be the same as tbr the corresponding chemical processes. However the tree energy profiles, especially (O + e-) show a strong dependence on voltage cp, which could be rationalized using a linear relationship

275 AG2~

~

AGo~ = (AGL)

....... ,,..~. -

+ aFo

(7.276)

(1- a)F(p

(7.277)

The parameter (z is the transfer coefficient (Polanyi parameter) and was previously discussed in connection to homogeneous (Bronsted relationship) and heterogeneous catalytic reactions. |ts value is typically found to be ca. 0.5 and provides an insight into the way the transition state is influenced by the voltage. A value of one half means that the transition state behaves mid way between the reactants and products response to applied voltage. The free energy on the right hand side of both of the above equations can be considered as the chemical component of the activation free energy change, i.e it is only dependent upon the chemical species and not the applied voltage. Substituting the activation free energy terms into the expressions for the oxidation and reduction rate constants gives (A(;~,.d),,o

k,.~,j = k ' e

,,oir.u,.

Rr

-o:FCO

(AG2,) . . . . . . :~,,~,,, (I a)/"CO

ko~. = k ' e

~

-aFCO

e Rr = h e

~r

e

Rr

~

=ke

(7.278)

~

/?/,'CO

Rr

(7.279)

with ~z+fl 1. It follows from eq. (7.278) and (7.279) that the rate constants for the electron transfer steps are proportional to the exponential of the applied voltage and consequently the rate of electrolysis can be changed simply by varying the applied voltage. The rates of reaction (7.271) in the ibrward and reverse directions are respectively given by r=

O,

(7.280)

r=

R,

(7.281)

Combining (7.280) and (7.281) with (7.278) and (7.279) we can arrive at an expression for the rate of reaction, given by (7.271) .

r=ke

c~hco

,,

fib'co

~r [ O ] , - k e

(7.282)

RT [R],

At equilibrium the rates in the forward and reverse direction are equal to each other, thus from equality -~"COo

ke

,.

flbcoo

[o] :ke

(7.283)

we get an expression for the equilibrium potential

-

RT

inks

RT

ln[O], - -R- T l n k +

RT,

[O]~. = c o n s t + R T , m [O]~ ....

m ....

(7.284)

276

or in a more general form for the reaction OZ(s)+e-(m/¢::>RZq(st

(7.285)

the number of transferred electrones (n) is taken into account in a following way q~o = c o n s t + R T ln [O], nF [R],

(7.286)

If adsorption/desorption of O and R are in quasi-equilibria, then the surface concentrations could be replaced by the bulk concentrations, leading to an expression similar to thermodynamic Nernst equation. q~o = c o n s t +

er nF

In

[o1

(7.287)

[R]

Defining the rate in the forward direction through the rate at equilibrium leads to ~

c~t@

~

al@o

a(q~0 q~)l,"

~

c~(~,0 q~)l,'

~

or@"

(7.288) where q = q~0 - q~ is the overpotential. Analogously

r = r e ~7

(7.289)

In electrochemistry values of current density i are applied instead of the rates of reactions (7.290)

i = nFr

which gives the Butler-Volmer equation

i=io(e~7

- e e7 )

(7.291)

At large overpotentials (q >> R T / c~F;-q >> R T / f l F ) for irreversible reactions eq. (7.291) could be simplified leading to the Tafel equations for using the absolute value of the current density q = a+_blgi

(7.292)

The -- sign holds for anodic and cathodic overpotentials respectively. A plot of electrode potential versus the logarithm of current density is called the Tafel plot and the resulting straight line is the Tafel line" The linear part (b 2.3RT/anF) is the Tafel slope that provides information about the mechanism of the reaction, and "a" provides information about the rate constant of the reaction. The intercept at t 1-0 gives the exchange current density io.

277 If we consider now a two step sequence 1.A++eA 2.A+e-c=>A-

(7.293)

The reaction rate can be expressed using the general treatment for the two-step sequence reaction

F=

0)10.)2__(O10)2

=

-~ltlF -cc~qF flltlF ]3~qF kl e R7 [A+~ge- R-~, [ A ] _ k l e R 7 [A]k2c'R~ [A-] ~ZF

~TF

F,~ZF

p~IF

/"7 , , . ~"3 . ,(~/I . ~ ,\

Different simplified expressions could be obtained form this general equation (7.294) for particular cases (i.e. irreversibility of steps, etc.). One should bear in mind, that when moving from the rates to exchange current density, that the eq. (7.294) should be multiplied also by the number of electrons transferred in the reaction, which is equal to 2 for scheme (7.293). Returning now to scheme (7.270) when all the steps are irreversible, the equation for the rate of the first step (adsorption) can be written as follows r~ = k, ((p)(1- O)C+

(7.295)

where 0 is the coverage of adsorbed hydrogen and the rate constant depends on potential. In a similar fashion the desorption rate (step 3 in scheme 7.270) is F3 = k 3 ((/9)~H+

(7.296)

The rate constant of the hydrogen recombination step (step 2 in scheme 7.270) does not depend on % thus

r2 = k202

(7.297)

The steady state conditions for scheme 7.270 imply that rt = 2r 2 + r~

(7.298)

leading to a quadratic equation for the surface coverage (7.299)

2k2 o2 + (k, (~o) + k3 ( ~ o ) ) c H o - k, (~o)c.~+ = o

Solution of this equation results in an expression for the reaction rate

-

r=r l=k l(~o)CH I1

(/q (~o) + k~ (q)))C.+ + x/((k~ (~o) + k3 (~o))C.+)2 + 8k2kl (~o)C.+ 4k 2

(7.300)

278 The treatment above demonstrates that the basic principles of catalytic kinetics can be also applied to electrochemical reactions.

7.8. Combined heterogeneous-homogeneous reactions The electrocatalytic processes considered above are examples of a combined process, when intermediates generated on a catalyst (electrode) then move to another phase and react either in solution or on another electrode. In fact, a similar generated process could also occur in the gas phase. There are numerous evidences of formation of free radicals on the surfaces of heterogeneous catalysis and their further desorption to the gas phase. The majority of the data that indicates an involvement of gas phase reactions into a heterogeneous catalytic process were recorded for total and selective oxidation reactions of hydrocarbons. The contribution of homogeneous radical reactions in the oxidation of alkanes could be neglected for reactions occurring at lower temperatures, however for higher temperatures the role of alkyl radicals becomes very important. Experimental evidence of chain reactions is provided by the dependence of the heterogeneous catalytic reactions on the reactor arrangements, e.g. surface to volume ratio of the reactor after the catalyst bed. Figure 7.14 demonstrates how the shape of the reactor behind the Ag/alumina catalysts affects the NOx to N2 conversion with octane as a reducing agent in a reaction 2NO + C8H~8 +11½ 02 ~ N2 + 8 CO2 + 9 H20

2cml

NO× to N2 conversion

,

ia ' ~

150

200

250

300

350

•

4(~

450

a "a

500

550

600

Temperature[ C]

Figure 7.14. NOx to N~ conversion over silver on alumina catalyst depending on reactor arrangements. Ag on alumina is an effiicient catalyst for deNOx removal but the drawback is the simultaneous tbrmation of CO, requiring an oxidation catalyst behind a bed of silver on alumina. The activity depends on the distance between the catalysts, e.g. residence time between Ag/alumina and oxidation catalyst. When the Pt-oxidation catalyst is placed immediately behind the Ag/alumina bed, a significant drop in the NO to N2 activity is observed in comparison with the single Ag/alumina bed. As expected the oxidation catalyst removes completely the produced CO. However, when the distance between the two catalysts is extended, the conversion of NO to N2 improves to levels close to those recorded over the single Ag/alumina bed (Figure 7.15).

279

>

Mix of Ag/alumina and ox.catalyst

Increasing >

ox.catalyst

Ag/alumina

NOx t o N 2

activity

>

ox.catalyst

Ag/alumina

ox.catalyst

Ag/alumina

Figure 7.15. NOx to N2 activity with octane as a reducing agent over silver on alumina catalyst and a platinum oxidation catalyst depending on the distance between the catalysts (K. Erfinen, L.-E.Lindfors, F. Klingstedt, D.Yu.Murzin, Continuous reduction of NOx with octane over a silver/alumina catalyst in oxygen-rich exhaust gases: combined heterogeneous and surface mediated homogeneous reactions, Journal of Catalysis, 219 (2003)

25).

If the Ag/alumina catalyst is divided into four layers with intermediate empty spaces between the catalyst beds (the total amount o f catalyst is equal to the single bed), such an arrangement exhibites higher N O to N2 activity at low temperatures compared with the single bed (Figure 7.16). 4-layers of Ag/alumina vs. single layer. Total m a s s of catalyst =

0A g a n d HC1/NO = 6

(octane).

lOO 80

~m~

60

40 ~3 0~--

8

~= 20

20 40 60 150

200

250

300

350

400

450

500

550

600

Temperature ( C )

Figure 7.16. NOx to N2 activity with octane as a reducing agent over a single bed and four layer silver on alumina catalyst (L.-E. Lindfors, K. Er~nen, F. Klingstedt, D.Yu. Murzin, Silver/alumina catalyst for selective catalytic reduction of NOx to N2 by hydrocarbons in diesel powered vehicles, Topics' in Catalysis, 28 (2004)

185). These results indicate that gas phase reactions, initiated over Ag/alumina, play an important role in the de-NOx process at lean conditions. Residence time is a key parameter behind the catalyst bed, as the gas phase reactions seem to be rather slow. It is reasonable to assume, that species generated on the catalyst surface desorb into the gas phase and enhance NO to N2 conversion.

280 The radicals desorbed from the surface should diffuse along the pores to the outer surface of catalyst grains and further through the catalyst bed with high probability to be trapped and terminated by the solid material, for instance the surface of the support. Hence only a small fraction of the radicals which left the surface eventually comes to the reactor free volume. Besides the variations of the relative sizes of reactors, kinetic evidence of the heterogeneous homogeneous nature of hydrocarbons oxidation was provided by also varying the ratio of an inert material to the reactor free volume. The impact of gas-phase radical chain reactions in heterogeneous catalysis was evaluated not only through kinetic analysis, but also by experimental detection of free radicals formed in heterogeneous processes. Such isolation could be done via some specific procedures, such as matrix isolation of radicals, combined with IR and EPR spectra, photoelectron spectroscopy, multiphoton and resonance ionisation methods to name a few. Let us consider a few mechanisms of heterogeneous-homogeneous reactions using an oxidation reaction as an example 1.02 + 2* ---~20* 2.A +* ---~A* 3.A*+ O*---~AO*+* 4.AO* +O*---~ AO2+2' 5.A*+ O2---~AO2* 6.AO2"----~ AO2+* 7. AO*+ Oz---~ AOz+O* A + O2--+ A + O2

1

0

0

1 1 1 0 0 0

1 0 0 1 1 0

1 1 0 0 0 1

(7.301)

where all the steps are occuring on the catalyst surface. This reaction can propagate to the gasphase by release of the radicals in several ways, either by a following sequence of steps leading to a final product by a recombination of two radicals 8. A* + O2--+AO*+O"

9.O'+A---~ AO"

(7.302)

10.AO'+ O'---~ AO2 or through a chain sequence 11 .A* + O2---~AO'+O* 12. AO" + 02 @ AO2+O"

(7.303)

Even chain branching is possible 11 .A* + O2---~AO'+O*

13. O'+A---~ A10"+OH"

(7.304)

14. OH" + A---~ AO" +H"

The specific feature of chain branching is that in elementary reactions, one reactive centre (atom or radical) reacts to produce two or more reactive centres thus increasing of

281 concentration of radical species with time. The overall rate increases with time like an avalanche until reactant consumption becomes significant enough that it decreases. Chain termination is also possible for the schemes (7.302) -(7.304), which occurs either on the walls of the reactor or on the other solid surfaces, including the catalyst itself. It is clear, that if the contribution of gas-phase reactions to the overall rate is not profound, than the presence of these reactions will be unnoticed. For radical reactions without formation of chains at certain conditions (the same amount of solid material in the reactor and the same walls surface area) the rate expressions for heterogeneous catalytic reactions are similar as for homogeneous-heterogeneous processes with the only difference being that the apparent rate constant is dependent on the surface-to-volume ratio of the reactor. For reactions with chain propagation, the difference becomes visible. For branched reactions with sufficiently long chain length, the kinetic regularities will correspond to homogeneous chain reactions. The approach to treat the kinetics of heterogeneous- homogeneous reactions can be summarized as follows. The rates of heterogeneous catalytic reactions occuring on the surfaces of solid materials are derived in a similar fashion as for the reactions without the formation of radicals. This gives a possibility to calculate the formation rate of radicals. Using the terminology of chain reactions the initiation rate is then computed. The rates of homogeneous reactions (radical or chain) are then computed taking into account the concentration of radicals generated in the initiation step. Usually from the steadystate approximation applied to chain reactions, the initiation rate is equal to the termination rate. An example of a termination reaction given below 2H" + M---~ H 2 + M demonstrates that a third body must be present to absorb the energy of reaction. Recombination of radicals in the bulk is thus impossible. Here M denotes the walls of a reaction vessel or a third particle. It also implies, that the termination rate depends on the geometry of the vessel, more precisely the surface to volume ratio. The termination process can be regarded as a conseciutive one and it involves diffusion to walls and the subsequent recombination. In the diffusion regime rate is determined by diffusion 8n / & = -div(-D

grad n) + v0 - v,

(7.305)

where n is the concentration of radicals, D is the diffusion coefficient, v o is the formation rate and vp is the consumption rate. Solution for cylindrical vessels in quasi steady-state gives v o =v,

=k~n,

=n*8D/r

2

(7.306)

where n* is the average concentration an.d r is the vessel radius. In the kinetic regime the rate is determined by the reaction of radicals with walls and is given by the following equation v o = v F = kgn* = n * £u * / 2r

(7.307)

wher.ec is the probability of reaction between a radical and a wall and u* is the average velocity of free radicals. After deriving an expression for the homogeneous reaction, the generation rates for components could be calculated.

282 As an example we can consider methane dimerization, where the most important catalytic reaction in the oxidative coupling of methane is the production of methyl radicals N (1)

N (2)

1.02 -- 2* 20*

1

0

2. CH4 + O*-+CH3" + OH* 3.2OH* 0 can be rewritten F : (0) 2 Jr- O)_2 ) ( G q- ( ~ -- G ) e-'/~) - 0)-2 = (0)2 + 0 ) 2)(

(01-}- 0)-2 +({90 __ (01-}- 0)-2 )e ' / : ) - 0 ) 601 + 0 ) 1 +0)2 + 0 ) 2 601 + 0 ) 1 +0)2 ~-0) 2

2 (01 Jr- 0)_1 Jr- (02 Jr- (0_2

(8.11)

0)1 + 0 ) 1 +(02 + 0 ) 2

and

F

0)10)2--0) 10)

:

~-(O) 2 +(0_2)(00 0)1 "l- 0)-1 "}-0)2 + 0 ) - 2

co, +0) 2

)e ' / : ) = G + ( r 0 - G ) e '/:

(8.12)

0)1 +0)-1 "}-0)2 "}-00-2

where ro~ = r at / = co , e.g. stationary rate 0)10)2 -- 0)-1 0)-2

G =

(8.13)

0)1 -j- 0)-I -j- O02 -j- (0-2

The time o f one turnover is defined as

U=L/r

(8.14)

and consequently

U~ = L / G A n expression for the relaxation time is

(8.15)

288

L

U~r~

CO1 -}- 09-1 q- 092 q- (0-2

0)1 q- 0)-1 -}- 602 q- 0)-2

U~

091092 --09 10) :

(8.16)

:~oG(1-o~)u~

:

091 +CO 1 +092 +CO 2 CO1 +CO 1 +092 +CO 2

where 09, +co :

O~ =

o91 + 09_ 1 + o92 + 09_:

,1-0~

=

°02 +09 I

(8.17)

o91 + o9 1 + 092 + o9 2

and ~o a function of the step frequencies (/3 =

001002-- CO 1CO 2

(8.1 8)

(co, +092)(09: +°)1) As the reaction rate is positive, then ~oN2+O*. The removal of the deposited oxygen by recombination is essential to regenerate the active sites and, thus to maintain a steady rate of decomposition. At higher temperatures, the oxygen desorption is thermodynamically favored and the subsequent mechanism step: 2 0 * - • 02 +2* no longer limits the reaction rate. The catalytic reaction sequence is described with the stoichiometry N l n +Mrm = 0

(8.28)

where the vectors for the chemical symbols as well as the stoichiometric matrices for the gas phase components and surface intermediates are

295

nS=[a202 N20], mT=[* N20* O*] N=

I zl o

1

° i]

N=

1

0

(8.29)

(8.30)

1

Here m is vector for chemical symbols of the surface intermediates, M T is the stoichiometric matrix of the surface intermediates, n is the vector for chemical symbols of the gas phase components, and N T is the stoichiometric matrix of the gas phase components. The overall reactions observed in the gas phase are obtained by multiplication with stoichiometric numbers o vTNTn = 0

(8.31)

The stoichiometric vector is o=[2 2 1]. The rates of the elementary steps are given by vector R and the generation rates of gas-phase (r) and surface components (r*) are calculated from r = NR

(8.32)

r* = M R

(8.33)

The step responses should be modelled quantitatively by using a transient plug-flow model. The isothermal plug flow- model for the components in the gas phase is written as dc_ dt

a ' d(c'~)+c~pB~ 'r dV

(8.34)

where c is the concentration vector for the gas phase components, V is the volumetric flow rate, V is the volume, e is the void fraction, 9~ is the catalyst bulk density, r~ is the specific surface area of the catalyst, and ris the rate vector for the gas phase components. After defining the dimensionless quantities z V/VR, ~ = V~/~/0, ® t/'c and 6 =@V0, where z is the dimensionless length coordinate, 8 is the dimensionless change in the volumetric flow rate, ® is the dimensionless time and replacing the concentrations by mole fractions, c = P0(RT0) ix the mass balance is converted to a dimensionless form dx -

d®

dx

dS

dz

- dz

~pB'cRT0 NR eP0

e

(8.35)

The change in the volumetric flow rate (8) is obtained by addition of all of the balances and assuming that Zdxi/d®=0 and £dxi/dz=0. The factor 8 is obtained by numerical integration of the sum z

8(z) = 1 + (•,oB'cRT0/P0)Ii r NRdz 0

(8.36)

296

where _ ix = [1,1 ... 1] . For the surface intermediates the mass balance can be written as dc* - - - MR dt

(8.37)

In practice, however, it is convenient to use the surface coverages (0.i) instead of surface concentrations. The relation c* Ojco, where Co is the total concentration of active sites and the dimensionless time is inserted in eq. (8.37). The final form of the balance becomes dO -= dO

(8.38)

MR

The initial conditions of the gas-phase and surface balance equations are x_ = x0 (z) _0 = 0_0(z)

6)3 CHBr (CO2H___12 - +3Br- +3 H +

Mechanism I : BrO3 + B r +3 CH2(CO2H)2 + 3H+~3 CHBr (CO2H)2 +3 H20

(8.69)

310

Comparison between (8.69) and (8.62) shows, that A and B = BrOw-, X = HBrO2, Y= Brand P - CHBr (CO2H)2 (brommalonic acid). The concentration of X is at the lower steady state. The reaction consumes bromide ions (Y) and the concentration of Y falls switching the system to the second process. 5.2 BrO3 + 2HBrO2 +2H + ~ 4BrO2 + 2H20 6.4BrO2 +4Ce 3+ + 4H+ ~ 4HBrO2 + 4Ce 4+ 7.2HBrO2 ~ BrO3+HOBr+H + 8.HOBr+ CHz(CO2H)2 ~ 3 C H B r (CO2H)2 +3 H20

(8.70)

where B = BrO3-, X = HBrO2, Z = 2 Ce(IV), P= CHBr (CO2H)2 and Q= HOBr. Note that reaction 7 is second order. Here cerium oxidizes from oxidation state (III) to oxidation state (IV). This gives the color change from red to blue. Step 5 constitutes an autocatalytic cycle. The autocatalysis causes the rate of this process to increase very quickly once it has switched on, so red changes rapidly to blue. The growth in the concentration of HBrO2 is limited by step 7. The bromide ion should be regenerated and the catalyst is reduced back to its lower oxidation state. Bromomalonic acid produced in steps 3 and 8 by bromination of malonic acid is then oxidized by the cerium (IV), leading to bromide and cerium (III) and some other products 9.4 Ce 4+ + CHBr (CO2H)2 + 2H20 ~ 2 Br- +4 Ce3++ HCO2H+2CO2- +5H +

(8.71)

As a certain concentration of CHBr (CO2H)2 is needed for reaction 9 to occm- long induction period for oscillations is expected, a phenomenon, which is also observed experimentally. During this induction period, the concentration of Br- is small and mechanism II dominates due to the slow conversion of Ce4+ into Ce 3+ and the accumulation of brommalonic acid (reaction 8). Step 9 (8.71) results in the change of the blue color of solution to red resetting the chemical clock for the next oscillation. In fact, the oxidized form of the catalyst can also react directly with malonic acid, so there may be less than one bromide ion per cerium (III) ion produced. 10.6 Ce 4+ + CH2(CO2H)2 + 2H20 ~ 6 Ce3++ HCO2H+2CO2- +6H +

(8.72)

Another model of oscillating chemical reactions, the so-called Brusselator model was proposed by I. Prigogine and his collaborators at the Free University of Brussels.

Ilya Prigogine The sequence of steps in the Brusselator is

311 1. A---~X 2. 2X+Y ---~3X 3. B + X---~Y + D 4. X ---~E A+B---~C+D

(8.73)

with a transient appearance of intermediates X and Y. The Belousov-Zhabotinsky reaction provides an interesting possibility to observe spatial oscillations and chemical wave propagation. If a little less acid and a little more bromide are used in the preparation of the reaction mixture, it is then a stable solution with a red color. After introducing a small fluctuation in the system, blue rings propagate, or even more complex behavior is observed. Study of the BZ reaction in a thin unstirred layer of reacting solution demonstrates that concentric waves ("target patterns") or spiral waves are developed. The reacting solution is normally spread out as a thin film with a few millimeters thickness in a Petri dish (diameter ca. 10 cm). After a certain time blue oxidation fronts which propagate on the red background (reduced ferroin) develop.

Figure 8.24. Propagating oxidation waves the Belousov-Zhabotinsky reaction.

The pictures from left to right (Figure 8. 24) show propagating oxidation waves in an unstirred layer of the ferroin-malonic acid BZ reaction. When the wave is broken at a certain point (for example by a gentle airflow through a pipette) a pair of spiral waves develop at this point. The Belousov-Zhabotinsky reaction is not the only one which displays oscillatory behavior. For instance the Bray-Liebhafsky reaction discovered in the 1920s by W. C. Bray and H. Liebhafsky is the decomposition of H202 in 02 and H20 with IO3-. I Q + H 2 0 2 + H ÷ - - ~ 1 2 + O 2 + H 20

(8.74)

The hydrogen peroxide also oxidizes iodine to iodate 12 + H 2 0 2 ~ IO 3 + H + + H 2 0

(8.75)

with the overall chemical equation being the iodate catalysis of the disproportionation of hydrogen peroxide. 2H202 --~ 02 + 2 H 2 0

(8.76)

Later on Briggs and Rauscher combined the hydrogen peroxide and iodate of the BL reaction with the malonic acid and manganese ions of the BZ reaction, and discovered the

312 oscillating reaction that bears their name. In the BR oscillating reaction, the evolution of oxygen and carbon dioxide gases and the concentrations of iodine and iodide ions oscillate. Iodine is produced rapidly when the concentration of iodide ions is low. As the concentration of iodine in the solution increases, the amber color of the solution intensifies. The production of I- increases as [I2] increases, and these ions react with iodine molecules and start to form a blue-black complex containing the pentaiodide ion. Most of the 02 and CO2 is produced during the formation of I2. The [I2] reaches a maximum and begins to fall, although [I-] rises further and remains high as [I2] continues to decline until the solution clears. Then the [I-] suddenly falls and the cycle I~

+

(8.77) (8.78)

2 H 2 0 2 + H + = HOI + 202 + 2H20

HOI + CH 2(C02H)2 = ICH(CO2H)2 + 2H20

begins again. This cycle repeats a number of times until the solution ends as a deep blue mixture that liberates iodine vapors. 8.4. 2. Enzymatic catalysis It is interesting to note that the original interest in the BZ reaction was inspired by biochemistry, and in particular the Krebs cycle, thus this reaction was intended as a model of an enzyme catalyzed reaction. This connection between enzyme kinetics and the B - Z reaction is often forgotten and rarely mentioned. The classical example of a biochemical oscillator is glycolysis. Damped oscillations were observed in the NADH fluorescence of yeast cell suspensions. Sustained oscillations were notable in yeast glycolysis (Figure 8.25) within a clearly defined range of substrate infusion rates, outside of which steady-state behavior was obtained. KCN

4S "1

glucose |

m~×

vv

0

5

~0

15

2tl

25 T~me {mill)

30

35

40

45

Figure 8.25. Sustained oscillations in yeast glycolysis (M. Bier, B. M. Bald~er,H. V. Westerhoff,How yeast cells synchronize their glycolytic oscillations: A perturbation analytic treatment, Biophysical Journal, 78 (2000) 1087). Oscillations were only observed when the initial substrate was a hexose, such as glucose 6phosphate or fructose 6-phosphate, not fructose 1,6-bisphosphate or subsequent metabolites in the pathway. This indicates that that the enzyme responsible for the periodic behaviour is phosphofructokinase (PFK), which is the key regulatory enzyme for glycolysis. It catalyzes the irreversible transfer of a phosphate from ATP Y ©

313 to fructose-6-phosphate o\ / d

o~P'~o_ . -CI-~

/OH H2C~

0

OH

HI

OH

giving fructose- 1,6-bisphosphate HO%_jO'-,.

[ &

ii

%o/~\-o. OH

and ADP 2e{!

according to the reaction: fructose-6-phosphate+ATP--~fructose-1,6-bisphosphate +ADP. It was shown, that phosphofructokinase plays an essential role in these oscillations. If PFK's substrate, fructose-6-phosphate (F-6-P), is added to cell-free extracts, the nucleotide concentrations oscillate. On the other hand, after the injection of PFK's product, fructose-l,6bisphosphate (F-1,6-bP), no oscillations are observed. PFK displays positive co-operativity, being allosterically activated by several metabolites, including one of its products, ADP. Therefore, the allosteric concepts were applied to explain the damped and sustained oscillations observed in experiments with intact cells, by taking into account non-linear feedback in a system held far from equilibrium. The model of Higgins was based on the activation of PFK by its product. The model by Sel'kov includes activation and inhibition properties of PFK by ATP, ADP, and AMP

The latter is formed by disproportionation of ADP to ATP and AMP. The mechanism of Higgins can be presented by the following sequence of steps 1. 2. 3. 4. 5. 6.

Go---~X X+E~XE1 X EF--~Y+ E~ Y+ E2B) is proportional to the coverage of 0 A, which is then equal to activity as defined by (8.94)). Integrating eq. (8.116) with the boundary conditions t=0, 0 ° =1 (the surface is initially totally covered by A) an analytical expression for the reaction rate is obtained (8.117)

r = to04 = a 3 + al e-"2t

where ro is the reaction rate at deactivation free conditions and parameters are given by

a1

ksr o --

-

-

ks + k s

;

a 2 = k s + k_ s;

a 3 --

k sro -

-

(8.118)

ks + k s,

In eq. 8.118 a3 corresponds to the reaction rate at infinite time. This rate is not equal to zero as catalyst self-regeneration was taken into account. Parameter a2 characterizes the steepness of activity loss, and the sum of al and a3 gives a value of the initial rate (rate at deactivation free conditions). An illustration of the application of eq. 8.117 to heterogeneous catalytic three-phase reaction is given in Figure 8.31. 0.04 0.035 0.03 0.025 v o~

0.02

o.o15 0.01 0.005 0 0

20

40

60

80

100

120

time-on-stream (min)

Figure 8.31. Comparison between experimental and calculated according to eq. 8.117 data for three-phase catalytic hydrogenation in a fixed bed reactor (E. Toukoniitty, P. M~iki-Arvela, A. Kalantar Neyestanaki, T. Salmi, D. Yu. Murzin, Continuous hydrogenation of 1-phenyl-l,2 - propanedione under transient and steadystate conditions, regioselectivity, enantioselectivity and catalyst deactivation, A p p l i e d Catalysis A: General, 235 (2002) 125).

323

I f deactivation is irreversible, then k s = 0 and eq. (8.116) can be simplified to r = roe k,,,

(8.119)

This treatment can be extended for catalysts with two types o f sites present in the catalyst that are different in their deactivation behavior. As an example, selective hydrogenation o f o~,[~- unsaturated aldehydes can be considered with involvement o f metal and interracial sites. Then deactivation occurs via the scheme shown in Figure 8.32.

~ ~ \ es \\,

e' S

F.,,.~jq 4

Figure 8.32. Mechanism of deactivation with two types of sites. Here Os and O's denote the fraction of coverage of two types of distinct sites that are covered by coke. Using a similar approach as for the catalyst with only one type o f site an analytical expression for reaction rate is obtained F = cl'3+al 8.a2t +

(8.120)

ast

a4e

where the parameters are ksro a I --

-

;a2 = ks + k s

-

ks + k s

k s,r0 k' a 3 -- - q '~'r'° k s,. + k_,~, k's. +k'-s a 4-

(8.121)

k'sr'° ;a s=k' +k's k, s +k,_s s'

with ro and r'o being reaction rates at deactivation free conditions. Eq. (8.119) can describe a rather steep decrease in activity (Figure 8.33). Activity

4o -

3s-:

3o--

2s--

2o~_

50

.

,

2O

,

,

4O

.

,

60

,

,

8O

.

,

IO0

,

,

120

.

,

140

,

,

160

,

,

.

180

time on stream

Figure 8.33. Comparison between experimental and calculated activity according to eq. 8.120 (K. Liberkova, R. Touroude, D. Yu. Murzin, Analysis of deactivation and selectivity pattern in catalytic reduction of a molecule with different functional groups: Crotonaldehyde hydrogenation on Pt/SnOe, Chemical Engineering Science, 57 (2002) 2519).

324 If in scheme (8.109) step 2 is considered to be reversible and deactivation (step 4) irreversible 1

2

3

A + * +-~ A* +-~ B* ++ B + *

(8.122)

,[.4

C* and steps 1 and 3 are in equilibria giving the relationship between the coverage of A, B and the fraction of vacant sites 0 r,

(8.123)

oA = cAX~o,~ , o~ = c~X~O,,

then the rate equations are r 2 = k+2CAKAO ~, -- k_2c ~ K~O~,

(8.124)

r 4 = k+4cAKAO ,,

(8.125)

The fraction of vacant sites can be solved from the surface balance 0 A +0~ +Oc, + 0 V =1

(8.126)

Inserting an expression for the fraction of vacant surface sites gives after rearrangement: 1"2 = (k+2KAc - k 2 K ~ % )-1

r4 = k+4KAc A

1-Oc*

(8.127)

+ KACA + K~c~

(8.128)

1 - Oc, 1 + KAc A + K~c~

From the mass balance for the adsorbed surface component dOc, dl

where

-

ar 4

=

(8.129)

denotes the accessible catalyst surface area in the volume

element, Amoa , is the catalyst mass in the volume element and Oj = e l / c ~ . Equations (8.128) and (8.129) describe how Oc, changes with respect to time-on-stream. The effect of deactivation, by the formation of the deposit (C*), on the main reaction rate is accounted for via the reduction of active sites (1- Oc, ). The change of the fluid-phase species (A and B) in time and spatial coordinates can now be calculated by combining the rate equation (8.127) and the reactor mass-balance equations. Rearranged mechanism (8.122) is presented in Table 8.3.

325 Table 8.3. Reaction routes and kinetic equations for mechanism in eq. 8.122. N(0 N (2) A + * - - A* A* - B* B*-- B+* A* - ~ C*

1 1 1 0

1 0 0 1

rA = l + K ~ c A +KÈcÈ

t3 = (k+2KAc - k 2K~cB).1

N (~) A - + B N (2) A ---~C*

r4 = k+4KAc,4

-Iv. lr4

= r= -Iv lr

re* =/'4

-- 0(7 *

+ K,4c,4 +

K~c,

IvJ,l=l, Iv l=o

1 - Oc,

1+K4c4 +K~c B

Analysis of Table 8.3 leads to an important conclusion, that deactivation can be treated using the general framework of the theory of complex reactions, simply considering deactivation as an independent route leading to coke on the catalyst surface. For the irreversible surface reaction G = k+2KAc.4

1-

(8.130)

1 + KAc A + Kec~

d O c , _ k+4KAc A 1 - Oc, dl 1 + KAc A + K~c~

(8.131)

From the definition of fraction of active sites, it holds that (8.132)

a =1-0,,, dcz

dt

dOIA E A --

-

(9.60)

Applying the relation in eq. (9.59) to a zero-order reaction gives, according to eq. (9.60), the enhancement factor,

E A = 1+ b

M K 4 C lbA

~

C~A - KACLA

The enhancement factor, EA, always obtains values larger than 1, i.e. EA_>I.

(9.61)

354

9.3.4. First order reactions For first order kinetics, the reaction rate for the component A is written as (9.62)

r A = v A R = vAkCLA

Applying this definition for ra to the mass balance, eq. (9.45), gives

d2cLA dz 2 -

v~kc~,,~ DI,,4

(9.63)

This second-order differential equation,

d~cz~ YAk dz-----7 + -~--~-fACL~4= 0

(9.64)

with constant coefficients yields the characteristic equation, r 2+

vjk

'

DLA

=0

(9.65)

with the roots

= +_E-

+_;

9.66)

The solution, the concentration profiles of A, can be written as

(9.67)

Ci.~~(_7) = Cle qz + C2 e'2:

By inserting the boundary conditions, c,~(g,)

b , c,,~(0)= = c,,A

~, c,,~

(9.68)

the integration constants, C1 and C2, can be determined. The result is CIA

-- C l A C

C1 = e ~a', - e g-a,, C2

CLAC z

" __ C L A

e 4 b'/' - e ,F,~

(9.69)

Insertion of the integration constants, C1 and C2, into eq. (9.67) gives the concentration profile in the liquid film c£A(z) = e "Ea~ _-1e -'Ea~ (c~A(eg-_,/a' - e-g~/a~)+c~A(e 4-(a~-~> - e-'f-(a"-~-)))

(9.70)

355

By defining a dimensionless group, M,

VA kDt.A

M =

2 kLA

(9.71)

we get

Mi/2

VAk]I/2 - D~A I

(9.72)

4• =

which can be inserted into eq. (9.70) (9.73) Eq. (9.73) can even be written in hyperbolic form

4))

"" sinh(M '/2 (1 z Cb,.Asinh( M'/2(z / 5,.)) + C,,A

c,. 4 (z) =

sinh M '/2

(9.74)

To calculate the flux, N)[~, the derivative of CLA(Z) is needed

dcr,A

1 =

/ cb

s i n h M ''~

c°sh[MVZ(z/~)]M"~+

~,A

"'

4

12

~ c°sh[MV2(1-z/c~')

~Tm~. ;)

(9.75)

-

c,,~

x

N1.4 is now- obtained

( dc, /

1 1 eL4 5' " ShHa tanh Ha cL4 co sh Ha

(9.88)

where Sh is the dimentionless Sherwood number

Sh-

kr4 a,,DLA

(9.89)

which expresses the ratio between the convective mass flux in the boundary layer and a pure diffusional flux. Note that eq. (9.88) was derived for the case of first order kinetics. For more complex reaction kinetics numerical treatment is required.

Thomas K. Sherwood

358 Expression (9.88) corresponds to the case of heterogeneous catalysis with diffusional limitations. The presence of the Sherwood number is needed to describe situations when a significant part of the reaction is occuring in the bulk of the liquid. At Sh 1, the reaction occurs simultaneously with diffusion throughout the complete liquid volume. Typical values are 10<Sh > 1 for porous catalyst particles. Equation (9.152) can now be rewritten as c i' =

b Ci

'

t nh(0)

;/

(9.154)

371

Insertion of the surface concentration, c[, into expression (9.151) gives

D~'cb'

tanh(~6)

N, = -

(9.155) R 1+
co)y*-0. Consequently, the effectiveness factor becomes F y'

,1 = ;/2 Irdy/

OLo

] °,

(9.195)

J

where yS is determined by the relationship y ' = 1 - ¢ 1- y'

]05

Bi., L2!r'dyJ

(9.196)

Calculation of the right-hand side of eq.(9.196) gives an algebraic equation from the viewpoint o f y s. From this expression yS can be solved iteratively. Thereafter, the effectiveness

378 factor can be obtained from eq.(9.195). If the diffusion resistance in the fluid film is negligible (BiM-->~), then yS=l and, consequently, eq.(9.195) is transformed into [- 1

~0.5

= 1/2 j'r'dy/ eL0 /

(9.197)

where ~0is given by eq.(9.194). By defining a generalized Thiele modulus ~0'

¢ ~ * - - [- 1

(9.198)

7 0.5

Lqr.j the asymptotic effectiveness factor, qi, can be expressed as */i = 1/gi*

(9.199)

This expression, in fact, is the same equation as the asymptotic effectiveness factor obtained for the first-order kinetics, in eq.(9.175). With the technique described above, the asymptotic values for the effectiveness factors can be comfortably determined for the slab form of catalyst particles, since the integral, fr'dy is usually rather simple to evaluate. The asymptotic effectiveness factor is a good approximation provided that the reaction order, with respect to the reactant, is positive, while a serious error can occur, if the reaction order is negative: in which case the reaction is accelerated with decreasing reactant concentration. Similar 'dangerous' kinetic expressions are also some Langmuir-Hinshelwood rate equations, in which the nominator is of a lower order than the denominator. For such reaction kinetics the effectiveness factor can, in extreme cases, exceed a value of 1. This can be intuitively understood: the reaction rate increases due to the fact that the reactant concentration inside the particle becomes lower than in the bulk phase. Such cases are discussed in greater detail in the specific literature. In general, for arbitrary kinetics, a numerical solution of the balance equation taking into account the boundary conditions is necessary. From the obtained concentration profiles one is able to obtain the effectiveness factor. This is completely feasible with the tools of the modern computing technology. Analytical and semi-analytical expressions for the effectiveness factor, tk, are, however, always favoured if they are available, since the numerical solution of the boundary value problem is not a trivial task. The solutions for different types of LangmuirHinshelwood kinetics were presented in the literature, for instance by R. Aris and P. Schneider.

Petr Schneider

Rutherford Aris

379 Plots for a family of curves (irreversible Langmuir-Hinshelwood kinetics) ranging between the zero order kinetics (in denominator KCA~ is much bigger than unity and approaches infinity) and first order kinetics (KCas~0) are presented in Figure 9.14. (zero ordeO ?

_0,1_

0 (t ~tordeO

0"I0.~I

t

10 = g

. [ kKC,,p,, -

•

~D~,i(1 + KCA,) Figure 9.14. Effectivenes factor as a fucntion of Thiele modulus for different kinetic expressions (NIOK Course Advanced Catalysis Engineering, Delft, 2003, handouts).

As it is clearly seen the catalyst effectivenes factors can be obtained for LangmuirHinshelwood kinetics by interpolating between 0th and 1 st order kinetics. From the mathematical viewpoint, the Langmuir-Hinshelwood form of kinetics is similar to MichaelisMenten kinetics. The influence of internal mass transfer on Michaelis-Menten kinetics will be discussed in the section 9.6. It is interesting to consider the effect of internal mass transfer limitations on the observed kinetics. Taking into account eq. (9.199) the reaction rate for n-th order kinetics at high values of Thiele modulus is given by r = k~,C'7/ ~b. Since the Thiele modulus for n-th order kinetics is [

~= ' ~1 R kvC;'~ . D~, [ r

C 0.5(;7+1)~

then the rate is given by r= k~C'~Dff-D-~-~/(R~,.C'-'), which finally gives / R oC C 0`5(n+l) . Hence the observed reaction orders in case of zero, first and

second order reactions are resepctively 0.5, 1 and 1.5.

9. 5.4. Non-isothermal conditions The heat effects caused by chemical reactions, inside the catalyst particle, are accounted for by setting up an energy balance for the particle. Let us consider the same spherical volume element as in the case of mass balances. Qualitatively, the energy balance can in steady-state be obtained by the following reasoning:

[the energy flux transported in by means of heat conduction] + [the amount of heat generated by the chemical reaction] [the flux of energy transported out by means of heat conduction]

(9.200)

Heat conduction is described with the law of Fourier, and several simultaneous chemical reactions are assumed to proceed in the particle. Quantitatively, the expression (9.200) is

380

(- A~ dT 4ar2 ), + ~_,R,(-AH,,)pp 4aT2Ar=(- Ac dT 43r21 dr j dr )o,,

(9.201)

where 2e denotes the effective heat conductivity of the particle. The difference, OvedT/dr47rr2)ut - (~. dT/dr 47rr2)in, is denoted as A (Le dT/dr 47rr2) . Equation (9.201) then becomes

( d r2/+ ) Z

2

a A,~ dr

Rj (-AH~j)p,r al = 0

(9.202)

After division of eq.(9.202) by r 2 Ar and letting Ar=#0 it is transformed to

J4

1

r2

a