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 - -
1 30
(1.41)
For larger catalyst particles, some gas by-pass close to the reactor walls is possible. When the reactants move along the reactor the compostion of the mixture changes and axial diffusion can become prominent. To avoid the possible impact of axial diffusion in kinetic experiments, the level of conversion should be below 10-15%. In general, nonisothermicity of a tubular reactor should also be taken into account, since socalled isothermal reactors are seldom isothermal (Figure 1.23), however for the sake of simplicity it will not be considered below. The mass balance for a plug flow reactor (Figire 1.23) is then given by dn A
r l x r x A V = An x + dt
(1.42)
18
where (1.43)
nA,,,- nA,,o,=AnA
t-.To T #t
.~%,
To X
Figure 1.23. Temperature profile in a tubular reactor. Replacing the amount infinitely small volume
in moles through concentration and volume and considering an
n• = c A A V , A V --~ 0
(1.44)
we arrive at
rlArA =
c/hA dV
dc A + -
(1.45)
dt
Further replacements at the constant volumetric flow /zA = cA~"
(1.46)
V //2o = r
(1.47)
lead to dc A dt
dc A -~- rlArA dr
(1.48)
At steady-state d c A / d t = 0, which means that the concentration is independent on time, therefore dc A
dr and
- rlArA
(1.49)
19
dc A
(1.50)
which could be transformed to
r =
C f- d ~ Ao Jo-lLIt~
(1.51)
For the first order reaction finally we obtain analogously to a batch reactor 1
r = -2ln(1-o0 k
(1.52)
with the only difference that here instead of astronomic time we use residence time r = V / ~ . For packed bed reactors (Figure 1.24) m :at
J~Aou,
Figure 1.24. Molar flows in a packed bed reactor. the balance is exactly the same as for homogeneous PFR except that the rate in eq. (1.48) is replaced by rAp~, where p~ is the catalyst bulk density, giving dc A dt
dc A - - ÷ pJ/Ar~ dr
(1.53)
For the steady state de"A / dt = 0 the design equation is
z- = c.j
dcA
(1.54)
1.4. 4. Gradient-free recycle reactors
Such reactors (applied for gas and liquid systems) are stirred flow or recirculation reactors, characterized ideally by very small concentration and temperature gradients within the catalyst region. For a recirculation reactor with an external mixing device (Figure 1.25) the pressure in the system is constant and gas (liquid) composition does not change with time or in space and each single pass of reactants through the bed results in very small conversion as the circulation rate is very high. The mixture of gases are fed through pipe 1 to the reactor 4 and then the outlet 2 for further analysis (GC). Pump 3 provided the necessary circulation of the mixture.
20
Figure 1.25. A recirculation reactor with an external mixing device.
The system is then free from concentration gradients along the catalyst bed, concentration gradients due to axial dispersion and temperature gradients. The treatment of data is simplified as the rate is extracted directly in the differential tbrm
r
= --
(1.55)
St
Efficient circulation pumps are needed and the inventory of reactants should be sufficient enough. The relaxation time (e.g. time to reach steady-state) can be significant. These disadvantages are however compensated by the easy mathematical treatment of experimental data. This reactor system can operate in a closed mode also. Then the entire kinetic curve can be measured in one experiment. At the same time, by-products can accumulate in the cycle. The concept of a circulating flow reactor was further developed in the Buss reactor technology (Figure 1.26). Large quantities of reaction gas are introduced via a mixer to create a well dispersed mixture. This mixture is rapidly circulated by a special pump at high gas/liquid ratios throughout the volume of the loop and permits the maximum possible mass transfer rates. A heat exchanger in the external loop allows for independent optimisation of heat transfer. For continuous operation, the product is separated by an in-line cross-flow filter which retains the suspended solid catalyst within the loop. Such a system can operate in batch, semi-continuous and continuous mode. self-inducing E shell and
tube heat
exchanger
reaction pump J ~
Figure 1.26. The Buss loop reactor
21 Some other types of gradientless reactors with stirred flow are the Berty reactor which circulates gas past a stationary catalyst bed (Figire 1.27)
Figure 1.27. Berty gradientless reactors and a spinning basket reactor (Figure 1.28). In the latter case the solid catalyst is retained in a spinning, woven wire mesh basket to allow gas-liquid circulation with low pressure drop. The large catalyst particles required for this operation mode often lead to conducting the reaction in the region of external diffusion.
Figure 1.28. Rotating basket By utilizing structured catalysts these problems can be avoided. Active carbon cloths for instance could be used in combination with a propeller stirrer, which serves at the same time as a catalyst holder (Fig. 1.29) and is coupled to a Rushton turbine tbr effective gas distribution.
Figure 1.29. Integrated gadget of a combined stirrer and catalyst holder.
22 1.5. Kinetics a n d t h e r m o d y n a m i c s
The rates of forward and reverse reactions are related through thermodynamics. While thermodynamics tells us the final product distribution for a given set of reactants concentrations, pressures and temperature, kinetics describes how fast the products will be generated. Expressing the rates of the forward reaction
r+ = k+ r [ c~ +-'
(1.56)
l
and the reverse (1.57)
c = k_Fic, I
it follows for the ratio of rates, that r+ _ k+ [ I C 7 ....... r ~
(1.58)
The equlibrium constant is defined similar to equation (1.3) k+ - K k
(1.59)
leading to the De Donder equation 1"+ = r
(1.60)
e .t/RT
where A is the reaction affinity, expressed by A = RTlnK•C;'"
~ '
(1.61)
l
The affinity A is defined as the derivative of the Gibbs free energy with respect to the extent of the reaction.
7",1)
At equilibrium the Gibbs free energy is at its minimum value and A = -~-~ v/l l l
(1.63)
23 where v is the stoichiometric coefficient for species and /l is the chemical potential of this species. According to De Donder formulation the relationship between the rates of forward and reverse reactions is expressed by the following equation
r=r+O_ e A,'~v)
(1.64)
or
r = C (1 - (?(P) / K )
(1.65)
where r is the overall rate, r+ is the rate of the forward reaction, q~ (P) is a function of only reactants concentrations, K - is the equilibrium constant, which is the ratio of rate constants of the forward and reverse reactions. As an example we will consider the following reaction A+B¢? C+ F. The rate of this reaction is given by
,.=k+P~P~-k P~.Pr =k+P~P~d k
8.P~_)
(1.66)
where Pa, is the partial pressures (in case of gas-phase reactions) or concentrations (for liquidphase reactions). Taking ito account eq. (1.59) one gets
,'=r+d
I K
8.P~ P~P~)
(1.67)
The temperature dependence of the reaction constants is expressed by the Arrhenius equation k = Ae
(1.68)
Eo/Rr
which is an approximation, since the preexponential factor, A, depends on temperature. Such dependence is rather weak compared to the exponential term, therefore is often neglected.
nkl s
j
InA slope~E.~R
l f r (K -1)
Figure 1.30. Determination of the activation energy
24 More rigorous treatment of the temperature dependence will be presented in a chapter on transition state theory. Taking the natural log of both sides of the equation (1.68) l n k = l n A - E° R ( (--1/ T)
(1.69)
A plot of In k vs 1/T (Figure 1.30) gives a straight line with a slope of-Ea/R and an intercept of In A. The activation energy is thus represented by a following equation E.
- R c31n___~k_ a(1/T)
=
RT 2
c31nk aT
(1.70)
Transition state theory is the theory behind the Arrhenius equation and should be applied to elementary reactions only. In the majority of catalytic reactions the reaction mechanism is more complex and contains several steps with different temperature dependences.
For example, the hydrogenation of o-xylene (Figure 1.31) is thermodynamically preferred up to 460 K (Figure 1.32).
~j
CH3 .,I C H 3
CH3
/
+3H2 \
CH3 ,,'CH3
Figure 1.31. Reaction networkfor o-xylene hydrogenation 100
1.1 1.09
80
\
1.08
60
1.07 1.06
e~
} 40
~ 3
20
~ N ~
=
4
±
5
Molar ratio trans/cis
%-+
1.05 1.04
Molar ratio hydrogen/o-xylene 0
I-.
423
-
443
463
483
503
Temperature, K
Figure 1.32. Thermodynamicsof o-xylene hydrogenation
1.03
523
25 At the same time the rate clearly passes through a maximum (Figure 1.33). 8O
60
~r
40
i
~20 ~ p
350
H2 J0.25 bar
360
370
~
p It2 J0.19 bar
380 390 Temperature, K
400
4D
Figure 1.33. Kinetics o f o-xylene hydrogenation
The apparent activation energy defined as E a =-R
81nk c~(1/T)
_
RT 2
81n C 8T
is negative, demonstrating an interplay between adsorption of xylene (diminishing with T) and reaction (increase with temperature).
(1.71)
on
the surface
27
Chapter 2. Catalysis 2. Homogeneous catalysis In case of homogeneous catalysis a catalyst is in the same phase as the reactants. Three types of homogeneous catalysis are usually considered: gas phase catalysis, acid-base and by transition metals. 2.1. Gas-phase catalysis The most common example of gas phase catalysis is the decomposition of ozone (03) into oxygen (02). This is catalysed by CFC's (chloroflourocarbons), VOC or nitric oxide (NO). These both catalyse the decomposition of trioxygen (03 or ozone). Let us consider the mechanism of the following gas-phase reaction in which hydrogen is oxydized in the atmosphere with an aid of NO2 catalyst NO2 2H2+O2 ---> 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
]
\)
SO 3"
\~_~
HOH2 C ~.tO HO~ / / ~OPh
ph2p,,
g3
"Rh- - - P P h / (solvent) n
~//C;O2Me
ca[. ~
\'NHCOCH 3
H2
~/CO2Me "~NHCOCH 3
Figure 2.14. Immobilization of organometallic catalysts. (C.Li, Chiral synthesis on catalysts immobilized in microporousand mesoporousmaterials, CatalysisReviews,46 (2004), 419-492).
2.4. Biocatalysis (catalysis by enzymes) Enzymes are proteins that function as biological catalysts. They mediate a vast array of biochemical reactions. Most reactions in a human body are too slow to sustain life without a catalyst, for instance the digesting (hydrolyze) of food, the oxidation of glucose, making ATP (useable form of energy for cells), the synthesis cholesterol for membranes, coping DNA for cell division to name a few. Biological catalysts have been postulated since the early 1800's. The term enzyme was coined in 1878 to describe the component in yeast involved in the fermentation of sugar into alcohol. The enzyme jack bean urease, which catalyzes hydrolysis of urea, was crystallized in 1926. Comparison between chemical and enzymatic catalysis demonstrates the specificity of enzymatic catalysis Chemocatalysts A variety of inorganic, organometallic substances Increase the rate of chemical reactions One catalyst can facilitate multiple reactions Could require high temperature/pressure
Enzymes Mostly proteins, few are RNA Rate increases of 106 to 1012 Specificity Mild reaction conditions
The catalytic activity of enzymes is dependent upon the native protein conformation. The primary, secondary, tertiary, and quaternary structures are essential for catalytic activity. Similarly to homogeneous and heterogeneous catalysis, enzyme catalyzed reactions occur at a specific active site, which is dependent on the arrangement of functional groups.
36 In terms of elementary reactions, there is no substantial differences between enzymatic catalysis and other types of catalysis as the same type of chemical reactions: breaking, ~brming and rearranging bonds are present. At the same time, the high specificity of enzymes is dictated by the enzyme active site, which interacts predominantly with one particular substrate, although some active sites allow for multiple substrates. Multi-point contact with the substrate, structural flexibility to undergo collective and rapid changes and the possibility to combine several catalytic features, like acid and base catalysis, hydrophilic/hydrophobic interactions at the same time, make enzymes so distinct from homogeneous transition metal complexes and heterogeneous catalysts. Some enzymes require cofactors, which are amino acids, vitamin derivatives, or metals (minerals) that bound as co-substrates or remain attached through multiple catalytic cycles. The specificity of enzymes is associated with their geometrical (special structure), as the substrates have to fit (geometry), and with their affinity provided by formation of hydrogen bonds, electrostatic interactions and hydrophobicity. Enzymes have been named by adding the suffix "-ase" to the name of the substrate or to a word or phrase describing their activity. Enzymes are classified according to reaction type. There are 6 major classes (with subclasses). Oxidoreductases catalyze oxidation-reduction reactions, the transfer of hydrogen atoms and electrons, for example dehydrogenation of lactate:
! debyd~ges~se Cr, O +NADii+~# i
I 1~3-Cti
+NAD+
!
cn~
c~
pyruvate
L-lactate
Transferases catalyze the transfer of functional groups from donors to acceptors t
~N..
%c.O
ol,ca,
%:,o
l
a minotransfera~
1
I
O ~ , O.
p~avatc
!
c-,., i
L-a|anine O,~/C.O~
gluta~e
a-ket0#u~te
Hydrolases catalyze the cleavage by the addition of water (hydrolysis), while lyases catalyze the cleavage of C-C, C-O, or C-N bonds (addition of groups to double bonds or formation of double bonds by removal of groups). Isomerases according to their name promotes isomerization reactions, e.g. the transfer of functional groups within the same molecule
, ltC! - o n HO - C H m
phosphogtucose isomerase ,~1
HC - OH HC - OH
glucose 6-phosphate
C =0 |
~,
HO -CHIn HC - OH HC - OH
fi'uctose 6-phosphate
37 Ligases use ATP to catalyze the tbrmation of new covalent bonds, i.e. C-C, C-S, C-O, and C-N bonds. Enzymes difl'er from ordinary catalysts, as the rates are typically 106 to 1012 times faster. Such higher rates are achieved at milder reaction conditions, e.g. body temperature, neutral pH, atmospheric pressure, and with greater reaction specificity for substrate and product rarely having side reactions or side products. Capacity for regulation is associated with a possibility to modify enzymatic reactions by various agents (e. g., modifiers, inhibitors). In addition the activity of enzymes can be regulated by allosteric interactions. This term refers to the ability of the enzyme to bind at a remote site, thus inducing changes in the protein structure, which finally influence the active site. The substrate can bind to the enzyme at the active site via noncovalent interactions (van der Waals, electrostatic, hydrogen bonding, hydrophobic), and with a specific geometric complementarity, as the surface of the active site of that enzyme is complementary in shape to the substrate. Electronic complementarities are due to the fact that the amino acid residues at the active site are arranged to interact specifically with the substrate. Although most enzymes are amino acids with hundreds of acids in the chain, the active site is the size of the substrate. Complementarity in the structure and charge between the enzyme and the substrate is illustrated by the so-called lock-and-key concept (Figure 2.15).
+
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii Figure 2.15. Lock-and-keyconcept. Moreover, the active site can be preformed, but still undergoes some conformational changes upon substrate binding via induced fit (Figure 2.16).
+
Figure 2.16. induced fit concept. Enzymes are highly stereospecific in binding chiral substrates and in catalyzing reactions. This stereospecificity arises because enzymes are made of L-amino acids and form asymmetric active sites. Similar to homogeneous and heterogeneous catalysis, chemical reactions proceed on active sites of enzymes, which represent a small part of the total protein.
38 Due to the complex enzyme structure the active site components are generally far apart in the linear aminoacid sequence. As with any other catalytic reaction, enzymes cannot change the overall thermodynamics of a reaction (i.e. AG) and cannot make an unfavorable reaction favorable, which means that general principles common for other catalytic reactions can be applied. Enzymes accelerate rates by reducing the activation energy. If the transition state can be stabilized the free energy barrier to reaction will be reduced. Binding energy can thus be used for rate enhancement. Enzymes make both the forward and reverse rates for a reaction faster - whether going forward or back, the energy barriers are lower either way. Enzyme catalyzed mechanisms represent fundamentally familiar reactions from organic chemistry (Figure 2.17). Acid-base catalysis is associated with the donation or subtraction of protons. Acid catalysis is a process in which partial proton transfer from an acid lowers the free energy of the reaction transition state, while base catalysis is a process in which partial proton subtraction by a base lowers the free energy of the reaction transition state. Conceited acid-base catalysis, where both processes occur simultaneously, is a common enzymatic mechanism.
.o.
o
:,
~13CsC%~C~Cll3
o
~t3c*c %*(:~cH3 v
S10w
-o
~
,,
.o
Figure 2.17. Illustration of the enzyme-substrateinteractions, leading to enhanced reaction rate. Covalent catalysis presumes formation of covalent (co)enzyme-substrate intermediates. One can imagine for instance the following sequence: nucleophilic reaction between enzyme and substrate to form covalent bond; withdrawal of electrons from the reaction center and finally elimination of the enzyme which is the reverse of the first step. Metal ions (Fe 2+, Fe 3+, Cu 2+, Zn 2+, Mn 2+, or Co 2+) participate in catalytic process by binding to substrates to orient them properly, mediating oxidation-reduction reactions through reversible changes in the metal ion oxidation state and by electrostatically stabilizing or shielding negative charges. Substrate alignment for the reaction is called the proximity and orientation mechanism. Catalysis by approximation is due to an increase in the rate as the binding energy is used to bring the two reactants in close proximity. If A@: is the change in free energy between the ground state and the transition state, then AG ~: =AH ~ -TAS :I:. The formation of a transition state is accompanied by losses in translational entropy as well as rotational entropy. If enzymatic reactions take place within the confines of the enzyme active-site wherein the substrate and catalytic groups on the enzyme act as one molecule due to proximity effect, there is no loss in translational or rotational energy in going to the transition state. Binding the substrate to the enzyme lowers the free energy of the enzyme-substrate complex relative to the substrate and is similar in that sense to adsorption on solid surfaces in heterogeneous catalysis (Figure 2.18). However, if the energy is lowered too much, without a greater lowering of the corresponding activation energy, then catalysis would not take place.
39
Figure 2.18. Energyprofiles for a) non-catalyticreaction, b) enzymaticreaction. A specific feature of enzyme catalysis is the ability of enzymes to bind the transition state of the reaction it catalyzes with greater affinity than its substrates or products. The more tightly an enzyme binds its reaction' s transition state, the greater is the rate of the catalyzed reaction relative to the uncatalyzed reaction. An essential feature of enzymatic catalysis is enzyme inhibition. Approximately one-half of the top drugs sold worldwide are enzyme inhibitors. Enzyme inhibitors block enzyme processes decreasing the concentration of products and increasing the concentration of substrates. Many diseases arise from either an excess of a product or a deficiency of a substrate, and the use of enzyme inhibitors can be extremely beneficial in treating these disorders. Enzyme inhibitors are often unreactive molecules which resemble the substrate (Figure 2.19). In the case of competitive inhibition they bind to the active site and block access by the normal substrate.
Leek & Key Mede~
eempe~i~ e I l:~loeksS M~diHg [e active si~e
iiii:iiiiii.i~..
Noncompetitive tnMbitor allows binding but blocks ~action
Figure 2.19. Competitiveand noncompetitiveinhibitors. Some inhibitors can bind more strongly than the substrate leading to irreversible inhibition. Inhibitors can also change the conformation of the enzyme so that it does not bind to the substrate (Figure 2.19).
40
2.5. Hetereogeneous catalysis 2.5.1. Classification Heterogeneous catalytic reactions constitute around 90% of all processes in the chemical industry. Different types of solid materials are used to catalyze a variety of reactions in the gas phase or in solutions. Some examples are given below. ...................
..............................................................
............................................. Typi
;i
y
t{;i
..................
...................
year4 Crude oil
Hydrocarbon fuels
SO 2, 02
Sulphuric acid
V205
1.4"108
N2, H2
Ammonia
Fe
9"107
NH 3, 02
Nitric acid
Pt/Rh
2.5* 107
CO, H2
Methanol
Cu/ZnO
1.5"107
C2H4, 0 2
Ethylene oxide
Ag
1"107
Hydrogenated
Ni
8"106
Unsaturated vegetable ~;I~,
ILT
u~n~f~hl~
Platinum/silica-alumina Platinum/acidicalumina Metal-exchanged zeolites
1"109
~ile
C2H4
Polyethylene
Cr(ll), TJ(III)
6"106
CH3OH, 02
Formaldehyde
Mixed Fe, Mo oxides
5"106
C3H6, NH 3, 02
Acrylonitrile
Mixed Bi, Mo oxides
3 * 10 6
o -Xylene, 02
Phthalic anhydride
V205
4"10 e
n-Butane, O 2
Maleie anhydride
V20 ~
4"10 ~
The catalytic properties of activity, selectivity and stability are closely related to the catalyst composition. Most catalysts are multi-component and have a complex composition (Figure 2.20). Components of the catalyst include the active agent itself and may also include a support, a promoter, and an inhibitor. For instance, metal catalysts are not in their bulk form but they are generally dispersed on a high surface area insulator (support) such as A1203 or SiO2. Sometimes, both the metal and the support act as a catalyst. These are referred to as bi-functional catalysts. An example is the platinum dispersed on alumina used in the gasoline reforming. The active agent is the component(s) that causes the main catalytic action. Without it, the catalyst will have no effect. A promoter is a substance added into the catalyst to improve either of the activity, selectivity or stability so as to prolong the life of catalyst. The promoter is often added in a small amount and by itself has little activity. There are various types of promoters, depending on how they improve the catalyst. Textural promoters are inert substance which inhibit sintering of the active catalyst by being present in the form of very fine particles, usually they have a smaller particle size than that of the active species, are well dispersed, do not react with or form a solid solution with the active catalyst and have a
42 solving applied problems in mathematical modelling, design and intensification of chemical processes. The necessity of kinetic investigations in heterogeneous catalysis is closely connected to the tasks, which a chemical engineer has to deal with, i.e. proper reactor design, evaluation of the side reactions and the impact of dynamic effects on reactor performance. Any reactor design thus starts from reaction kinetics and, therefore, from the reaction mechanism, which means understanding of the reaction on a molecular level. Reaction kinetics is the translation of our understanding of the chemical processes into a mathematical rate expression that can be used in reactor design. Kinetic modeling used for process development and process optimization has a historical tradition. Quite often power law models are still used to describe kinetic data. Such phenomenological expressions, although useful for some applications, in general are not reliable, as they do not predict reaction rate, concentration and temperature dependence outside of the range of the studied experimental conditions. Thus, in catalysis, due to the complex nature of this phenomenon, adsorption and desorption of reactants as well as several steps for surface reactions should be taken into account. Models based on the knowledge of elementary processes provide reliable extrapolation outside of the studied interval and also make the process intellectually better understood. Activity is an important property of the catalyst. It tells whether the catalyst is able to affect the rate of a thermodynamically feasible reaction. For example at a temperature of 200oC-400oC CO and 02 can react completely over Pt catalyst while the same gas mixture hardly reacts in the presence of substance such as SiO2 alone. Therefore the activity of Pt is much higher than that of SiO2 for the given reaction. The activity of a chemical catalytic reaction may be expressed using the conversion or the turnover number. The turnover number is the number of molecules reacted per site per unit time and depends on the kinetic (concentration of reagents, T). A m o u n t of reacted A TONmass of catalyst or o f active site/time
mol
g (mol) s
The turnover number is a useful concept, but is limited by the difficulty of determining the true number of active sites. The situation is somewhat easier for metals, as a chemisorption technique could be used to measure the exposed surface area. For some reactions (e.g. so called structure insensitive reactions on metals) the rate is independent on size, shape, other physical characteristics, while for structure sensitive reactions - the rate depends on the detailed surface structure. The temperature dependence of heterogeneous catalytic reactions was partially addressed above where it was demonstrated that in general activity increases with temperature, however, increased temperature shortens the catalyst life time and typically increases the rate of undesired reactions. A good catalyst must possess both high activity and long term stability. But the most important attribute is its selectivity, which reflects its ability to direct conversion of reactant(s) along one specific pathway to the desired product. For many reacting systems various reaction paths are possible and the type of catalyst used often determines the path that will be followed. A catalyst can increase the rate of one reaction without increasing the rate of other reactions. In general selectivity depends on pressure, temperature, reactant composition, conversion, and nature of the catalyst. Therefore selectivity should be referred to under specific conditions.
44 The potential energy diagram (Figure 2.22) demonstrates the importance of adsorption in heterogeneous catalysis, which could be related to some extent to effect by approximation in enzymatic catalysis. In the similar way as with enzymatic catalysis the catalyst does not affect the thermodynamics of the reaction. Theretbre first the reaction conditions (temperature, pressure and reactant composition) must be optimized to maximize the equilibrium concentration of the desired product. Once suitable reaction conditions have been identified, the catalyst screening starts to find a suitably active and selective material.
F~H J
catalytic
~actlon
~
Figure 2.22. Potential energy diagrams in catalytic processes.
The first kinetic approach in catalysis based on a physical chemical understanding is due to I. Langmuir, the Nobel Prize in Chemistry winner in 1932, who applied the mass action law to reaction on solid surfaces, e.g the rate of an elementary reaction was supposed to be proportional to the surface concentration (coverage) of reactive species adsorbed on the surface.
Sir Hugh Taylor
Irving Langmuir
The approach most often used to treat the surface is the Langmuir model of uniform surfaces. This concept assumes, that all the surface sites are identical and binding energies of the reactants are the same independent on the surface coverage. The interactions between adsorbed particles may be neglected. The ideal adsorbed layer is then considered to be similar to ideal solutions with fast surfhce diffusion, allowing an application of mass action law with the introduction of surface concentrations and concentrations of free sites into rate expressions of elementary steps.
45 Below we will consider adsorption on heterogeneous surfaces assuming ideal adsorbed layers. Table 2.1. Comparison between physisorption and chemisorption
Properties
Chemisorption
Adsorption
virtuMly unlimited range
temperature
tion nero: or below Tbp of adsorbate (Xe < 100 K. C(): < 200 K)
Adsorption enthalpy
wide range (40 -800 kJmol 1) heat of liquffaction (5-40 kJmol 1)
Crysiallograpbic specificity
marked difference between cwstal planes
independent of surface
often dissociative and ilTeversible in many cases
non-dissociative and reversible
S a tu ration
limited to a monolayer
multilayer occurs often
Adsorption kinetic
activmed process
fast, non-activated process
Nature of adsorption
geometDT
In case of chemisorption a chemical bond is formed between molecules and the surface. For chemisorption, the adsorption energy is comparable to the energy of a chemical bond. The molecule may chemisorp intact or it may dissociate. The chemisorption energy is 30-70 k J/tool for molecules and 100-400 k J/tool for atoms. In physisorption the bond is a van der Waals interaction and the adsorption energy is typically 5-10 kJ/mol. This is much weaker than a typical chemical bond and the chemical bonds in the adsorbing molecules remain intact. However, the van der Waals interactions between adsorbed molecules are not much different from the van der Waals interactions between the molecules and the surface. For this reason many layers of adsorbed molecules may be formed.
E(d)
a Phgsisarption
Figure 2.23. Potential energy curves for chemisorption and physisorption.
For chemisorption, the PE curve is dominated by a much deeper chemisorption minimum at shorter values of d , which is the distance between the surface and adsorbed species (Figure
2.23). The structure of solid surfaces is fairly complicated (Figure 2.24) and the adsorbed species can form different types of complexes on the surfaces (some examples are given in Figure
2.25).
46
Figure 2.24. Structure of solid surfaces O
ii
/H\ M M
/C\ M M
\
/ /\C /\C M M M M
''-C
/
C
C C i i M M
ii ii M M
I M"~'M M
~1 M"~'M M
0 --O
MI MI
I I
\/ \/
/C M M
C --.-_ Fx//', "M M M
""-C/- ~C / I I M
\
C--C
\
/ C=C
i
I
M M
M
M ~H M M
~ "
M-
M
M
Figure 2.25.Examples of the complexes which could be formed on the solid surfaces (M stands for a metal site). The classical treatment of adsorption and kinetics assumes one to one binding of the adsorbates to the surface sites, the same adsorption strength for all sites and no interactions between adsorbed species. 2.5.4. A d s o r p t i o n i s o t h e r m s - I d e a l s u r J a c e s
An adsorption isotherm describes how the partial pressure of an adsorbing species varies isothermally with the fraction of surface covered. In the associative adsorption on the surface site Z A+Z~-~ AZ
(2.5)
The desorption rate can be expressed by
r_ -
dOA -- k_O A
dt
(2.6)
47 where 0 Ais the coverage (surface concentration), while the adsorption rate is given by dOA r+ - - - k+P40 o dt
(2.7)
where 0 is the coverage of vacant (empty) sites. Taking into account the balance equation, which relates occupied and vacant sites 04 + 00 = 1
(2.8)
the adsorption rate is expressed as dOA - k+ PA (1 - 0 A) dt
(2.9)
In case of equilibrium the rate of the forward reaction is equal to the rate of reverse one (2.10)
k+PA(1-O~)=k_OA
leading finally to an expression for the coverage of A at equilibrium 0A -
k +PA
KPA
- - -
k_ + k+PA
(2.11)
I + KPA
where the equilibrium constant is the ratio between adsorption and desorption coefficients (2.12)
K = k+/k_
Plotting the surface coverage as a function of partial pressure according to equation (2.11) it becomes clear that two limiting regions exist. At low values of partial pressure the surface coverage is approximately equal to 04 ~ KPA, while at high pressures the coverage is approaching unity. The value of K is increased by a reduction in the system temperature, as the adsorption is predominantly exothermic, and by an increase in the strength of adsorption.
08 06
04 02
/-f
0. t
0 0,2
04
0~6
08
10
& ~ar Figure 2.26. Surface coverage as a function o f partial pressure at different values o f the equilibrium constant.
48 Dissociative adsorption requires several sites on the catalyst surface. For a diatomic molecule (H2, 02) adsorption occurs on two adjacent sites A2+2Z <-+ 2 AZ
(2.13)
Adsorption and desorption rates are expressed as (2.14)
r+ = k+PAO~, r = k O~
The balance equation is the same as for associative adsorption, thus at equilibrium k+PA(1-0A) 2 = k 02
(2.15)
giving an equation for the coverage at equilibrium for dissociative adsorption
0~-
~
(2.16)
At low- pressures the surface coverage is proportional to the square root of partial pressure, while at high pressures it approaches unity, which corresponds to the monolayer coverage. In case of competitive adsorption equlibria constants for the following reactions A+Z<---> AZ, B+Z<---> BZ
(2.17)
could be directly written K A-
04
POo
;K B -
0~
(2.18)
POo
resulting in (2.19)
0 A = KAPAOo;O ~ = KBPBO o
The balance equation for the surface sites requires
1=o
+Oo = Oo(1+ KAP (2.20)
leading to the expressions for surface coverage OA
KI'PA ;OR 1 + KAPA + K , P B
KBPB 1 + KAPA + K,P B
(2.21)
In a more general case for multicomponent adsorption analogously to the derivation above we can write directlly
49
KAPA 0a-- I + Z K , p ' ....
KL,PI, ,OD-I+~K:p,""
i
(2.22)
J
When adsoption requires several sites on the catalyst surface A i + J Z <=>jA Z
(2•23)
similarly to dissociative adsoption of diatomic molecules we can generalize
OA
(K4pA)t/:
(2.24)
1 "Jr-( K A P A )1/1
Adsorption on ideal surfaces from a mixture of two molecules A2 and B, one of them dissociates, can be then easily written
m
OA
•
I+K~Ap~+K~p,O~
z
KBG 1+ K~-~--Ap~+KBpB
(2.25)
Mathematical treatment of adsorption on ideal surfaces, formation of complexes in homogeneous catalysis and substrate bonding with enzymes assumes in the most simplified form a constant number of sites. Therefore the form of rate equations for enzymatic, homogeneous and heterogeneous catalytic processes for several cases (two-step sequence for instance, which will be discussed later in the text) is the same, although different from gasphase non-catalytic reactions. As we will see later in the enzymatic, homogeneous and threephase heterogeneous catalytic reactions for the two-step sequence the reaction rate expressions are exactly the same. For these reactions, concentrations in the liquid-phase are used, while for heterogeneous gas-phase reactions it is more convenient to operate with partial pressures•
2. 5. 5. Adsorption isotherms- Real surJhces Although the Langmuir theory of adsorption is used frequently for technical process development it is a crude approximation, as surface reconstruction frequently occurs. Adsorbed molecules change the structure of the surface layer and the catalytic properties of surface sites are not equal in the ability to bind chemisorbed molecules• The rate is dependent on spatial arrangement and the heat of adsorption depends on coverage (Figures 2.27, 2•28)• Two different assumptions are generally used for the description of the physical chemistry of the real adsorbed layers: either surface sites are different or there is a mutual influence of the adsorbed species. The first case is defined as biographical nonuniformity and the second one - induced nonuniformity. On biographical nonuniform surfaces a certain distribution of properties is considered. Such nonuniformity can be either chaotic, when adsorption energy on a given site is independent on the neighbor site or discrete• However, if
50
in an elementary surface reaction only one adsorbed particle is involved the difference in these distributions cannot be observed. IEOICO-
127-
1(9E
%
co-
\
~0-
......
2o-
d0'
d2'
d4'
0!6'
d8'
.h: : 110'
112'
1!4
n H(O)/n Ir
Figure 2.27. Differential calorimetric isotherms of hydrogen and oxygen adsorption on lr/SiOt (cited in D.Yu.Murzin, On surface heterogeneity and catalytic kinetics, industrial and Engineering ChemistO, Research, 44 (2005), 1688-1697)
140 130 120
110
1
e .......* .... ......e .....
--e
e. ........e.
......e ,,
100: E
so j
"" " ~\' \
•
"----.
eo:
~,
\ e
4oJ
ao-
; ~o ~'~
; /~ ;~ CQ/(H2)coverage
~
~o~
Figure 2.28. Differential heats of adsorption of hydrogen and CO on several Pt catalysts (cited in D.Yu.Murzin, lnd~¢srrial and Engineering Chemistry Research, 44 (2005), 1688-1697)
The physical nature of the biographical (intrinsic or a p r i o r i ) non-uniformity, advanced originally by Langmuir and Taylor can be attributed to the difference in the properties of the different crystal faces and the occurrence of dislocations, defects and other disturbances. The treatment below originated from M.Temkin, who in the late 1930s developed the theory of adsorption and kinetics on real (nonuniform) surfaces and proposed a special model of nonuniformity. Contributions by Zeldovich and Roginskii to this field in the same years were also essential.
Yakov Zeldovich
Simon Roginskii
Michail Temkin
51 In the special model of nonuniformity, several assumptions were made. The Gibbs activation energy AG~ is considered to be a linear function of the standard Gibbs energy of adsorbed species AG°~ AG~=oc(AG°~)+const,
(2.26)
where the value of transfer coefficient ct (Polanyi parameter) is 0_
dl = Ce ~o d(AG o)
(2.27)
where C and ® are constants. The values of ® could be positive, negative or infinity, reflecting cases of Freundlich, Zeldovich-Roginski and Temkin isotherms respectively. In case when ®---~m instead of (2.27) the following distribution is valid dl = C d ( A G,,o )
(2.28)
Such a distribution is called even and means that the number of sites with the same adsorption strength is the same. For each site the probability that the surface site is occupied at equilibrium is defined as f) -
ap
(2.29)
l+ap
where p is the pressure corresponding to adsorption equilibrium and a is the adsorption coefficient. Defining ~=ln b = -ln a , where b is desorption coefficient, leads to ~=AG°a/RT, as AG°a=-RT In a. Numbering the sites in the sequence of increasing ~ (decreasing adsorption strength) and defining the relative number of site s-l/L, where L is the total number of sites (0<s ~o
q,G) = 0 q~(~)=A exp (7~) ~G) = 0
(2.3o)
where {o is lowest value of ~ and ~ l is the highest value of {, A is a constant, ?=T/® and for even distribution 7=0, e.g. 9(~.)=A. The value of A is determined by the normalized condition
52 oo
I~o(¢)d~ = L
(2.3 1)
where L - total number of sites on the unit surface. Taking into account the limiting values of the logarithm of desorption coefficient instead of eq. (2.31) the normalization condition is
[e(
)d4 = L
(2.32)
4
Finally the expressions for the constant A are A-
yL
(2.33)
er~ _er4O
if y~0 and for evenly nonuniform surfaces L A- - -
?,
(2.34)
-
A
For discrete nonuniformity the total surface coverage is determined by summation of equilibrium adsorption isotherms of Langnmir type 0 = , ~ , l +a,p a, p
(2.35)
while for continuous nonuniformity when all values in a range of adsorption energies are present integration gives the surface coverage
0 = jf- - aa ,sp 0 l+a,P
,
(2.36)
When the pressure is small, then a0p<
53 1
0 = p Iads
(2.37)
0
At high equilibrium pressure, alp>>1, where a] a at s=l (lowest a) the fraction of vacant sites is given by
' Js
1 - 0 = 1/ p J ~ -
(2.38)
being proportional to the reciprocal value of partial pressure similar to ideal surfaces. An important conclusion follows from these considerations, e.g. a nonuniform surface is not manifested in the region of small and large coverages. For evenly nonuniform surfaces it follows from eq. (2.28) that I = C
G~ "~'
where C'
(2.o9)
C'
is an integration constant. Then the Gibbs energy is defined as
L
C'
(2.40)
aG2 = 7 * - 7 leading to an expression for the adsorption coefficient L lna
-
-
C--~s
C' +
CR----~
(2.41)
Introducing )'and ao f-L/CRT
(2.42)
in a o = C ' / C R T
(2.43)
one arrives at
(2.44)
a = aoe
where for the smallest value of a at s-1 a 1 = aoe
f
(2.45)
where the value o f f is called the nonuniformity exponent f = In a0 O1
(2.46)
54 Integrating now eq. (2.36) 1
- f~.
a°e p ds O= l + aoe J'~p
(2.47)
and keeping in mind an expression for the standard integral EtX
I ~
1
(2.48)
dx = - l n ( b + c'eox
we arrive at s=l and s=0 correspondingly to the following values
a°P ln(l+a0e J p) fpao aop - --ln(1 fpao
(2.49)
+ aop )
(2.50)
and finally to the quasi-logarithmic isotherm 0 = l l n 1+ aoP j" 1 + alp
(2.51)
For strongly nonuniform surfaces a0>> ai, there is a region of medium coverage where a0p>>l, alp<
(2.52)
lO
08 AH 06
04
02 00
02
04
0
06
08
Figure 2.29. Differential heats of adsorption as a function of coverage for 1) ideal, 2) evenly nonuniform surfaces.
55
Comparison of the experimental data for ammonia adsorption on titanium silicate and that calculated using the quasi-logarithmic adsorption isotherm is presented on Figure 2.30.
25-
~20E
~
/
15-
ii /
~10QS
/
I
,
0
I
50
,
I
,
10) Pressure, Torr
I
I£0
,
I
,
I
~0
Figure 2.30. Comparison of the experimental data for ammonia adsorption on titanium silicate and that calculated using the quasi-logarithmic adsorption isotherm (N.V. Kul'kova, M.Yu.Kvyatkovskaya, D.Yu.Murzin, Liquid-phase ammoximation of cyclohexanone. 3. Ammonia adsorption on ammoximation catalysts, Khimicheskaya Promyshlennost, (1997), 28). For derivation of adsorption isotherms in the case of exponential nonuniformity, the desorption exponent ~=ln b = -lna should be preferably used as an integration variable. The probability that a site is occupied is given by eq. (2.29) ap o- - - -
(2.53)
l+a p
and adsorption equilibrium is given by a re-arranged form ofeq. (2.36) 1
1
0
Ol+__
(2.54)
ap
Ase ~=l/a,then
(2.55) P
For this distribution function, e.g. exponential nonuniformity
1 ~iAeY~d~ P
(2.56)
MCKC02.fm Page 56 Friday, August 19, 2005 6:03 PM
57
AH
0 Figure 2.31. Differential heats of adsorption as a function of coverage corresponding to Yemkin (linear decrease) and Frendlich adsorption isotherms.
2.5. 5.1. Adsorption isotherrns- "induced" nonunijbrmity On both polycrystalline surfaces and single-crystal faces the isosteric heat of adsorption depends on the coverage, thus indicating nonuniform behavior of adsorbed species as a function of coverage, in order to describe such data other types of models can be used; induced nonuniformity or uniform surfaces with a varying of the binding energies of the adsorbed particles with coverage due to adsorbate-adsorbate interactions. The most complete mathematical model of a nonuniform adsorbed layer is the distributed model, which takes into account interactions of adsorbed species, their mobility, and a possibility of phase transitions under the action of adsorbed species. The layer of adsorbed species corresponds to the two-dimensional model of the lattice gas, which is a characteristic model of statistical mechanics. Currently, it is widely used in the modeling of elementary processes on the catalyst surface. The energies of the lateral interaction between species localized in different lattice cells are the main parameters of the model. In the case of the chemisorption of simple species, each species occupies one unit cell. The catalytic process consists of a set of elementary steps of adsorption, desorption, and diffusion and an elementary act of reaction, which occurs on some set of cells (nodes) of the lattice. The interatomic and intermolecular interactions of adsorbed species and their state on the catalyst surface are the basis of all elementary steps of the catalytic process. The importance and reliability of the modeled results depend on the correct choice of the potential of the interatomic interaction. The question about the type and nature of interatomic forces between adsorbed species is the focus, interatomic forces are diverse and usually anisotropic. Adsorbed species do not form structures at low coverage. When the number of adsorbed species and the rate of their surface diffusion increase, the probabilities of their interaction and formation of surface polyatomic structures increases. These structures can be rather stable and they form islands of adsorbed species. The theory of interatomic interactions on the catalyst surface is far from being complete. Only some specific mechanisms for the formation of interatomic forces are known. Exchange forces act at short distances and result in repulsion. Long-distance dispersion forces always result in attraction between surface species. Therefore, the repulsive interactions between nearest neighbors and attractive interactions between next-nearest neighbors are considered in the modeling. Taking into account a critical effect of interaction parameters on the rates of elementary steps and the fact that it is currently impossible to theoretically determine the true interaction parameters, their apparent values are obtained as already stated above from the whole set of experimental data on adsorption, desorption, and thermal desorption, including
58 the analysis of TPR spectra. The assumption of the perfect mixing of adsorbed species makes it possible to develop rather simple models for complex reactions. The importance of lateral interactions between chemisorbed molecules has been demonstrated for many cases, i.e. lateral interactions result in ordering of adsorbed particles. With the development of the low-energy electron diffraction technique, this phenomenon has been observed in thousands of adsorbate/substrate systems. The analysis of LEED data obtained at different coverages and temperatures makes it possible to construct adsorbate substrate phase diagrams. Comparing measured and calculated phase diagrams, one can obtain values for lateral interactions. Alternatively, lateral interactions can be assessed from a judicious analysis of desorption kinetics. Information on lateral interactions can also be extracted from high-resolution scanning tunnelling microscopy (STM) data by analyzing distribution of adsorbed particles. A common way to evaluate lateral interactions is based on analysis of desorption kinetics, as the desorption rate usually decreases rapidly with a decrease in coverage. The value of lateral ineteractions between two nearest -neighbour species are often in the range of 4-16 kJ/mole, which is less than the energies of adsorption. These interactions could be either attractive or repulsive and are not prominent at low coverage as the adsorbed particles then have no neighbours. Commonly applied simplified models nevertheless do take into account lateral interactions in the adsorbed layer. In the widely used lattice gas model the relationships between the rate of an elementary reaction and coverage is complex and cannot be written in a closed form when this model is used. In the model each adsorbate is assumed to be localized on a twodimensional array of surface sites and a site is assumed to be either vacant or occupied by a single adsorbate. The regular lattice systems with nearest - neighbor interactions (Ising models) are among the simplest models, still able to reflect many characteristics of real systems. As closed solutions cannot be obtained, physically reasonable approximations were proposed. In the Bragg-Williams approximation, configurational degeneracy and nearestneighbor interaction energy are treated as though molecules were distributed randomly among the sites. The simplest representation of adsorbate - adsorbate- adsorbent interactions is the Fowler Guggenheim isotherm which is Ibrmally similar to the Bragg-Williams approximation. This isotherm differs from the Langmuir one in the exponential term, accounts for the interactions. aP =
0
1-0
e
vO
(2.63)
For multicomponent chemisorption we will consider the monolayer of N i species of molecules (or atoms) of type i and Nj species of molecules (or atoms) o f t y p e j with the total amount of J sorts of molecules (or atoms) (i ¢ j). For multicomponent chemisorption the canonical partition function Q in the one-dimensional case is given by J
Q = Q,(Ni,L,T) I--I j-l,i¢j
-Ql
Qj(Nj,L,T) = q;~"Z e 1
~)~
e~/~rqiNi 2 e e,/kr
(2.64)
1
where qi is the single-molecule partition function without interactions, E i is the total energy of nearest-neighbor interactions of N i molecules (or atoms), f2i is the total number of configurations of Ni molecules (or atoms) on L sites. Equation (2.64) can be rewritten:
59 J
;\:, 'K''~
(o.N u/kT+ Z
o)::N!//kT
':~
*
: ~r r
Q = qi 2., guv~ , L, N , ) e .G
(2.65)
.i
d
• rI
-os:N//kT-o/i:V://kl+ Z - O ) lNfl/kT
q)~"Z g ( N j ' L ' N ,
1
)e
':'"~'"-~'
N~s
where g is a degeneracy factor allowing the possibility that several physically distinguishable states may have the same energy. In eq.(2.65) o)~j can be either positive (repulsive interactions) or negative (attractive). Within the framework of the Bragg - Williams approximation the energy of interactions is replaced by an average interaction energy. Therefore we arrive at ,/
co~iN/i/kl + Z Q z
q,N'e
co!:N!:/kT
~'g(N,,L,N,)* _u.
.1 1 ....
(2.66)
,! -(o;:N~7/kT-oJm:V:~/kT+Z-o):iA::s/kT
d
,[-I@:,e
l=id,~i I,~/
1
Zg(Nj,L,N/:) N,/
The degeneracy factor for adsorbed species of /-type is given by .: L!
~"g(Y:,L,X:i ) =
".
(2.67)
.:
( L - u , - Zu,)!u,! [-[u.,! .j=l,i~.j
1=13,aj
Similar equations can be derived for species of ./'-types. An expression for the average interaction energy follows from the Bragg-Williams approximation (2.68)
N_ii = (cNi/L) (N i/2)
and N/y
(2.69)
(cNi/L) (Nj/2)
where the factor two is used for preventing the counting of each pair ii or /j twice. Using In y!= yln y - y the chemical potential can be obtained (2.70)
t.z, = - k T ( 8 In Q / aN,),v.:. T
and therefore J
I~,= - k T
j
In q, - ln(L - N, - ~ N j) + In N, - ~ j N , / 2 L k T + Y - c o j j N j / 2 L k T j=l,i~ j
j=l ,i:~i
(2.71)
60 The chemical potential of a gas molecule (/_lg i ) is expressed, setting the lowest energy state as energy zero (2.72)
¢tg, = - k T l n ( G T / P,d)
The equilibrium condition is the equality of chemical potentials in the gas phase and in the adsorbed layer, therefore an adsorption isotherm follows naturally ,]
a,P, =
Oi j
e ''~ I--I e~'~'oj
(2.73)
j=l,j~i
where a i is the adsorption coefficient of i, 0 is the degree of covering (0 i =Ni/L) and v. = co.c / 2 k T
(2.74)
vii = coilc / 2 k T
(2.75)
Equation (2.73) can be derived using a thermodynamic approach within the framework of a surface electronic gas model. Although simple this model gives a physically reasonable explanation of the interaction parameter, as it is based on Sommerefield' theory of metals. The surface electronic gas model explains mutual interactions between adsorbed particles and their interaction with the catalyst determined by changes in the position of the Fermi level. The model is based on the assumption that a complete or partial ionization of the adsorbed particles takes place during chemisorption, with electrons being transferred to the surface layer. It means, that chemisorbed particles feed their electrons into the surface layer of the solid or take electrons from it, forming at the surface a kind of two-dimensional electronic gas. The changes in electron concentration in the solid proceed only in the subsurface layer. The model can be used only when surface coverage is not small. A peculiar characteristic of the model is that the energy of the adsorbed layer is determined only by the total number of adsorbed particles and does not depend on their arrangement. Another essential feature of the model is the statement that electrons in the subsurface layer can be treated as isolated from other metal electrons, as if they are put in a narrow and deep bath. Distinct from the more refined, and thus much more complicated lattice-gas model, the tbrm of the model of the surface electronic gas provides the possibility for its application to chemisorption of gas mixtures and thus to modeling of kinetics of complex reactions. A simple calculation in the spirit of Sommerfeld's theory of metals for the twodimensional case leads to the equation g =
go rl2h2L0
(2.76)
4~m * where rl is the effective charge acquired by an adsorbed particle, h is the Plank constant, e is the heat of adsorption, m* effective electron mass. For the treatment of multicomponent chemisorption let us assume that on the surface exists N i adsorbed species A i with the effective charge qi and Nj species of A i with the effective
61 charge ~lj, where i,j. It can also be assumed that q is not a function of the degree of covering. The total quantity of electrons in a subsurface layer N e can be expressed as follows J
N~ = N~° + IkN , + ~ ,]iN,
(2.77)
j=l,j~i
where ~
is the value of N~, at 0 0. The value of maximum surface kinetic energy E s is
therefore
h 2 N C0
h2N,
- - - + q , - - + (E~) .... 4am * s 4am * s
J
h2Nj
~ q/ i<,.i< " 4errn * s
(2.78)
where s is the surface area. When an additional amount of particles N i is adsorbed
an
additional amount of electrons ~li Ni will be in the surface layer. They will change the kinetic energy of the surface in comparison with the adsorption on a clean surface and the adsorption energy will be
+ ~ 2-h-Ni N,_ ~
Ei = g°N, - 7/, 4ran * s
j=l,j,~i
q, rl/
+_h2N./ 4ran * s Ni
(2.79)
where g0 is the value of E at 0=0. The sign "+" corresponds to the case of repulsive interactions, and sign "-" for attractive interaction. A thermodynamic approach results in in Pa
TM
(ttCai/RT)-SCai/R + (H e ai/RT
)- S ¢ai/R
(2.80)
where H___Cai is the configurational enthalpy and S C a is the configurational entropy of the adsorbed layer. SCa coincides with that lbr the simple adsorption, thus S__Ca= R in 0°0''*
0,0o'
(2.81)
where 0,'t is the standard coverage, which can be defined in the following way 0[ t = 00't = 0.5. The site balance is given J
00=1-0 ,-
~'0j
(2.82)
j =], j ~i
From eq. (2.79)-(2.82) and I-ICai/R (e°-e)/kT, where e F E i / N i the adsorption isotherm for the adsorption of i component of gas mixture can be obtained •
62 d
aip,=
O, J l-t), -
e <'¢~°'~y H e <,~'~°,c'q ~]Oi j=l.j¢i
(2.83)
j =l,.j~i
where h 2
C- - 4knm *
(2.84)
and aLi(ari ) can be either +1 (repulsive interactions) or -1 (attractive). Eq. (2.84) is of the same type as eq. (2.73), however it gives more possibilities for predicting (or estimating) the values of the interaction parameter, which, within the framework of the Bragg-Williams lattice gas remains unclear. In the case of formation of clusters on the solid surfaces when the surface of adsorbent has N J 0 molecules and Nj (0 clusters of i molecules, where i varies from unity to /max, /maxbeing the number of molecules in the larger cluster, j, and n represents the number of chemically distinct species. The canonical partition function Q is then J
Q Qn(NjO,L,T) H
Qj(N/jj,L,T)
./=Ln~j
± 1
,1
H
exp (-E__jO)/kT)
2=l,n~-]
(2.85)
1
where qn is the single-cluster partition function without interactions, En (p is the total nearestneighbor energy of interactions of Nn 0) clusters, ~n OJis the total number of configurations of Nn 0) clusters on L sites, T-temperature, k- Boltzmann constant. Replacing the energy of interaction by average interaction energy we arrive at .!
Q=qn Nn exp (- wnnNnn/kT +
- wnjNnj/kT) ~ j= I ,n ;~/
.[
g (Nn, L, Nnn)*
Nn~~
J
I-I qjNj exp(_wjjNjj/kT_wjnNjn/k T _ ~ 1=1
_wjlNjl/kT) ~
I=1,1 ~en, I~.j
g(Nj,L, Nnj)
(2.86)
Nm
where g is a degeneracy factor allowing for the possibility that several physically distinguishable states may have the same energy, Nnn is average interaction energy and w is the potential energy of an interaction. For the degeneracy factor of adsorbed clusters of ntype it holds that
., L!
~., g(N,, L, N,,,,) = (L- N,-
.,
~ Nj)?N,,? 1--I Xj? J=l,n~-j
.j=l,n• ]
(2.87)
63 Similar equations are valid for clusters of j-types. The average interaction energy is expressed within the framework of the Bragg-Williams approximation, which finally leads to an adsorption isotherm 0 (1) n
a,,P, =
(I)
I
J
(i)
exp(v~,,O;, T ) 1--Iexp(v~jN, T
j
1-O~t)- Z N¢')T 1/L
I
/L)
(2.88)
.l=l,]~n
/=l,]~-n
where a n is the adsorption coefficient of n and, 0 / 0 and
is the degree of covering of singlets
vnj wnjc /2k
(2.89)
vjj u{/jc/2k
(2.90)
Defining the coverage of singlets through the total coverage of species of n-type 0/1) = On/c~'n
(2.91)
where '
:aT(i) /
an = ~ " ~ ,
XT(1)
"~'n
(2.92)
i=l
we finally arrive at ,l
a,,P, =
ON/ ai',J l_Onan/a2_
ZOlat/a: j=l,j~-It
exp(vn, T -' O,,a,,/a;,) 1--I e x p ( v J -' Ojaj / a'j) }
(2.93)
]=l].~,,
where !
an = Z..,~N('~--n / N~ '1
(2.94)
i=l 1
a, = ~ Nj( 0 / N,O)
(2.95)
/=1
The system of adsorption isotherms (2.93) allows for the discussion of the critical phenomena in the multi-component adsorption layer. When there is an interaction between the adsorbed particles and the adsorbate segregation occurs. For the sake of clarity, as an example we will present below the derivation tbr a two-component layer, however the conclusions are qualitatively the same for this case and for the cases where more than two components form the adsorbed layer.
64 Each point of the bifurcation set for the two-component case corresponds to two types on the surface 0 a and Oa+AO a ,0 b and Ob+AOb, where a and b indicates two different types of adsorbed molecules. Using the logarithmic form of the system (2.93) ln(O. + dO..) _ ln(l - (0. + dO.)o~. / c[. - (0 b + dOb)cc b / o:'b.) 0. 1 0.~. / c~'~,- Ob~ b / o(h -
(2.96)
= Van(a,,/a~,)dO/T + V,,h(ah/al,)dOh/T and ln(0h + dOh) _ ln(1 - (00 + dO,)a, / a', - (Oh + dOh)a h / a'h 0a 1 - Ojz. / a'. - Ohah / a'h : v ,,(~./~'o)do/r
(2.97)
+,'h,,(~h/~;)dO~ IT
Resolving eqns. (2.96) and (2.97) in series and dropping all terms higher than the first power we obtain
dOa/Oa+(~xa/~'adO a+~Xb/~X'bdOb)/(1-Oa~ a/~'a-O b~ b/a'b )= =V aa( CZa/~'a)d Oa/T +Vab( CZb/ C~'b)d Ob/T
(2.98)
dOb/Ob+(Ota/(z'adO a+Otb/Ot'bdOb)/(1 -Oa~a/(X'a-Ob(Xb/~'b )= =V ab( C~a/~'a)dO a/T +Vbb(C~b/C~'b)dO b/T
(2.99)
and therefore dO a/O a- dOb/Ob=(V aa- Vab )(O~a/O~'a)dOa/T+(V ab- Vbb )((Zb/O~'b)dO b/Y
(2.100)
Further manupilations give 2 +(~a/(~'aOa+(~b/O~'bOb)/(1-Oao~a/~'a-ObO~b/(~'b)=
=%ha + Vab)(c~a/OCa)dO a/T+(v aa+Vbb)(c~b/c~'b)dOb/T
(2.101)
At the critical point the derivative of the concentration with respect to density equals zero. In a two-dimensional phase, the coverage is the equivalent of the density, hence it holds that
dOa/dO b- Oa/Ob=O
(2.102)
and O~b/O~'bOb=O~a/~'aO a (Vab-V aa)/ (Vab- Vbb)
(2.103)
Introducing the quasi-total coverage O~b/ O~'bOb + O~a/ (z 'aOa = 0
(2.104)
65
we arrive at 2/0+ 1/(1- 0)=(Tc0q( 2V2ab - Vaa Vab - VaaVbb )/( 2Vab- Vaa - Vbb ) The left side of eq.(2.105) gives a minimum at 0cr=0~,. = ~ f 2 / l + ~ , equation for the critical temperature 2V~t )
-
VaaVah
(2.105) thus leading to the
-- V aaVbb
T~r -- (1 + t~)2 (2v. b _ v.. - vh,,)
(2.106)
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 has been used for the description of TPD spectra as well as for reaction of gases on metal surfaces in some instances. The heat of adsorption within the framework of the model, developed by D.King is given by
q.j~(O)=qo +
zoo 1
{1_40(1_0)[1_e~,,% r
(2.107)
where m is the heat of adsorption at zero coverage and z is the maximum possible number of nearest neighbours.
Sir David King
2. 5. 6. Adsorption is'otherms- multicentered ads'orption
A complication, which arises in the treatment of complex reactions, is that kinetic analysis is usually simplified by neglecting the polyatomic nature of the reacting molecule. However, for instance hydrogenation of hydrocarbons requires a patch of sites for adsorption, while atomic hydrogen is attached to only one metal site. A semi-competitive model was advanced based on the assumption that in such a situation the molecules do not absorb either with full competition, neither with non-existing competition. In liquid-phase reactions, when the bulk concentration of the organic species is relatively high, the catalyst surface is easily covered by these bulky organic molecules. At the same time, geometrical restrictions prevent organic molecules from completely covering the catalyst surface. Thus, there are potential interstitial spaces between the larger organic species adsorbed on the surface sites that cannot accommodate these species, and on which smaller (by comparison) atoms (hydrogen or oxygen) are able to adsorb. This situation is schematically presented in Figure 2.32.
66
idrogen ly s m a l l )
Figure 2.32. Selni-competitiveadsorption
The larger (organic) molecule adsorbs on a cluster of primary sites on the catalyst surface. An arbitrary number of sites can be assumed for the adsorption of the organic molecules, but a preferable way to estimate the size of the adsorption cluster is based on molecular considerations, i.e. the size of the organic molecule and the structure of the catalyst surface. Interstitial sites remain for the smaller molecule (here: hydrogen) to stick between the adsorbed molecules and the maximum coverage of the larger molecule is less than unity because of the larger molecular size. Here we shall derive the rate equation for a general case where an organic molecule (A) adsorbs on a cluster consisting of m primary sites. The small molecule (B, typically hydrogen) adsorbs either in molecular form or dissociatively. A+'= A*'
(2.1o8)
B + i* = i B *
where *' denotes a cluster consisting of m primary sites (*). For the rapid adsorption steps, the quasi-equilibria are expressed by K4-
cA*'
(2.109)
C * CA
and i
K~ - ca° i
(2.110)
C,C B
The total balance of primary sites is given by C O = nICA,, + CB, + C,
(2.111)
where Co is the total concentration of primary sites on the surface. For the larger molecule (A) the site balance is written as cA,. + c,. = a ( c o - c , , ) / m
(2.112)
67
where (x is the adsorption-competition parameter of the larger molecule and c*' is the concentration of vacant clusters of sites accessible for the adsorption of A. The adsorption semi-equilibria are inserted in the balance (2.112), and the relation of vacant clusters of sites (c*') and vacant primary sites (c*) is obtained: =
-(K~c~)
(2.113)
c,)/(1 +
This expression is inserted into the balance of primary sites. After straightforward algebraic steps, the coverage of free sites becomes (2.114)
G
+ X(Xor.,:Cor.,, ) +(l -
0, e. =,+(x.c.l"
k.C.tJ'X(Xor.,:Co
.,j
)
This concept did not include any shielding (screening) of sites with increasing coverage and is limited to ideal surfaces. However, it is known that adsorbed polyatomic molecules have a specific arrangement which leads to screening of the surface. The general approach of deriving such types of isotherms was proposed by Yu. Snagovskii. The surface was characterized by the number g of elementary sites adjacent to any particular site. If a molecule has an axis of symmetry (c; order) the number of arrangements is lowered by a factor of c~ , leading to the number of arrangements of a first particle equal to q)L, where cp is the multiplicity of the system ((p=g/G). Calculation of the number of arrangements for a second particle requires subtraction of the number of positions that are screened by the first particle. As the number of molecules increases, there is an overlapping of forbidden configurations of adsorbed species. Hence the coverage dependent screening parameter n needs to be introduced. Below we will consider multi-centred species and will take into account, interactions between adsorbed species. The number of possible positions for N molecules is given by N
M = (1 / N ! ) l ~ ( r l L - ( j - 1)n(j))
(2.115)
j=l
Consequently the partition function for the adsorbed layer is given by r •
r
Q = ( xNc'v) ~i ~L ((JL1)"(J))exp(-Ej / k T ) N~
(2.116)
j=l
where qN (gN!) is the partition function for one adsorbed particle and E I is the total nearestneighbour energy of interactions of Nj molecules (atoms). The free energy of the adsorbed layer is given by A n =-kTlnQ=-kT{-coN/kT+Nlnq-lnN!+NlnL+~"ln
NFJ L
1
n(j) }
q-
j =1
L
(2.117)
In eq. (2.117) the total interaction energy is replaced by multiplication of the potential energy of interactions by the number of molecules. Now we introduce a new variable x:
68
j-1
; Ax=m/L
x=----~m
(2.118)
which satisfies the following boundary conditions j=l ; x = 0 j = N ; x = (N-1)m/L = 0
(2.119)
where 0 is the surface coverage and m is the number of elementary sites that one molecule occupies. Summation in eq. (2.117) can be replaced by integration since N and L are very large numbers:
A H =-kT{-coN/kT+Nlnq-lnN!
+NlnL
L Ex 1
+--fdxlnm q---m n(x)
}
(2.120)
The differential of a definite integral at its upper limit is equal to the intergrand at this limit. As In y ! ~ In y - y, then d l n y ! >>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..
(2.129)
An expression similar to (2.128) can be formulated for many-centred species. The number o f ways o f distributing a single centred species follows directly from statistical mechanics /klt
Ma
(L ~rn, N~, ~rn N~:)! Ch
M~
;='
:vla
/='
M h
MIj
(2.130)
/=1
h=l
h=l
The chemical potential o f activated complexes o f one-centred molecules and adsorbed species are given by
70 t~lri
,-Z o,,,-Zo,, /=l
q"-ZH~-aj'/'/)(OP)I=I x
/x, =-kTln
h=l
Ma
1 - ,~____.0~, ,
(2.131)
j=l
/14~,
-w.CNt/2LkT
M F, "V
- ~.
u.'~,i,(.N ' i / 2 L k T - z__, wh, C N h 2 L k T n
j=l
h=l
(2.132) Ma
-ws~CN.i/2LkT-
;wrh
~
w,,;,,CN;/2LkT-w~,;CN~/2LkT-~
]=l,],at
h=l
Ma
ixh = - k T l n
Mh
(fh (1- ~2 0~, - ~ A ) { oh ,=1 ~=1 Ma
-Z
}
w~,;,CN h / 2 L k T
U',CNh/2LkT-Z
(2.133)
M;
Wo,hCN;/2LkT-Z j=l
We,,CN;/2LkT }
i=1
The total number of ways of distributing the species is independent of the order in which the species are attached. In transition state theory applied to heterogeneous catalysis the coverage of activated complexes is neglected. Consequently, interactions between a particular type of adsorbed species and activated complexe are also neglected. Equilibrium condition for adsorption again implies equality of the chemical potential in the gas-phase and the adsorbed phase: M~
o~ (l- ~ o~ )"", ]=1 a j ~ jp
=
.g
~
mo (;7o,
-
Z
M.
"' ma /
M<,
,,o.o(oo ;
'
M;,
o. ) ) . ( 1 - y o o , - y ~ o ; , 3 '
j=l
"°;
(2.134)
h=l
M,
* exp(r°zi0.j ) ~ I exp(c°jr0., ) l ~ exp(Oojt,0h) r=l,t*,aj
h=l
As an example, adsorption isotherms for a competitive adsorption of two molecules will be considered. A bulky molecule (A) requires several sites m for adsorption. Substance B is a one centred molecule, which adsorbs onto a single site. aAP A =
OA exp(c010~ + (°20~) (~om - k " O A + k ' " ov A 2 )(1 - 0• ) ";
(2.135)
71
e
-
G exp(o)20A+O)30e) 1--OA--G
(2.136)
Numerical calculations for multi-centered adsorption over nonuniform surfaces revealed that the multi-centered nature of adsorbed species masks the influence of non-uniformity, thus a seven-centered species obeys an almost "classical" profile. This indication in principle supports the utilization of models of ideal adsorbed layers to treat the adsorption behavior of large organic molecules. Other types of isotherms are reported in the literature for multi-centred adsorption on nonuniform surfaces with inclusion of lateral interactions with an average energy aP = 0 exp 0/2 (1 - 0 ) exp (~O/kT) / rlmlx/~O-O
(2.137)
This isotherms is reduced, for ideal surfaces, to aP = 0 exp 0/2 (1 - 0 ) / ~Tm1 ~ - 0
(2.13s)
2. 5. 7. Different adsorption modes In catalysis involving complex organic molecules with different functional groups, both the number of sites and the mode of adsorption are important. The changes of adsorption geometry are a common phenomenon and is also discussed in connection with simple molecules like carbon monoxide adsorption on metals. For instance, CO adsorbs with its molecular axis perpendicular to the surface, however it can tilt over some surfaces. The mode of adsorption can be dependent on the concentration, as demonstrated in selective hydrogenation of a,[3 unsaturated aldehyde - cinnamaldehyde. It is supposed that the adsorption mode of cinnamaldehyde at high concentrations differs from that at lower concentrations, more precisely, cinnamaldehyde adsorbs at high concentrations perpendicular to the catalyst surface with the aromatic rings in parallel arrangements. Another example is catalytic asymmetric hydrogenation on supported metal (Pt) catalysts in the presence of a chiral modifier- cinchonidine (D. Yu. Murzin, P. Mfiki-Arvela, E. Toukoniitty, T. Salmi, Asymmetric heterogeneous catalysis: science and engineering, Catalysis Reviews 47 (2005) 75-256). This reaction over non-chiral catalysts produces racemic mixtures of optical isomers when the substrate contains a prochiral center. Cinchonidine (Figure 2.33), being bulky molecule, reduces the accessible active platinum surface as it adsorbs and causes some deactivation with respect to racemic hydrogenation. The decrease in formation rate of the main product after the maximum can be a result of poisoning by adsorbed spectator species, which inhibits the enantiodifferentiating substrate-modifier interaction. Adsorbed cinchonidine in parallel mode (active form) provides enantioselective sites and when the reactant is adsorbed in the vicinity, interactions between the reactant and modifier lead to such orientation that hydrogenation forms the preferred product. However, when a tilted form of modifier (spectator) is adsorbed in the vicinity of actor species, the site becomes poisoned and the overall activity decreases. At low coverage, cinchonidine adsorbs mainly via ~-bonding of aromatic quinoline rings, the ring system being almost parallel to the metal surface (Figure 2.33). The adsorbed parallel form of the modifier would be involved in formation of enantiodifferentiation substrate-modifier complexes on the catalyst surface. At higher
72 coverage, two additional tilted adsorption modes of cinchonidine were observed. The parallel and tilted adsorption modes of cinchonidine require different amounts of primary Pt sites for adsorption, the tbrmer occupying more active metal sites than the latter and being active in enantiodifferentiating interactions. Therefore, with increasing cinchonidine coverage, the surface is occupied more by the ineffective form of the modifier and becomes unfavorable for the substrate-modifier interaction thus lowering the selectivity. This would imply that after reaching certain optimum modifier coverage, a further increase in the modifier concentration would result in a gradual decline in the enantioselectivity. In a tilted adsorption mode, the catalyst surface accommodates more modifier onto it and less substrate and as a result a lower reaction rate is obtained. This results in a maximum in enantioselectivity as well as reaction rate, when the modifier concentration is increased.
b
/
Tilted
Parallel
Figure 2.33. Different adsorption modes (a) of cinchonidine (b). It is important to remember when dealing with bulky organic molecules adsorbed on supported metal catalysts, that in the classical kinetic treatment the surface is treated as having infinite size. In the majority of organic catalytic reactions over nanometer-sized transition metal clusters dispersed on oxide supports it is definitely far from reality. The differences between extended surfaces and a nonometer-sized cluster can be profound, which requires special approaches.
73
Chapter 3. Elementary reactions 3.1. Reaction rate theory
The transition state theory (TST) developed by Eyring and Polanyi represents a theory which is capable of predicting the rate of a catalytic reaction, more specifically the pre-exponential factor.
Michael Polanyi
Henry Eyring
It is supposed that elementary reactions go through a path with the lowest energy barrier, or in other words minimum energy path (Figure 3.1) Potential
Figure 3.1. Mhmmumenergypath. Assume there exists a surface in phase space that divides it into a reactant region and a product region. It is assumed that this dividing surface is located at the transition state (Figure 3.2), e.g. maximum value of the potential energy surface on the minimum energy path (MEP) that connects the reactant(s) and product(s). Any trajectory passing through the dividing surface (or bottleneck) from the reactant side is assumed to eventually form products: non-recrossing rule. In the TST the reactant equilibrium is assumed to maintain a Boltzmann energy distribution. Activated complexes are assumed to have Boltzmann energy distributions corresponding to the temperature
74 of the reacting system, while the whole system is not at equilibrium. The motion of the system over the col is separated from other motions associated with the activated complex. Classical motion over the barrier is considered ignoring quantum effects and adiabatic processes are assumed, e.g. no changes in the potential energy of electrons. I__
Products
Reaction coordinate
Figure 3.2. Potential energy surface
No special equilibrium between activated complexes and reactants is assumed, it is supposed however, that within the space between q j and q,+Aql, q2 and qz+Aq2, configurations (activated complexes) have impulses (motions) between Pl and pl+Apl,p2 and p2+Ap2 respectively. These configurations are computed in accordance with Maxwell-Boltzmann distribution. Recall from the gas laws that an energy profile for molecules can be describe by the Maxwell-Boltzman distribution diagram (Figure 3.3). As the temperature goes up, the population of molecules with more energy also increases. Several features are specific for this distribution. The higher the temperature the higher the average kinetic energy and the broader the range of energies. The most probable kinetic energy is at the maxiumum of the curve. The average kinetic energy is slightly to the right of the maximum and is slightly greater than the most probable kinetic energy. Note the difference between statistical and chemical equilibrium. Statistical equilibrium assumes no changes in distribution of states with time, while at chemical equilibrium there is a constant ratio of concentrations of reactants and products. Chemical equilibrium leads to statistical equilibrium, while the latter does not necessarily mean chemical equilibrium, otherwise there will be no chemical reaction at all. lower temperature temperature lower 0.04
/
fraction
0.03
/ f
.~
\
\
m o r e species species have have more energy>E e n e r g y > EAAat at higher h i g h e r TT
\
/
0.02
oolI
0.01
higher temperature
0.00 0
20
40
60
80
100
120
activation energies energies activation energy energy
Figure 3.3 MaxwelI-Boltzmann distribution
140
160
76 freedom and 3n-6 and 3n-5 for vibrational in case of nonlinear and linear molecules respectively. Thus an atom (n=l) has 3 translation degrees of freedom; a molecule of two atoms 3 translational, 1 vibrational, 2 rotational degrees of freedom. For a linear molecule of 3 atoms besides 3 translational degrees of freedom, there are 4 vibrational and 2 rotational degree of freedom, while for a nonlinear molecule there will be 3 vibrational and 3 rotational degrees of freedom. There is one specific feature in the partition functions for the transition state as the one vibrational degree of freedom is replaced by the movement along reaction coordinate. Thus the total numebr of degrees of freedom is 3n-1. The translational partition function can be calculated knowing the particle mass, temperature and is conventionally calculated per unit volume
(3.6)
q/tans
or per dimension qmm.~, = 1 (
Hk
]2,aI/2 rg.r,
(3.7)
h where bt is the reduced mass, for instance for a molecule AB it is mAm B / ( m A + m~ ). The vibrational energy levels are usually calculated using the harmonic approximation and is equal for a diatomic molecule per degree of freedom
1 q,,i~, -
@ hv
1 -
e
k~r
oscillator
(3.8)
@
IR spectra are applied for calculations of vibrational partition functions, which obviously represents a challenge in calculation of vabration frequence of short-lived (10 -12 s) transition states. However vibrational partition functions are usually equal to unity, unless vibrations are of low- frequency. For calculations of rotational partition functions the moment of inertia (e.g. the molecule structure) should be known. In case of a diatomic moleule the rotation partition function is
qAB
-
8k~ a'2 IT h2
J ~
(3.9)
The electronic partition function is usually equal to unity. In the transition state theory developed by Polanyi and Eyring, the transition complex is located at the top of the energy barrier (Figure 3.2) and the reaction can be presented a movement along a potential energy surface where the transition state is located at the saddle point.
77 For a reaction A + B ~ [ X *] the activated complex moves along the imaginable reaction coordinate (motion over the col is a translational motion) with a speed defined from classical mechanics
v = [x#j x_/5
(3.1o)
where V - is the rate, x is the speed, [X~] is the concentration of activated complexes, .Sis the length of the col. The concentration of activated complexes is calculated from the MaxwellBoltzmann distribution
[x
[AI[&
(3.11)
The partition functions are given in eq. (3.11) with respect to the same zero energy level. Taking into account the energy difference between ground state and the transition sate we have instead of eq. (3.11) E #
[X ~]= [A][B]q~/q~qB e
E,,
Rr
(3.12)
The average speed of the particles moving from left to right over the barrier is calculated from from kinetic theory x (kT/2rtm--)05
(3.13)
TM
Placing these in the expresion for the rate we arrive at E~
V
[X~] x / 8 =[A][B]q ~ / qAq~e
Eo
Rr
(kT/2r~m±)°s/8
(3.14)
In the activated complex the translation motion along the reaction barrier is separeted from the other degrees of freedom leading to an expression for a modified partition function of the transition state q* = (2am~kJ)°s&L / h
(3.15)
and finally to an expression for the reaction rate according to transition state theory AE
r=[A][B]-k~ T q~ e Rr
(3.16)
Closer inspection of this equation reveals that although it resembles an Arrhenius equation for the temperature dependence of the rate constant, such dependence is more complicated
78 k=-ATn exp (-AE/RT)
(3.17)
Recalling that thermodynamic equilibrium is expressed by K s - ([X "] / [A][B]) eq- (q./qAqB) exp (-AE/RT)
(3.18)
and that the relationship between the equilibrium constant and the Gibbs energy is (3.19)
A G ~ - - R T In K ~
expression for the rate constant is k = (kBT/h)/~=(kBT/h) exp (-AG'/RT) = (kBT/h) exp (AS'/RT) exp (-AH*/RT)
(3.20)
where AS* is the standard enthropy and AH* is the standard enthalphy of activation. The activation energy (kJ/mole) is calculated from
(3.21)
E~ = R T + A H * - A * n R T
where A~n is change in the number of molecules when the activated complex is formed, which is important to take into account in case of not unimolecular reactions. As an example we consider the reaction A+AB=A2+B with formation of a linear activated complex. For the atom A there are 3 translational motions qt,x = (2a-m_~kBT)3/2 / h3
(3.22)
The total partition functions for AB are q,; AB =
(3.23)
qt, AB qr, AB qv,AB
q~,A = (2a-(m~ + m B)k~T) 3/2 / h 3 ; q,.,/B
8k~zc2 I ~ T . -
h 2
'
_
q,.,,/B
1
1 - e h,,ik;J
(3.24)
The partition functions for the linear activated complex are q,,.4AB = (2~-(2mA + r o B ) k S ) 3/2/h3;q,.,AAB -1
8k~-I
9
A/BT .
h2
(3.25)
qv,~,._~B = ( 1 - e h"/k~l')(1--e ~,,2/k~1')2 The moment of inertia for the diatomic molecule depends on the masses and the distance between the atoms
79 I = roam B/(mA+m~)
(3.26)
dAB 2
Similarly for the linear activated complex A ..... A . . . . c . . . . B , the moment of inertia is calculated from
where c is the center of mass and A c
x
I mAX+mA(x-AA) + mB (AB-x) with
(3.27)
maX+ mA(x-AA ) - m B (AB-x)
(3.28)
For a nonlinear activated complex in case of reaction A + B ~ A C ~ with nAc=nA+n B the partition functions are expressed by partition functions per degree of freedom 3
3
3n 1 - 6
q~ = q ~ , A q r j q v /4
3
3
3n,-6
, qB = q~,,qr,Bq,,,,
3
3
3n 'c'* - 7
, q<,~ = q,,A(,~q,.j~c~q,,.~
(3.29)
As already deiscussed above, the probability of an activated complex to go into products is considered within the framework of transition state theory to be equal to unity. In fact, such probability is < 1, if there are changes in the electronic state of the system. For instance, if systems of atoms move from one potential energy surface to another, which means that the reaction is "nonadiabatic" in the quantum mechanical sense. So far qualitative treatment of transmission coefficients is available only for simple cases with very little practical application to reactions on surfaces. Formally the expression for the rate constant should be completed with a transmission coefficient Z, the value is often around 10 -6. k = Z (kBT/h)/C
(3.30)
After considering the transition state theory the major contributions of it to the field of chemical kinetics can be briefly summirized. It gave the possibility to calculate the rate knowing characteristics of reactants and provided a link between geometry of reactants (configuration) and kinetics, as well as between the structure of matter (molecules) and kinetics. The notion of reaction coordinates was introduced and it became possible to calculate rather accurately the values of constants k for for mono-, bi-, and trimolecular reactions and to predict the temperature dependence for these reactions (Table 3.1). At the same time transition state theory requires information about the activated complexes, assumes equilibrium only for reactants, but not products and requires introduction of a special partition function (minus one degree of freedom). Another question which remains is the applicability of statistical thermodynamics, if the life time of activated complexes is ca. 10-13 s. For instance the application of transmission coefficient contradicts the basic principles of TST, namely statistical equilibrium between reactants and activated complexes.
80 Table 3.1. Temperature dependence of pre-exponential factor according to transition state theory* Reaction type Linear activated complex Non-linear activated complex T05 A+A T-05 TO A+L T-05 A+N T-I Z-i 5 L+L T-I T-I.5 L+N T-2 T-25 T-2 N+N T-0.5 A+A+A T-I T-1.5 T-2 A+A+L T-2.5 T-3 A+L+L T-3.5 T-4 L+L+L T-2.5 T-2 A+A+N T-3.5 A+L+N T-3 T-4.5 T-4 L+L+N T-4.5 T-5 L+N+N T-5.5 T-5 L+N+N * A- atom, L madN- linear madnonlinearmoleculesrespectively A recent advancement in the transition state theory was made by the winner o f the 1999 Nobel Prize in Chemistry A. H. Zewail, who showed that it is possible to see how atoms in a molecule move during a chemical reaction using a rapid laser technique. In his studies o f the transition states o f chemical reactions using femtosecond (10~5 s) spectroscopy, the original substances are mixed as beams o f molecules in a vacuum chamber. The pump pulse is the starting signal for the reaction while the probe pulse examines what is happening (Figure 3.4). By varying the time interval between the two pulses it is possible to see how quickly the original molecule is transformed. The new shapes the molecule takes when it is excited - perhaps going through one or more transition states - have spectra that may serve as fingerprints.
Ahmed H. Zewail
Figure 3.4. Principle of femtosecond spectroscopy
81 The time interval between the pulses can be varied simply by causing the probe pulse to make a detour via mirrors. An ultrafast laser then injects two pulses: first a powerful pump pulse that strikes the molecule and excites it to a higher energy state, and then a weaker probe pulse at a wavelength chosen to detect the original molecule or an altered form of this.
Figure 3.5. Potential energy paths for formation ethylene from cyclobutane The left-hand side of Figure 3.5. shows for the reaction of cyclobutane molecule to two ethylene molecules, how- the state energy varies if both bonds are stretched and broken simultaneously. The right-hand figure shows the case where one bond at a time breaks. Zewail demonstrated with femtosecond spectroscopy that the intermediate product was in fact formed, and had a lifetime of 700 fs.
3.2.Reaction mechanism
A mechanism of a catalytic reactions is a sequence of elementary steps, the rate of which can be described by; for instance, transition state theory. Catalytic specie react with a catalyst forming complexes as described in Chapters 5-7 respectively for homogeneous, enzymatic and heterogeneous catalysis. From these rather complicated reaction sequences rate laws should be derived which could be then compard with experimental data. The method, which greatly helped in simplifying the kinetic treatment, is the so called steadystate approximation (SSA) originally developed by Bodenstein in 1913, who investigated bromination of hydrogen to produce HBr. In general a reaction is considered to be at steady state if the concentrations of all species in each element of the reaction space (volume for a homogeneous reaction or surface tbr a heterogeneous reaction) do not change in time (Figure 3.6.) [ nterrsed a e]
Rate of p oduc ng the ne;medate s h e same as s rae of consumpton
Figure 3.6. Concentration of a reaction intermediate as a function of reaction time.
82 Such conditions in general are fulfilled in open systems, like tibular reactors or in flow circulation reactors. Grouping the reaction species in two categories: reaction intermediates in one and substrates and products in another one, it can be stated, for reaction participants in the steady state, that if a species enters an element of the reaction space its concentration does not change with time, while for intermediates the generation rate of them is equal to the consumption rate. Bodenstein proposed that there are intermediates in chemical reactions that are present in "inferior" amounts, e.g. in much less concentrations than the major species in the mechanism. When this condition is met, the rate of change of the concentration of the intermediate X can be considered negligible. In mathematical language
(3.31)
d X / dt = O
As an example we will consider a catalytic mechanism 1. A+B<=>X 2. X<=>Z A+B - Z
(3.32)
where X is a reaction intermediate. The rate of the reactions 1 and 2 in (3.32) are given by r l = r+l - r 1 = k I C A C
B -
k 1Cx,
~ = r+2 - r 2 = k 2 C x - k 2 C z
(3.33)
The overall rate of the product formation Z is equal to the rate of the second step in the reaction mechanism. In the steady state hypothesis the rate is r x = 1+1 - r
I --1+2 + r 2 = k I C A C B - k
1Cx -k2C
X
(3.34)
+k 2Cz = 0
giving the concetration of the intermediate cA, =
klCACI~ + k
2Cz
(3.35)
k l+k2 Substitting the intermediate concentration in the equation for the rate r = k 2 klCAC# +k-2Cz k 1 +kz
k-2 (k-1 + k2 ) C z = k l k 2 C 4 C ~
k 1 +k2
k 1 +k2
k-lk-2
Cz
(3.36)
k 1 -}-k2
The application of the steady-state approximation to heterogeneous catalysis does not require that the surface concentrations are low, but implies that they do not change with time. The steady-state approximation applied to chain reactions by Bodenstein cannot describe several time-dependent phenomena, for instance branched chain reactions, leading to explositions. For such cases the semi (or "quasi") steady-state approach, developed by a Nobel prize winner N. Semenov, assumes that concentrations of all radicals except one (the greatest) are considered as steady state.
83
Max Bodenstein
Nikolai Semenov
Imagine that one step is much slower than the other ones. In such a case other steps could be considered as being close to equilibrium, as their rates will be close to each other, e.g. i+/~ ~ 1
(3.37)
In the example of mechanism (3.32) if the first step is rapid k 1 >> 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/k<. As will be demonstrated later for heterogeneous catalytic reactions, the quasi-equlibrium approach is often utlized. This limits the description of catalytic kinetics as it leaves out transient processes. For catalytic processes the reaction mechanisms can be simplified to consist of three steps: binding of the reactant by the catalyst, catalytic reaction leading to the product and release of this product. Quite often the first and the last steps are considered to be fast (quasi-equlibrium) while the catalytic reaction is supposed to be rate determining (Figure 3.7). i I
•
i"c
quasi- equilibrium
1I
I~ 1 1 I,t
~
rate determining step quasi- equilibrium
F
Figure 3.7. Illustration of quasi-equilibrium and rate determining step concepts.
In the case of complex reactions several steps are possible and rigorous treatment is required, as the quasi-equilibrium approach cannot be applied. Such cases will be treated in Chapter 4. In the case of homogeneous catalytic reactions we can consider the following mechanism 1. A + K <=>AK 2. AK + B ~ A B ] K (activated complex) 3. A B I K ~ K + C
(3.39)
84
The overall rate if we assume reversibility of only the first step the reaction rate is
r -
k l k ~ C 4CB - • Cx k_ 1 + k 2 C B
(3.40)
For the case of so-called Arrhenius intermediates (Figure 3.8) the second barrier along the reaction coordinate determines the rate and thus k<>>k2[B], leading to (3.41)
r - klk2CACB C
k~
x
while for van't Hoff intermediates (Figure 3.8), when the first barrier is the highest k.l<< k2[B] (3.42)
r = klCAC x
Comparsion between eq. (3.41) and (3.42) illustrates the apparent danger of deriving reaction mechanism from kinetic data obtained in a narrow domain of reaction parameters, as depending on the concentration domain of B the rate can be either first order or zero order with respect to B. Energ Y
Energy
Reaction coordinate
Reaction coordinate
Figure 3.8. Potential energydiagrams for Arrhenius (left) and van't Hoff intermediates(right).
3.3. Quasi-equilibrium approximation Below we will consider surface decomposition of a molecule A A (g) ~ A (ads) ~
Products
We assume that the decomposition reaction occurs uniformly across the surface sites at which molecule A may be adsorbed and is not restricted to a limited number of special sites. Moreover the products are supposed to be very weakly bound to the surface and, once formed, are rapidly desorbed. The rate determining step (rds) is considered to be the surface decomposition step. The molecules of A adsorbed on the surface are in equilibrium with those in the gas phase and we may predict the surface concentration of A from the Langmuir isotherm
85 0=KP/(I+KP). The rate of the surface decomposition (and hence of the reaction) is given by an expression of the form
r = kO
-
kKP - I+KP
(3.43)
Two extreme limits are possible, the first when the partial pressures are low and KP<>I; then (I+KP) KP and the rate is r - k, i.e. we arrive at a zero order reaction (with respect to the partial pressure of A). This is the high pressure (or strong binding i.e. large K) limit. Under these conditions the steady state surface coverage, 0 , of the reactant molecule is almost unity. The rate shows the same pressure variation as does the surface coverage in the Langmuir adsorption isotherm. Rate
/
/
J
high pressure limit: rate - k
limit:
Pressure
Figure 3.9. Pressure dependence o f reaction rate (eq. 3.43)
Consider a Langmuir-Hinshelwood reaction of the following type • A (g) +Z = ZA (ads) (quasi-equilibrium) B (g) +Z = ZB (ads) (quasi-equilibrium) ZA (ads) + ZB (ads) ~ C (g) +2Z where the surface reaction between the two adsorbed species is the rate determining step. According to the mass action law- if the adsorbed molecules are mobile on the surface and assuming competition between A and B for the surface sites the reaction rate is r = kOAO = k
KAP KBP = kKAP K PB 1-t-KAPA-t-KBPB I + K A P A + K B PB (I+KAPA+K P ) 2
(3.44)
In case that KaPA << 1 and KB PB << 1 both coverages 0A and 0B are very low, and the rate is
86 r = kK APAKB PB
(3.45)
which means that the reaction is first order in both reactants. In case that KAPA<
(3.46)
Depending upon the partial pressure and binding strength of the reactants, a given model for the reaction scheme can give rise to a variety of apparent kinetics. This highlights the dangers inherent in the reverse process - namely trying to use kinetic data to obtain information about the reaction mechanism. In the treatment above it was assumed that two adsorbed species react forming a product, the so-called case of Langmuir-Hinshelwood reaction (Figure 3.10).
Figure 3.10. Langmuir-Hhlshelwood (left) and Eley-Rideal (right) reaction mechanisms.
If the product C adsorbs on the catalyst surface the rate equation is becoming slightly more complicated, as the coverage of C should be included in the denominator r =
kK~PAKBPB
(3.47)
(I + K4p4 + KBPB + K e P t ) 2
Sir Eric Rideal
Sir Cyril Hinshelwood
If the reaction is reversible, then r = k+OAOe - k O<:& =
k+K/P/KBPB (1 + KAPA +KBPB +K( PC ) 2
(1 + KAPA + KBP~ + Kcpc:) 2 (3.48)
k+KAPAKBPB(1 =
k K(,Pc ) k+KAp4KBPB (I+KqPA +KBPB + K e P t ) 2 =r+(1
k K<:Pc
1 Pc K PAP~)
87 which is similar to equation (1.67). Another possibility is that a molecule from the gas phase reacts with an adsorbed molecule without adsorbing itself on the surface. Such a case is denoted as an Eley-Rideal process. For the mechanism
A(g)+Z =ZA(ads) 13(g)+ZA=:>ZC(ads) me(ads) = C(g) +Z
(quasi-equilibrium) (quasi-equilibrium)
the reaction rate in case of an irreversible reaction is given by r = k+OAP B =
k+Kj~PJ~PB
1 + K A P / + K c Pc
(3.49)
In the case that KAPA<
r --
kinetic_ factor * driving _ force (adsorption _ term)"
(3.50)
where the adsorption term includes KAPA, etc. for molecular adsorption or (](APA)1/2 for dissociative one and the power in denominar corresponds to the number of species in the rate determining steps, while the driving force is (1- P(,PI/Keq PAPB). The classical treatment of Langmuir kinetics on uniform surfaces can be extended to the reactions that occur on two types of sites in close proximity. The reaction proceeds via interactions between species adsorbed on such distinct sites. Examples of such reactions are various oxidation reactions on metal oxides, which contain two types of sites, metal atoms sites and oxygen atom sites, each having a specific function. Another example is hydrogenolysis of haloarenes where spillover hydrogen reacts with an organic molecule adsorbed on the metal surface:
A(g)+Z = ZA(ads) (quasi-equilibrium) B(g)+Z' = Z'B(ads) (quasi-equilibrium) ZA(ads)+ Z'U(ads)~ C(g) +Z +Z' rate determining step. Similarly to the consideration presented above we can write for the reaction rate r = kOAO' ~ = k
K-~PA KBPB 1 + K A PA 1 + K ~ P B
Dual site formalism can be easily extended to other cases, e.g. for dissociative adsorption.
(3.51)
88
3.4.Relationship between thermodynamics and kinetics Linear free energy relationships (LFER) are widely used in homogeneously catalyzed reactions. These relationships bind reaction constants k with equilibrium constants K in a series of analogous elementary reactions: k= g K ~
0
(3.52)
where g and o~ (Polanyi parameter) are constants. One of the typical examples are Bronsted or Hammett-Taft equations. Such semi-empirical relationships axe based on a supposition of existence of a certain relation between reaction thermodynamics and kinetics, e.g. between the Gibbs free energy of activated complexes and the Gibbs energy of a reaction A G" - c~ A G
(3.53)
Evans and Polanyi introduced this relationship to organic reactions, Semenov extended the application to chain reactions and Temkin applied linear free energy relationships to heterogeneous catalysis.
Louis Hammett
Johannes Bronsted
Linear free energy relationship was applied by Hammett to predict the influence of substituents in the aromatic ring on the reaction rates, involving reaction center Y Y
d>
R
and relates it to dissociation constants of substituted benzoic acids 0
0
//
//
C-OH
C-O-
+H +
R
R
The Hammett equation takes the form
89
lg(~o) = p lg(~-~o)
(3.54)
with k0 and K0 corresponding to R=H, and p is the reaction constant characteristic of the given reaction of Y. Introducing the substiment constant K lg(~-o ) = cr
(3.55)
one arrives at l g ( ~ ) = per
(3.56)
For the substituent constant, if c~ >0, than the functional group R attracts electrons stronger than H, in the opposite case c~ <0, R attracts electrons weaker than H Table 3.2. Values of the substituent constant R
o-
R
(Ol l)p (NO2)p (OCH3)p (C1)p (CN)p
-0.37 0.78 -0.27 0.227 0.66
(Oll)m (NO2)m (OCH3)m (C1)m (CN)ln
0.121 0.71 0.115 0.373 0.56
If p is positive there is a removal of electronic density from the reaction center, which facilitates the reaction, while if the value is negative the reaction rate is slower, than the dissociation of the acid. Values of,o for some reactions are presented below Ar COOH ~ArCOO-+H + Ar Br+C2HsOH ~ ArOC2Hs+HBr
1
ArCOO C2H5 + H + + O H - ~ ArCOOH+ C2HsOH ArCH3+Br2 ~ A r C H 2 B r + H B r
4.9 0.144 -1.085
A r H + N O 2 + ~ ArNO2+H+
-7.29
The effect of the rate and substituent constants on the rate could be summirized in the way presented in Table 3.3. Table 3.3. Ratio of rate constants depending on the substituent constant crand reaction constant p. p>0 p>0 k 0 k>ko k>k0 cr < 0 k
90 For alifatic systems an equation proposed by Taft is often applied that involves the polar substituent constant cy* and the steric substituent constant E,
lg(k/k ) =,o* cr* +c~Es
(3.57)
The complication in heterogeneous catalysis arises from the fact, that due to the influence of the reaction milieu, catalytic surfaces can undergo transformations. An intrinsic property of catalytic reactions is their complex, multi step (i.e. adsorption, surface reaction, desorption) nature. Therefore, LFER can be applied only to true, but not apparent kinetic constants and should be attributed to the same number of catalytically active sites. Another essential limitation arises from the generally complex behavior of organic catalytic reactions, more specifically a variety of products with diftbrent selectivity dependencies. The consequence of this behavior is the possibility of LFER application only to specific (for example, zero or first order) reactions. Following successful application of the Hammett equation in homogeneous catalysis, several publications appeared about the direct utilization of the Hammett relationships to heterogeneous catalytic hydrogenations, as well as acid catalyzed reactions. However, relatively few studies (compared to homogeneous reactions) have been reported, where LFER in heterogeneous catalysis are applied. One example is worth considering which shows the possibilities of application of LFER to elucidate reaction mechanisms in the case of heterogeneous catalytic reactions. Synthesis of organic esters, which are used as perfumes, flavours, pharmaceuticals, plasticisers, solvents and intermediates, by esterification can be performed using either mineral acids or solid acid catalysts Rc~O
+
R'OH
~
RC ~O
~
%H
+
H20
"oR'
In the former case the Taft equation was applied to describe the influence of substituents both in alcohols and acids. However, for a solid acid catalyst (J.Lilja, D.Yu. Murzin, P. M~iki-Alwela, T.Salmi, M.Sundell, Esterification of different acids over heterogeneous and homogeneous catalysts and correlation with the Taft equation, J.Molec. C a t a l y s i s A, 182-183 (2002) 555) it was demonstrated, that although there is a correlation between rate constant and the chain length in alcohols (Figure 3.11) there are no correlations with the chain length in acids (Figure 3.12), indicating that for a second order reversible reaction the equation for the rate
,-
=
k(C~,,lCAlcoh,,l - CK~,c~Cw,,,~,~/ K) (1 + KA,,,CAc,, + KA,~,~,,,/CA,~,,,o , + K~,,,,.Cs,,,, ,. + KH,~,,,~C~v,,,,,.) 2 -
(3.5s)
should be modified to exclude adsorption of alcohols (and consequently of esters), which is weak, as the correlation between the chain length in the alcohol and the rate is the same for homogeneous and heterogeneous catalysis. At the same time, the nonexistance of any corrrelation between the rate and the acid chain length points out to more complex behaviour due to acid adsorption on the catalytic active sites and thus calls for inclusion of the acid adsorption terms in the denominator
lO/O+
91
i-qHT-
C]tj-
C4HgGHTGH5-
-O.8
l
,
-0.2
-0.1
-1.6 0
Figure 3.11. Taft relationship of alkyl substituents in alcohols in esterification of acetic (at), propanoic (11) and pentanoic (0) acids.
i4H9
i2H50.4
0~
-0.15
-0.10
-0.05
-0.4 0.00
Figure 3.12. Taft relationship of alkyl substituents in acids in esterification of methanol (11), ethanol (at), (x) -1propanol and 2-propanol (0).
3.5. Transition state theory of surface reactions Transition state theory of surface reactions was developed independently by Temkin and Laidler, Glasstone and Eyring shortly after the more general treatment of Eyring and Polanyi appeared. It was supposed that each molecule occupies one elementary space and there is a random distribution of molecules. Activated complexes occupy several adjacent elementary sites (s) and there are several possible positions of activated complexes (g). For a reaction of A and B there is a possibility to have several equivalent positions of the activated complexes (here g-4)
B
B A B
B
The number of activated complexes Nt is a small number in comparison with L ( the total number of sites). In general according to statistical thermodynamics the ways of distributing activated
92 complexes on the surface and adsorbed molecules (reactants or products) among the remaining L-sN t places are given by (gL) N' wt - - -
(3.59)
Nt!
and wo =
(L - s N t )!
(3.60)
NI~N 2~(L-sN t-N~ - N 2 -..)t
By combining these two probabilities the various ways o f covering the surface are obtained by multiplication (L- sNt)!LN'g N'
w=
(3.61)
N t !N l !N 2! L - sN t - N I - N 2 -..)!
An expression for the Helmholtz free energy relates it with the canonical partition function and the partition function o f adsorbed molecules and the transition state A=-kT in Qs + Eo
A~ =-kTln@"qi~2..qJ v'
(3.62)
(L - sNt)!LN'gN' Nt!NI!N2!(L_sN
t_N l_N 2_..)!
+E
(3.63) 0
The chemical potential o f activated complexes in the adsorbed state bh (c~ As/8 Nt)l'v,Yl,N 2
(3.64)
is expressed after some manupilations as
,tq = - k T l n q , g L ( L - N 1 - N 2 . . ) ~' F-gt N t Ls 0
(3.65)
where the coverage o f vacant sites is 0o -
L - N I - N2.. L
For a reaction
(3.66)
93 n i X 1 +n2X 2 +..pl~ +P2Y2 + . . = Z
(3.67)
where X1, etc. are adsorbed species and Yl are molecules reacting from the gas phase. At equilibrium the chemical potential of adsorbed species and gas molecules participating in the reaction is equal to the chemical potential of the transition state ~tt - Znctga+Zpg~tg
(3.68)
leading to an expression for the reaction rate in the spirit of conventional transition state theory g
t e 770x,. . ...(/F, . ,~ ^, y,, r = khT flt~llfin2gLq' U0 fiPlflP2
(3.69)
where q't - is the modified partition function with one degree of freedom extracted (as explained before in 3.1). The consequences of this treatment are immediate. The rate is proportional to the number of sites, e.g. surface area, e.g. amount of catalyst and depends on the surface structure and configuration of activated complexes. For the indirect adsorption of an atom, which can move freely in two dimensions over the surface but does not have any rotational or vibrational degrees of freedom the rate constant is given by 2~mk;3T k = kBT h2 kBT h (227nkBT) 3/2 (22-mkBT) 1'2
(3.70)
h3 Assuming the ideal gas behavior p-kBT, the rate of collisions with the surface for such a case will be r -
PA
(3.71)
( 2 s i n k S ) '/2 If, however, the molecule adsorbs via a state in which it is flee to move in two-dimensional space still keeping rotational
and vibrational degrees of freedom
94
/
/
T
T
than the adsorption rate is
r =
PA PA -q ,.o, - q ,,~ - S (22.mkBT)l/2 q~o, g~,q~Jh ga,,, ( 2 ~ n k B T ) I/2
(3.72)
where S is the sticking coefficient. The transition state is associated with the decrease of entropy, therefore the sticldng coefficient is less than unity and is usually a function of surface coverage, temperature and of the details of the surface structure of the adsorbent. Transition state theory applied to surface reactions provides possibilities to calculate the rates of elementary reactions and the entropy of activation, thus providing a framework ibr analysis of the transition state structure and estimating the number of sites if the entropy is known. Transition state theory, which is extremely important, can be applied to complex reactions, theoretically predicting the values of preexponential factors for multistep reaction mechanisms (Figure 3.13), where due to overparametrization direct numerical fitting of rate constants results in values with a large error interval. Adsorbed state
Transition state
Desorbed state
Preexponential factor
O
~ 1015 S 1
mobile
@ immobile
- 1013 S i
c
i 1 0 1 4 i6 S i
mobile
@ immobile
~
1013 S
t
Figure 3.13. Values of pre-exponential factors depending on the adsorption/desorption and transition state (1. Chorkendorff, J.W. Niemantsverdriet, Concepts of Modern Catalysis and Kinetics, Wiley, 2003).
3.6. Rates o f reactions on n o n i d e a l surfaces 3.6.1. B i o g r a p h i c a l n o n u n i f o r m i t y
In real adsorbed layers the kinetics is more complicated, as the active sites could have different activity associated with various defects. Similarly to the treatment of adsorption, we will present
95 here a special model of surface nonuniformity, which was developed by Temkin. The rate of sorption on an ideal surface is given by r+ = z + P ( 1 - 0 )
-
Z+P 1 + ap
(3.73)
where p is the pressure in the ideal gas phase that would be in equilibrium with the surface at the given coverage, called fugacity or virtual pressures and Z÷ is the rate constant. When there is no equlibrium with the gas phase, p is not equal to the actual gas pressure P and at p < P the amount of adsorbed particles increases with time. According to Boudart, fugacity or virtual pressure is sometimes regarded with suspicion for the simple reason that it is called virtual.
Michel Boudart The concept of virtual pressure is however very useful whenever an adsorption- desorption step in a catalytic cycle at the steady state is not at equilibrium. Each site on a nonuniform surface has an adsorption rate
z+P L l+ap 1
(3.74)
For nonuniform surface the adsorption rate will be the sum of all the local adsorption rates
P ~[Z+(p(~)d~
(3.75)
Making use of the transition state theory
k = k S Lge -Ac~/kBr h
(3.76)
and the relationship between the adsorption coefficient a and the Gibbs energy as well as linear free energy relationship (eq. 3.53) the preexponential factor is defined Z+ = keT Lge+~l.~- kBT Lge ~" h h
(3.77)
96 is defined as ~=ln b=-ln a, where b is desorption coefficient, thus ~=AG°dRT, as AG°~=-RTln a. The sites are numbered in the sequence of increasing ~ (decreasing adsorption strength) and their distribution is given by eq. 2.30. At the sites with the most strong adsorption ~=~0 the rate constant is Z+ =
kBT
(3.78)
L g e ~#o
h
Then the rate constant is ~z+
= ,FOe /t,+ --
6~ Y
; G ~o)
(3.79)
where 2`0 is the value of 2`+ at the most strongly adsorbed sites ¢=¢0. Remembering the distribution function qff~)- A exp (7~) the reaction rate is given by
r+ = L ; J
-1@
L
~ l + p e -~
(3.80)
The parameter 7 is defined as y-T/® and for the so-called even distribution 7-0, e.g. (p(¢)-A. Introducing u-1/ap-(1/p)*exp(¢), which leads to e ~ = l / u p , differentiating du=d(~)u and remembering the expression for the constant A (eq. 2.33) we have
zz+°e° 'P
,,,
e-,,¢ du
r+ - e e ' _e,< ,0u(l+
(3.81) )
/,/
where m - a - 7. Further manipulations introducing n - l - m give
r+
=
o < u n ldu z+P e J - 1 (aoP)" ,
r
(3.82)
Changing the integration limits at medium coverages to 0 and 0% taking into account 0
J u+l
~r
sinn~
and that sin (nx) = sin (mx) we arrive at the equation for adsorption rate
(3.83)
97
r+
=
0
Y
Z+ P
;2-
(3.84)
(aoP) m sinm~r
e j-I
At low coverage the adsorption rate is proportional to the actual gas pressure P. For evenly nonuniform surfaces (m=c0 7/(exp(Tf)-I ) is replaced by its limit at 7=0, (e.g. l/f) z+°P
r+ = - sin(m a-) f ( a o p ) ~
(3.85)
giving finally r+ -
k+P
P
(3.86)
m
where k+ is a constant (the expression for the desorption rate also contains the same constant). At low coverage the adsorption rate r+ is proportional to P and does not depend onp. This result can be obtained at m=0. At high coverage the rate is proportional to P / p , this is obtained at m=l. Equation (3.86) relates the rate of adsorption to the fugacity of adsorbed layer. Expressing fugacity via surface coverage, using for instance logarithmic isotherm at medium coverage 0 = ( 1 / f ) l n ( a o p ) we have aop = e ")
(3.87)
leading to the adsorption rate
r+ =
;,c
z +° P - - =
sin( a~- ) Jb ~to
~
Z+ °
sin( 6~7r )
f
which can be further presented in the form ofZeldovich r+ = k + P e
<jo
(3.88)
Pe ~jo
Roginskii equation (3.89)
sometimes erroneously called the Elovich equation, who was working in the laboratory of Roginskii and applied the Zeldovich -Roginskii equation to treat adsorption kinetics on some particular systems. In eq. (3.89) the rate constant is given by 0
k+ -
;r Z+ sin(~;r) f
(3.90)
98 If the Freundlich adsorption isotherm is valid than the adsorption rate is proportional to pressure and inversely proportional to a fractional power of surface coverage, giving the Bangham equation P
r+ = k+ 0 .....Y
(3.91)
Illustration of the adsorption rate dependence on coverage is presented in Figure 3.14, demonstrating that for ideal surfaces the rate does not vary with coverage while for nonideal surfaces such dependence could be very pronounced. Rate 10
'
y>O ,'(
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+SO2<-->Z+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+2Z<---~2ZH 3. ZO+ZH---~ZOH+Z 4. ZOH+ZH---~H20+2Z 5. ZO+H2---~H20+Z N (1), N (2) : 0 2 + H2-2H20
1 0 0 0 2
1 2 2 2 0
The route N (2) above cannot be obtained by multiplication of ix w't a certain number, which means that the routes N (I) are N (2) different, although their overall equations are identical. Denoting the stoichiometric number of a stage of a route N (3) in the ~bllowing way 1/!3) = C V (1) + C 1/(2) S "1 S "2 5' '
(4.1)
where Cj and C2 are arbitary constant numbers and v~3) is the stoichiometric number of step s in the route N (3), any linear combination of routes of a given reaction (infinite combinations) will also be a route of the reaction
N.~'3) =
C~N(1)s+
C.2 N(2),s'
(4.2)
In terms of linear algebra the reaction routes form a vector space. If in a set of reaction routes none can be represented as a linear combination of others then the routes of this set are linearly independent and a set of such routes is called the basis of routes. Although the basis of routes can be chosen in different ways the number of basis routes for a given reaction mechanism is determined in a unique way, being the dimension of the space of the routes. There are also relationships between inrtermediates giving balance or conservation equations. For example, such an equation could relate the total concentration of intermediate species bound to enzyme. The enzyme could be then either in free form or bound to substrate and the sum of these two forms is equal to the initial concentration of enzyme that was introduced in the system. For heterogeneous catalytic reactions a balance equation relates the surface coverage of adsorbed
113
s p e c i e s a n d v a c a n t sites. I f there are t w o t y p e s o f the sites o n the s u r f a c e there w o u l d be t w o b a l a n c e e q u a t i o n s . It w a s s h o w n b y H o r i u t i that the n u m b e r o f b a s i c r o u t e s P is d e t e r m i n e d by,
P-S-I
(4.3)
w h e r e S is the n u m b e r o f stages a n d I is the n u m b e r o f i n d e p e n d e n t i n t e r m e d i a t e s in the sense u s e d in the G i b b s p h a s e rule. S i m i l a r e x p r e s s i o n w a s p r o p o s e d b y T e m k i n
P S+W-J
(4.4)
w h e r e P is the n u m b e r o f routes, W is the n u m b e r o f b a l a n c e e q u a t i o n s , J is the n u m b e r o f i n t e r m e d i a t e s , S is the n u m b e r o f steps. L e t us illustrate an a p p l i c a t i o n o f the T e m k i n rule to h e t e r o g e n e o u s c a t a l y t i c h y d r o g e n a t i o n o f 4 - t e r t b u t y l p h e n o l ( F i g u r e 4.1)
)3
C CH )3
A
C CH )3 a6s
E
Y
OH ~ C C(CH3)3
H
3
T
OH )
3
C
~ C
Figure 4.1. Hydrogenation of 4-tertbutylphenol w i t h o u t g o i n g d e e p e r in the m e c h a n i s m itself, w h i c h is e x p r e s s e d in a f o l l o w i n g w a y
1.ZA+H 2
Nil)
N (2)
N (1)
N (2)
¢::>ZAH 2
1
1
1
0
2 . Z A H 2 + H 2 ¢::>ZAH4 3.ZAH4 ~ ZE 4 . Z E ¢:~ Z Y
1 1 1
1 1 0
1 1 1
0 0 0
5.ZY+H 2
** Z Y H 2
1
0
0
0
ZC
1
0
0
0
l
0
0
-1
0 0 0 0
1 1 1 0
0 0 0 ]
0 0 1 0
6.ZYH 2 ~ 7.ZC+A
-
ZA+C
8 . Z E + H 2 ¢:> Z E H 2 9.ZEH 2 ~ ZT 10.ZT+A --- Z A + T ll.ZY+A ¢~ Z A + Y 12.ZC+H 2 ~ 13.ZH2C
ZH2C
0
0
0
1
¢::> Z T + H 2
0
0
0
1
N(I):A+3HE=C
N(2):A+3Hz-T N(3):A+2H2=Y
N(4):C=T
(4.5)
114 The total number of steps is 13, there are 10 intermediates (ZA, ZAH2, ZAH4, ZE, ZY, ZYH2, ZC, ZEH2, ZT, ZH2C) with one balance equation, which relates the coverage of adsorbed species ZA+ ZAH2+ZAH4+ZE+ZY+ZYH2+ZC+ZEH2+ ZT+ ZH2C=I
(4.6)
leading finally to 4 basic routes. Note, that in route N (4) which is cis-trans isomerization of alkylcyclohexanols, hydrogen is a stoichiometric compound, as it does not appear in the overall equation for this route. A step can have a stoichiometric number equal to zero along all the routes. Imagine a situation when there is a possibility tbr inert species to be bound to the catalyst (adsorption of solvent Y of the surface), then a mechanism can take the form 1 .Z+A <=>ZA 2.ZA ¢=> ZB 3.ZB <=> Z+B 4.Z+Y E ZY A=B
1 1 1 0
(4.7)
The stoichiometruc number of step 4 in eq. (4.7) is zero, which is logical as the solvent does not take part in the chemical transformations. At the same time the solvent concentration can influence the reaction rate by blocking some catalytic sites, which will be otherwise accesible to reactants. Linear independence of routes does not mean that corresponding overall equations are linearly independent. Considering the oxidation of ethylene on silver we will see that the number of overall equations is 2 according to stoichiometry, while the number of routes is 3. The route N 0) is an empty one, as it does not result in any chemical transformation. N0)
N (2)
N (3)
2 0 1 1. ZO2+C2H4--+ ZO+C2H40 0 1 0 2. ZO2+C2H4----~ZO+CH3CHO 0 1 0 3. CH3CHO+5ZO2-+ 2CO2+2H20+5ZO 0 0 1 4. Z+ C2H40 - Z C2H40 1 3 0 5. Z+O2K----)"ZO2 1 3 1 6. 2ZO--->Z+ ZO2 0 0 0 7. ZO2+ C2H4 - Z 02 C2H4 0 0 0 8. ZO2q- H20 - Z 02 H20 0 0 0 9. ZO2+ CO2 =Z O2 CO2 0 0 1 10.ZC2H40 ---~ZO+ C2H4 N tl) " 2 C 2 H 4 + O 2 C 2 H 4 0 , N (21 " C2H4+302 2CO2+2H20, N (3) " 0 0
(4.8)
In deriving the kinetic equations of heterogeneous catalytic reactions the surface concentracions are assumed to be steady-state (or stationary). A reaction is at steady-state if the concentration of all species in each element of the reaction space (i.e. volume in the case of homogeneous reaction or surface in the case of heterogeneous reaction) does not change in time.
115 In a steady-state reaction the concentration o f an intermediate does not change in time because the sum o f rates o f formation o f this species in elementary reactions is equal to the sum o f the rates o f its consumption in other elementary reactions. N o w each route o f the reaction can be represented as a linear combination o f its basic routes, therefore a run along any route can be expressed as a corresponding linear combination o f runs along the basic routes. The notation "rate along a basic route " has a rigorous sense only for a steady-state reactions. Let us denote the rate o f the s stage as r(s), which for an elementary step is ~,1 = r, - r ,
(4.9)
where r~ and r_~ are the rates o f the forward and reverse elementary reactions comprising the stage. The rate along the basic route N (p) is denoted as r (p), according to its definition there are vl'tr ('~ runs o f the s-th stage per unit time per unit reaction space. Here v lt') is the stoichiometric number o f the s-th step along the route N (p). The total number o f runs o f the s-th stage per unit time per unit reaction space is obtained by summation over all P basic routes, hence,
• p=l
v("~r ~' ('~ = G I
(4.10)
If for some stage r, -~ r , ~+-li.,i the stage is called "fast". It may also be called a quasiequilibrium stage. In addition, strictly equilibrium stages are also possible, namely if the stoichiometric numbers o f a stage for all basic routes are zero, then ~ = r , . Reversible adsoprtion o f a catalytic poison or solvent may serve as an example o f such a stage. Applying equation (4.10) to nonequilibrium steps we obtain P
~ v(~/r (P~ = r, - r ,
(4.11)
p=l
where the rates elementary steps are determined by the mass action law. For nonlinear steps there could be several solutions for (4.11). Sometimes the steady state solution cannot be reached and the reaction system starts to oscillate. Such a case will discussed in Chapter 8. For a one route mechanism in case o f the steady state reaction the overall reaction rate follows immediately from (4.11) r - ~r+.,m- r _ Vs
r+l - ~ l _ C2 - ~ 2 _ r+3 - ~ 3 _ F+4 --F4 --... VI
V2
V3
(4.12)
V4
If the stoichiometric coefficient o f an intermediate Xj in the chemical equation o f a stage s is defined as xjs, and N~>0 when Xj is tbmed, xjs<0 when Xj is consumed and xj~-0 when Xj does not take part in the reaction, the stoichiometric numbers for a route N (p~ by definition satisfy the equation
116 S
~-~,"V s,(P)" A'js = 0
(4.13)
s= 1
as they should not appear in the overall chemical equations. Let r(Xj) be the rate of formation of an intermediate Xi, i.e. the number of molecules Xi formed per unit time per unit reaction space, then the formation rate is given by S
~-'ji,lXj., = r(Xj)
(4.14)
x=l
After substituting r(~) from (4.10) the formation rate is S
r_Xj_(
)=,.~ s=l
P
5
~,~
P
,
<~
p=l
s=l
P
S
p=l
s=l
,
p=l
j,
,
(4.15)
Taking into account eq. (4.13) the formation rates are equal to zero, theretbre S
S
0 = ~--~x,,q,~.I =22x.j~.(r ' - r . ) s=l
(4.16)
s=l
In a number of cases steady state conditions (4.16) are more convenient to apply than the Bodenstein conditions for intermediates r(Xj) = 0
(4.17)
As an example of the derivation of kinetic equations we will consider the hydrogenation of butadiene (J.Goetz, R.Touroude, D.Yu. Murzin, Kinetic aspects of selectivity and stereoselectivity for the hydrogenation of buta-l,3-diene over a palladium catalyst, Ind. Eng. Chem. Res., 35 (1996) 703). The overall reaction network, which was used for kinetic modeling is presented in Figure 4.2. In this scheme the addition of hydrogen to anti and syn - adsorbed diene molecules is assumed, producing but-l-ene: trans- and cis but-2-ene are formed from anli and syn adsorbed diene respectively. There could also be a conformational interconversion of adsorbed buta- 1,3-diene. Syn~
+H2
Cis-butene-2
~ +H2
Bten
%.~ti Trans-butene-2
Butane
Figure 4.2. Scheme for buta-1,3-diene hydrogenation. For but-l-ene isomerization it was proposed that the isomerization reaction takes place by at least two different mechanisms: an intramolecular hydrogen shift, and an associative mechanism
117 with the formation of a half-hydrogenated intermediate. Both pathways were taken into account in kinetic modeling. Starting from trans-but-2-ene and cis-but-2-ene negligible amounts of but-1ene were initially observed. Therefore the double bond migration in but-l-ene could be considered irreversible. A mechanism similar in some way to that proposed by Horiuti and Polanyi for olefin hydrogenation suggests that two successive steps involving the addition of adsorbed hydrogen atoms to adsorbed hydrocarbon molecules brings about the hydrogenation reaction. It is assumed in the derivation below that hydrogen and hydrocarbons are adsorbed on different types of sites without competition and that the addition of a second hydrogen atom is fast. The rates of cis and trans-but-2-ene formation from but-l-ene are higher than the contbrmational isomerization rate. Therefbre, tbr simplification, the trans and cis but-2-ene conformational isomerization was neglected. Then in the case of buta-l,3-diene when overall reactions are irreversible, the reaction mechanism can be described by 11 reaction routes (22 reaction steps) and written as presented below, where Z and Z' represent two types of surface sites; A, B, C, T, D represent molecules of buta-l,3-diene, but-l-ene, cis-but-2-ene, trans-but-2ene and butane respectively; and AH, etc. represent intermediate complexes. On the right hand side of these equations are stoichiometric numbers for the different routes.
1.2z'+ H2--- 2 Z ' H
N I
N2
N3
N4
N5
N6
N:
N8
N9
N ~°
1
1
1
0
0
0
0
1
1
1
2 Z A a E ZAs 0 3 ZAa+Z'H~ ZAaH+Z' 1 4 ZAaH+Z'H~ZB+Z' 1 5 . Z A a H + Z ' H ~ ZT+Z' 0 6.ZAs+Z'H ~ Z A s H + Z ' 0 7. Z A s H + Z ' H ~ Z B + Z ' 0 8.ZAsH I Z ' H ~ Z C I Z ' 0 9 Z B + A ¢:> ZAa+B 1 1 0 ZT +A¢::> ZAa+T 0 11 Z B + Z ' H ~ Z C + Z ' H 0 12ZB +Z'H~ZT Z'H 0 1 3 . Z B ~ ZC 0 14.ZB=ZT 0 15,ZD+A > Z A a+D 0 16.ZC+A <=>ZAa + C 0 17ZB+Z'H~ZBH+Z' 0 18ZBH Z'H~ZD+Z' 0 19ZC+Z'H~ ZCH+Z' 0 20.ZCH Z ' H ~ Z D + Z ' 0 21.ZT Z ' H ~ Z T H + Z ' 0 2 2 . Z T H + Z ' H ~ Z D Z' 0 N1,N2:A+H2=B; N3:A+Hz=T;
1 0 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 I 0 -1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N 4, N s : B = C ; N 6, NT: B = T
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -I -I -I 0 I I 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; Ns: B+H2=D ; N9: C+H2=D;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 -I 0 0 0 0 0 0 0 0 0 0 0 I I 0 -I 1 0 1 0 0 1 0 1 0 0 0 0 N 1° : / ' + H 2 = D ,
N ~l 1
0 1 0 0 0 0 0 0 0 I 0 0 0 1 0 0 -I 0 0 0 0 0 0 0 0 0 I 0 0 I 0 0 0 0 0 0 0 0 l 0 I 0 Nl1: A+H 2 = C.
The reaction order in buta-l,3-diene is close to zero, indicating that the fraction of vacant sites is very low, and at the total consumption of buta-l,3-diene the mole fractions of butenes are not equal to zero. The assumption of equilibrium adsorption of the intermediate compound (but-1ene) in the case of irreversible butadiene hydrogenation and but-l-ene isomerization and hydrogenation cannot explain the latter observation. Therefore, adsorption/desorption steps for buta-l,3-diene; but-l-ene, but-2-ene are thought to be reversible and have an "adsorption-assisted desorption " nature. The desorption of butane step 15 is assumed to be irreversible and fast. For conformational isomerization (step 2) a quasi-equilibrium approximation will be used. For the mechanism of butadiene hydrogenation, assuming two types of sites on the surface would mean two balance equations. It follows from equation (4.4) that in the mechanism there are 22 step equations, 13 intermediates, 2 balanced equations, and therefore 11 basic routes. According to equation (4.11) we have for the step 3
118
VO)/"(1)3+ V(2)p(2)3+V~3)r(3)~ + .... +V~ 11)F01) = 1' r(1) + O* r(2) + 1' r(3) + . . + O* r(H) =
(4.18)
and correspondingly for steps 4 and 5 V~I)F(1) _1_V~2)F(2) _}_V~3)F(3) _}_.... _}_V~I 1)F(11) = 1 * F (1) -}- 0 * F (2) -}- 0 * F (3) -}-,.-1- 0 * t"(11) =/'4
(4.19)
V(1)F(1)5 q- V(2)F(2)5 q- V(3)F(3)5 q- .... q-V~ ll)F(ll) = 0 * F(1) q- 0 * F(2) + 1"/'(3) q-"q- 0 * F(11) =F5
(4.20)
Under steady-state conditions it follows from (4.18)-(4.20) that the rate of step 3 is equal to the sum of the rates of steps 4 and 5 (4.21)
r~ = r 4 + 1 ; therefore, k3OAa = (k 4 + ks)O4~H
(4.22)
and OAa H = k 3 O A a / ( k
4 -}- k5)
(4.23)
In case of the quasi-equilibrium approximation for step 2 OA ~ = K2OA, ~
(4.24)
The surface coverage of AsH can be obtained from
l'r(2) +l*r°~) = G
(4.25)
1*
(4.26)
r (2) = r 7
l * r ~11) = r 8
(4.27)
and from the steady state approximation r6 = r7 + r 8
(4.28)
leading to G , = (k7 + k )G,H Hence it holds
(4.29)
119 OA~,H = k6K2OAa /(k 7 + ks)
(4.30)
The surface coverage of B can be derived from the steady-state conditions r9 = r4 +rv -rll -~2 -rl3 -~4 -r17
(4.31)
which gives k90~PA - k_9OA~PB = k3k4OA~O), /( k4 + k s ) + k6kTK2OA~O), /( k7 + k 8) - kllOBO)~ -
(4.32)
- k, zOBO'H - k,30 B - k,40 B - kl70~O' H
Hence the surface coverage of B is expressed as 0B = O~.(k'+k9PB)/(LP A + k")
(4.33)
where the lumped constants k' and k" are given by k'= k3k40'H / ( k 4 + ks) + k6kTK20'H / ( k 7 + ks)
(4.34)
and k"=kl~O'~ +k120'H +k~3 +k14 +ka70'H
(4.35)
The surface coverage of C is expressed from the equality 1"16 =
Fs -}- rll -~- El3 --/"19
(4.36)
and kl6OcP A - k_160A.~: = k6ksK2OA.O'H / ( k 7 + k s ) + (knO' H + kl3)OA.(k'+k_gPB) / ( k g P A + k " ) - klgOcO'~
(4.37) leading to 0 c = OA~(k"'+k"(k'+k9PB)/(kgP A + k " ) + k l 6 ~ ) / ( k , 6 P A + k .... )
(4.38)
where k " ' = k 6 k s X 2 0 ' H / ( k 7 + k s ) , k .... =k, eO'H, k" =(kll0' H +k,3 )
A similar derivation for trans-but-2-ene results in
(4.39)
120 rio = r~ + r12 + r14 - i~1
(4.40)
and k, oOrP , - k .oG~P, = k3ksG,fl' H / ( k 4 + ks) + (k,20'H + k , 4 ) G , , ( k ' + k 9P,)/(k9P,, ,
+ k") -
k2,0rS'H
(4.41) Therefore the expression for the coverage of t r a n s - b u t - 2 - e n e is 07, = OA~(kV'+k""(k'+k 9P#)/(k9PA
+
k") + k , 0 P 7 )/(k~0PA + k2,0',)
(4.42)
where k"'= k3ksO'. / ( k 4 + k s ) k " " = (k,20'. +kH)
(4.43)
The site balance equation for the first type of site is given by 0Aa +0A~, +OAa . + 0A.,, +0~ + 0 c + 0 j +01~ =1
(4.44)
Taking into account that the surthce coverage of butane is negligible after a straighttbrward procedure an expression for the surface coverage of 0Aa can be obtained W = OA,, =
1
(4.45)
+(k'"+k"(k'+k ~P~)/(k~PA + k " ) + k ,~:)/(kl~P~ + k .... ) + + (k"'+k""(k'+k ~P~)/(k~e~ + k") + k ,o~)/(k, oP~ + k~,G) As we suppose two types of active site, for the adsorption of hydrogen the following equation is valid :
-
(0'H)2
K, (O,)2PH2 where
K 1
(4.46)
represents the adsorption constant, and 0'H and 01. represent the degree of coverage of
hydrogen and unoccupied sites on the second type of sites, respectively. The site balance for the sites where hydrogen is adsorbed is given by 0'H + 01. = 1 finally giving the surface coverage of hydrogen as a function of the vacant sites
(4.47)
121
0',-
(4.48)
(K'P'2)°5
1 + (K, P.~ )05 Butadiene is consumed along routes 1, 2, 3 and 11 therefore the rate of butadiene consumption rA, is given by r d) + r (2) + r (3) + r (11) = r 3 + r6 = r l
(4.49)
leading finally to l"A : ( k 3 q- k 6 K 2 ) W O '
(4.50)
H
where W and o'H are given by Eqns. (4.45) and (4.48), respectively. The rate equations for B, C and T are analogously obtained l"B = k ' W - k " (k'q-k 9PB ) /(k9PA - } - k " ) W
(4.51)
/"( = kl6WP4(k"'+k"(k'+k 9PB)/(k9PA + k " ) + k 16P()/(kl6PA + k .... ) - k l 6 W P ~
(4.52)
fr = NloWPA (k"'+k"" (N'+N_9P[~) /(NgP d + k " ) + k_10g)/(kloP• + N210' H ) - NlOWPr
(4.5 3)
The set of rate equations can be further used for analysis of selectivity and parameter estimation.
4.2. Basic routes of c o m p l e x reactions Applying the theory of complex reactions to a consecutive reaction of A (+m___~C (+m__~D type which is presumed to occur via the following mechanism N (~)
1 .A+Z ~ AZ 2.AZ+B---~ CZ 3.CZ+B---~ DZ 4.CZ <--->C+Z 5.DZ <---> D+Z N (~) N (2)
N (2) 1 0 1 0 0 1 1 -1 0 1 A+B=C C+B=D
N0) '
N¢2)' 1 1 1 1 0 1 1 0 0 1 NO)' A+B=C N(2)' A+2B=D
(4.54)
with Z as a site on the surti~ce of the catalyst and AZ, CZ and DZ as adsorbed species we arrive at a conclusion that according to Horiuti's rule the basic set of pathways in scheme (4.54) should contain two pathways. Two sets of pathways are shown in the scheme: N (~), N (2) and N 01', N (2)'. The final equations of the first set describe the formation of products C and D as sequential, and the second set descibes it as parallel. However, if a steady-state or quasi-steady-state reaction is under examination both descriptions are equivalent. If the rates r(1) and r(2) along the routes 'N0) and N (2) are known,
122 then by algebraic manipulations the rates r (~)' and r(2v for the r o u t e s N (I)', N (2)' can be computed. Stoichiometric numbers vm's and v~, 2)'. of each step along the routes N (~)', N (2)' are expressed through the stoichiometric numbers v!S1) and v!5'2) of the same step for the routes N 0), N (2) in the following way v S0/' --- 1 * v Sm + 0 * v!S2)
(4.55)
1 * v S(1) + 1 * v S!2)
(4.56)
V (2)' =
S
In a generalized form for N m) routes analogously to eq. (4.55) and (4.56) we have l/(I)' S = Cll * V~,I) -[- C12 , V~,2) ~- ..-}- C1p , 1 / 7 ) (2)+..+(__72t , * V (v)
v(2)'
, ,v0)+6,22,v ,~' = 621 s
s
~'
(4.57)
v(p)' S = C p I , VS(1) _}_Cp~_ * v S(2) ~-..~-Cpp * v S(p)
Then the set of equations for the rates in the basic r o u t e s N (I), N (2) .. N (p), will be expressed by the rates for another basis of routes N m', N (2)' .. N (p)' r(I) = C11 , roy q_ C21 ,/~(2), q_ .. -l- Cp1 * r (P)' r (2) = C'12 * F (I)' _}_6'22 *F( 2)' q_..q_ 6'p2 * F (P)'
r(e) = Clj, * t o ) ' + C 2 v * r(2)' + .. + C1,1, *
(4.58)
r(e)'
It is composed in such a way, that coefficients of each row in (4.57) m'e located in a corresponding column of the set (4.58). In the particular case of mechanism (4.54) we get r (1) = 1 * r °)' + 1 * r (2)'
(4.59) r (2) = 0 , r d )
' +l,r(2)
'
From (4.59) we can easily get that r ( 1 ) , = r(1) _ r ( 2 ) r(2) ' = r(2)
(4.60)
The more generalized equation can be obtained from (4.58) by summation of all rates multiplying consequently each equation by the corresponding stoichiometric number
123 P
Z v { P ) r (1) :v~.-(l)'~,,,'A1)' _1- F { I ) c ) I F (2)' +..-I- Y,II)CpIF(P)' qp=l q_V. s"(2)f~%12r'*(1)' _}_vs"(2)f,%22 r(2)' + " +
v"(2)Ps ~ P 2 t" (P)' +
(4.61)
(P) (1)' . (P)[~ ,~(2)' . (P)[~ ,,(P)' -}- l"s C I p F -}- l/ s '~2P" + " + l/s '~PI° l
which can be further transformed
P
/ (1)[, , {2)f, "2 P ) f , Nr(1)' = {,Vs u I I q-Vs k-'12 q- "" q- ~ S t-~ I," J q-
Zv(P)v(1)
s
p=l . ( P ) ~ c'2P ) r ( 2 ) ' _}_ \Vs("(l)f~t~21 _}_v12)C22 q_..q_vs
_}_
(4.62)
Jr- (~.ll)cp1 Jr- v(2)Cp2 -[-.. Jr- 121/0)Cpp)F(P)'
giving finally the relation of rates for two sets of basic routes at steady state P
P . (P),~{P)
l~
.
p=l
= ~T'v(~)'r p=l
(4.63)
4.3. Single-route steady state reaction General expressions for the reaction rate can be constructed knowing the expressions for the rate of steps if the following equation is used s
s'
I ~ r+, - l q r , = (121 - C~ )122...r+~ + F 1 (r+2 -- C2 )1+3"" "F+s q - " " q- F IF 2""(l+s' -- ]"v ) i=l
i=l
(4.64)
The number of steps in (4.64) is chosen in a aribitrary way. It is easy to check that the equality (4.64) is correct by opening the parantheses. For a four -step reaction from (4.64) one gets 4
4
~ I r+i -- ~ F_ i = (r+l -- F_ 1 )r+2 ...r+4 -}- F_ 1 (r+2 -- r_ 2 )F+3 ...1%4 -1- "'" -}- F-1F-2'"(F+4 -- F-4 ) i=l I=l
(4.65)
which is transfbrmed to r, = r+i - r_i F+lr+2F+3r+4 -- F_IF_2F_3F_ 4 = rlIV+2F+3F+4 q- F_IF21F+3r+4 q - . . . q-/~_lF_2F_3F41
(4.66)
124 At steady-state the global rate is expressed by the rate of a step and its stoichiometric number ri = o-, r , then from (4.66) it follows F+IF+2F+3F+4 --/~_lF_2/~_3t'_4 z (V1F)F+2P+3F+4 q- F_1 (v2r)r+3r+ 4 + . . . + F_1/~_2/~_3 (V3r)
(4.67)
giving the expression for the rate r+lF+2/~+3F+4 -- C 1 C 2 C 3 C 4
r =
(4.68)
VIF+2/~+3F+4 -'}-F_IV2C3F+4 -t- F_IF_2V3F+4 -'}-F_IF_2F_BV3
which is in more general tbrm s
r =
(4.69) V1F+2 ...F+s q- F1V2F+3...I+s q- FllL2 ...V~,
The reaction rate is expressed by the difference of the rates in the forward and reverse directions (4.70)
F=F+--F
where l~I r+i
1Sir,
i=1
1-+=
;r
V1F+2 ...F+s "4- F1V2F+3 ...F+s "4- F_IC2 ...V s
=
i=l
(4.71)
v~r+2...G,~ + Clv2r+3...r+, + C~C2...G
The ratio of rates in the forward and reverse directions is then expressed F
t'+
_ FIt~ 2F3...Fs
(4.72)
qr2q.., q
Let us consider as an example a gas-phase reaction occuring according to the following mechanism 1 .N2Os<=vNO2+NO3 2. NO2+NOB~NO2+O2+NO3 3. N O + N O ! ~ 2 N O 2 N2Os-2NO2+l/2 02
1 0.5 0.5
(4.73)
125 If the rate of the third step in this sequence is fast, than we have to take into account only steps 1 and 2 and the reaction rate from equation (4.69) is reduced to an expression of the two-step sequence r -
F1F 2 - - F 1 F
vir 2 +
2
(4.74)
Civ2
Further assuming that step 2 is irreversible gives r~r2
(4.75)
F --
vl I"2 + C1 v2 Taking into account the expressions for the rates r=kl [N205], r-l=k-1 [NO2][NO3], r2=k2[NO2] [NO3]
(4.76)
the final expression for the reaction rate is
F --
r, G r 2 + 0.51t,
k, k 2 k 2 + 0.5k,
[NzQ ]
(4.77)
For a two step catalytic sequence 1. * +A~++*I +B1
2. *I+Az++Z +Bp AI+A2 ~-> BI+B2
(4.78)
where * is an active catalytic site (hetereogeneous, homogeneous or enzymatic), the rates of elementary reactions are given in the following way (4.79) Following the expression for the steady-state reaction rate (4.74) and numbering the step 1 as the first one and the step 2 as the second one we get (4.80)
Equation (4.80) can be rewritten [*]=r
k2CA2+k 1C~ kICAk2CA2-k ~C~k 2C~2
(4.81)
126
Numbering step 1 as the second one and step 2 as the first from equation (4.74) another expression for the rate follows
F= N2[* I]CA2NI[*](~AI-N_2[*]CB2N_I[*Z](~BI
(4.82)
which results in the expression of the intermediate concentration
[*I]=r
+ I%C 2
(4.83)
Summation of (4.81) and (4.83) results in klCA~+ k~CA, + k tCBI + k 2CB~ [*d+[*]=r
-
-
klC A k2~-'A2- NICB k2CB)
(4.84)
The conservation equation for the catalytic sites is ['1+ [* I 1 : ['1~o~= C
(4.85)
in case of homogeneous (enzymatic) reactions the total catalyst concentration is equal to the initial amount of catalyst (enzyme) introduced in the reaction mixture. For heterogeneous catalytic reactions the sum of surface coverage is equal to unity. Finally the expression for the reaction rate is
kICAlk2CA2 - k ~C~lk 2C~, r = C~,~ k~CA, + k2CA2 + k ~CB, + k 2C<
(4.86)
In case of gas-phase catalytic reactions equation (4.86) takes the form
ktPAlk2PA2 - k ~P~Ik 2p& r =Q,, k~PA, +k2PA2 +k ~PB,+k 2P<
(4.87)
which can be generalized using the frequencies of steps mi r=
0910)2 --(.0 ~0) 2 (-01 + 0 ) 2 + 0 9
C,
(4.88)
I +0) 2
In the following text (Chapters 5 and 6) we will demonstrate how eq. (4.88) can be applied to specific cases of homogeneous and enzymatic catalysis.
127 In this relatively simple case of a two-step sequence an expression for the rate can be also obtained directly from the steady-state assumption I"1- C1 = I"2 - r_2
(4.89)
kl [~]~'Al-k ,[~ I~-~'B, =k2[ :'@] ] C A 2 - k 2[~]~B2
(4.90)
which in combination with the balance equation (for heterogeneous catalysis [*]+ [* I ] : 1) gives
kick11 + k 2 C B 2
['1]-
kRC& + k ICB, [,]
0)2
k2C42 + k
=
(4.91)
- °91 + ° 9 2
"t'-O) I
ICB,
klCA, + k2CA2 + k IC~, + k 2C<
=
°)2 + °9-1
(4.92)
091 + 0 ) 2 + 0 . ) 1 + 0 . ) 2
In a steady state the rate is given either by the rate of the first or the second step (1)2 + 0 ` ) 1
F : T 1 --F_1:0.) 1
O)1 + 0) 2 -4- 0)_1 + 0)_ 2
0)_ 1
0) 2 -J- O.) 1
~91 -J-O) 2
09] -I- 0.) 2 -I- 0)_] + 09_2 CO2 + 0)_]
(4.93)
O) 10) 2 -- 0)_1 0.)_ 2 0.)1 q- 0) 2 q- 0.)_1 -I- 0)_2
4.4. Topological analysis of complex reactions
The reaction mechanism can be shown graphically. The vertices of the graph can be shown as circles and represent intermediates. The edges of a primary graph can be shown by complete lines and represent stages. In the case of linear stages the edges join the vertices which correspond to the intermediates taldng part in the stage. The edges are supplied with ordinal numbers of the stages. The arrow at the number indicates the direction of the stage which is arbitrarily taken as the forward one when writing the mechanism. In order to depict the non-linear stages, secondary edges can be introduced and shown as dotted lines. If all the stages are linear then the mechanism will be called linear. A graph of such a reaction contains only primary edges. A set of edges which continue one another are called a chain in the theory of graphs, and a chain whose beginning and end coincide is called a cycle. The cyclomatic number of the graph i.e the largest number of independent cycles is the number of basic routes. An example of a mechanism when not all the cycles are independent is for instance, the mechanism of the interaction between water vapor and coal, where Z is an active site on the coal surface
128 N (l)
N (2)
N (3)
1 1 0 1. C+Z+H20 =H2+COZ 2. COZ = CO+Z 1 0 1 3.COZ + CO-CQz+C+Z 0 1 -1 N (~)H20 + C - C O + H 2 ; N (2) C O + H 2 0 - C O 2 + H 2 ; N (3) C O 2 + C - 2 C O
(4.94)
According to the Temkin-Horiuti rule the mechanism (4.94) contains 3 steps, two surface intermediates Z and COZ and one balance equation, therefore only two independent routes, as v(~.~) = v~.') - v/~,2)
(4.95)
In the kinetic graph vertexes denote intermediate surface species; edges represent reversible reactions. It contains three cycles, of which two are independent. 3
@
Figtu-e 4.3. Graph of a catalytic mechanism 4.94.
Examples of graphs corresponding to different reaction mechanisms are presented in Figure 4.4.
4
2
8
i
4
3
Figure 4.4. Examples of the graphs of different catalytic mechanisms. The example on the right hand side represents a mechanism for ammonia synthesis with a hanging vertex or a dead-end (step 5) with a stoichiometric number equal to zero. 1. N2+Z=ZN2 2. ZN2+H2 = ZN2 H2 3. ZN2H2+Z- 2ZNH 4. ZNH+H2 -NHB+Z 5. Z+H20=ZO+H2
1 1 1 2 0
Step 5 represents catalyst deactivation by the oxygen in water vapor.
(4.96)
129 An interesting example of a graph is a case which contains steps when no intermediates are formed. Such graphs give a possibility to describe homogeneous or chain reactions. For instance a mechanism of a reaction between hydrogen and bromine 1. Br2++2Br 2. Br+H2++HBr +H 3. H+Br2~--~HBr +Br is expressed by a kinetic graph, containing an empty circle (Figure 4.5)
Figure 4.5. Graph with an empty circle. As an example we will consider stereoselective catalytic hydrogenation of dialkylbenzenes with the formation of dialkylcyclohexene and c i s and t r a n s dialkylcyclohexanes and dilkylcyclohexenes. The most intriguing question in elucidating the reaction network is the origin of the formation of c i s and t r a n s - dialkylcyclohexanes, as only c i s is expected if an aromatic molecule is lying flat on the surface. Either desorption -readsorption of some reaction intermediate (Figure 4.6) or some kind of "rollover"of an adsorbed intermediate (Figure 4.7) has been proposed to explain the formation of t r a n s isomers. The difference between these two mechanisms is briefly sketched in the Figures 4.6 and 4.7, where A-dialkylbenzene, B dialkylcycloolefin, C - c i s , T - l r a n s ~dialkylalkanes A " ~ + 2 H2 +H2 T ~-
+H2 (B1)
(B2)
~
%// B
Figure 4.6. Desorption-readsorption mechanism A NN,+2H2 +H2 T ~
+H 2 (B1)~
(B2)
// B
Figure 4.7. Rollover mechanism.
~-- C
C
130 Every route of a chemical reaction corresponds to a graph's circle and vice versa. To simplify the discussion, we will use suppressed graphs in what follows. They are shown in Figures 4.8 and 4.9 respectively for the mechanisms given above. Generally speaking edges 5 and 6 are complex and should include circles corresponding to coverages of dialkylbenzene and partially hydrogenated species, which in turn depend on the partial pressures of aromatic compounds and hydrogen.
6
5 Figure 4.8. Graph tbr desorption-readsorptionmechanism
5
6 Figure 4.9. Graph for rollover mechanism. The surface coverages of the intermediates can be computed through the vertex determinants, which are the sums of the weights of all trees containing the particular vertex (spanning trees). A spanning tree is a sequence of edges going through all the vertices of the initial graph and it suffices to add at least one edge to the spanning tree to obtain a circle. Spanning tree weights can be obtained by multiplication of consecutive edges and summation of parallel ones. Following this procedure we arrive at the following equations for the rollover mechanism. DzB~ = co,co2
(4.97)
DzB2 = colco4
(4.98)
Dz = co2co5 + co3co4 + co4co~
(4.99)
where co are the frequencies of steps (e.g. the ratio between reaction rate and concentration). Vertex determinants tbr the adsorption-desorption are
131
DZ]J1 : 0)30); -4- 0,)00);
(4.100)
Dz~2 = 0)10)4 + ~°5°-)1
(4.101)
Dz = 0)50)0 + 0)30)4 + 0)5093 + 0)60)4
(4.102)
The intermediate concentration (surface coverage) is the ratio between the vertex determinant and the graph determinant (the total spanning tree weight for a graph). Therefore, the ratio of the vertex determinants is equal to the ratio of the surface coverages. Hence, for the rollover mechanism it holds: (4.103)
O~I / OB2 = DzB, / DzB2 = 0)2 / o94
Similar considerations for adsorption-desorption mechanism result in O, I / OB2 = DzB,/DzB2 = 0)'2 (0)3 q- 0)6)/0)1 (0)'4 q-~05)
(4.104)
Eq.(4.104) can be modified to O; / OB2 = DzB ~ / DzB 2 =0)'2 (1 + 0)6 / 0)_0/0)1(0)'4 / 0)3 + 0)5 / 0)3)
(4.105)
The ratio of the coverages determines the ratio of trans and cis dialkylcyclohexanes. It follows then, that in general the cis to trans ratio for the desorption-readsorption mechanism versus the rollover mechanism have different dependencies on the concentrations of the reactants. However, if the frequencies of steps 1-4' are very high (desoprtion and readsorption are equilibrated) analysis of eq.(4.105) leads to O p,I / G 2 = 0)'2
(4.106)
0)3 /0)10)'4
If steps 2 and 4 in rollover mechanism also are equilibrium steps, eq.(4.103) results in the same dependence of the stereoselectivity on the reactant pressures as eq.(4.106). The method described above is essentially based on the general King-Altman method, the application of which we will describe below for a case of enzymatic reactions occurring via the following sequence
kl[A] Ef
~
% '
k2
EA,
ks ' EP
k4
,
.
k6[P]
Ef+P
(4.107)
132 where the total concentration of enzyme Et is split between a free form (E or equivalently El) and a bounded substrate and product. After defining the intermediates Et = Ef + ES + EP, a reaction scheme in a cycle with each enzyme form occupying a node can be drawn (Figure 4.10).
k,[A] @ " kz - @ Figure4.] 0. Graphof an enzymaticmechanism4.] 07. The scheme is used to set up a series of line patterns connecting the enzyme forms but in which the cycle is broken (spanning trees) E
EA
E
EA
~EP
E
EA
E /
Figure 4.11. Spanning trees for the graph in Figure 4.10.
Each enzyme species is proportional to the sum of rate constants and concentration terms (frequences of steps) that lead to the enzyme form from each enzyme pattern (proportional to vertex determinants) E: ~ck3k5 +k2~ +kzk4;ESock4k6P+klksA+klk4A;EPocl@~P+kzk~P+kJqA,. The fraction of the particular form of enzyme to the total enzyme is expressed El _
Et
_
k3k 5 + kzk 5 + k2k 4
(4.108)
Er
Ej, + E S + E P
ES
ES
k4k6P + klksA + kak4A
Er
Ef + ES + EP
k3k 5 + k2k 5 + k2k 4 + k4k6P + klksA + klk4A + k3k6P + k2kGP + klk~A
EP _ Er
EP
k3k 5 + kzk 5 + kzk 4 + k4k6P + klksA + klk4 A + k3k6P + kzk6P + klk3A
=
Ef + ES + EP
(4.109)
(4.110)
k3k6P +k2k6P +k~k3A k3k 5 + kzk 5 + kzk 4 + k4k6P + klksA + klk4A + k3k6P + kzk6P + kak3A
The rate of the reaction is the sum of the rates of the steps that generate one particular product. The rate of P formation is then given by r = k 5( E P ) - k 6 ( E / ) P
(4.111)
leading to r=
k5 (k3koP + kzk6P + k~k3A) - k6P(k3k 5 + k z k ~ + k2k 4) k3k 5 + k2k 5 + k2k 4 + k4k6P + klksA + klk4A + k3k6P + k2k6P + klk3A
E 1,
(4.112)
133 which is further transibrmed to klk3ksA - k2k4k6 P
F=
Er
(4.113)
k3k 5 + k2k 5 + k2k 4 + k4kGP + k~k~A + klk4 A + k3k6P + k2k6P + klk3 A
This equation rewritten in a ibrm with frequencies (010.)2 (~O3 -- O) 10)20.) 3
r =
C,, t
(4.114)
(2)20)3 +(0_3(02 +(0_30)_ 2 -}- O.)3(01 -}- O)_I(Q3 +(-O_1(O_3 -t-(01(09 +0)_2(01 +(0_20)_ I
can be derived directly from a general equation for complex reactions
r,>; ooo,: -r,
r =
Z
V/~++I/~+2ooo /~,s:+ Z
ooo,:.
(4.115)
Vi'~ 11~+1ooo/~+ + Z Vi''i 11~+1°°°/~':
when changing the counting order in the reaction mechanism (4.107) for the steps 1 through 3. In the case of equation (4.107), the terms in the denominator are F2F3,"r_3r2f F-3F-2
s=l,
If we start counting from the second step on the mechanism (4.107) then the denominator should include s=2,
F3FI,"F_IF3," F_IF_3
Finally if the count starts from the last step we add the tbllowing terms to the denominator s=3,
FLY2,"r_2rl," F_2F_1
which in combination with other terms leads to equation (4.113). Another slightly more complicated case for enzymatic reactions
kl[A ] Ef
k3[B] ' EA
,
k2
,
'EAB
k4
k6~
k9 Ef
kg[Q]
k7 , EQ
'
ks[P]
,
EPQ
can be simplified if the species ABC and EPQ are treated as single species (Figure 4.12)
(4.116)
134
kl[A] ks[QI ]k7
k41k3[B]
@ ~" k6[p]
"
Figure 4.12. Graph of a simplified enzymatic mechanism 4.116.
Considering the tbllowing patterns as acceptable (Figure 4.13)
I
I
Figure 4.13. Spanning trees for the graph in Figure 4.12.
the concentration of the free form of enzyme is proportional to (4.117) In analogous manner each of the other three enzyme forms are EAoc kak6ksPQ+ k~kskvA + k~k4kTA + k~k4k6A P
(4.118)
EAB+ EPQ~c k3k~ksBPQ+k2k6ksPQ+k~k3kvAB+ k lksk6ABF
(4.119)
EQoc k3k~kSBQ+kzk6ksPQ+~ k2k4ksQ+klkgksAE
(4.12 O)
leading to r = k~k 3k5k 7 AB
-
k 2k 4 k 6 k sPQ Er
(4.121)
D
where D = k3ksk7B + k2ksk v + k2k4kv + k2k4k~P + k4k6ksP Q + k~ksk7A + k~k4kvA + k~k4k6AP + + k3k6ksBP Q + k2k6ksP Q + k, k3kTAB + k, k3k6ABP +
(4.122)
+ k3ksk~BQ + k,_kok~PQ+ k2k4ksQ + k, k3ksAB
An important relationship between the concentration of an intermediate (surface coverage in heterogeneous catalysis) and the catalyst concentration follows fi'om the topological considerations
135
x.,
Dj
D1 z)
(4.123)
i
where D is the total weight of all spanning trees in the graph. Equation (4.123) can be applied for derivation of kinetic equations in case of homogeneous, enzymatic or heterogeneous catalytic reactions. In the last case the sum of total catalyst concentration is equal to unity.
[X]z
4.5. Kinetic aspects of selectivity One of the most important requirements for catalytic reactions in fine chemicals applications is proper selectivity, which in a broad sense should be understood as chemo-, regio- and enantioselectivity. Kinetic analysis of complex reaction schemes, where the proper selectivity dependence is the key point of analysis, is still more an exception, than a rule. The main objective is to bring the knowledge of chemical reaction engineering of catalytic reactions to organic chemistry, in particular stereochemical and enantioselective reactions. In what follows heterogeneous catalytic reactions are considered. Selectivity is defined as the ability of a catalyst to selecttively favor one among various competitive chemical reactions. Intrinsic selectivity is associated with the chemical composition and structure of surface (support). Shape selectivity is associated with pore transport limitations (Figures 4.14 and 4.15).
V
Figure 4.14. Reactant selectivityin catalysis by zeolites
Figure 4.15. Product selectivity in catalysis by zeolites In the literature one can also find the following notations: chemioselectivity to describe the ability of a catalyst to discriminate among the different and the same functional group, regioselectivity - selection between several orientations, diastereoselectivity- control of the spatial arrangement of the t)nctional groups in the product and enantioselectivity, - the catalyst ability to discriminate between mirror-image isomers or enantiomers.
136 Differential selectivity is defined as the ratio between the rate of accumulation of the desired product and the total rate of conversions of one of the original substances in various directions, including those leading to the formation of the desired product at a given temperature and with a given composition of the reaction mixture S
(4.124)
rdejr
Integral selectivity is the ratio of the desired product concentration to the total concentration of all of the products formed. It describes the course of the reaction up to a given point. In a gradientless system differential and integral selectivity coincide. As will be shown below, selectivity depends on conversion and on other reactions that occur in the system. Therefore it is extremely dangerous to compare selectivities for different catalysts at just one end-point or at a certain period of time, as, for instance, very often done in the field of fine chemicals. In case of parallel reactions 1. A ~ B 2. A ~ C
which are described by the rate equations r I = k~CA;r 2
(4.125)
= k2(f A
selectivity is independent of the concentration of A and therefore of conversion (Figure 4.16). rl
kIC A
kl
rl + G
k I C A +k2C,4
kl + k2
SA - m
(4.126)
08
06
8
04
02
oo
02
04
06
08
1o
Conversion
Figure 4.16. Selectivity vs. conversion dependence for parallel reactions.
In case of catalytic reactions, the substrate A has first to bind to the active site of a catalyst (metal complex, enzyme, solid material). It is possible that the binding energy or binding strength is different for the reactant A depending on the direction of the reaction. It is easy to
137 imagine a molecule with two different functional groups which adsorb or bind to the active site with binding constants K1 and K2, then rearanging eq. (4.126) we have S A/S B -
k~K~
(4.127)
k2K2
with selectivity still independent on conversion. For consecutive reactions 1. A--~ C 2. C-+ D the expression for selectivity assuming equilibrium binding of the reactants A and C is Sc _ r 1 -1~ _ l _ r 2 =1 rl q
(4.128)
k2K2C c klK3C ~
Selectivity for the intermediate product declines with conversion (Figure 4.17), with the steepness of the decline depending on k 2K 2 /k~ K~. The lower this ratio is, the higher the selectivity to C can be obtained at higher conversions. Selectivity [%] 100
0
0
J
t
i
25
50
75
100
Conversion[%] Figure 4.17. Selectivity vs. conversion dependence for consecutive reactions.
Analysis of eq. (4.128) shows that a strong complexation of A in comparison with C leads to high selectivity. The strength of complexation also depends on the influence of i) promoter, ii) additives in the feed, iii) solvent, iiii) dispersion of the active metal in the case of heterogeneous catalysts. In the explanation of the influence of promoters or additives on relative complexation (adsorption) properties, ligand effects, electronic effect, as well as the changing of the ensemble size of the active metal are invoked. When complexation (adsorption) properties of A and C are similar, the way to change the selectivity is to prevent readsorption of C onto the catalyst surface by applying the right additive (spectacular substance, e.g. a substance that does not appear in the overall chemical equation), that displaces C on the surface or by impeding readsorption through formation of strong
138 complexes of C with this additive (or solvent). The possible difference in the number of binding (adsorption) sites for A and C and thus their different density could in principle be responsible for high selectivity. Thus various crystal faces might have different selectivity if the geometry of adsorption of A and C is different. It is also self-evident that a low reaction rate of the second step results in high selectivity. In general the ratio of rates of the first and second reaction depends on the catalyst used. In selective reactions of compounds with two identical functional groups, especially when there is no conjugation, big differences cannot be expected. However, the reaction conditions, like pressure and temperature, can influence the selectivity. For parallel -consecutive reactions, as examplified (Figure 4.18) by cinnamaldehyde hydrogenation over a Ru-Sn sol-gel catalyst,
rly- ~ O H /
Citmamylalcohol " ~
Cinnamaldehyde (A)
~/"'/~O
/~r4
3-PhenylprOpanO1 (D)
3-Phenypropanal (C) Figure 4.18. Parallelconsecutive reactionsin cinnamaldehydehydrogenation. selectivity to intermediate products follows similar behavior as for consecitive reactions, but with lower (<100%) values of initial selectivity (Figure 4.19) 100
75
~
COL
50
25
PPAL °
0 0
25
,
,
50
75
100
C o n v e r s i o n [%]
Figure 4.19. Selectivity vs. conversion dependence in cinnamaldehydehydrogenation. (COL - cinnamylalcohol, PPAL- phenylpropmaal) It is important to note here that selectivity values are more accurate than the values of reaction rates, even with comparatively large errors in these reaction rates, therefore selectivity analysis is crucial for elucidation of reaction mechanisms in complex reactions. Let us consider this reaction (Figure 4.18) assuming different adsorption sites for organic compounds and hydrogen. The role of hydrogen adsorption in the hydrogenation processes is interesting: on one hand, hydrogen is known to adsorb on noble metals; on the other hand, hydrogen molecules are much smaller than the organic ones, particularly in the present case. This
139 implies that interstitial sites between adsorbed organic molecules remain accessible for hydrogen adsorption. Thus the adsorption behavior is non-competitive. The adsorption steps o f the organic compounds are assumed to be fast compared to the hydrogenation steps, which implies that quasiequilibria can be applied to the adsorption steps o f all organic molecules. Following the overall reaction thermodynamics, hydrogenation steps are presumed to be irreversible and determine the rate o f product formation. The mechanism can be then depicted in the following form
1.ZA + Z, H2
k, >ZB+ Z'
N (~) 1 -1 0 0 1 1
2. ZA + Z, H2
k, >ZC + 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 <=5 Z * 13. Z * A H 2 ~ Z * + B
0 0 0 0 0 1 1 1
0 1 0 -1 0 0 0 0
0 0 1 0 -1 0 0 0
1 0 0 1 -1 0 0 0
ZBH 2
AH 2
(4.136)
N ~1~ • A + H 2 ~ B; N (2) : A + H 2 ~ C; N (3) • B + H 2 ~ D; N ~4) • C + H 2 ~ D The route N (1) corresponds to hydrogenation o f carbonyl groups on the interfacial sites, while the route N (2) describes hydrogenation o f C=C bond on metal sites. This reaction mechanism describes that at high hydrogen partial pressures the reaction order towards hydrogen and the organic compounds is equal to zero by assuming formation o f a complex AH2 on the surface o f the catalyst that slowly reacts to C. At high hydrogen pressures the surface o f the catalyst is completely covered with the AH2 complex and the reaction rate ceases to depend on either hydrogen pressure or on the concentration o f A. At lower hydrogen pressures, mechanism (4. ! 36) is o f first order with respect to hydrogen pressure. Another specific feature that is taken into account is the effect o f promotion on selectivity. In general, a promoter (in this case Sn) should activate the carbonyl group, but not bind the olefinic one. In terms o f the reaction mechanism it would imply that cinnamaldehyde can be adsorbed by an olefinic bond on the metal sites as well as by the C=O bond on the interfacial sites. In a rather simplified case it can be assumed that the conjugation o f bonds in cinnamaldehyde leads to a higher adsorption constant compared to other reagents. Assuming that the complex o f cinnamaldehyde with hydrogen is the most abundant species both on metallic sites and interracial sites t h e / b l l o w i n g rate expressions are obtained tbr mechanism (4.136)
142
k12k13KHcAPH:
k tk 2K 7c,4 PH2
klcAPH~ 1 + k~"Zc4pH ; r2 =
k 12 +k13
=
1 Jr k~2KIICApH'~ k_~_ + kl3
ksk6K7C 4PH~ IV
k_3 + k 4
k c~pH~ . K cApH~ ,
klK7cAPH,
1+
kHcAPH~ 1 + kZHc~pH~
k_1 + k~_
k3k4K7CBPH2 r~ =
k2 ktK7cApH k 1+
1 + ,m
-
~
k_t + k 2
r4 =
1+
k_s + k 6 klKTcAPH,
-
kVc~pH~ I+kH~cApH.
(4.137)
k_t + k 2
where k I to k vI are combinations of constants. Similarly to eq. (4.129) description of concentration vs time behavior performed by nonlinear regression was sufficiently good (Figure 4.22). 120 4 3 3 K, 14 b a r
'~
80
.=
40
0v-
,
0
100
,
,
200 Thne[nfin]
300
Figure 4.22. Reactants concentration as a function of time in cinnamaldehyde hydrogenation- mechanism (4.136) ( J. Hfijek, J. W~irnfi, D.Yu. Murzin, industrial & EngineeringChemistry Research, 43 (2004) 2039).
However, most interesting is the selectivity analysis. The selectivity-conversion dependence is fitted (Figure 4.23) in a much better way by mechanism (4.13 6) than by mechanism (4.129). 0.5 Model
445.
DD
0.4 gO.3 ~
~
~
O O
0.2
0.1 433
0
K 0.25
0.5 Conv~sion
0.75 [%]
Figure 4.23. Selectivity w~ conversion at different hydrogen partial pressures in cinnamaldehyde hydrogenationmechanism (4.136). (J. Hfijek, J. Wfirnfi, D.Yu. Murzin, Ndustrial & Engineering Chemistry Research, 43 (2004) 2039).
143 This is not surprising since the ratio of initial selectivities to (B) and (C) at low conversions defined as
r~
1 HI
k t
1 + tC CAp..
1 + k HcApars
kl/
rB =
(4.138)
and thus depends on both conversion (cimlamaldehyde concentration) and on hydrogen pressure. Such selectivity analysis gives the possibility of discrimination among rival reaction mechanisms in cases that the statistical data fitting provides very similar results. Heterogeneous catalytic hydrogenation of cinnamaldehyde over a Ru/Y zeolite shows quite different behavior (Figure 4.24) contrary to the situation illustrated above for a Ru-Sn sol-gel catalyst. Namely selectivity to cinnamylalcohol is increasing with conversion which cannot be explained by conventional treatment of parallel-consecutive reactions calling for some other explanations. One possibly is in-situ formation of active sites selective for hydrogenation of the C-O bond. 100 Cinnalnylalcohol 75 L~ i~ 50
25
1trine Cyclohexane
0 25
50
75
100
Conversion [%]
Figure 4.24. Selectivity vs conversion in cinnamaldehyde hydrogenation- mechanism over Ru/Y zeolite. (J. Hajek, N.Kumar, P. M~iki-Arvela, T.Salmi, D.Yu.Murzin, Selective hydrogenation of cinnamaldehyde over Ru/Y zeolite, J. Molecular Catal. A. 217 (2004) 145).
4.6. Parallel reactions: Kinetic coupling
Kinetic coupling between catalytic cycles is responsible for the observed phenomenon, if when two catalytic reactions take place simultaneously, their individual rates will not be the same as those prevailing if they are run separately with the same catalyst under the same conditions. The reason for such behavior is that reactants and products in each reaction compete for the same catalytic sites. The simple catalytic cycle shown in Figure 4.25 describes the kinetically important steps for parallel reactions giving two enantiomers R and S. A
A,s
7
'
A,R
Figure 4.25. Kinetic coupling between cycles.
144 There are two cycles which are coupled by the availability of the free form of the catalyst A. Spectroscopic investigations for such a parallel homogeneous catalytic reaction (hydrogenation of enamides using cationic Rh complexes with chiral phosphine ligands) showed that the intermediate species A *R was present in ten times higher concentration than the intermediate A *s. However, kinetic studies ultimately revealed that hydrogenation of the minor species A *s proceeded nearly 600 times faster than that of the major species A *R, yielding an enantioselectivity of 98% towards the S-product. Product enantioselectivity was dictated not by the difference in concentrations of two intermediate species but by their relative reactivities. The ratio of rates for producing different enantiomers is expressed by r ~ _ l + e e _ k2~kl ~ • k is +k2sP,:
rs
S
1 - ee
S
k2 k1
k 1R q- k2 R pH 2
(4.139)
where ee is enantiomeric excess (R-S)/(R+S). At high pressures of hydrogen assuming k s << k2spm ;k ~ << k2~PH~ instead ofeq. (4.139) we obtain r l~
kl R
12,
kl x
(4.140)
and the selectivity depends on the ratio of the forward constants of the first steps. At low pressures of H2 the first steps could be considered to be at quasi-equilibria k i s >>k2SpH2 k_t e >> k2~PH: and the selectivity is determined by the equilibrium constants of the first steps and the rate constants of the second steps F~
k2R k~ ~ • k i s k2Sk( ~ k i~
_
k2R Kl R k2SKi s
(4.141)
In the case of a heterogeneous catalytic hydrogenation reaction following the same network the reaction rate can be expressed in a following way:
r=rl~+r s
=
(k~2 IO~,eA +
k 2
SO~',s )0,
(4.142)
where 0 and OH denote the coverage of two intermediate species and adsorbed hydrogen correspondingly. Under steady-state enantioselectivity could be related to the ratio between the formation rates of the different enantiomers FR
l+ee_k2Rkl ~ ,k
- --
1-ee
S
S
k, k~
S+k2sOH k R + k2~OH
(4.143)
145 which is in fact very similar to eq. (4.139) and differs only in the term describing the hydrogen pressure dependence. Similar analysis for consecutive reactions A (+m~C (+B)~ D
1 . A + Z ¢-->AZ 2 . A Z + B ~ CZ 3.CZ+B ~ D Z 4.CZ <--->C+Z 5.DZ +-~ D+Z N (u A+B=C N (2) C+B=D
N (1)
N (2)
N(1)'
N(2)'
1
0
1
1
1 0 1 0
0 1 -1 1 N(U' N(2)'
1 1 0 1 1 0 0 1 A+B=C A+2B=D
(4.144)
where steps 1, 4 and 5 axe reversible gives the selectivity dependence
dC~ dC'4
1 kaK 4(1 + k2C~ / k 1) Cc ~ l + k 3 C B/k 4 k2Kl(l+k3C B/k4) CA
(4.145)
where Ki kiMi and K4 k_4/k4. At a low concentration (pressure) of B, the selectivity will be independent of the concentration of B (in other words hydrogen, in hydrogenation reactions or oxygen in oxidation reactions) and will depend on conversion
dCcdC4 _
_
Cc 1+ kaK4 k2Ki CA
(4.146)
For high concentrations (pressure) B k3 CB/k4 )) 1; k2 CB/k-1 )) 1 and the initial selectivity will be dependent on the concentration (pressure) of B, thus influencing the slectivity.
dC( - -
dC A
-
k4 k 4 Cc -~ k3C B k I C~
(4.147)
If component A is much more strongly bound (adsorbed) to the catalyst than C is, or the rates differ significally, i.e. 1 )) k3 CB/k4 ; k2 CB/k-1 )) 1 and
Cc dCc~'- 1 +k3K4CB - dCx k 1 CA
(4.148)
The initial selectivity will be independent on the pressure of B. Moreover, if the ratio k3 K4 CB/ kl is small enough, the selectivity remains essentially equal to unity up to high conversions, decreasing thereafter. The character of selectivity beyond the point of high conversions will be pressure dependent, so that at higher pressures (concentrations) of B, the selectivity towards C will be lower. Such dependencies have been observed for isoprene hydrogenation over a palladium catalyst (Figure 4.26).
146
98-
2~ 3"
96'95 92-
s°o
0.2
o,~
s,s
o,s
~,o x
Figure 4.26. Selectivity as a function of conversion in isoprene hydrogenation at different ratios PHJPI: 1- 0.7; 2 -6.5; 3- 10. (S.L. Kiperman, Foundations of chemical kinetics in heterogeneous catalysis, Moscow, Chimia, 1979)
A special case of the mechanism (4.144) could be considered taking into account the different number of binding (adsorption) sites required for the reacting molecules. Such a situation was thought to be a possible reason for high selectivity to the products of monohydrogenation of organic compounds with double and triple carbon-carbon bonds, in the treatment below, mechanism (4.144) will be simplified and the adsorption of D will be neglected. Applying the considerations presented previously for eq. (2.125); the adsorption isotherm for an ideal adsorbed layer for molecules occupying more than one site on the adsorbent surface is given by the following equation: aalPa =
Oa
~gX. -- O.n(O. + O )
(4.149)
where q~ is the multiplicity of the system, n is the screening parameter, x is the amount of elementary sites that one molecule occupies., .0 is the coverage and P is the partial pressure. Although the screening parameter n can depend on the coverage, for the simplification in the following treatment we will consider only the case when it is coverage independent. In a similar way the adsorption isotherm of C is expressed as a~P~ =
O~ ~o~xc - O~n(O. + 0~)
(4.150)
From eqns.(4.149) and (4.150) and the balance equation 0a+0 c+0o=l
(4.151)
where Oo denotes the fraction of empty sites, one arrives at o -.c,g,(e,,xc,
.c,) .~L.aOo, 0 =acg(~o, x c - n c ) - a c ~ n c O o
which after a straightforward procedure leads to
(4.152)
147 Oo
= 1-a~P.(~Gx~,
- n~,)-a~P~(~o~x,,
(4.153)
- n~)
1 + a P, n + a~.P~.n~.
For case above it holds for the rates of elementary reactions (in a similar way as for monoatomic molecules) that the rates of elementary reactions can be described as (4.154) where re denotes the rate of step 2 in mechanism (4.144) and k2 is the rate constant of this step. Defining the selectivity as the ratio of the rate of the desired reaction to the sum of the rates of all reactions in the system, we have for the selectivity to C
S(, -
r(~/_ r(2~ k30 c r(l) - 1 k20,
(4.155)
and thus Nc, =1
k3a~,P ~ q~.x~ +GP,(Gcpcx~. - n ~ G G
) k2a oP~, q~,x , + a cP~ (n, G x ~, - G~o~x ~)
(4.156)
Eq.(4.156) can then be transformed into
dP,/dP. =-1+
f +f2P" P. l+A
(4.157)
with f~=k3acGxc/k2a,(o.x.f
2 =a~(ndp~x~-n,(Gxa)/~Gx~);
~ =a~.(nzoax~-n~(p~.x~.)/(p~x ~
(4.158)
Numerically it can be demonstrated that for relatively low values of fl (fl-0.5), relatively high selectivity to the intermediate product C is observed when parameters f2 and f3 are equal to zero (e.g. the geometrical arrangement of A and C on the catalyst surface is identical). When f2 is negative (f2 =-0.5) and f3 is positive (f3=l), which is valid, for example, if C occupies less sites on the surface than A, the selectivity to C increases in comparison with the case when f2 and f3 are equal to zero. The further selectivity increase is observed with the increasing of adsorption strength of A (f~ -0.5, f2 =-1, f3 =0.5). On the contrary, the occupation of more surface sites by the intermediate product C than the initial substance A results in a selectivity decrease (fl =0.5, f2 =0.5, f3 =-1). In that instance, we see that the difference in the adsorption strengths does not have such a profound influence and the selectivity pattern is almost the same for the following values of parameters f~ -0.5, f2 -0.5, f3 --1 and fj -0.5, f2 -1, f3 --0.5. Similar observations tbllow tbr relatively high values of fl (-5) and thereby low selectivity. However, the effects of the polyatomic nature of compounds in question is not as pronounced as in the case of low values off1 (Figure 4.27).
148 Mole fraction
of C Fig2
1,0 0,8
1
,';
0,6
"
2
J~
5""-
?-..
0,4 0,2 0,0 i 0,0
i 0,2
i 0,4
i 0,6
Mole fraction
i 0,8 of A
i 1,0
0,2
0,4
0,6
0,8
1,0
lVIole f r a c t i o n of A
Figure 4.27. Numerical analysisof selectivitydependence (eq. 4.155) a): Mole fractionof C as a functionof mole fraction of A for fl=0.5:1)f2 =-1, ~ =0.5; 2)f2 =-0.5, f~ =1; 3)f2 =f3 =0 ; 4)f2 =0.5, ~ =-1; 5)f2 =1, t~ =-0.5. b): Mole fractionof C as a functionof mole fractionof A for f1=5: 1)fz=-0.5, f3 =1; 2)f2 =f3 =0 ;3)~ =0.5, f3 =-1; 4)fz =1, f3 =-0.5.
4.7. Reduction of complexity The theory of complex multistep catalytic reactions discussed in this chapter can be applied to homogeneous, enzymatic and heterogeneous catalysts. Sometimes there is a need to reduce the complexity of the systems in considerations and there axe several tools available for that. The Bodenstein steady state approximation is widely applied in catalysis. At the same time this approximation is often not valid and the dynamics should be taken into accont. Transient kinetic modelling as well as oscillation reactions will be considered in Chapter 8. As we have seen, the derivation of kinetic equations in general does not require the concept of the rate control by a single step with all other steps at quasi-equilibrium, though such a concept was very helpful when computers were not at hand to do sophisticated numerical parameter estimation. A concept of a most abundant catalyst-containing species (macs) or most abundant surface intermediate (masi) is often used, which could reduce the complex mutistep reaction to two-step sequence. Several rules which could used for reducing the complexity were formulated by Boudart for heterogeneous catalytic kinetics and extended by Helfferich for homogeneous reactions. In a catalytic sequence of any number of irreversible steps if the surface intermediate involved in the last step is the masi, there axe only two kinetically significant steps, the first and the last. In a catalytic sequence of steps, all steps that follow an irreversible step involving masi (macs9 as reactants are not ldnetical significant, e.g. have no effect on the rate. All equiulibrium steps following a step involving masi (macs) as a product can be combined in a single overall equilibrium step that regulates the concentration of this intermediate. Similarly to this statement if the forward and reverse rate constants of a step in a reversible reaction are much smaller than all others, the other steps are quasiequlibria.
149
Chapter 5. Homogeneous catalytic kinetics 5.1. Homogeneous acid-base catalysis Many homogeneous catalytic reactions are catalyzed by a single chemical species, usually by solvated protons. Bronsted general acid catalysis begins with the addition of a proton whilst Bronsted general base catalysis begins with the removal of a proton. This produces a low energy intermediate step, which then forms the products and releases the proton. Lewis acid catalysts are acceptors of electron pairs (+ A1C13, BF3, etc), while Lewis bases are donors of electron pairs. The kinetic scheme of homogeneous catalytic reactions is often discussed in terms of a twostep sequence
1) A + K<=>AK 2) AK + B y K+C A+BmC
(5.1)
where K is the active catalyst form. Following the general derivation (see eq. 4.93) the reaction rate can be directly written
r=
klk2[A][B][K]
(5.2)
k 1 -}-k2 [B]-}- kl[A ] Usually two limiting cases are considered (Figure 3.8). The first one corresponds to the socalled Arrhenius intermediates, when the activation energy in the second step is higher than in the first one. As a consequence the denominator in eq. (5.2) could be simplified as kq+kl[A] >>k2[B], which leads to the equation for the reaction rate r =
k~k2[AIB][K]
(5.3)
+
demonstrating first order dependence in B. At low substrate concentration the rate is first order in A, and at high concentrations it becomes independent on A. Alternatively, in the case of van't Hoff intermediates it is the first step that determines the rate. It holds that kq+kl[A] <
[A][K]
(5.4)
150 A generalization of the two step sequence for acid-base catalysis still keeping the second step as irreversible is given below 1. C + S<=>X+ Y 2. X + W ~ P + Z
(5.5)
where the reactants and catalysts for different types of acid-base catalysis are Catalyst C Y W Z Acid HA AH20 H30 + Acid H20 OHB BH + Base B BH + H20 OHBase H20 H30 + HA AAs an example consider an acid catalysis when X in (5.5) can be written as SH + 1. BH + + S <=> SH + + B 2. SH + + H20 ~ P + H30-
(5.6)
The second step is considered to be irreversible. In eq. (5.6) the proton is considered to be transferred to solvent (water) and is an example of a protolytic mechanism. In a prototropic mechanism, a proton is transferred to the solute ( S H + + B ~ P+BH+). The concentration of the BH + species is defined from the protonation equilibrium BH++H20 =B + H30-
K
BH+ [ ~ A
(5.7)
If step 1 in eq. (5.6) is essentially in equilibrium then
kl = [BH +Is]
(5.8)
At the beginning of the rection before any transformation of the substrate
(5.9) (5.10) Considering for simplification that BH + is in excess of S, than
(5.11) leading to the following equations
B]~f
I~BH+ A
(5.12)
151
k,
_[sH + lIB l [BH+lo([S]o _ [sH+ b
(5.1.3)
The concentration of intermediate follows from eq. (5.13) NI[BH + ]o [S]o
kl[H + ][S]o
(5.14)
The reaction rate is equal to the rate of the second step
(5.15) showing specific hydrogen - ion catalysis, as catalysis by other species could not be detected, although in the first step the proton can be transferred from any acidic species. At a high concentration of [H +] the rate will attain a limiting value equal to k2[H+][ S]0. For the case of case of van't Hoff intermediates the concentration of the intermediate SH + is considered to be negligible in comparison with BH +, therefore a steady state treatment of mechanism (5.6) requires that the rate of steps 1 and 2 are equal
k~[BH+IS]-k_~[B][SH+]= k~[SH+]
(5.16)
Replacing [BH +] by [BH+]0 and [S] by [S]0 - [SH +] the concentration of the intermediate [SH +] is (k.~[B] is negligible)
[SH +I z
kl[B-H+~[S]0 kl
kliBH+lo[S]o k2
(5.17)
leading to an expression for the reaction rate
l, = k2[SH+]= klk~[BH+ ]o[S]o
(5.18)
which is an example of general acid catalysis, as the rate depends on the concentration of any acid catalyst [BH+]. If kl [BH+]0<< k2, then the rate is controlled only by the first step
Equation (5.7) contains the acidity constant which strictly speaking should be expressed by activities, rather than concentrations
152
K =
[&,[H+ L, [BH+~/H ÷
(5.20)
Activity coefficients of the free and protonated form of B and the activity of the proton cannot be determined directly, however, their combination aH+y~H+/YB can be calculated from the known value of K by experimentally measured ratio [B]/[BH+], which can be done by spectrophotometry. Values of K can be obtained from dilute solutions (YBH+=YB= yH+=l). Hammett proposed to use electroneutral bases of various strength (nitroanilines) as indicators for nonaqeous solutions p-NOzC6HsNH2+ H + <=>p-NO2 C6HsNH3+ The acidity function Ho determined by this method characterizes the tendency of the solution to protonate a neutral basic indicator H 0 = - l g aH+YB
(5.21)
YtIH ÷
As an example of the application of the acidity function, let us consider the following scheme 1. H + + S <=> SH + ; 2. SH + ~ P
(5.22)
If the first step is at quasi-equilibrium and the second is slow the reaction rate is expressed with an aid of the transition state theory as r = k'(SH+) ~
(5.23)
The concentration of the activated complexes is defined from ( S H +)" = K " ( S H +)ysH+ / YsH+~-
(5.24)
The equilibrium constant for the first step gives K,. =(SH+)y~.,+ /[(S)ysa +] , (SH+)y~,+ = Ks[(S)7.,aH+ ]
(5.25)
The reaction rate is thus r = k* ( S H +) = k ' K * K s [(75 / 7,,.H+) a + ~S) = k ( S )
(5.26)
where the apparent rate constant is k=k'K~Ks[(ys/y,.H+)aH+]
Assuming that (Ys/YsH+~) ~ YBH+/ YB we arrive at Zucker-Hammett equation
(5.27)
153 lg k - lg (kS K S Ks ) - Ho -lg k ~- Ho
(5.28)
which relates the rate constant with the Hammett acidity function (Figure 5.1)
lgk 0
1
0
-1
Ho
Figure 5.1. Dependence of the rate constant on acidity function (eq. 5.28)
From eq. (5.28) it follows that there is a linear dependence between lg k and Ho, however experimentally it was observed that the slope is not equal to -1. Therefore, various modifications of Zucker-Hammett equation were proposed. As an example of a more complicated case than the two step sequence, we will discuss the esterification of a carboxylic acid with an alcohol. This is a very old and well-known category of homogeneous liquid-phase reactions. The esters of carboxylic acids are of an enormous practical importance; for example, millions of tons of polyesters are produced via the reaction of dicarboxylic acids with diols and a wide variety of mono- and di-esters are used in the production of fine and specialty chemicals, such as pharmaceuticals, herbicides, pesticides and fragrances. The esterification reaction is a homogenous liquid-phase process where the limiting conversion of the reactants is determined by equilibrium. Typically the equilibrium constants of esterification reactions have values of - 1-10, which implies that considerable amounts of reactants exist in the equilibrium mixture. The esterification reactions are very slow. Homogeneous and heterogeneous acids act catalytically in the esterification because the initiating step in the reaction mechanism is the protonation of the carboxylic acid. Efficient homogeneous catalysts are mineral acids, such as H2SO4, HC1, HI and strong organic acids, such as HCOOH. As heterogeneous catalysts the ion- exchange resins containing sulphonic acid (-SO3H) groups are active in the esterification. The acids used as catalysts often undergo side reactions since they can in principle be esterified by the alcohols present in the system. Two overall reactions will be considered in the kinetic model: esterification of acetic acid with methanol and esterification of hydrogen iodide to methyl iodide. The overall reactions can be written as follows: CH~COOH + CH~OH -H~O* H~O÷
,o C~r'I3.
The second reaction is a side reaction which destroys the catalyst. The hydronium ions acting as catalytic agents in the esterification are created through protolysis of HI and CH3COOH; the protolysis is enabled by the presence of water in the system. The protolysis equilibria are
154
HI + H 2 0 ~ I -
+ H30 *
a
CH~COOH + H 2 0 ~ C H 3 C O O - + H~O + b
(5.29)
Both reactions are rapid in forward and reverse directions and therefore could be considered to be at equilibrium, but the acid strengths of CH3COOH and HI are very different. CH3COOH is a rather weak acid (pK- 4.75), whereas HI is a very strong acid (pK <-7). To obtain rate equations for the esterification reactions (1) and (2), a detailed knowledge of the underlying mechanisms is necessary. For the acid-catalyzed esterification of carboxylic acids with alcohols, the mechanism was proposed (Figure 5.2). The nucleophilic substitution, step (2), is generally believed to be rate determining, whereas the proton donation step (1) as well as the subsequent steps (3)-(6) are assumed to be rapid. Consequently, steps (2-6) can in the kinetic treatment be lumped to a pseudo-step and the simplified mechanism becomes as depicted in Figure 5.3. +H~O+ ~
CH,-- C -- OH
OH+ I)
+~0
~ - - C -- OH 1
.,OH" ~ It + CI~OH ~ CH3-C--OH OH
OH I CH3--C-OH )+ HO CH~
2
OH
|
Cn,--C--Olq +H,O~ ! HO*CH3
C~--(~-- OH+ H~O~ ~H3
OH
3
OH
~ - - C - O' H +
H~O+ ~
)
!
~ CI43-- --OI'!a* + HzO
~
OCH3 4 OH
OH*
6CH3 o~
.,-,-~ , , , _ ~~- q
+ ~o ~
o ~ - ~ . -)o c H ~
+ ~o'
6
Figure 5.2. Acid-catalyzed esterification of carboxylic acids with alcohols.
155
(I)
,u C H i C + H30~ ~ OH
CH3C*
,OH ,OH + H20
0I) CHaC+ ~OH + CH~OH~ CHIC:O ÷ H30* OH ~H~ Figure 5.3. Shnplified mechanism of acid-catalyzed esterification of carboxylic acids with alcohols. The rate-determining step II can be expressed as
r n = knCACc,,3on - k nC~.C,2 o
(5.30)
where A = CH3C+(OH)2 and E = CH3COOCH3. The concentration of the intermediate (A) is obtained applying the quasi-equilibrium approach to the step I.
CH,o+ [ ~ ~jj = K]kjj @[C(.H3(,()OHC(.H30H H20 L
CEC,~o
]
~ "] K i ( k I I / k II)
C____/
(5.31)
The product K1 (kn/k n) equals the equilibrium constant of the overall reaction. In addition, the product K j k n is denoted by a lumped constant k'.
[
r n = k' CH3°+ C(:n~coonC(:n~oH Cn2o
C~:C<'°] Kn
J
(5.32)
The concentration of protons depends on the protolysis equilibria (5.29) and on the side reaction, which can be written as III. CH3OH + H30 + <:::>CH3OH2++ H20
(5.33)
IV. CH3OH2++ I- <=>CH3I + H20
(5.34)
The rate of step (5.34) is considered to be rate determining, therefore ri~: = kl~ C 1Cc~3oH2 - k_l~,Cc~ J CH~o
(5.35)
The concentration of CH3OH2 + can be calculated from the quasi-equilibrium (5.33), giving instead of (5.35)
G" = k~v KH~ Ccm°H C H~°+ C~ - k ~l,C(w~,~CHad CH20 Taking into account the protolysis equilibrium of HI
(5.36)
156
C l C H30+
Ka,
-
-
(5.37)
-
the rate becomes C<,.~I C.~o
C<,H
)
(5.38)
If the contributions of carbenium ions are discarded due to their low concentrations, the protolysis equilibria link the concentration of cations and anions CHso, = Q + Cooo , where Coo o = C~H
protolysis equilibria CH~°+ -- I K~ICm + Ka2CCH~COOH CH~O
~
(5.39)
CH~o
The concentration of undissociated HI is calculated from the total balance of iodine (5.40)
C°i = CHI + C I + Cc~j
where C% is the analytical concentration of added HI to the reaction mixture. The concentration of I- is calculated from (5.37) giving C°t
-
(5.41)
) CCH;Z = C m (1 + K~'C"2° - -
CH~O÷ the ratio CH2 o /CH~o+ is derived from (5.39) finally giving a cubic equation with respect to the concentration of HI. (
)3+ I a CCH3COOH o 2(1--~-CH± ) - K t
CH1
C~,H I (1-~)(1
CcH~Z cO~
)(
)2 + (5.42)
(~HI 2a C~ ~ o H . ) (~~ ) + a(1 - ~ ) 2
C( ,H3 ( OOH CHz 0
=0
The generation rates of components are expressed by rC,H,COOH = -rzz ; r~,H~OH= - r H - riv ; r~'H~C:OOC'H~= r H ; r~,Hj = r w ; rH2o = rzi + r w
(5.43)
The model can be solved numerically in the following way: the concentration of CH~ is solved analytically from eq. (5.42), the ratio Cu2 o/CH~o+ is then obtained from (5.39), the rates are calculated from (5.32) and (5.38) and then the generation rates of components are
157 computed I?om (5.43). In general, the generation rates should be related to the liquid-phase mass balances, giving for a batch reactor a set of sifferential equations, which should be solved numerically during the course of the parameter estimation. The examples above considered acid catalysis in a protonic solvent. However, this type of catalysis can also operate in aprotonic solvents. For a reaction Sl+S2 ~PI+P2 the following mechanism could be proposed, where HA is an acid catalyst. For this type of catalysis, only general acid (or base) catalysis can be imaginable as there are no protons or hydroxyls present in the reaction milleau. 1. H A + S~ <=> S~H+ + A 2. SIH + + $2 ~ PH + + P~ 3. PH+ + A ~ A H + P z
(5.44)
S1+$2 ~PI+P2 Assuming that the second step is rate-determining and the third one is fast, the reaction rate is expressed by
~x~l -- k2 [SIH+ Is9 ]
(5.45)
The concentration of SjH + species can be computed from the equilibrium of the first step
K,-
[A-Is'H+] [s,][HA]
•
(5.46)
nto ccountthat [*,H+]: [ ]we a ' ve
[SIH+ ]:
(K[SI][HM~ 0'5
(5.47)
and finally r =
2[S2I(K[S
I[HAb °5
([HA) °5
(5.48)
which is a very interesting result as it show-s that the reaction rate is proportional to the square root of the catalyst concentration. Note that in quite many cases acid-base catalysis is usually simplified. The literature often considers only the two-step sequence. Finally we discuss a classical example of the base-catalyzed aldol condensation of aldehydes based on this concept. The mechanism can be presented in the following form 1. A+B-<=>XI-+B 2. X~-+A¢=>X23. X_~_-+B~P+ B2A~P
(5.49)
The reaction starts by a proton transfer from the reaction molecule (aldehyde) to the catalyst base B- forming an intermediate X1-, which can further react in the second step with
158 another molecule of aldehyde giving the second intermediate X2-. This second step is usually in the literature considered as irreversible, while the last step (abstraction of proton from a coniugate acid leading to aldol) is assumed to be fast and thus the rate of this step does not appear in the rate equation. Following the general considerations discussed in section 4.3 we can directly write the expression for the reaction rate if the second step is irreversible r =
0),02
k, CAk2C A
Cc~~ =
0)1 + 0)2 + 0) i
k~CA + k2CA + k 1C,
C
(5.50) B
If klC A + keC A << k 1C,, then the rate is second order in aldehyde and we have a case of specific base catalysis as the reaction is first order in hydroxyl concentration (C B / C B is proportional to Coil ). Interestingly, the rate is first order in OH- although that ion is not a reactant in any of the steps. If k1C4 + k2C A >> k 1CB then it is a case of general base catalysis and the rate is first order in B and in aldehyde.
5.2. Nucleophilic catalysis Nucleophilic catalytic reactions are usually addition and substitution reactions. A diverse array of Lewis bases (e.g., tertiary phosphines, tertiary amines, pyridines, and imidazoles) have been shown to serve as nucleophilic catalysts. Nucleophilic reactions typically occur at C - X and activated C - C multiple bonds. In a general form for a reaction Z=X+YH ~ Y - X - Z - H the nucleophile is an anion, posessing much stronger nucleophilic properties than YH, therefore the catalytic mechanism can be expressed by a two step sequence 1. Z = X + Y <=>Y-X-Z2. Y-X-Z- + Y H ~ Y-X-Z-H+Y-
(5.51 )
Nucleophilic substitutions RZ+Y ~ R Y + Z can be also expressed as a sequence of two steps 1. RZ+Nu <=>RNu+Z 2. RNu+Y ~ RY+Nu;
(5.52)
As an example we can refer to nucleophilic catalysis of hydrolysis of benzoate esters by a phosphate buffer (Figure 5.4)
159
/nt~
X,d~\r O.4~D ~ \O~K
k~
D
×
<
"OH
+..o:
Figure 5.4. Hydrolysisof benzoate esters by a phosphate buffer which could be simplified with a two-step sequence (Figure 5.5)
tnt2
k3
X
~OH
* N~]
Figure 5.5. Acid-catalyzedesterificationof carboxylicacids with alcohols. Hydrolysis can be perfomed not only by phosphate, but also by some other nucleophiles, like imidazole and the o-iodosobenzoate anion, having essentially the same (from kinetic viewpoint) mechanism. The acylation of alcohols by anhydrides, catalyzed by 4-(dimethylamino)pyridine (DMAP), is one of the most frequently described in the literature examples of nucleophilic catalysis (Figure 5.6). O °
M
Me
P
Me
s/ow
~AP
"Me
Figure 5.6. Acylationof alcohols by anhydrides. It is usual that in both schemes (5.51) and (5.52) the first steps are considered to be reversible, while the second are irreversible. Then it is possible to directly apply equation (4.88) derived for the two-step sequence. For instance, in case of scheme (5.52) it takes the form
160 c°l °)2
r =
C~; =
k 1C~z k 2Cy
CN,`
(5.53)
k~C~z +k2C Y+ k ~Cz
co~ +co2 +co ~
Several limiting cases can be considered. For instance klCRz is simplified to
<<
k2Cy
>>
k_lC z then eq. (5.53)
(5.54)
r = kiC~zCN,
with the rate having zero order dependence in Z and Y and the catalyst is almost entirely in the free from. When k~C~z >> k2C }, ;k~C~z >> k ~Cz , the reaction rate takes the form (5.55)
r = k2CyCN,
with the catalyst existing in the form of a catalytic complex with the nucleophile. The rate is determined entirely by the second step being independent of the concentration of RZ. Assuming quasi-equlibrium of the first step, which means that the rates of step 1 in the forward and reverse direction are faster than the rate of the second step klCRz >> k2C r << k_tC z from (5.53) were arrive at F = (kl IN 1) N2CyCRz C~N~¢= K1 N2CyCRz Oh), Cz Cz
(5.56)
when the reaction rate has a negative order in the product Z and first order in both reactants RZ and Y. Eq. (5.53) can be analyzed furher assuming for instance that k 1Cz << k2Cy, klC, l~z
>>
klC Z
or
ktCRz << k2C~ ;klCl~z << k iCz , etc.
With the advance of numerical methods it is possible to use the general form of eq. (4.88) for the treatment of experimental data. At the same time, the analysis presented above is still important for the preliminary evaluation of the correspondence of experimental data with the proposed reaction mechanism. If the experimental data is acquired in the domain when one component (for instance Y) is in excess and its concentration does not change during reaction and if the first step is irreversible we get directly from eq. (5.53) r =
kl C ez k2 Cr k~C;~z + k 2Q,
CN,, -
kC ez
(5.57)
1 + k' CRz
with the reaction order in RZ ranging between 0 and 1 depending on the concentration of RZ. The constants in equation (5.57) can computed numerically or easily calculated from its linearized from 1
1
k'
- - - + -
r
kCRz
(5.58)
k
161 Although such linarization is seldom used in homogeneous and especially in heterogeneous catalysis, due to historical reasons this approach is frequently utilized in enzymatic catalysis and will be discussed in detail in the corresponding chapter. An interesting example of nucleophilic catalysis is the case when the rate increase with conversion, e.g. catalytic self-acceleration (autocatalysis), is observed. A hypothetical single step reaction with the stoichiometry A+P~2P
(5.59)
shows, that product P acts as a catalyst. Nucleophilic addition of mercaptanes to epoxides is an example of an autocatalytic reaction (Figure 5.7) CH 2
/
CH2
RSH "'" / i
+ RSH
0
H2
"'oH2
/ /CH2 RSH "'" O/
+RS-
~ RSC2H4OH +RS-
"~112 Figure 5.7. An autocatalytic reaction: nucleophilic addition of mercaptanes to epoxides.
which can be formally written RZ+YH<=>YH..ZR YH..ZR+Y-~BH+ Y-
(5.60)
The alcohol BH, which is formed in this sequence, is a stronger nucleophile than the mercaptane YH. This leads to a faster reaction than the first step in (5.60) leading to the following reaction mechanism RZ+BH<=>BH..ZR BH..ZR+Y-~2BH+ Y
(5.61)
this gives an overall stoichiometry as (5.59). The reaction rate is a sum of the two pathways, where the second is the autocatalytic one. r =
k ICRz k 2Cy kiCez +k2C ~ + k 2
C~w -~
k I ' CRz k 2 ' Cy
CBH
(5.62)
k~'Cez +k2'C Y + k 2'
The reaction of ethylene oxide with n-decylmercaptane in the presence of the catalyst triethylamine can serve as an example of an autocatalytic reaction. As one can see from Figure 5.8 after the initial induction period the rate increases indicating a typical autocatalytic behaviour.
162
04
•
os~
Ccat--0.25 m o l A Ccat--O.015 m o l A
g
Ol
93
103
150
~me, min
Figure 5.8. Reaction of ethyleneoxidewith n-decylmercaptanewith triethylamineas the catalyst.
5.3. Catalysisby metal ions In the section 2.3. we presented catalysis by complexes of transition metals and argued that this type of homogeneous catalysis is gaining importance in industrial processes. Kinetic models of several reactions will be considered in the next paragraph. Here we would like to address reactions where the metal ions act as catalysts. As an example of such a process, hydrogen peroxide decomposition will be considered. Transition metals, mainly manganese, iron and copper, are effcient catalysts for this reaction. We will consider a simplified kinetic treatment following the reaction mechanism proposed by Kobozev for the decomposition of hydrogen peroxide. 1.2 H202+K<=>
2. ~
K~
+2 H20
=:>K+O2
3. H202+K<=>K=O+H20
1
]
(5.63)
0
where K is the metal ion catalyst. In terms of the graph theory the third step is a hanging vertex, e.g. the stoichiometric number of this step is equal to zero. Such a step can be considered only as an equilibrium one. The expressions for the rates are if the second step is taken as irreversible
~, -- ~, [H~O:
]~[K]- ~ [H~O]~[K, ]
where k-.~ I
is denoted as [K1].
(5.64)
The steady state treatment of the mechanism (5.63) gives the concentration of the catalytic complex [K]]
[KI]= ~'[H20~12 [K] ~_,[H2012+k2
(5.65)
163
Equilibrium of step (3) in (5.63) relates the concentration of inactive catalytic complex (K=O, denoted below as [K2]) to the catalyst in its tree tbrm [K]
[K2]=K3~[K]
(5.66)
where K3 is the equilibrium constant for step 3. Taking into account the balance equation
[K] +[K1]+ [K2] =[K] 0
(5.67)
we arrive at
klk2[H2O2] 2 r=/42 = k 2 [ K l ] _
(1 + K 3
k l[H2012q-k2 kl[H20212 +
k
[K]O
(5.68)
)
l[H2012+k2
which can be further rearranged to
r =
r....
(1 + K 3
1
klk2[H202] 2
[K] °
(5.69)
~)(k<[H20] 2+k2)+ k, [H202 ]2 LHeU]
For the case when the third step is very slow from eq. (5.69) we arrive directly to an expression tbr the two-step sequence, which is analogous to eq. (5.53). The concentration of water can be considered as constant, because the reaction takes place in an aqueous solution, which gives then
r =
k[H202] 2
1+
'[H202]+
"[H202]:
[K] °
(5.70)
where k, k' and k " combinations of constants. Equation (5.70) can describe experimental data showing reaction order in hydrogen peroxide from zero to two. For the Fe(III)/H202 system, H202 is catalytically decomposed by Fe(III) at an acidic pH and the reaction mechanism is much more complex as it includes also radical reactions, i.e. formation of hydroxyl and hydroperoxyl radicals. Symbol ° in eq. (5.71) denotes a radical. Fe 3+ + H202 ~ F e 2+ + HO2 • + H + Fe 2+ + H202 ~ F e 3+ + ° OH + OH Fe z+ + t O H ~ F e 3+ + OHH202 + " O H ~ H O 2 " + H 2 0 Fe 2+ + HO2 ° + H + ~ F e 3+ + H202 Fe 3+ + HO2 • ~ F e 2+ + 02 + H +
(5.71)
164 In fact the reaction of H202 with Fe 3+ leads to the formation of Fe(III)-hydroperoxy complexes FeH~(HO2)2+ and FeH~(OH)(HO2)+. At very high concentrations of H202, the formation of diperoxo complexes has also been suggested. Kinetic modelling of such a case is complicated and can be found in special literature.
5. 4. Catalysis by organometallic complexes Organometallic catalysis is very diverse and many different mechanisms have been proposed in the literature tbr various reactions. Interestingly it is rather seldom that kinetic models are derived for the proposed catalytic cycles and the kinetic equations are even compared qualitatively with experimental observations. We will discuss several examples which represent typical cases in homogeneous catalysis by metall complexes.
5. 4. 1 Single catalytic cycles Catalysis by transition metals can often be represented by reaction mechanisms which correspond to the case of single catalytic cycles. Only one route exits but it could contain several steps. If the reaction mechanism can be simplified to only one intermediate besides the free catalyst form (the most abundunt surface intermediate or catalytic species) while other intermediates even if present are in interior quantities, then the mechanism corresponds to the two-step sequence described in detail in section 4.3. As an example heterolytic oxidation can be considered (Figure 5.9). The catalyst can exist in two forms with different oxidation states and the catalytic cycle consists effectively of oxidation-reduction steps. The reaction rate in such a case is described by the kinetic equation (4.88). L ...
n+
M
H ~.,dE,,,y
Figure 5.9. Heterolyticoxidation mechanism If the number of catalytic species is three, then the reaction rate becomes more complicated, but still managable to derive using a steady-state approximation. However, this laborous excercise is not needed, as we can directly use the derivation presented for the three-step catalytic sequence with linear steps, i.e. equation (4.114). Examples of direct and indirect hydrogenation mechanisms when the reaction mechanism can be expressed by a cycle with three intermediates are presented in Figure 5.10 and Figure 5.11 respectively.
165
CH3-CH 3"~~
2
L
L.,. [./H L/M "~'CH 2
L\ /H L jM "H
CH /
~
C
2H4
Figure 5.10. Three step direct hydrogenation mechanism Lx
CH3-CH3. ~
/X
H2
L/M'-x ~ .
L.. ./CH2
L\
L/M "--C[H2
L/M
~ C 2 H
4
Figure 5.11. Three step indirect hydrogenation mechanism
The general form of the eq. (4.114) can be used in the case of the direct hydrogenation mechanism shown above giving
r =C,,, k2k3 + k 3Ccjtk2 + k
kiC,2k2Cc~_H k 3 k ik 2k 3CczH/, 3Ccy~k2+k3klC~2 +klk 3 +klk 3C(,2H. +klC~2k2CqM4 +kzklC~2
+k2k 1 (5.72)
Steps 1, 2, and 3 are counted from the top of the figure clockwise, e.g. step one corresponds to addition of hydrogen to ML2 complex, where L is a ligand. For an irreversible reaction eq. (5.72) is essentially simplified to r =
klCH2 k)Cc H, k3 - e Cc,, k2k 3 + k3k I CH2 + klCH2k2C(,2H 4
(5.73)
The kinetic expression for the indirect hydrogenation mechanism can be written in a similar fashion.
166
HNR 2 [M ]-N R2
y
:::::::::
[MI
wwwwwwwwwwwwwwwwwwwwwwwwww :~:~:~:~:~:~:~:~:~:~:~: ',', ,~:~:~:~:~ '~ ~5555:~ ,,:,~,,~:~~2 :~:~:~:~',~,:,,4:~2 ~ :~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~
Figure 5.12. Three step mechanism of o lefins hydroamination
Besides hydrogenation, some other reactions like hydroamination of olefins as examplified in Figure 5.12 for ethylene, can be expressed by eq. (4.114) resulting in the rate equation when all the steps are irreversible (steps 1, 2, and 3 are assigned starting from the intermediate M clockwise)
kiCH,~ k~Cc< k3
r=
: - :
Co,,
(5.74)
k2C~'~H~k3 + k3kl CH~, : + klCH~R' : k~C~,_ 2fl~
which gives orders in reactants between zero and one depending on the reactant concentrations. Hydrogenation mechanisms catalysed by transition metal complexes could be somewhat more complicated combining direact and indirect hydrogenation containing a single cycle with 4 catalytic intermediates (Figure 5.13) CH3-CH3
L\ L/
L... / X
~..X M
~
HX~-J-
"C H2
/ ~
L. ~ H \ / L"M " X
02H4
L ~M . . ~ - , ~ "~
Figure 5.13. Four step hydrogenation mechanism
Mechanistically different from hydrogenation but kinetically similar is double bond migration in olefins (Figure 5.14) and catalytic epoxidation (Figure 5.15).
167
L\
Hc//CHR
/\
/R
H#~~,CH2R
L,. / L~'N "'\'C HR
CH Figure 5.14. Four step mechanism for double bond migration in olefins .2/~GH R__ R'OH O '% T L~MIIIo=O
/ .'o;a,o ~
R'OOH
~:~,o:o
i R.c
/ /
H o~/MO'~O'~-
CH=CH2
+ R'OOH
Rx
Mo
C~CH2 O
-
+ R'OH
'k~
RH ^ : / O ...... ~)~ .
X
R\
.o~ ,o-o,R.
L.
Figure 5.15. Four step mechanism for catalytic epoxidation An example of the equation tbr tour steps was considered using an enzymatic reaction (4.116). Analogously to (4.121), applying the general form of a single route equation (4.115) we will get for directly from a general equation for complex reactions + + F~tp ~ o o o
r =
Z
/,+
--
o o o r~7
.,,, Q,, ++ or,.+ + Z g v,.,(~,+ ooor +,,,,+ Z r~,C v< ooog + +...+ Z r~,r~ r,. ooov, v~g2r~.~ 11, •
.2
,,,
r,, ~.2
m
m
'
' 2
2
(5.75)
'
with frequencies as rates of the corresponding steps
r =
0.)10)20.)30.) 4 - - 0 3 10) 2 ° ) 3 0 ) 4
where
D
(5.76)
168 D = 0)20)30)4 + c0-]0)3 0)4 + 0)-10)-2 0)4 + 0)-t0)-2 0)-3 + -}-0)10)20)3 +0)-40)2 0)3 -1-0)-40)-i 0)3 -}-0)-40)-1 (2)-i q-
(5.77)
-}-0)40)10)2 -j- 0)-30)1 0)2 -}- 0)-30)-4 0)2 "}-0)-30)-4 0)-I "}+ 0)30)4~O1 + 0)-20)4 0)1 + 0)-20)-3 0)1 + 0)-20)-3 ~O-4
In general, equation (5.77) should also include the stoichiometric numbers of the steps. In eq. (5.77) they are all equal to unity. The terms in eq. (5.77) are calculated by changing the counting order in the four step reaction mechanism tbr the steps ~?om 1 through 4. In case of a four step mechanism (Figures 5.13-5.15) the terms in the denominator are s=l,
(5.78)
O)20)30)4;O)-10)3 0)4;0)-10)-2 0)4;O")-10)-2 0)-3
If we start counting from the second step on the four step mechanism then we can make the following assignement 1=>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 CCH
(5.83)
169 ~o ~, RCH2CH2C\x"~
~
L
\M ~ -
f ]~:
R-CH=CH2 + CO + HX
k,. L/ --.e~o
X
" R CH2CH2C.~
7H
Figure 5.16. Carbonylation of olefins. Several mechanisms ibr homogeneous catalytic reactions by transition metal complexes could be even more complicated and consist of more than four steps. Carbonylation of olefins (Figure 5.16) and olefins methathesis over a complex catalysts WC16 + EtA1C12, Mo(CO)6/SiO2 where n is the oxidation state of Mon+:w n+ and n-2-4 (Figure 5.17), can be described by catalytic cycles with 6 intermediates.
PhClI=CII,-~. "~j'~
CH2 ~/ ~
PhCH=CHPh
H2C CH2 PhHC=CH Ph W
~
H2C PhHC
OH2 CHPh
W ~ P h
I P,.c Figure 5.17. Metathesis of olefins The derivation is essentially the same as before leading to O.)LO)20.)30)4(/)50.) 6 --O) tO) 2(I) 30.) 4(2) 50.) 6 ~7 F z
cat
(5.84)
D'
where D' is calculated according to the same principles as for the 4 step sequence. We start counting from the step 1 taking into account the stoichiometric number of steps
170
1710)20)30)40)50) 6 + 0).1V20)30)40)50)6 + O).10).21/30)40)50)6
s=l,
(5.85)
+ 0)-10)-2 ~;0-31/40)50)6 + ~;0-1(0-2 0")-3(0-41/50)6 + 0")-10)-2 0)-3~;0-40)-5V6
Such counting should be done 6 times starting always with another step and reassigning the step numbers in a similar fashion as for the four step mechanism. As an example if s-2 than we do the following assignement 1=>6; 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 Col
(5.93)
giving a variation of reaction order in CO between zero and two, while orders in olefin and hydrogen vary between one and unity. Here we do not want to discuss how this kinetic equation can describe experimental data of olefins hydrofbrmylation and the derivation just serves as an illustration. For a mechanism with s steps, the general form of the kinetic equation will be O)lO),O)3...OOs.--O)
10,) 200 3...O0 s [,~-.~cat
D where the denominator is the sum of all elements of a square matrix of order s
t~)~z'[')l~'~*\
172
Yl(020) 3 ..COs
09.1V2(03(~94 ..O9 s
........
09.]09. 2 ..Og.(s.l)Y s
(/)lY2 (O3..O.)s
O)10-)_ 2 V3 ~ 9 4 . . ( O s
........
V 1 (O_2 CO_3 • .0-)_ s
(Z)l ~02 V 3 ..O.) s
O.)1(Z)2 (-0_3 V4..(-0 s
........
O.)_ i V20.)_ 3 , .O.)_ s
(2)lO')2"'('Os-lYs
YI (/)2 O)3 ""(/)- (s - I ) O')s
........
(5.95)
O)-10)-2 "'Fs-I (/)-s
This matrix, if we omit the stoichiometric numbers of steps, is reduced to the so-called "Christiansen matrix", named after the famous Dannish scientist Jens Anton Cbxistiansen. C02 O.) 3 ..CO s
(D.I ~03 O) 4 ..0.) s
........
O.).i (A).2 • .O-).(s.1)
(2)1 (A)3..O.) s
(Z)l (Z).2 (A)4 ..(Z) s
........
(2).2 (2).3, .(2).s
(2) 1 0.) 2 . .O) s
CO1O32 (-0_ 3 . .O.) s
........
(2)_10.)_ 3 . .O.)_ s
(-01032 • .O)s_ 1
(/)2033 ..(D_(s_l) O) s
........
(5.96)
(O_10)_2 ..(A)_ s
in this matrix the sum of the elements of each row is proportional to the concentration of one of the members in the catalyts cycle. In other words, if we start the cycle with the free catalyst concentration taldng part in the first step, then the first row is proportional to the free catalyst conentration (coverage of vacant sites in heterogeneous catalysis) - which we will consider as "first intermediate" and the second row is proportional to the concentration of the second intermediate, i.e. intermediate which is bound to the free catalyst, etc. From these mechanistic considerations we arrive at eq. (4.123), presented in the previous chapter. The ratio of the concentration of the intermediate to the total catalyst concentration is
Xj
Cx/ I
[X]z
Dj
(5.97)
I
C[x]~
D
where Dj corresponds to the sum of all elements in rowj.
5.4.2. Ligand-deficient catalysis and inhibition Catalysis by organometallic complexes often involves as a first step the loss of the ligand from the catalyst (Figure 5.19). It results in freeing the coordinative site which is then able to bind the reactants
X3
X2
Figure 5.19. Ligand defficient catalysis.
173
The external parthway involves no species that are formed or consumed during the reaction. This is the case of the reaction mechanism with a hanging vertex, which means that this additional step is in equilibria. A quasi-equilibrium of this step gives
X -
Cx, CL
(5.98)
C'x0 thus
Cx~' = Cx C L / K
(5.99)
the total concentration of the catalyst is then
Q o, = Czv +Cx, Q . / K
(5.100)
The traction presented as Cxl is given by eq. (5.97), e.g. it is the ratio of the sum of the firstrow elements to the sum of all elements of the Christiansen matrix of the reaction cycle Q~,, = Cra-(1 + D'Cr~ ) DK"
(5.101)
For a mechanism with s steps we replace the catalyst concentration in eq. 5.94, which is in eq. (5.94) just Czn by Q ~ we arrive at
r=
,=t
*=t
D
CzAr =
J=l i=l Qc,' (D + D IC L / K)
(5.102)
As an example we continue to consider a case with three steps. Then, D~ consists of three elements leading to an equation for the rate 1" =
0)10)20)3 -- 0) 10) 2 (93
Ccat
0)20)3 -- 0)-30)2 q- 0)-30)-2 q- 0)30)1 q- 0)-10)3 q- 0)-10)-3 q- 0)10)2 q- 0)-20)1 q- 0)-20)-1 q- (0)20)3 q- 0)-10)3 q- 0)-10)-2 )CL / K
(5.103) which is somewhat more complicated than eq. (4.114). The external parth can be more complicated as examplified in Figure 5.20 for the hydroformylation reaction, as it requires not only the release of one ligand, but also hydrogen and CO should react in this external pathway.
174
OO@"
H Ph3P...... I Ph3P
"~ PPh3
R
~//
I CO
Ph3P/~. . . . . OC
~k
Ph3P,'~~,,,..Rh ( .,,,~H OC~"
CO ~
~,,H PPh3
\
+CO
\,,-.,-PPh~
*
H~"\\
-
CO
R
:
+ PPh3 \k
0 ,
\'~
PhaP~,, ~ ,,~H
Ai 0
OC~ " Rh,~pph3
; o O?, , . ... .h, H
Ph3P,. "Rh
Ph3Pt~""
,~C(O)CH2CH2R ....
o
I'~H CO
1L Ph3P.,
.
.
.
.
,,.~CH2CH2R
OC
Ph3P~/,. . . .
~',C(O)CH2CH2R
PhaP
CO
PPh 3
/
Ph3P.,,,,. t~H2CH2R Rh CO Ph3P I CO
j
CO
Figure 5.20. Mechanism ofhydroformylation with an external path.
Moreover the external parthway can consist of more than one step as it is for instance in the Heck reaction
L
~d
R
~
......... J
L
R
Figure 5.21. Mechanism of Heck reaction
In the previous section we were discussing the case o f a ligand-deficient catalyst, as the reaction o f the ligand with a catalytic species from the active cycle syphon off the catalyst material from the cycle. This can be considered as a case o f inhibition.
175
[HBaselX Ba~,
+ElL"%L
L
I
0
HipdiX
+CO
I
At'.--- Pd i
I
[
L
',,
L
",.
X
L
/X Pd
Al~C/
~L
II
L,,,,
Ar I Pd I
0
CO
X
Figure 5.22. Mechanism of carbonylation reaction. Considering Pd catalyzed carbonylation (Figure 5.22) we will see that equation similar to (5.102) could be directly applied with the only difference that the equilibrium is defined as (5.104)
C xo = C x~CI~ / KCco and then the rate equation is just slightly modified to
~9i--H F =
i=l
(D
C.Oi
i=I
Cc,a t
+ D1C L / KCco )
(5.105)
This is a case of competitive inhibition, as the inhibitor and reactant compete directly for the active site. Ligand-deficient catalysis is then a special case of competitive inhibition when the ligand itself acts as an inhibitior, although it is a necessary ingredient of the catalyst system. For the noncompetitive inhibition, if the inhibitor does not react with the free sites (XI) , but with some other catalytically active species (Xj) by a reaction Inhibitor + Xj = S
(5.106)
for which the quasi equilibrium constant is K± - - CXX -
CiCr,
this leads to
(5.107)
176 C x s = K, C t C Vj
(5.108)
Analogously to the treatment of ligand deficient catalysis we introduce the concentration of
Cx, into the total catalyst balance and by defining Cx, through Di, the reaction rate is
n
O) i - -
r = ~=l
ni=1
n
(O i
COi - -
C~. = ,=l
D
n
09 i
i=l
(D + DjK~C~)
C~,l
(5.109)
Comparison between (5.109) and (5.102) reveals, that in (5.109) D1 should be replaced by Di and equilibrium adsorption of inhibition in case of ligand-deficient catalysis was defined in a slightly different way. As an example let us consider a mechanism, where the two step sequence with a second step being irreversibe is combined with the third step, which corresponds to inhibition. The stoichiometric number of this step is equal to zero: 1.2A+ Xlc=>X2+2B 2. X 2 ~ X~+C 3. A+ X± E X~ +B 2A=2B+C
1 1 0
(5.11o)
Writing down an expression for the concentration of inactive species C XS = ( K 3 C
'
4/C~)C
"
(5.111)
"V1
we can get directly from the general form ofeq. (5.109) that (2)10-)2 F--
091 -}-W: -I-09_1-}-(0.)_1 + 032)(K3(.: A / C B )
k,k2A2
eta t =
(5.112)
Ccu~
(I+K~
; ) ( k LB2 +k~) +k~A:
Careful comparsion between eq. (5.112) and eq. (5.69) reveals that we get exactly the same description where A stands for hydrogen peroxide and B for water.
5. 4.3. Multiple cycles Catalyst systems in general can contain several reaction routes as discussed in chapter 4. The same is valid also for reaction with organometallic complexes. An example is the hydroformylation reaction (Figure 5.23). A general form of the kinetic equation was derived by Temkin in the following way
177 r+r + o o o r,,,, + - r, r, o o o r,:,,,, : st ,~: EF(N)~
(.~) + + o r + r + + ' ~ tVSI ~'2 ~'3 Sm ~
- (N) + ~'1 VS2 /~'3000
(5.113)
F+ + . + Sm
N
which then relates the rates along different routes. For a single route reaction eq. (5.113) is simplified to eq. (4.115)
CO
H Ph3Pn,,, Ph3P~Rh--CO I
I
~,. /
CO
~
R
Ph3P/~,,'Rh" ,~H / e~t,¢~ ~PPh3
CO
"I ~
R
~
R H Ph3P,,,
Ph3Po,,,, - ,~H
Ph3P~v,. J .,,,,~( Rh" " Ph3P~¢"10~CO
OC
PPh3
,aC(O)CH2CH2R
Ph3P~¢" I
H Ph3P/t,....
fl
OC
H2 ~-
,~CH2CH2R PPh3
/
Ph3P. . . . . . . . . . C(O)CH2CH2R
Ph3P,,,,, TH2CHzR
Ph3P
Ph3P
CO
I CO
~.. ~
\ CO
Figure 5.23. Mechanism ofhydroformylationreaction with multiple cycles. Another example was discussed in section 4.5 when addressing kinetic coupling between two cycles. Very similar to it is the situation when two reactants are reacted in two otherwise independent cycles, but are however connected by a reaction between them. Dynamic kinetic resolution is then different from a case when there are two parallel reactions (Figure 5.24). Classical resolution
A
p
B
Q
D y n a m i c kinetic resolution A
~
p
kpglc k8
B
kB,
O
Figure 5.24. Classical and dynamic kinetic resolution. In the mathematical treatment below we will use the slightly more complicated case o f parallel c o n s e c u t i v e asymmetric catalytic reactions and will use the reaction model presented in Figure 5.25. k~ k3 A1
~ R-B
f*K
~
R-C
k4
S-B ~ S-C k2 Figure 5.25. Dynamic kinetic resolution for parallel -consecutive asymmetric catalytic reactions.
178
Reactant A exists in two forms Al and A2, the latter giving S- and the former R enantiomers o f B, respectively. B can further react to C which exists in two enantiomeric forms, i f we use mole fractions instead o f concentrations then one arrives at the dependence o f the mole fraction o f B as a function o f A (boundary conditions (t=0, NA=I, Np.3=0, NsB=0, NRc=0, Nsc=0),
N ~
L 1 1- M
-
(1-L) N
sB
~
(N
(N
1-M
' -
N A)
(5.114)
AM~ _ N ~ )
(5.115)
2
with
k~ K
L 1 = (k
,•
M
k 2
1 +
1
k3K~(K =
+ I)
(k~ K + k 2 ) K A
k4KsB
;M
2
(klK
(K
+ 1)
+ k2)K
(5.116)
4
where Ni - mole fractions, Ki are binding (adsorption in case o f heterogeneous catalysis) constants and k~ are kinetic constants. Enantiomeric excess can be defined in the general form y+ -- y
ES -
-
-
(5.117)
Y+ +Y_ where y+ is the yield o f the (+)-stereoisomer and y_ is that o f the (-)-stereoisomer. In the case o f B, it is independent o f the conversion o f A, thus the ratio o f R- and S-enantiomers o f B is independent o f the conversion o f A. aS,-N NR~ _ (1 - M 2)L1 (N A M N sB (1 - M 1 )( 1 -- L~ ) ( N A : _
N
A) = D (N~I-N (N A '~'~: ~)
N
~) A)
(5.118)
Enantioselectivity in formation o f R-and S-enantiomers o f C is expressed via NR(, -
ee . . . .
Nsc
(5.119)
Nl~c + N s c
with Nk c N sc
=
DI
[
= D z[
( N A~ '
l
MI-1
-
~)-
(klK
1 M 2 -1
k 3 + k2)
~)]
-
(5.120) ( N
M2 ~1
_
and
O 1
(NA
K ~ L1 K ,4 1 - M 1
1)
-
(N
~
-
1)]
179
D 2 = (klK
k4 + ke)
K.~,. K A
(1 1-
-
L,) M 2
(5.121)
It can be demonstrated numerically (Figure 5.26) that for certain values of the parameters M~ and M2, the ratio of R and S enantiomers of B is independent of conversion and enantiomeric excess is negative, while for the same values of parameters enantiomeric excess in enantiomers of C will be positive and dependent on conversion. enantiomeric excess, % 20
-/
10
..... J ......
w
o-
•
•
B
~
C
......................
-lO
•
...... I!
-20
....
•
-30 -40
•
0.2
0.4
0.6
0.8
1'.0
conversion
Figure 5.26. Numerical analysis ofeqns. 5.119-5.121. The case of kinetic resolution in asymmetric catalysis with enantiopure ligands was discussed in detail in section 4.5, which was devoted to kinetic coupling. In the case where the first steps are considered to be in quasi-equilibria, the enantioselectivity of two enantiomers is expressed by eq. (4.144), implying that selectivity is independent on substrate conversion. When a kinetic resolution is carried out using enantioimpure catalysts which do not interact with one another, the reaction network (Figure 5.25) must be expanded. Recently, the concept of kinetic resolution has been extended to the case where enantioimpure catalysts are used. Kagan discovered the first examples of nonlinear effects in asymmetric catalysis, where there was no proportionality between the ee of the auxiliary and the ee of product (Figure 5.27) and gave some mathematical models to discuss these effects. The nonlinear effect (NLE) originates from the formation of diastereomeric species when the chiral auxiliary is not enantiomerically pure, either inside or outside the catalytic cycle. The observed effects were classified as (+)-NLE and (-)-NLE where "asymmetric amplification" and "asymmetric depletion" respectively occured. t
0,6
0.4
0.2
0 0
0.2
0.6
~4
&8
1
ee~t
Figure 5.27. Illustration of nonlinear effects in asymmetric catalysis
180
Henri B. Kagan The origin of nonlinear effects is due to formation of dimeric or higher-order catalyst species. For a reaction system R ~ P a , S ~ P s the Kagan's ML2 model could be used where three separate dimeric catalysts, [catRR], [catSS], and [catRS], are formed (Figure 5.28). The third catalyst, the m e s o species [catRS], binds each enantiomer equally with its own binding constant KRs, and it will react with each enantiomer equally with its own rate constant kRs.
R *Rs
R *K~
R~
s,Rs
S~R
eatRS
ca~S
S.SS
Figure 5.28. Scheme for kinetic resolution mechanism with two homochiral (catRR and catSS) and one heterochiral (catRS) catalyst species (D.G. Blackmond, Kinetic resolution using enantioimpure catalysts: mechanistic considerations of complex rate laws, Y. Am. Chem. Soc., 123 (2001) 545). The existence of three types of separate catalytic species means, that there are three cycles with three balance equations Cf,,~ol c'a[ l?]? = C~ OCa[ l¢R + C ~*~.,~,~,~+ C s* ~.c,f,~,~ = C~ Oc a l R i e + K ~ C ~ C
o ~.,t,~,~+ K s.sCsC°~.,~ ....
(5.122)
181
~c,,,,, = (., 0 c<,~(1 + KssC ~ + K ~ C s )
(5.123)
C '°"'o,..~ = C°.,,.~ (1 + K,~s,(C s + CR) )
(5.124)
ctotal
where C ° corresponds to the concentration of the free dimeric catalytic species. Now the reaction rate along each cycle can be computed as well as the rates of generation for the products. For instance, for product PR, starting from R, the overall rate is the sum of the rates along the three different routes (cycles) k
(
r~&
l~ (-~ total R R * ~ R R ~ ~al Rg~
q-
]¢u (~ total "~ R S * ~ R S ~ catR5
]e- J ~ ( ~ total " %~,'S* ~ .YS ~ " cats. s
(5. 125)
l+Kzes(Cs +C/e) + 1 + KssC,e + K,~,eCs
'1 + KmeCte +KxsCs
Analogously
k ~YS k* "~ ~Y.'~"r~'
total COIRR
rR=e" = ( I + K ~ C s +KxsC s
4
lr l~ ~ , total '~ R S * m R S ~ catz{.s
l+Kes(C s +Ce)
+
leK ( , total " ~ R R * = R R ~ cat&s.
l+Ks:s,C R +KR~C s
-)Cs
(5.126)
The enantioselectivity is the ratio of rates showing rather complicated behavior, as all three denominators are functions of the concentrations of reactants, and thus conversion. An observed nonlinear effect in catalyst enantiopurity in kinetic resolution may thus be the consequence of partitioning of the total catalyst concentrations into homochiral [catRR]total, [catSS]total, and heterochiral [catRS]total.
M + LR +
Ls - - ~
~au×
MLR
K MLsLs ~
X
2 MLRLs
y
Rss 1 product eemax
pr~uct -
e%~x
Z
IKRs pr~uct racemic
Figure 5.29. ML2 model in asymmetric catalysis (H.B. Kagan, Nonlinear effects in asymmetric catalysis: a personal account, Synlett SI (2001) 888).
The scheme in Figure 5.29 shows the reaction network for the ML2 model developed by Kagan which describes an asymmetric catalytic reaction based on two enantiomeric chiral ligands, LR and Ls, and a metal center, M. Three different catalyst species, two homochiral complexes and a meso complex, may be formed with relative concentrations which are fixed by an equilibrium constant K = z 2 / x y . Parameter [3 (fl = z / ( x + y ) relates K to the optical purity of the chiral catalyst eecat
z - Keec2,, + ~-4Kee~2,, + K ( 4 + Kee2o,) /7 - - 4 + Kee~,, x+y
(5.127)
182 The two enantiopure catalysts MLRLR and MLsLs give identical reaction rates (rRR -- rss) and reaction products of opposite enantioselectivities (eeo and -eeo, respectively). A racemic product is formed from the m e s o catalyst MLRLs, which exhibits a reactivity of g relative to the enantiopure catalysts (g = k m , / k ~ ) . The enantioselectivity of the reaction product eeprod, will vary with eecat in the following way (5.128)
e e ,~,,d = e e o e e ca, - l + gfl
The parameters K and g give information about the relative distribution of ligands between the three complexes. Eq. (5.128) provides a description of the nonlinear effect in asymmetric catalysis (Figure 5.30). eepr ~ t0" (%)
8 6 ML2 model K=64
4,
0 ee ~= (%)
Figure 5.30. Dependence of the enantiomeric excess of the product on the ee of the chiral auxiliary (H.B. Kagan, SI (2001) 888).
Synlett
An interesting example of the reaction mechanisms with multiple cycles is coordination polymerization. The most important industrial coordination polymerization processes are Ziegler-Natta and metallocene catalyzed olefin polymerization processes (Figure 5.31).
Figure 5.31. Metallocene catalyzed olefin polymerization. The reaction requires as a first step the attachment of the monomer to the metal of the hydride complex typically at a free coordinative site as a ~ complex of the olefinic double bond. Insertion of the monomer between the metal and hydrogen is following after that, while
183 the coordinative site is released. That site accepts another monomer molecule. The chain can be terminated in several ways. The mechanism is depicted in Figure 5.32.
Initiation
+
+
--1
Propagation
v¢ Termination '~-~ ".~ '---'
~
•- H
+ []~,,t]-H
+ [M].-R
~ R
+ [l~-[-Z R
[ ~ t ] @ N R
+ D,f]--~,,~.,~
Figure 5.32. Polymerizationmechanism.
If the steps of monomer addition and insertion are lumped into one and termination is simplified than the reaction network is as presented in Figure 5.33. M
M
M
M
cat .~ Figure 5.33. Polymerization mechanism. M is the monomer, cat is the catalyst, P* is the metal-complex alkyls, while Pi are dead polymer molecules, produced in chain termination steps.
Another way to present this mechanism is 1.* + M ~
PI*
2. Pl* + M ~ P 2 *
chain propagation chain propagation
kp kp
184 3. P2* + M ~ P3*
chain propagation
kp
n. Pn_l*+ M ~ Pn* ti. Pi* ~ Pi +*
chain propagation chain termination
1% kterm
tn. Pn* ~ Pn +*
chain termination
kterm
(5.129)
.......................................
All growth steps from 1 to n are chemically the same therefore the rate constant of them are equal to each other. The rate coefficients of termination are assumed to be equal to each other as well. The steady-state approximation for propagating centers P*i+l gives kpC~ICP*•
~ = kpC~ICP*~J
i+l
-I-
k~,.r,,,Cp,,
(5.130)
from which the ratio between the concentration of adducts is ki, Q~ ~/(kpC~.s + ktc~,,) = Cp,+, / Ce,
(5.131)
This ratio is equal to the ratio of mole fractions of the respected polymer molecules, as the formation rates are expressed by
r~ -
dCe'+~ - k,~,,~Cp,+~ dt
(5.132)
Therefore Ce+~/Ce = k¢,CM/(kpQ~ + kt~,.m) = g
(5.133)
This ratio is independent of the number of monomer units in the polymer molecules and describes a declining geometrical progression, which is similar to the one obtained in freeradical polymerization with termination by disproportionation or terminating chain transfer. The progression factor g is constant if the monomer concentration is constant. A mole fraction distribution described by a declining geometrical progression is called a Schulz-Flory distribution
Paul J. Flory The mole fraction of polymer with i monomer units as a function of that number i (if the maximum number is i) can be expressed (the monomer concentration is included in the counting of polymers) as
185
i-1
Ce, g C~, xp, = Cp, + C ~ + C p,+Cp...+Cp, = C p , + g C ,p , + g 2C,p , + g C~p,, + . . . + g ,l~, Cp, l+g+g2
(5.134)
g i-1 +g3 +...+ g~
In eq. (5.134) C~, denotes concentration of P*1. For the case of the fraction of polymer with i monomer units if the number of molecular units is n one gets similar to (5.134) g xe'
J I
(5.135)
~ gn n=l
g" and ~ ' g " -
Taking into account that as g
n=l
n=l
g 1- g
from (5.135) the mole fraction of polymer with i monomer units is expressed as x~ = g, 2 (1 - g)
(i>2)
(5.136)
The weight of the polymer with i monomer units is iN, M W M where N i is the number of moles of that polymer and MW M is the molecular wieght of the repeating unit. The total weight of the reaction mixture is equal to the initial amount of monomer NoMW~¢, where N O is the initial molar amount of monomer. Then the weight fraction is
iNiMW ~ w,-
'
iN i -
N oMW,~
(5.137)
No
Although the mole distribution declines with increase of repeating units in the polymer (Figure 5.34), the molecular weight distributions have maxima (Figure 5.35). In the low-polymer range the weight increase with the number of repeating units overcompensates the decrease in mole fraction. 0.0040 0.0035
0,0030 ~ ,D.0025 0.0020
.~ 0.0015 0,0010
0.0005 0.000%
20
40
60
80
iO0
DegreeofPolymerization
Figure 5.34. Mole distribution as a function of the degree of polymerization at different g.
186 0,025
I
I
\
i
I
0,94
0,020
g
0,96
e 0.015 o
0.010 0,98
¢2r}
0.005
..... 0
20
40 60 Degree of Polymerization
80
I 00
Figure 5.35. Weight fraction as a function of the degree of polymerization at different g.
(MWM ), N, + (MWa,)2 N2 + (MV/M)3 N3 + (J~(J/V~/~)4N4 +"" + (MWM)~ N,, ix i IM ~ ]42,
ix i l
M
9
MW~4 + 2 gMWA,~ + 3 g - MWA,I + 4g3 MWA~ + ... + n g
n-1
• ~nrTr
M rG~
(5.139)
co
Zngn
(5.138)
1
n=l
Rearanging the denominator in (5.139) will give co
co
d
~ - 2 n g n , =--d 2 g , , + , _ d 2 g g , , n=l dg ,,=1 dg ,,=1
~
=TgZg ag n=l
,
=
d ~-g
g
g
(5.140)
1-g
and finally
,,,
~-2ng .<
-
d g2
_
@ 1- g
2g(1-g)-g2(-1) (1 --
,~0-)2
g(2-g)
(5.141)
(1 - g)2
and then igi I Wi m
_
igi I
_ igi
2(1_g)2 (i>2)
co
~-2n g~, ,
g(2-g)
n=l
(1
-
-
(5.142)
2-g
g)2
The average molecular weight is ~--~ng~, 1 ,,=1 N pol - -
co
Zgr, I n<
g(2-g) _
( 1 - g)2
g 1- g
_ 2-
g
1-g
(5.143)
187 Replacing g in eq. (5.143) with of (5.133) one arrives at Npol
2-g - - - 1-g
/(kl, CM +k,~,r,,,) '" = 2 + k p C v / k ' ...... = 2 + v 1-k¢,Q~e/(kpQ~e +k,,,,.,,)
2-kpCu
(5.144)
where v is the chain length. In polymerization reactions this value can reach 10 6, thus the average molecular weight is simply equal to the chain length. The steady-state approximation for catalyst concentration gives the equality of initiation and termination rates G = r t k MC,s~C p , =
k~,,,~ C,,,
(5.145)
where kM corresponds to the addition of the monomer to bare catalyst
Cj,o = C1,**,~,k M C v
(5.146)
/ k, .....
From the mass balance ~ Ce, ' + C e , ~"Ce, =
= Q,a, the concentration of propagating centers is
k ~ C H / k , ...... Q or k'~l Ca: Q.a, 1 + k MC~ / k~...... k M C M + k, ......
(5.147)
The polymerization rate is equal to consumption of the monomer dC M dt
km CM
(5.148)
When the rate constant of termination is sufficiently smaller compared to k ~ C v than the rate is first order in total catalyst concentration and first order in monomer.
189
Chapter 6. Enzymatic kinetics 6.1. Enzymatic catalysis We will consider the following reaction mechanism k+l )
E+S
(
ES
k, > E + P
(6.1)
kt
in which binding of the reactants called substrates to the active site forming the enzymesubstrate complex allows the substrates to react and to form the product.
Figure 6.1. illustration of an enzymatic reaction. The substrate is denoted by S , the enzyme by E, and the product of the reaction by P (Figure 6.1), and k+l , k_l, and k 2 are the reaction rates in the corresponding reaction steps. Michaelis and Menten assumed the formation of an enzyme-substrate complex ES complex is in rapid equilibrium with free enzyme and breakdown of ES to form products is assumed to be slower than 1) formation of ES and 2) breakdown of ES to re-form E and S. This assumption is equivalent to assumption of the quasi-equilibria for the first step. In fact in order to derive a kinetic expression tbr enzymatic kinetics this assumption is not needed.
Maude Menten
190 In the following derivation we will apply the concept of steady state approximation, which was introduced to enzymatic catalysis by Briggs and Haldane (1925), who had proposed that the rate of formation of ES = k~ [E][S] balances the rate of breakdown of the complex ES = (k_l + k2)[ES], or in other words (Figure 6.2) d ( E S ) / d t = 0
Time
Figure 6.2. Illustration of the steady-state concept in enzymatic catalysis From equality of the rates k I[g] [S] = k_l [ES] + k 2[gs]
(6.2)
after introducing the equation for the balance of enzyme [Ev] = [ES] + [E]
k, ([ET ]- [ES])[S] = (k., + k2)[ES]
(6.3)
and some rearangements [ESI(k_, + k 2 + k, [S]) = k, [ET][S ]
(6.4)
an expression tbr the concentration of the enzyme -substrate complex is obtained [ES]
k, [E~ ][S] kl+k2+k~[S]
[E T][S] k l+k2 t-[S] kl
(6.5)
Defining the Michaelis constant Km k l+k2 K., - - k,
(6.6)
we get
[ES] [Es][S] Km+ IS] Since the reaction rate of product formation (sometimes defined as V (velocity)) is
(6.7)
191
r =V-
dP
- k2[ES ]
(6.8)
dl
we finally arive at
r=k2[ES] k2[ET][S]- Vmax[S] Kin ÷ [ S ]
(6.9)
Kin + [ S ]
where
V .... =k2[ET]
(6.10)
In fact this equation can be derived directly from the general equation for the two step sequence (4.93), which in instance of irreversible second step is 1~
= Cca !
0)1002 -00
I032
kl[S]k2
k2[ET][S]
00'+002+00t+002=kl[S]+k2+kl[Ef]-k'+k2t-[S]
-
V"ax[S]
(6.11)
K,,+[S~
kl The concentration dependence of reaction rate shows two regions, the region of the first order dependence in substrate concentrations at low values of S (below Kin) (Figure 6.3) r = V -
Vlll~x[S] - Vma×[S] = K m + [S] K,,,
const*[S]
(6.12)
and zero order in substrate at high concentrations (Figure 6.3)
r -
V.~x[S] - [s]
V,,a~
= const
(6.13)
Vm~ j
....
~1ti* <=~ ES ~ E + P
KM
[S]
Figure 6.3. Dependence of the reaction rate on substrate concentration.
Km is a constant derived from the rate constants and is the ratio of the rates of decomposition and formation. Therefore it differs from equilibrium constants in a pure chemical sense, which are defined as the ratio of formation constants to decomposition ones
192 of a particular reaction (e.g. K=k+/k_). Small Km means tight binding; high Km COlTesponds to weak binding. In the original derivation of Michaelis and Menten it was assumed that the first step is rapid and reversible, thus this step can be considered as a quasi-equilibrium with a c o n s t a n t Kcq. Note that it is defined as the ratio of constant in the reverse direction to the constant in the forward one (or the ratio of reactants to products) following a special "bio"logic. Km
k, + k 2
.
.
.
k~
k I + k+~
.
.
.
k+~
k I
+
k+2
k+~ k+~
= Kl,eq
k+2
+ - -
k+t
(6.14)
If there are several enzyme-bound complexes during the reaction, then Km is considered as apparent dissociation constant Km=
[El[S] sum of all complexes [E - S]
(6.15)
If we set the rate at half of the maximal rate (r =V, lax/2) then it can be shown that Km = [S], i.e. the Michaelis constant is the substrate concentration when the rate is half of the Vmax. Km has the same units as [S]. Vmaxis the theoretical maximal velocity, although never achieved in reality, as it would require that all enzyme molecules are tightly bound with substrate. Vmaxis asymptotically approached as the concentration of substrate is increased. The ratio between the theoretical maximal rate and the total enzyme concentration is the turnover number which indicates how good a catalyst an enzyme is. k 2 - Vm~x - k~., ET
(6.16)
Turnover number is a measure of catalytic activity and is the number of substrate molecules converted to product per enzyme molecule per unit of time, when E is saturated with substrate. Values of koat range from less than 1/sec to many millions per sec. The catalytic e f f i c i e n c y kcat/Km is an apparent second-order rate constant and measures how the enzyme performs when S is low. It is sometimes useful to transform the Michaelis-Menten equation into a linear form. One of the best known is the Lineweaver-Burk plot.
Dean Burk
Hans Lineweaver
From the Michaelis-Menten equation taking the reciprocal values one gets 1
K.,
,"
E o,[s]
+
1
Vm ,
(6.17)
193 For enzymes that obey the Michaelis-Menten relationship, a plot of 1/r versus 1/[S] the socalled double-reciprocal plot yields a straight line, from which one can obtain more accurate values for the Michaelis constant, Kin, and the maximum velocity, Vmax(Figure 6.4).
I
~slope
r
KM
Vmax
!
\
1
1
V .....
lSl
KM Figure 6.4. Double-reciprocal plot for Michaelis-Menten kinetics. From eq. (6.17) it follows that at 1/[S]=0 the reciprocal value of rate is equal to 1/Vmax and at 1/r=O the value of abscissa is (-1/Km). Interestingly, when Lineweaver and Burk submitted a paper on the plot, the referees were opposed to its publication in J A C S for a variety of reasons. Some of the reviewers considered the paper "just a mathematical exercise and not really chemistry at all". The method is primarily used nowadays for initial examination of kinetic data and then the data are fitted to an appropriate rate equation using computer programs. Another common plot is that of G.Eadie and B.Hofstee. From the same Michaelis-Menten equation one gets F(Kln + [ S ] ) =
(6.18)
Vinax [S ]
Further rearrangement yields to r[Sl =V~.~x[S]-rK,,,
(6.19)
giving F
(6.20)
r = V,,,ax - - ~ i K,,,
This time the variables are r and r/S, so a plot of these two would give a straight line of slope Km and intercept Vm~x.The intercept on the horizontal axis would be V m J K m . (Figure 6.5)
ii,------------Vmax
Vmax
II r&. [Sl
Figure 6.5. Eadie-Hofstee plot for Michaelis-Menten kinetics.
194
For the Eadie-Hofstee as well as for the Lineweaver-Burk plots there is usually a scattering of the data, resulting in some errors for the values for Km and Vmax. In the Lineweaver-Burk plots, the weight of experimental values obtained at low concentrations of substrate is overemphasized, and those are the values which are associated with the largest experimental error, at the same time the Lineweaver-Burk plots have the advantage of separating the variables. The error is less severe tbr the Eadie-Hofstee plot than with the Lineweaver-Burk plot, thus this is generally regarded as being a better technique. Another problem with the Eadie-Hofstee plot is that the values plotted on the horizontal axis, which should be the independent data, are not truly independent as they contain an element of the velocity of the reaction. Any error which occurs in the experiment will be present on both axes. Another way of converting the Michaelis-Menten equation in the form of eq. (6.17) to give the equation of a straight line is the method of Hanes-Woolf
4,is]
Km
1 )*IS]
(6.21)
or
{s]_ Km - -
r
[S]
(6.22)
~
V,,,aX V,....
A plot of substrate concentration divided by velocity against substrate concentration will give a straight line with an intercept on the horizontal axis of-K,, and on the vertical axis of Km/Vmo~.The slope is 1/V,,~x(Figure 6.6.) $/r
// J
f
J
J
J
S~- Km0Vmax
Figure 6.6. Hanes-Woolfplot lbr Michaelis-Menten kinetics. This graph gives a direct readout of Kin. Calculation of V,~ax can be done from the Kin~Vmax intercept. The Hanes plot avoids the problem of the Eadie-Hofstee plot of having velocity (dependent) data influencing the data on the horizontal (independent) axis. Although computational methods are available for numerical treatment of enzymatic kinetics and estimation of parameters, still plotting methods are rather widesperad in enzymatic catalysis, while there are rather seldom used by researchers in heterogeneous catalytic kinetics. It is believed, that via plotting methods it is possible to recognized unexpected behavior and to better design experiments.
195 If reaction (2) in eq. (6.1) is reversible S+E+-~P+E then from the general equation for the two step sequence we get the expression for the reaction rate
kiSk 2 F=Cca l
O")1O)2 -- O') 1O") 2
=
0)1-}-0")2-[-0)1-]-0)2
kp~k2~k~lk2P klS+k2+kl+k2pt
[E ]=
rl
k ik 2P k2-}-kl [ET] k ~ - m ~ P ~-1
k2 -}- k 1
kl S k2 + k ~
(6.23)
k2 + k i
from which we arrive at (further we will also S instead o f [S], etc. to denote concentrations)
r =
(re,/KF)P
(v,/K,)S-
(6.24)
I+S/K~ +P/Kv
with
kI
k ~v, ;Kr, _ k~K,
.
(6.25)
Equation (6.24) is a general reversible form o f the Michaelis-Menten equation. When the reaction is at equilibrium the rate o f the forward reaction is equal to the rate o f the reverse and it therefore it follows from (6.24) (6.26)
(v,. / K,.)S o - (vr, / K~,)P~ = 0 Denoting the equilibrium constant o f the reaction as
Ke~~ = P~ / S~
(6.27)
we get the Haldane relationship
(v,/K,) (vp / Kp)
- K~v
(6.28)
which relates the kinetic parameters with the equilibrium constant o f the overal reactions, e.g. reaction thermodynamics. Quite often in enzymatic catalysis not only one substrate takes part in the reaction (isomerization or epimerization), but several reactants are involved, for instance transfer o f functional group from one substrate to the other (Figure 6.7). ATP + Glucose In general:
A + B
hexokinase ~
•
~
•
ADP + Glucose -6-phosphate P + Q
Figure 6.7. An example of a two substrates and two products (BiBi) reaction.
BiBi
196 Two types of reaction schemes are usually considered. Below we will use the notation of Cleland for denoting reactants and products. In sequential mechanism both substrates bind to the enzyme betbre any product is released. Such mechanism can be "ordered" with an obligatory order of addition of substrates and release of products. Substrate A binds E first, giving an EA complex. Then B binds to EA, giving a "ternary" complex (EAB). A and B are converted to products P and Q on (EAB), giving an (EPQ) complex. P dissociates first from (EPQ), then Q from EQ, regenerating E. A
B
.[
E
EA
'+TERNARY
P
l_+°2 t (EAB)
(EPQ)
Q
t
EQ
E
Figure. 6.8. Sequential "ordered" mechanism for bisubstrate reactions. Another possibility is the random sequentiual reaction with no obligatory order of binding and release (Figure 6.9). a B P Q
+
A
Q
P
Figure. 6.9. Random sequential mechanism for bisubstrate reactions. In a ping-pong mechanism (Figure 6.10), a product is released before all substrates are bound, and a functional group can transiently become covalently bound to E
AN " \ V / / ~ P IE
(EA-FP)
B~ / ~ F
(FB-EQ)
Q E
Figure. 6.10. Ping-pong mechanism for bisubstrate reactions. First substrate A binds to E (forming EA first), and a functional group is transferred from A to E giving a complex of modified enzyme "F" and first product, P ("FP"). P dissociates from FP, leaving free F. Second substrate B binds F (forming complex FB), and F's functional group is transferred from F to B to tbrm second product Q, with reversion of F to E. The resulting complex EQ dissociates, releasing Q and free E. Previously, while discussing the general theory of complex reactions, we have considered some other mechanisms with lineal+ steps, such as one given by eq. (4.107) corresponding to three-step sequence or eq. (4.116). in a similar way kinetic expressions could be derived for more complicated reaction networks, as presented for instance in Chapter 5 (see equations 5.76 for 4 step sequence, eq. 5.84 for 6 steps eq. 5.88 and 5.89 for a mechanism with 8 linear steps and the general form for n-step mechanism eq. 5.94). Ordered sequential bisubstrate reactions can be expressed by eq. 5.76 for the 4 step sequence (Figure 6.11)
197
A
E
B
P
Q
I I ' I EA
X
EQ
E
Figure 6.11. Four step sequence in bisubstrate reactions. ((0,(0_~(0/°4
F
(0 ,(0 ~(0 3(0 4)E7 '
(6.29)
(02(03(/) 4 -F (0-1(03 (04 -- (0-1(0-2 (04 -F (0-1(0-2 (0-3 + (01(02(03 -t- (0-4(02 (03 -t- (0-4(0-1 (03 -t- (0-4(0-1 (0-1 -F
+ (5040)1(02 + O)-30)1 (502 + 0)-30)-4 (502 + (0-3(0-4 (O-1 + (O-3('O4(O1+ ('0-2('04 (01 + (0-20)-3 O)1 + (0-20)-3 0)-4
with 0), = k , A ,
0), = k 2 B , 0 ) 3 =k3,(.o 4 = k4,0)_ 1 = k_,,0)_ 2 =k_2, 09 3 = k 3P 0)_4 = k _ 4 Q
(6.30)
Substituting values of step frequences in (6.29) we get (kAB - k PQ)E~
F= C O+
(6.31)
CAA + CBB + CA~AB+ CpP + CQQ + C~,QPQ+ CA~AP+ CBQBQ + CABpABP+ C~BQABQ
A special case of the ternary complex is a situation when the first two steps in the reaction SI+S2=P are considered to be "quasi-equilibria" (Figure 6.12). In the example below we will apply the notation of Dalziel to denote reactants and products. S1 E
S~
~ _ _
ES l ~ _ _
K1
K2
k ES18 2
~E + P
F i g u r e 6.12. T h r e e step s e q u e n c e with t w o first q u a s i - e q u i l i b r i a steps.
The reaction rate is given by r = kES~S 2
(6.32)
From the conservation equation Ev = E + ES + ES1S 2
(6.33)
and expressions for quasi-equilibria of the first two steps E S 1 _ E * S I ;ESv~2 _ E * S I * S 2 Ki KIK2
the total enzyme concentration can be defined via the concentration of free enzyme
(6.34)
198
E~, = E(1 +
S t + S 1 *S 2 ) K1 Kl K2
(6.35)
leading to an expression for the reaction rate
kETS1* $ 2 / ( K 1 K 2 )
r =
(6.36)
(1 + S1/K 1 + S 1 * S 2/(K1K2) ) Eq. (6.36) can also be derived from the general equation for the three-step sequence. In the treatment above, the quasi-equilibrium steps were defined as the ratio of reactants to the ratio of products following the "bio"logic to have an analogy with the Michaelis Menten constant. In the Lineweaver-Burk coordinates, eq. (6.36) becomes 1
1
1 ~K )
r - kE r +-~2(-~r
K1K 2 + kETS1 )
(6.37)
Plotting the reciprocal velocity as a function of the reciprocal concentration of $2 and different values of S1 will result in slopes dependent on S1 (Figure 6.13).
Ino~azing S1 Y
b'S2
Figure. 6.13. Double reciprocal plot for sequential mechanism.
In the general case of a ping-pong mechanism (Figure 6.14) the theory of complex reactions could lead directly to the rate equation (6.29) with gO1 = N1S1, O)l=k2,(03=k3S2,0)4=k4,(Ol = k l , 0 )
SI
E
' ES I , 1
' X 2
82
X
, XS 2 , 3
,
E +P
4
Figure 6.14. The general form o f the ping-pong mechanism.
2 =k
2,
o) 3 = k 3 0 ) 4 = k 4
P
(6.38)
199 Special literature on enzymatic kinetics contains different cases of the ping-pong mechanism as well as an ordered sequential mechanism for bi-substrate and tri-substrate cases which can in fact be rather easily derived from eq. (6.29). For the special case when binding of the substrates to either the free form of enzyme or the complex X is quasi-equilibrated (Figure 6.15) the reaction rate is (6.39)
r = kES 1 $1
kl ES t ~
X
KI 82
k2
X
' XS 2
E +P
K2
Figure 6.15. Ping-pong mechanism with quasi-equilibria binding steps and irreversible reaction steps. The balance equation relates the total concentration of enzyme with its different forms (6.40)
E 7 = E + ES~ + X + X S 2
From the steady state approximation the rate of two nonequilibrium steps is equal to each other q=r 2 k 1ES 1
(6.41) =
k 2X S 2
From equilibria the tbllowing relationships hold E *S1
ES l = --;XS
Kl
X *82
2 = --;XS K2
2= E
k~ '~'t
;X = E
k2 K1
K2 kl
$1
(6.42)
$2 k2 Kl
thus leading to
u =eo+s, +k, s, +x2 k, s, Ki
k2 KI
(6.43)
$2 k2 Ki )
and finally klErS1
K~ (1 +
Slq_klS1 K2klS1 q ) K 1 k2 K l $2 k2 K1
A plot of 1/r versus 1/[Sl] (or 1/[$2] )
klEr
1+
~
(6.44)
k1
~b'l + k 2
+ (1
K2) S2
200
1-
-- 1
r
k 1E T
+
1 - -K,
~-k~- (1 +
S 1 k 1E r
_
K2) 1 S2
(6.45)
k 1E r
yields a straight line, with the slope independent on the concentration o f the other substrate (Figure 6.16). -...
Figure. 6.16. Double reciprocal plot for ping-pong mechanism. In case o f the r a n d o m sequential m e c h a n i s m (Figure 6.17) there are two routes leading to the ternary c o m p l e x . S~
E
S2
, ES I kl
, ES~S2
rE +P
k2
S2
E
k,
Sl
• ES2 k3
k~
• ES1S2
~-E +P
k4
Figure. 6.17. Random sequential mechanism. The m e c h a n i s m can be presented also in the following f o r m
1. E + S I ~ E S 1 2. E S l + S2~ES1S2 3. E+S2=>ES2 4. ES2+ S ~ E S ~ S 2 5. ES~ S_2_~E+P N °), N (2) $1 + $ 2 ~ P
N (1)
N (2)
1 1 0 0 1
0 0 1 1 1
The reaction rate is given by the rate o f step 5, w h i c h is the s u m o f the rates for the routes N (~) and N (2) r = k5ESIS 2
(6.46)
201 The steady state approximation gives respectively for the routes N (1) and N (2) r1 =r2;
(6.47)
r3 =r 4
and expressions for the concentration o f complexes ES 1 z
k- I E- * S 1 "~E S 2 k2S2
~
k 3E *7 S 2 . k4Sl
(6.48)
-
From the theory o f complex reactions we can relate the rates o f the two routes r = r 5 = r x' + r -~'2 = r
(6.49)
I +~
which gives an expression for the concentration o f ESiS2 ksES1S 2 = k 1 E
(6.50)
* S 1 -}- k 3 E * S 2
(6.51)
E S 1 S 2 - kt St + k3 $2 E
k5
From the balance equation E r = E(1 + -k, -E *+S,
k- 3 E- *+S 2
k, S,
+ k 3S 2 )
(6.52)
k5
k4Si
k2S2
and equations for the reaction rate (6.46) and (6.51) finally we get kIS I +
r = (14
k1 E * S 1
k 3$2
Er
(6.53)
k 3 E * S 2 k 1S t + k 3S 2) F - F k4S1 k5
k2S2
In some cases to be efficient the enzyme requires a cofactor (Figure 6.18) as discussed in more detail in chapter 2. apoen~Tme(protein) A+C ,
AC enzyme
cofactor
II +S P+free enzyme
,
substrate
ACS enzyme.~substratecomplex Figure. 6.18. Mechanism with a cofactor.
202 Despite the chemical specificity of this type of enzymatic catalysis, a rate equation can be derived in a similar manner as before. In the general case for all the steps being reversible one should use a general form of a three-step sequence, however the mechanism could be essentially simplified if the first two steps are considered to be in quasi-equlibria and the last one irreversible. The reaction rate is (6.54)
r =kACS
From the equilibrium the following relationships follow AC -
A*C
AC*S . . Ks
;ACS .
Kc
A*C*S . Ks Kc
(6.55)
Taking into account the balance equation A~. : A(I+ C + C * S , K,,
(6.56)
K sK, )
the rate is k r =
*Ar*S*C
K~,K(,
k'A, t * S*C
" = C (__7* S (KsK c + CK s + C * S) (1+ + ) K(:
(6.57)
KsK(:
A plot of 1/r versus 1/[S] gives a straight line 1
- :
r
( K s K c + CK~.) 1 ,
kA r * C
S
1
+ - -
(6.58)
kA r
with an intercept independent of the cofactor concentration, while the slope shows a more complex behavior. At low concentration of cofactor it is inversely proportional to cofactor concentration, however, when the concentration of cofactor increases, the slope starts to be independent of the cofactor concentration.
6.2. Cooperative kinetics Not all enzymes obey Michaelis-Menten kinetics. The experimental dependence of the reaction rate on substrate concentration is different from Michaelis-Menten type of concentration - substrate curve and follows a sigmoidal, or S-shaped, curve. Binding of oxygen to myoglobin(Mb)/hemoglobin(Hb) can serve as an example. The oxygen binding curves for Hb and Mb are significantly different. In contrast to Mb (Michaelis-Menten kinetics), Hb binds 02 less tightly and displays so-called co-operativity (i.e. binding of one molecule of 02 increases the affinity for subsequent 02 binding) (Figure 6.19).
203
Lungs
Tissue
1.0
I..I .............Y
Hb
r
B,
0.0013 0.026 0.13 Oxygen partial pressure (atm)
Figure 6.19. Binding of oxygen to myoglobin/hemoglobin.
The key element of this curve that distinguishes it from a hyperbolic graph is the "toe" at low substrate levels (Figure 6.20). 12 10
8 U i0
.........................
iiiiiiiT~iiii~iiiiii
6
iiiiliiieu~iiiililil
I
I
I
I
2
4
6
8
10
[Substrate]
Figure 6.20. An example of cooperative kinetics with a toe at low substrate levels.
At this point an increase in the concentration of substrate causes only a very small increase in rate - the graph has a very shallow slope. At slightly higher levels of substrate, the increases in substrate begin to bring about a much more dramatic increase in rate and the graph becomes much steeper. At high substrate levels the graph is very similar to a hyperbolic plot and it flattening out towards the maximal rate in the usual way. The empirical Hill equation V Ah
r .... K~;5 + A;'
(6.59)
where A is the substrate concentration, and Ko.5 is constant was proposed to be capable of explaining cooperative behavior. At h=l the curve is a typical hyperbolic plot, while at h=2 and 4 lines are clearly sigmoidal, more so in the case of the h - 4 line. In other words these are typical plots showing positive substrate cooperativity, with cooperativity increasing as the Hill coefficient rises. The logarithmic form of this graph can be used to obtain the values ofh.
lnl r---L--l=hlnA-hlnK~'5 LV,,,
(6.60)
204
In tact this graph will usually deviate from a straight line at very low and very high substrate concentrations. The central portion of the plot, which should be linear, is used for determination of h and K~'5 (Figure 6.21).
lag
Figure 6.21. Logariphmic form (eq. 6.60) of the Hill equation (6.59).
Several mechanistic explanations were proposed to explain deviations from Michaelis Menten kinetics. Fo example for the synthesis of pyruvate the binding of one molecule of phosphoenol pyruvate (PEP) to pyruvate kinase increases the affinity with which pyruvate kinase binds subsequent molecules of phosphoenol pyruvate (Figure 6.22). ATP
-~AAT %i_-°o
C--O~p.O o a2e'S
OH3
O=
phosphoenol pyruvate
?
re
j
%o,"0.
p pyruvate
re
ADP
2
4
6
8
[PEP]
Figure 6.22. Synthesis ofpyruvate catalyzed by pymvate kinase.
In general for cooperative enzymes, the binding of one substrate increases the affinity of the enzyme for other substrates K4
[Es3 ][sl K4-[ES4
]
X Figure 6.23. Binding in cooperative enzymes.
[El[s],x
-[Es]
[~sl[s] x~ -
~ - [Es~]'
-
[Ex2I[s]
[Es~]'
205
In cooperative enzymes KI>K2>Ks>K4, since the binding of substrate increases the affinity of the enzyme for binding additional molecules of substrate. In noncooperative enzymes KI-K:-Ks-K4. in the literature two types of effects leading to deviations from ideal kinetics are treated separately: cooperative effects per se, e.g. changes in affinity or activity of an enzyme with changes in the concentration and allosteric effects, binding of the substrate at a site other then the active site affects affinity or binding, by stabilizing the high affinity (R) state of the other active site. In positive substrate cooperativity, substrate binding is "cooperative" in a sense that the binding of the first substrate at the first active site stimulates the active shape, and promotes binding of the second substrate (Figure 6.24) T
state active
T state active site
I I
Figure 6.24. Positivecooperativity The concerted hypothesis advanced by Monod, Wyman and Changeux is based on the assumption that one enzyme molecule could contain only one type of subunit - R or T.
Jacques Monod
Jean-Pierre Changeux
An allosteric enzyme was supposed to consist of a number of subunits which can exist in two different conformations, or three dimensional structures. These are usually referred to as the relaxed, or R-state, and the tense, or T-state (Figure 6.25). .................................................................................
....................................................................................
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iiiilililililil :ili~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!~i~ili
Figure 6.25. Illustrationof R- and Y- states. The subunits within a particular enzyme molecule interact in such a way they must all have the same conformation. In other words a molecule of enzyme must consist purely of R subunits or T subunits, it cannot contain a mixture of the two. In solution the two forms are able to convert from one to another and so will exist in equilibrium. Since a mixture of types of an individual enzyme is not possible, it will be a very simple equilibrium between enzymes consisting purely of R subunits and ones purely made of T subunits.
206 If a small amount of substrate is added to this solution it is more likely to bind to the R enzyme as this has a higher affinity. This would form a complex in which an R-state enzyme had one substrate molecule bound and this enzyme form would be added to the existing equilibrium That combination of substrate with R-state enzyme removes some free R enzyme from the medium. This would pull the R/T equilibrium to the left (Figure 6.26). As a result of this there would be more R enzyme and less T enzyme in the solution. Since R is the high affinity form the overall enzyme/substrate affinity is increased. The result is positive substrate cooperativity the binding of one substrate increases the ability of the enzyme to bind other substrates. -
~i
iF~
i!
ii.........~
'~qiiililililililililililililiRiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiill lili~ql liliBlili~lili~lil'ilililililililililililililili~'
Figure 6.26. Equilibrium between R-S, R- and T- states. The concerted model kinetic model Monod, Wyman and Changeux makes a number of assumptions. The enzyme can exist in 2 states: Tense (T) and Relaxed (R), the first one Tstate has low affinity for the reactant (KT large), while R-state has high affinity (KR small). As mentioned above all the subunits of any one molecule are either in the T-state or the R-state. In the following treatment we will consider an enzyme with 4 subunits (Figure 6.27)
s:t l
_: T
s Binding
s
Tt o R conversion
Figure 6.27. Concerted model with four subunits.
207 Here S is substrate, KT is the dissociation constant for the T-state, KR is the dissociation constant for the R-state and L is the equilibrium constant for the conversion of T to R, e.g. L = [T01/[R0]; LI = [Td/[R~I, L2 = [T2]/[R21, etc.
(6.61)
For the binding of S to R0 (Figure 6.28), e.g S+R0*-->R1
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~<--~R2
KR1L
Figure 6.29. Binding of S to R1
In the equation for the constant KR 3 is in the numerator (because there are 3 free binding sites on R 0 and 2 on the denominator (because there are 2 occupied sites on R2, either of which might dissociate to yield R0: 3[RI][S]
2[R21
(6.63)
The binding equilibria involving R3 and R4 (S+R2<---~R3 and S+R3<--->e4) in the same way give respectively
2[e2][s] 3[R ]
(6.64)
208
[R3][S] K~ = 4[R4]
(6.65)
Exactly the same approach is used to analyse the binding equilibria involving S and T-states K~
4[T°][S] 3[T~][S] 2[T2][S] [T3][S] [T~] - 2[~] - 3[T~] - 4[T41
-
(6.66)
Defining Y as the fraction of occupied sites Y=
number of occupied sites total number of sites
(6.67)
one can obtain
y=
[R,]+2[R~l+3[R3]+4[R4]+[T,l+2[T~l+317~l+4[T41
4([e01+ [& ]+ [&~]+ [R31+[e4]+ [T0]+[~ ]+ [r2]+ [T~]+ [T4])
(6.68)
Substituting Ri and Ti in this expression and applying the following definitions
H ; L - it0 ] 1% [Ro]
a =~
-
K~
(6.69)
where L is the allosteric constant, c is the ratio of dissociation constants, c<
(6.70)
(1 + a) 4 + L(1 + ca) 4
which gives the following dependence, closely matching experimental results (Figure 6.3 0)
/--
Y 1,0
~. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Z
Figure 6.30. Dependence of the fraction of occupied sites as a function of a = [S]/K~ (eq. 6.70) In a similar way as for the tetrameric enzyme it can be demonstrated that in a general case with n-subunits eq. (6.70) will take a form
209 y = a(1 + a) n 1 + Lca(1 + ca) ~ 1
(6.71)
(1 + a) '~ + L(1 + ca)" Eq. (6.71) can be written in another form y = a(1 + a)" ~ + Lca(1 + ca) '~ 1
a(1 + a)" ~
ca
z
(1 + a)" + L(1 + ca)" (1 + a)"
a
l- -
(l+a)"+L(l+ca) ca
~
,
L(1 + ca) ~
-
l + ca
(1 + a)" + L(1 + ca)"
(6.72)
(1 + a)"
= (1 + a)" + L(1 + ca)" (1 - +- a) + 1 + ca (1 - (1 + a) ~' + L(1 + caT' ) and finally a
-
ca
F= R (l+a) +(1-R)l+ca
(6.73)
with -
R =
(1 + a )
(6.74)
(1 + a) n + L(1 + ca)"
At low values o f a (low substrate concentration) and sufficient equilibrium constant R ~ 0, since (1 + a) '~ << L(1 + ca)"
(6.75)
giving an expression for Y y~
ca
[S]
(6.76)
l+c.- [sl+K,
which is a hyperbolic equation describing low-affinity binding (Figure 6.31)
(z
Figure 6.31 Dependence of the fraction of occupied sites as a function of a (low affinity binding) At high substrate concentration (a >> 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
Kee,~K~,KsS ° + KeaeK 2r K 2s S o2 2 9 2 1 + 2Kle~K~,K,,S o + K ~ K r K - s S o
(6.81)
which is similar to an equation derived from the concerted model. There is however a difference between these concepts. For instance, negative substrate cooperativity which is not very common but still occurs in some enzymes cannot be explained by the symmetry hypothesis which depends purely on the law of mass action pulling the R:T equilibrium towards the high affinity side. The sequential hypothesis explains it by assuming that the interactions between the subunits are such that conversion of one of them to the R-~brm by induced fit makes it more difficult for the others to change. More detailed analysis of the difference between symmetry and sequential models could be found in specialized literature (V.Leskovac, C o m p r e h e n s i v e e n z y m e kinetics, Kluwer Academic/Plenum publishers, New York, 2003).
212
6.3. Inhibition
The rate of enzyme catalyzed reactions is also affected by the presence of inhibitors, which are compounds that lower the enzyme efficiency. Inhibitors do not necessarily destroy all of the enzymes activity. The inhibitors can be divided into two groups on the basis of the kind of interaction, which occurs between the enzyme and the inhibitor. Covalent inhibitors are often referred to as irreversible inhibitors or known as inactivators and react covalently with the enzyme to lower its efficiency. The second case are reversible or noncolvalent inhibitors. If an inhibitor binds reversibly at the same site as the substrate, the inhibition is referred to as competitive. In other words for competitive inhibition - inhibitor (I) binds only to E, not to the enzyme substrate complex ES. For noncompetitive inhibition - inhibitor (I) binds either to E and/or to ES. A further type of reversible inhibition, uncompetitive, occurs when the inhibitor binds only to the complex enzyme-substrate ES and not to the free enzyme. This is a very rare case and sometimes is even referred to as a hypothetical case. Competitive inhibitors bind at the substrate binding site, i.e. they compete with the substrate for the active site. In pure competitive inhibition, the inhibitor is assumed to bind to the free enzyme but not to the enzyme-substrate (ES) complex. The enzyme is unable to bind both S and I at the same time and in competitive inhibition, the enzyme-inhibitor complex EI does not react with substrate S. Competitive inhibitors often resemble substrate structurally. As an example we can mention malonate, which is an inhibitor for dehydrogenation of succinate of a enzyme succinic dehydrogenase and resembles the structure of succinate (Figure 6.35) S uccinic Dehydrogenase
,-FAL C02
~ H2 cot
malonate
succinate
j
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E-FADH2
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fumaratc
Figure 6.35. Illustration of competitive inhibition Competitive inhibitors compete with substrate for active sites and while in the active site, prevent binding of the substrate, therefore as they combine with E to ibrm a complex, competitive inhibitors do not lead to catalysis. Combination of E + l lowers the efficiency of the enzyme and such a case is similar to what was discussed previously as a reaction cycle with a hanging vertex. Competitive inhibition can be analyzed quantitatively by steady-state kinetics (Figure 6.36). 1
E + S
~
2 ES
> E+P
El
Figure 6.36. Scheme of competitive inhibition
213 In Figure 6.36 K~ is the dissociation constant for the EI complex. An equation for the linear Christiansen sequence with an extra cycle was presented before (equation 5.109) and scheme 5.110 for a similar case
klk2S
r =
k,S + (k 2 + k
I
E~.
(6.82)
l)(l "l- ~/-/)
which after some simplifications leads to r =
-
[S]+K (1+ [II)[S]+ K,,,<,~
(6.83)
Ks with KI-[E][I]/[EI] and K,,,is given by eq. 6.14. If one raises the concentration of the substrate high enough the effect of the inhibitor will be overcome. That is, the Vmax can still be reached in spite of the presence of the inhibitor (i.e. Vm~xis unchanged) (Figure 6.37). : Inhibitor
itor
.-~ ~
-
7',
n-
i
I I
[Substrate]
Figure 6.37. Dependence of the reaction rate on substrate concentration for competitive inhibition. On the other hand, the K,,, is changed and apparently increased to K,,,(1 + [1]/Kr), while Vmax is not changed. It is characteristic of competitive inhibitors that they cause an alteration of the K,, with Vmaxunchanged, e.g. Vmax,app=Vmax , but Kmax,app >Km. The apparent increase in Km is due to a distribution of enzyme between the ES and El complexes. Adding inhibitor shifts the reaction towards the left (formation of the EI complex decreases the amount of ES complex present, thus Kmax,ap > Km). There are several graphical methods for detecting and analyzing competitive inhibition. The Lineweaver-Burk and Eadie-Hofstee equations can be modified to include a term that describes the inhibition by I. The Lineweaver-Burk equation for competitive inhibition takes the ibrm
1_ 1~
K,.
[/]
1
(6.84)
Vmax[S~(lq-KI)q-Vma~
and the double reciprocal plot shows that the slope depends on the inhibitor concentration (increasing with an increase of [I]), while the intercept is the same as the case for no inhibitor (Figure 6.38).
214
1
7-
Vmax 1
K~,~pp
[S]
Figure 6.38. Double reciprocal plot for competitive inhibition. The Dixon plot relates the reciprocal value of the rate and inhibitor concentration and follows directly from eq. (6.84). Analysis of this plot reveals that the higher the substrate concentration is the lower the slope is. The y-intersept is also lower at higher substrate concentrations (Figure 6.39).
l/r
ingis]
slop~~~l
(1+I~rjS)/Vmax ~"0 -
KI
l/Vmax
[I]
Figure 6.39. Dixon plot for competitive inhibition (1/rvs [1]). The Eadie-Hofstee equation for competitive inhibition takes the form r
r : V , .... - [-~K,,,(1 + Kz )
(6.85)
From this equation it can be seen that if a competitive inhibitor is present, the slope of the Eadie-Hofstee plot is Km(I+[I]/KI), and not any more Km as in the case without inhibitors. The slope increases as [I] increases. The y-intercept is %,1,× and is unchanged tbr competitive inhibition. The Hanes-Woolf equation can be easily derived from (6.83) leading to [ S ] _ K ( 1 + [ I ] ) + [S] r V,.... K~ V,....
(6.86)
The slope is the same as in the case when no inhibitors are present, while the intersept increases with an increase of concentration of inhibitor.
215 Plotting
[S]/r
as a function of inhibitor concentration leads to Comish-Bowden plots. The
slope of this plot is K., /V,,1.xKz and is thus independent of the inhibitor concentration, while the y-intercept
is
([S]+K.,)/V,n.x
and
increases
with an
increase
in the
substrate
concentration.
Athel J. Cornish-Bowden If a reversible inhibitor can bind to the enzyme at a site that is distinct from the active site, it is described as a "noncompetitive inhibitor" (Figure 6.40).
Figure 6.40. Illustration of noncompetitive inhibition. In pure noncompetitive inhibition, the inhibitor binds with equal affinity to the free enzyme and to the enzyme-substrate (ES) complex. In noncompetitive inhibition, the enzymeinhibitor-substrate complex IES does not react to give product P. A kinetic scheme for noncompetitive inhibition is given in Figure 6.41
E +S~ IIKI EI ~
[ES]
• P+E
IIKI [ESI]
Figure 6.41. Scheme of noncompetitive inhibition. The mechanism can be presented also in the following fbrm NO} N (2) E+S-ES 0 1 E+I-E1 1 0 EI+S-ESI 1 0 ESI ---ES+I 1 0 ES~E+P 1 1 N (~), N(2): S ~ P
(6.87)
216
The reaction rate is given by the rate of the last step r =
k[ES]. From
the quasi-equilibria
K, = [EI[II/[EII, K,,, = [EI[Sl/[ESl the concentration of [ESI] complex is
[Esi]--EE]Ez]/K,K,,
(6.88)
Taking into account the balance equation
[ESI]+ [E] +[Eq+ [ES] ET
(6.89)
the reaction rate can be easily derived
(k2/Km)[S]E T r=
1+
ts]
+
K,,
[zl
+
Kt
(6.90)
[slizl KmKI
which can be written in the form
.......,,Es]
(6.91)
r_ [s]+ K,,, where
IT1 v, ..... ,,,, = v, .... /(1+ ' - ' )
(6.92)
K1
'
Here KI is the dissociation constant which is the same for the EI complex and the IES complex. Comparison of eqs. (6.91) and (6.92) with Michaelis-Menten equation reveals that the l+[I]/K~ term is now in the denominator, therefore increasing [I] decreases the Vmax,app. It can be concluded, that noncompetitive inhibitors decrease the Vmax, app, but does not affect the Kin. (Figure 6.42).
- inhibitor
Vmax n-
"~ ¢"
0
Vmax,app ½Vmax
Vmax,app
Km
Km,app
[Substrate]
Figure 6.42. Dependence of the reaction rate on substrate concentration for noncompetitive inhibition.
217 The reason is that the inhibitor binds equally well to free enzyme and the ES complex without altering affinity of the enzyme for the substrate. The Lineweaver-Burk equation for noncompetitive inhibition 1_
i-
~]
K,,
V~nax[S](1 +
)
+ 1 (l+~j]) VIIlaX
(6.93)
can be presented in the graphical lbrm (Figure 6.43) 1/r I
.
- Vmaxj~pp Vme~×,app Figure 6.43. Double reciprocalplot for noncompetitiveinhibition. The x-intersept (at 1/r=O) gives [S]=-Km, thus all the plots are coming to a single point. The slope is increasing with an increase in inhibitor concentration, the same holds for the yintersept. Analysis of eq. (6.91) in Dixon coordinates (reciprocal of rate vs inhibitor concentration) shows for noncompetitive inhibition, that with an increase in subsU'ate concentration the slope is decreasing. The same is valid for the y- intersept (Figure 6.44). lk
slope=(I+Km/[S])V~xKI
/
~-KI
~ing'
' IS] ( l + Km/S )/Vmax
[I]
Figure 6.44. Dixon plot (1/r vs. [1] ) for noncompetitiveinhibition. The Eadie-Hofstee equation for noncompetitive inhibition:
Vmax _ r K (1 + [S] "
(6.94)
218 demonstrates that the slope is -Kin, and does not change for non-competitive inhibition; and the y-intercept is Vmaxand is decreasing for non-competitive inhibition with an increase of [I]. In the Hanes- Woolf plot for noncompetitive inhibition [S]_ K., ( 1 + [ I ] ) + [ S ] ( l + [ I ] ) r V,.... Kr V,nax Kr
(6.95)
Km is unchanged for noncompetitive inhibition so the lines pass through the same xintercept. The Y-intercept and slope increase due to the reciprocal dependence on Vmax, which decreases for non-competitive inhibition. If the same equation (6.95) is replotted in the Cornish-Bowden coordinates the slope increases with an increase in the substrate concentration Contrary to other types of inhibition uncompetitive inhibitors bind only to the ES complex and cause alterations in both the Km and the V.,ax (Figure 6.45)
Figure 6.45. Illustrationof uncompetitiveinhibition. The reaction rate for this type of inhibition can be easily derived from the approach discussed in Chapter 5 devoted to homogeneous catalysis taking into account the hanging vertex (Figure 6.46). 1 E + S<
2
@'
> ES ~
> E+P
IES
Figure 6.46. Schemeof uncompetitiveinhibition. Kinetically speaking this case is very similar to competitive inhibition
r= s
or
<11 x[s] l+[I] K)+<
(6.96)
219
(6.97)
r _ [s]+
where V ,
_
Km
m_~x
"K
(1 + ~+a_)
-
(6.98)
]-i]--
(1 +
Uncompetitive inhibitors decrease both the Vmax,avp and the Km,app at low substrate concentrations, and Vm,x, app
Vmax
o n~ 0C k:
O
.
Vmax'app 1
+ inhibitor
T v~
TVm~,app
Kr~,~p
l'Kr~
[Substratel
Figure 6.47. Dependence of the reaction rate on substrate concentration for uncompetitive inhibition.
The Lineweaver-Burk plot is a diagnostic for uncompetitive inhibition. The slope of the lines is independent on the concentration of inhibitor, while the y-intersept increases with an increase of [I] (Figure 6.48).
glax
~°i ~,~pp
1/[S]
Figure 6.48. Double reciprocal plot tbr noncompetitive inhibition.
In a similar fashion as for other types of inhibition the Hanes-Woolf and the Eadie Hofstee equations for uncompetitive inhibition could be used respectively for analysis if this type of inhibition is present
220
_ K,,,
[s] (1 + [ / ] )
(6.99)
r
r -
Vmax ~ (1 + KI
r
K
"~/)
(6.100)
[S] (1 +
An interesting case of inhibition is uncompetitive inhibition of the susbtrate itself (Figure 6.49)
E +
"-
-~
ES
__at> E + P
s ES2
Figure 6.49. Scheme of the inhibition by substrate. The reaction rate is expressed by
r =
kK[1Eo S 1 + K[1S + KI1K21S 2
(6.101)
where the equlibrium constants are defined as the ratio of decomposition to formation. According to (6.101) the reaction rate passes through a maximum with an increase in the substrate concentration.
6.4. Effects of pH The essential catalytic groups in the active-site are often ionizable and frequently act as acid, base, nucleophile or electrophile catalysts, and thus are only functional in one of their ionization states. The activity of enzymes can therefore have a very strong dependence on pH, leading to sigmoidal shape or bell shape ((Figure 6.50) behavior.
/ 5
7
kcat IKrn
pH
10
5
7
pH
9
10
Figure 6.50. Dependence of the rate constant on pH : a) sigmodial, b) bell-shape "behavior.
221 We will consider the following reaction scheme (Figure 6.51), which accounts for the pH dependence of enzyme action E-+H +
K2
EH
, H+
KI
kl
S '
EHS
.
E +P
I~tn
EH2 + Figure 6.51. Mechanism of enzyme action with involvement of H .
The scheme is similar to the case of inhibition, therefore the final equation should also resemble the one of the many given cases when an inhibitor is present in the reaction mixture. The equilibrium constants in the spirit of Michaelis-Menten are defined as
,,-
[EHS l K, = [EH21
K2-
[EH]
(6.102)
From the balance equation (6.103) and the reaction rate
r=k,[EHS]
(6.104)
we arrive at r=
(k,/Km)[S]ET 1+ [ ~ ; ] +
(6.105)
+ K,,[S]
or in another form V',........,,[S] r - [S]+X'm,,~ with V'ma~ k f r a n d
(6.106)
222
K' m , a l ) l ) -- - K '
1+
tpl
~-
(6.107)
It follows from (6.105) that for zero order kinetics the rate is proportional to r=klET and there is no dependence on pH. At the same time for the first order kinetics at low concentration of H + the value of the apparent Michaelis-Menten constant is high leading to lower values of kl/Km,app = =kl/(Km(l+K2/[H+])), while at high concentration of H + the value of apparent Michaelis-Menten constant is high once again Km,app=Km(l+[H+]/K1), leading to lower values of kl/Km,app, which means that the kinetics follows a bell-shape behavior.
6.5. Heterogeneous systems/immobilized enzymes So far we were considering reaction systems where enzymes and the substrate are in the same phase. One can imagine a situation where an enzyme in solution reacts with an insoluble compound by adsorbing onto the substrate (A) surface k1
E + A~(=~ EA---~ E + P k_~
(6.108)
k2
We have to consider now a different mass balance equation, which should relate not the total concentration of enzyme, but rather the substrate surface. In heterogeneous systems quite often the enzyme is in great excess compared to substrate. This is different from reactions in solutions, when it is the other way around [A]0 = [A] + [ E A I
(6.109)
From adsorption equilibrium
K d~ - -
[EA] -
[EI[A]
;[EA] = K z,[E][A]
(6.110)
and the balance equation [A]o = [A] +
[EA]
= [A](1 + K~,d,[E])
(6.111)
an expression for the substrate concentration is [AI -
[A]° (1 + K~j, [E])
(6.112)
Finally the reaction rate is given by r = k 2 [ E A ] = k 2 K~,~,
[El [A] == k2Ka'~[E][A]° (1 + Ka~ [El)
(6.113)
223 Often the enzyme is in great excess compared to substrate, as there are small changes in the enzyme concentration due to reaction, which leads to [E]0 = [E] and
r =
k2Ka~z,[E][A]o (I + K~d,,,[E])
=
k2K,~z,[E]o[A]o
(6.114)
(I + Ko~,,[E]o)
Analysis of eq. (6.114) shows, that Lineweaver-Burk double reciprocal plot (1/r vs 1lEo) is linear 1
1
1
1
r - k=[Ao] ~ k2[Ao] K a,[Eo]
(6.115)
Another case of heterogeneous systems refers to immobilized enzymes. The kinetic behaviour of a bound enzyme can differ significantly from that of the same enzyme in free solution. The properties of an enzyme can be modified by suitable choice of the immobilisation protocol, whereas the same method may have appreciably different effects on different enzymes. These changes may be due to confonnational alterations within the enzyme, immobilisation procedure, the presence and nature of the immobilisation support. The advantages of immobilised enzymes are for instance in reusability and possibility to use continuous mode. By immobilsation, the pH and temperature profiles of the enzymes may be shifted and the stability of the enzymes altered and in most cases the objective is towards enhancement of these properties. Methods used for the immobilization of enzymes fall into the following categories: physical adsorption onto an inert carrier, inclusion in the lattices of a polymerized gel, cross-linking of the protein with a bifunctional reagent, and covalent binding to a reactive insoluble support (Figure 6.52). The selection of the carrier depends on the nature of the enzyme itseli, as well as the particle size, surface area, molar ratio of hydrophilic to hydrophobic groups and chemical composition.
entrapped in a matrix
entrapped in droplels
Figure 6.52. Methodsfor enzymesimmobilization Physical adsorption relies on non-specific physical interaction between the enzyme protein and the surface of the matrix. In this method only weak bonds are involved, e.g. hydrogen bonds, multiple salt linkages, van der Waals forces, which on one hand is less desruptive towards enzyme, while on the other hand leads to desorption from the surfaces once there are changes in temperature, pH, ionic strength etc. Physical forces are non-specific and further adsorption of other proteins or other substances can occur as the immobilized enzyme is used which may alter the properties of the immobilized enzyme.
224 Stabilization of enzymes temporarily adsorbed onto a matrix has been achieved by intermolecular cross-linking of the protein, either to other protein molecules or to functional groups on an insoluble support matrix. Confining enzymes within the lattices of polymerized gels is another method for immobilization. Such occlusion does not lead to any bond formation between the enzyme and the polymer matrix, thus there is no disruption of the protein molecules. Another possibility is the formation of covalent bonds between the enzyme and the support matrix. The choice of reactions is limited by the fact that the binding reaction must be performed under conditions that do not cause loss of enzymic activity, and the active site of the enzyme must be unaffected by the reagents used. As with cross-linking, covalent bonding should provide stable, insolubilized enzyme derivatives that do not leach enzyme into the surrounding solution. Immobilization influences reaction kinetics, promiting mass transfer resistance. The influence of internal diffusion, e.g. the diffusion of substrate from the bulk solution to the micro-environment of an immobilized enzyme becomes very important and will be treated separately in Chapter 9.
225
Chapter 7. Heterogeneous catalytic kinetics 7.1. R e a c t i o n s on ideal surfaces
In Chapters 3 and 4 the Langmuir-Hinshelwood-Hougen-Watson approach to heterogeneous catalysis was discussed. Such an approach supposes that usually there is one rate determining step (adsorption, surface reaction or desorption) and that the other steps are in quasi-equilibria. As an example, if we consider the reaction A+B<=>C+D, the reaction mechanism could contain adsorption of A on the catalyst surface A+S<=>AS, and adsorption of B either on the same type of sites B+S<=>BS or on some other catalytic sites S' ccording to a following reaction B+S'<=>BS'. The surface reactions could be linear ( A S ~ C S ) or nonlinear. A special situation of the former case is Eley-Rideal mechanism AS+B(g)~CS+D¢g), where a molecule from the gas-phase reacts directly with adsorbed surface species. For nonlinear steps, one can imagine a reaction of the following type AS+BS<=>CS+DS, or alternatively a case when adsorbed A species requires an empty site AS+S~BS+S. The general form of kinetic equation for such LHHW cases was presented in Chapter 3 (eq. 3.50). Different cases of LHHW equations for reversible reactions with different adsorption (competitive, noncompetitive adsorption, adsorption with dissociation) and different rate limiting steps (adsorption, surface reaction and desorption) are summarized in textbooks on chemical reaction engineering and will not be presented here. For instance in the case of the reacton mechanism 1.A+S=AS 2.AS<=>BS "~ + o.BS=B S
(7.1)
if the surface reaction is the rate determing step for a reversible reaction the rate is
r4 =
k2K~ (P~ - PB / K~,q) (_.7 co~ I + K1PA + P~ / K 3
(7.2)
If either adsorption or desorption steps are rate determining and the other steps are quasiequilibria then the rates respectively are
k~ (PA - P ~ / K ~ q ) Cc~, r4 = 1 + (1 + 1/K2)P~ / K 3
(7.3)
226
rA =
k3K, K 2 ( P A - P~/ K~,~) I + (I + K2)KIPA
(7.4)
Q,,,
Notice, that eq.7.2 is similar to Michaelis-Menten kinetics. In tact it is a more general case as it implies also binding of the product to the catalyst. As in enzymatic kinetics, linearization in coordinates 1/r vs 1/PA in the framework of Lineweaver-Burk approach can be quite easily done and leads to the same conclusions for a simple case of irreversible reaction without product binding as discussed in the previous chapter on enzymatic kinetics. Some other cases for the rate determining surface reactions and overall irreversibility are collected in Table 7.1. Table 7.1. Rate expressions for various catalytic mechanisms Surface reactions [ Rate expression A+S=AS kKAPA rA = AS+S~BS+S r.ds. ( I + K A P A + KBP#) 2 BS=B+S A+S A B+S B AS+BS~CS+S r.ds. CS C+S A+S=A AS+B(g)~CS+D(g) r.d.s. CS=C+S 1.A2+S' 2AS' 2.B+S B 3.AS'+BS~ABS+S' r.ds. 3'. AS'+ABS~CS+S' (fast) 4. CS C+S
G =
F~4
kI£ APAK~,P~
(1 + K A P 4 + KIxPI~ + K e r r ) 2
kK~P~P~ I + K APA + K e P t
k ~
KePa
I~4 z
( 1 + ~
+ K a P ~ +KcPc) 2
Derivation of these rate expressions is straightforward and follows the pattern examplified below for the mechanism 1. A+S - A S 2. B+S - B S 3. A S + B S ~ C + 2 S
(7.5)
From the equlibrium of the adsorption steps 1 and 2 in eq. (7.5) one gets
oA = I+K
C +K PB
(7.6)
and 0~ =
K~P~
I + K AP~ + K B P B
thus the overall rate is equal to the rate of the surface reaction
(7.7)
227
r = kOAOB =
kKAPAK'P"
(7.8)
(1 + K A PA + K~PB) 2
Two limiting cases can be considered. If KAPA<< 1 and KBPB << 1 then {~A and 0B are both very low, and the rate is r = kK
P
K,P, = k'CP
(7.9)
i.e. first order in both reactants. If KAPA<1, and
(7.10)
k,P
i.e. first order in A, but negative first order towards B. Once again we stress, that depending upon the partial pressure and binding strength of the reactants, a given model for the reaction scheme can give rise to a variety of apparent kinetics: this highlights the dangers inherent in the reverse process - namely trying to use kinetic data to obtain information about the reaction mechanism. Sometimes in the literature LHHW models are borrowed ~}om the textbooks and used for parameter estimation without preliminary qualitative analysis. It is obvious that it does not make any sense even to try to do some fitting for a model, which does not explain theoretically the experimentally observed behavior. For instance, if LHHW predicts maximum first order in some component and experimentally second order is observed, then it is evident apriori that such a model is not applicable. The two-step sequence is a frequent case in heterogeneous catalysis and can be presented in the form
1. Z+A~->ZI +B~ 2. ZI+A2+->Z +B2
(7.11)
AI+A2 ~-> BI+B2 where Z is a surface sites and ZI is the surface intermediate. Catalytic mechanisms in heterogeneous catalysis with two steps can be for instance examplified by a Mars-van Krevelen oxidation mechanism (Figure 7.1)
o C
J O
Figure 7.1. Illustration of Mars-van Krevelen oxidation mechanism Such a redox mechanism takes into account possible reduction of the oxide, which occurs at a rate dependent on for instance organic reactant and takes place in the absence of gas phase oxidants. Isotope labeling studies confirmed that oxygen in product comes from lattice oxygen, not gas phase species directly. Reduction of oxides happens independently of the
228 reduction step. Dissociative adsorption of oxygen occurs often at a different site from oxidation and diffusion through the bulk refills the surface vacancy. Several oxides like NiO, C0304, CuO, Cr203, MnO2, Fe203, SnO- Sb203, CdMoO4 follow this mechanism for a number of reactions, like CO oxidation, homomolecular isotopic oxygen exchange, ammoxidation reactions, etc. It was demonstrated in Chapter 4, that the general derivation for the two-step sequence results in
r =
(01 (02 -- (0-1 (0-2
(0, +(02 +(1) 1 +(0 2
C~,,, =
NpPAIN2])A2 - k
lJ°Blk 2& 2
C,,,
(7.12)
k, P4~ +k2PA~ + k IP~ + k 2P,2
which is the same type of equation as eq. (4.93) which application was discussed for homogeneous and enzymatic catalysis. As an example for heterogeneous catalysis we can consider the water-gas shift reaction Z+H204--~ZO + H2 ZO+CO~-~Z +CO z H20+ CO - H2+CO2
(7.13)
and write the reaction rate directly from eq. 7.12
r =
(7.14)
If there are more than two steps and the mechanism of a heterogeneous catalytic reaction follows the linear Christiansen sequences, then the general kinetic expressions are given by eq. (5.94-5.95). Some particular cases of Christiansen sequences with 3, 4, 6 and 8 steps were presented for homogeneous catalysis by metal complexes, e.g., equations (5.72, 5.76, 5.84) and (5.88) respectively. It should be stressed that in the case of heterogeneous catalysis equations for the reaction rates are exactly the same, which is not surprising as similar kinetic steps describe the reaction mechanisms. For a three-step sequence a general equation (4.114) can be adopted. For the case of mechanism 7.1 assuming that all the steps are reversible 1 .A+S<=vAS 2.AS<=vBS 3.BS<=vB+S
(7.15)
the rate is (01(02(03 --(0 1(0 2(0 3 0)2(03 +(03(02 +(03(02 +0)3(01 +(01(03 +(010) 3 +(01(02 +(020)1 +(02(01 z
k l P / k 2 k 3 - k_,k_2k_3P ~ k2k 3 + k_3P~k 2 + k_3P~k_ 2 + k 3 k l p ~ + k_lk 3 + k_lk_3P ~ + klPAk 2 + k_zklP A + k_2k_ 1
This equation can be presented in a more general form
(7.16)
229
kP~-k'&
r =
(7.17)
l+k"Pa +k'"p A
and is in fact quite similar to eq. (7.2). An example of a four step catalytic cycle is the steam reforming of methane which occurs via the following mechanism 1 .CH4+Z<=>H2+CH2Z 2.H20+CH2Zc=>ZCHOH+H2
3.ZCHOH<=>H2+ZCO 4.ZCO<=>Z+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(,ok
(7.20)
+ k3k4ktP(,H4 + k_2P~k4klP(,< + k_2P
NJ~('lq4Plq20 = k'PH2 o + k"PcHPH~
(7.21)
+ k ' " P ( , ~ C,~
An example of a reaction mechanism with five linear steps and only one catalytic cycle is the hydrogenation of benzene, which is supposed to occur via the following mechanism where all the steps are non quasi-equibria (e.g. reversible or irreversible) 1.C6H6 +* ¢:>C6H6 * 2.C6H6 * +H2<=> C6H8 * 3. C6H8 * +H2 <=> C6H~o * 4. C6Hlo * +H2 <=>C6H12" 5. C6H12 * <=> C6H~2+* C6H6 +3H2 =C6H12
(7.22)
230 In (7.22) * denotes a surface site. In some other mechanisms presented in the text the surface site is denoted by S or Z reflecting in fact the current practice in the literature, which lacks unification. The general form of the kinetic equation is (7.23)
r = c°10)2~0)40)5 -0)-10)-20)-30)-40)-5 C.~, D
where the denominator D in its most general form consists of 25 terms, i f reaction occurs far from overall equilibrium then we can assume all the steps are irreversible and obtain the equation
r
~
0)10)~°330)40)~ 0)10)2 (03 094 + 0)10)2 0930)5 + 0)1 0)2 0)4 0)5 + 0)1 0)3 0)4 0)5 + 0)2 0)3 0)4 0)5
Ccat
(7.24)
which leads to 3
k,k:k~k~ksP,~HPH~ C~o,
(7.25)
k~k2k3k4Pc,,n~Pn~_ 3 + k~k2k3ksPc,,n~Pn~ 2 + k~k2k4ksP~n, Pn2 2 + k~k3k4ksPc~H PH~2 + k2k3k4ksPH2 3
Eq. (7.25) can be simplified
r =
klk&3k4ksPc~n, Pn2C c~t ' ' k k2k3k4P(,n ' Pn~ + k k2k3ksP(,n ' + k k2k4ksP(.n ' + k k3k4ksPc~n ~ + k2k3k4ksPH~
(7.26)
finally resulting in
r =
Pc~n~Pn~ C .,,
(1/ ks)8 . PH2
+(1/k4 + l/
+1/ k2
+ (1/ k,)P 2
(7.27)
with the reaction order in hydrogen and benzene varying between zero and unity. Other examples of rate derivation for complex reactions with one route can be constructed in the same fashion as illustrated above for heterogeneous catalytic reactions and for homogeneous metallo-complex and enzymatic catalysis in the two previous chapters. Often in heterogeneous catalysis, the reaction mechanism is rather complex and cannot be represented by a one-route multistep reaction sequence, as there are several routes leading to a variety of products. An example of the rate derivation for a multi-route mechanism of butadiene hydrogenation was presented in Chapter 4. Often for the derivation of kinetic equations in such a case it is assumed that the adsorption/desorption steps are in quasiequilibria. The rate of tbrmation of a certain component in the reaction mixture is then defined as dC/dt = rate o f f o r m a t i o n
rate o f consumption
(7.28)
For the liquid phase citral hydrogenation reactions (Figure 7.2) let us take as an example the formation of citronellol
231
O H2 =
OH 3H,
Citral. trans ~
Geraniol' ' i
o4a ..
H~.~--.~ ~..OH H2 _
H2
Citronellal
Citronellol
7
h, o
OH
3.7-Dim ethyloctanol
OH
C itral, cis Nerol Figure 7.2. Scheme of citral hydrogenation (P. Mfiki-Arvela, L.-P. Tiainen, M. Lindblad, K. Demirkan, N.
Kumar, R.SjOholm, T.Ollonqvist, J. V~iyrynen,T.Salmi, D.Yu.Murzin, Liquid-phase hydrogenation of citral for production of citronellol: catalyst selection, Applied Catalysis A: General 241 (2003) 271).
dt
.
.
.
.
.
.
.
In eq. (7.29) the rate constants are apparent ones as they include also the hydrogen pressure dependence. Surface coverage can be expressed by cq. (7.30). O~ =
(7.30)
K,C,
1 + K~C~+ ~ Kj C j ]ej
For citral hydrogenation, the coverage of citronellol is O~i~ro~,~llol=
(7.3 1)
K c"~°""ll°lC c"~°~'"ll°l 1 + K..,../ ~.,,C.~../ ~.,.~+ K,.../
f~.~,.,Q..,../ f....... + K
o/Cn,,,.ot +
+ Kge,.aniolCge,.amol + Kc#,,oCdm o + Kc~tronellolCc.itror, ellol + Kcit,.or, ellalCc,tror, ellol
Then from (7.29) and (7.31 ) we obtain dC~ dt
llol
- (k3Kg~,.,,,~otCg~o,,,o/+ k4K¢,,.o,~H,/C~.,,.o,,~u, / + kTK,,~,.otC,,~,.ot - ksK,,.o,,~uo/C~.,,,.o,,~t/ot)O°
(7.32)
where 0o =
1
(7.33)
1 + K~i,,..t_ci , Q.i,r,,t-~i, + Kci,,.,,l-t ...... Cci,,-,,l-.-,,n,, + K,,~r,,t C,,~,,I +
+ K~,r,,io/C~,,.,,,io/+ K,~,,,oCj,,o + K~.if~o,,~,Ho~C~.~fro,~4tol+ K~.it,.o,~zt,t C~.~o,,~,lt~,z
Equations for the reaction rate analogous to (7.33) can be easily constructed for the other intermediates
232
dC.itra/ trans _ dt
(k 1 +
k2)Kcjlral
IranvCc./lra l Iran,s~)
(7.35)
dQ.,,..z ~.,.~ _ (k 6 + ks)K~.,.o~ ~.,,.C~...t ~.~0~, dt
d C g~"~"'°l -
dt
(k2K cit~al-,.on,~.C~m.ol-, ....... -
(7.34)
k~ K ~,.on,o IC ~,~,,,,o l ) 0,,
(7.36)
(7.37)
dt
d C ...o,,~,., _ ( & K .,,~, c,~C .,~., . . + k~ K .,,o, ,,..,,. C . , , . , ,...... - & K~.,o,,,,,,~,C.,ro,,~.., )Oo
dt
(7.38)
(7.39)
dQl"'° - ks K .,,.o,,~.HozC~j,ro~,~4,oflo dt
The system of differential equations (7.32, 7.34-7.39) is solved numerically together with the balance equation for the surface sites giving a dependence of the concentration of reactants as a function of time, if the data are collected in a batch reactor. More details on numerical treatment will be given in Chapter 10. Another option in analysis kinetics of complex reactions is to separate activity from selectivity and to eliminate time as a parameter in the system of differential equations. For instance, in the case of o~-pinene isomerization over a natural zeolite (Figure 7.3) it was observed that selectivity is independent of temperature or catalyst pretreatment. Therefore, the selectivity analysis was carried out separately from the numerical treatment of activity. ~dW A¢~ ~Pinene ( A ) A1
¢ a m p l ~ (E)
A2 ~N.,. _H+
YricydeRe(C)
/
Aa
( ~
P~H~d~es ~) "~terplnene
Limone~e(D)
Tei~olene
i~-te~inexe
Figure 7.3. Scheme of c~-pinene isomerization (A.l. Allahverdiev, S. lrandoust, D.Yu.Murzin, lsomerisation of c~-pinene over clinoptilolite. J.Catal. 185 (1999) 352).
The system of differential equations describing c~-pinene isomerization can be written in the following tbrm
233 dC B dC'----~-
C~. f CA
dC c
(7.40) Ch
(7.41)
Cc
dCD - f5 + .f6 C----~-D dCx Cx
(7.42)
dC~: _ CH C(: dC A ./~ + L ¢'A j~ (JA
(7.43)
dC~,
(7.44)
-
Cl) + f C~,
where A -(z-pinene, B -phellandrene, C tricyclene, D limonene, E camphene, F-o~ and 7terpinene, a n d f are combinations of rate and adsorption constants. Figure 7.4 demonstrates a dependence of the reactants concentration of conversion for this reaction. The numerical treatment performed for the system of differential equations (7.40)-(7.44) gives values of parametersf
0.4-
B
•
phellandrene, e~o. phellandrene, calc. N cycl erie, exp. N cycl erie, calc. :<~ limonene, e~o. limonene, calc. camphene, e~o. camphene, calc. • terpinene, e~o. terpi nene, cad.
m
0.3 c 0 'M
0.2 E . . . .
l
£
0.0
:x
' '
(~0
(~2
+ i
'
0.4
i
0.6
7 2 7 ~ - ~72T:?L '
-
-i
0.8
~
'
i
1.0
pkt~e mole fraction1
Figure 7.4. Product distribution as a function of c~-pinene mole fraction. Comparison between experimental (points) and calculated data.
The complex reaction mechanisms presented above mainly describe the cases of linear surface steps, e.g. the surface reactions do not involve two adsorbed species (or adsorbed species and the surface site). For reactions involving nonlinear steps, the expression for the reaction rate can be obtained in a closed formed in a few cases by assuming quasi-equlibria (eq. 7.5). In a more complex case, e.g. for the following reaction sequence, where the first two steps are reversible
234 1. A+S <=:>AS 2. B+S caBS 3. A S + B S ~ C + 2 S
(7.45)
from the steady state conditions r1 - r 1 = ~ - r 2
(7.46)
one has (7.47)
klP f l o - k ~0A = k2P~Oo - k 2 0 ~
An expression for the surface coverage of A (7.48)
klPAO0 -- k2PBO° -[- k - 2 0 B = 0 A k_ 1
should be inserted into r2 - r 2 = t~
(7.49)
giving k2PBOo - k 20, = k 3 klPAO° - k2PBO° ~- k 20B OB k -1
(7.50)
and a quadratic equation which relates the coverage of B to the fraction of vacant sites k3 k 20, 2 + (k 2 +~-~3(k,P A -k2PB)Oo)O B -k2PBO o = 0
k<
(7.51)
and subsequently
0 B =
- ( k 2 + k k3 (kiP~-k2P~)Oo) + I ( k 2 + ; 3 (k,PA-k2pe)Oo) 2 + 4 k 3 k2k2p~Oo k, 2k3 k 2 k_l
(7.52)
with only one real solution which is then inserted in eq. (7.48) giving oA - k, PA -k2P~, o,, +
kl
+1,,(
kl
-(k2+~--~-i(ktPA-k2P~)Oo)+I(k2+~--~-(ktPA-k2P~)Oo)2+4~k2k2P~j 2k3k 2 k~
(7.53)
235 The fraction of vacant sites can be solved from a balance equation 00 + 0 B + 0 A : 1
(7.54)
where the expressions for coverages of species A and B are expressed from eq. (7.52) and (7.53). The reaction rate is given by the rate of the third step. It is clear that for such a reaction mechanisms, an analytical expression for the reaction rate is not tractable and the reaction mechanism should be either simplified or treated numerically. More examples on reaction mechanisms with nonlinear steps will be discussed further in this chapter and in Chapter 8, which is devoted to dynamic catalysis.
7.2. R e a c t i o n s on n o n - i d e a l surfaces
As discussed in Chapter 2 the concept of ideal surfaces is often too simplistic to describe adsorption and kinetics, since adsorbed molecules are not equal in the ability to bind chemisorbed molecules. Two different assumptions used in the literature, e.g. intrinsic nonuniformity and induced nonuniformity will be discussed in the sequel.
Z 2.1 Intrinsic nonuniformity If the two-step sequence discussed in the previous section (eq. 7.11) is occurring on nonideal surfaces, than the Langmuir adsorption isotherm cannot be applied. In the framework of the Temkin theory of biographically nonuniform surfaces discussed in detail in Chapter 3, we introduce Z= ~-~0 with its highest value Z-f-~l-{0. The value of (equal to the AG°/RT) depends on the surface site. If the Polanyi transfer coefficient (z is equal for both steps from the general expressions of the rate constants of adsorption and desorption
k+ = k+°e ~(¢ ¢o)
(7.55)
k = k °eP(~ ¢o
(7.56)
we can write the reaction rate constants of steps 1 and 2 (eq. 7.1 l) in the forward and reverse directions k1=
kl°e "~, k 1 = k l°e pz , k 2 = k2°e ~J, k 2 = k 2°e ~z
(7.57)
where k I = kl°,• = 0. For each surface site (there are overall L sites) the reaction rate is expressed in the same way as for ideal surfaces p+ =
1 L klPA, +k
IP ,k2P
1PB, +k2PA~ +k
2P~
Since the rate in the forward direction is the sum of all the rates on each catalytic site
(7.58)
236 gl
r+ = ~p4,(¢)d4 ¢o
(7.59)
where the distribution functions are given by eq. 2.30, changing the variable from ~ to )~ (~=)~+~0) the reaction rate is t r+ - e~¢ - 1
kl°PA, k)°PAe (..... >d2 (k OpA, +k_20p<)e mX +(k_opB, +k20pA~)e,,X
(7.60)
where n [3+7; m o~-7. Rearranging eq. (7.60) we have
kop ion I &to2 YA, e(n ,.)%m~d2 y
/fkl°PA, + k 2°P.~
r+ - eZ/ -1 J
k ° IP., +k2°P4~ e.%,,, ~ 1 Jr klopA, + k_2Ope.,
(7.61)
N o w a new variable U is introduced which is equal to the ratio of the probabilities for a site to be free and to be occupied U=[Z]/[ZI] with
k~PA~k2pA~ + k
[ZI] = t
k,P., + k 2P.2 "
klPAIN2PA2 -}-k IPB1k 2PB2
(7.62)
(7.63)
This variable is defined as U -
k 1P~, + k2I'A~ -
~
klPA, + k-2P~,2
(7.64)
It then follows from eq. 7.57 and 7.64 that
U=e
kl° PA, + k 2°PB~
(7.65)
and 2 =lnU+ln
From (7.66)
k° 'Pr~, + k2°p~2 k,°P~, + k 2°P<
(7.66)
237
e *'z =U" k°
(7.67)
As n+m=l, then in the denominator ofeq. 7.61 we have l+u. Taking into account that 1-n=m, and d)~=l/u du, eq. (7.61) is transformed to y
r+
=
kl°P&k°p& -
&g rr,, I [~--~---dg e j - 1 (kl°PA~+ k 20p&)" (k°-lP~l + k2°PA2 )" r~!l + U
(7.68)
For the range of medium coverage, integration limits are changed U0=0, Ul=oo. Making use of the standard integral f d'-ldu
7r
J TT; - sin
) - sin&
(7.69)
)
we arrive at an expression for the reaction rate in the forward direction =
r+
7"
.,r
kl°P4k2°Px~
(7.70)
e j -1 sinm~- (kl°P4~ +k 2°P~)m(k ° IP~, +k2°P~) '-"
Note, an important step in the derivation above is that the integration limits were changed, which is only valid for the region of medium coverage. If the reaction is reversible than the following equation relates the rates of the forward and reverse reactions via an equilibrium constant ~2 _ K
1
PqP~
(7.71)
From eq. (7.57) klk2
_
klk2
kl°k2 °
kl°k20
(7.72)
Denoting kl ° k2° 0 - KI' 0 - K2 k I k2 we have the product of equilibrium constants of the steps
(7.73)
238
kl°k2 ° k ~°k 2o -
(7.74)
K1K2 =K
identical on all surface sites and equal to the equilibrium constant of the overall reaction /--z-= ( kl°k2 ° ,i r+ "k_,°k_2 °"
i /)BI/)B2 PAP&
(7.75)
which leads to an expression of the rate for the reverse reaction y
Y2"
r =- e ~ - 1 sinma-
k _ l ° & , k_2 ° & 2 (k~°p~
+k
-Op~2 ),,(k
0 tP.~ + k 20 PA2)l
*?1
(7.76)
In case of evenly nonuniform surfaces 7=0, m=(x and 1
limj~
~ 7
(7.77)
7 ---~ 0 the expressions for the rate in the forward and reverse directions are respectively
r÷ =
1
, Op , Op Ki A1K2 ,42
;r
f s i n a z (k,°PA, + k 2°P~,)~(k °
r =
1 ~ f sina;r
1PB,-1-k2°eA2 )lee
k<°P~,k-2°P~2 (k~°PA,+k_2°iPB2)~(k ° IPB, +k2°PA23~1
(7.78)
(7.79)
The reaction rate, which is r = r÷ - r takes the form
r =)"
;r
kl°PAk2°P& - k ,°PBk 2°P< e j -1 sinmer (kl°P~,. -}-k 2°Pt32)m(k 0 IPBI q_k20pA2)lm
(7.80)
This equation also holds for m=0, e.g. low coverage and m=l, e.g. large coverage coinciding with the expression for the two-step sequence on ideal surfaces at low and large coverage respectively. For evenly nonuniform surfaces eq. (7.80) is slightly simplified
r=1
~r
kl°PA,k2°p~ - k ,°PB,k 2°PB2 f sinaTr (kl°p~, + k 2°PBz)~(k ° IPBI +k2°P&) 1 ~
(7.81)
Further simplifications are possible when both steps in the two step sequence are irreversible. Then we arrive at a power law model
239
=
I~P"P
(7.82)
1-"
where 0_
kl P.~,k2 P r
-
~
~2
(7.83)
'
k~PA~ + k2PA~
shows the difference between the reaction kinetics on uniform and nonuniform surfaces. We will illustrate the difference for a simple isomerization reaction which occurs through the following sequence of steps •
1 .A +Z--+ZA 2.ZA ---~Z+B A=B
(7.84)
For homogeneous surfaces the reaction rate is given by
r -
ktPAk~ kip ~ + k2
(7.85)
In the instance of apriori (biographical) nonuniform surface it holds that
r = k'p~
(7.86)
where c~ is the Polanyi parameter with a typical value of 0.5. Now let us compare the reaction rates as a function of partial pressure of reagent A for some particular values of parameters (Figure 7.5) for ideal and nonideal surface models. "----
reaction rate
Ideal Nonideal
]
1,6 1,4 1,2 1,0 0,8 0,6
/
0,4 0,2 r
0
i
i
2
i
4 partial
6
i
8
|
10
pressure
Figure 7.5. Dependenceof reaction rate on partial pressure for someparticular values of parameters. We can easily see, that in a certain range of partial pressures the descriptions offered by the two models: for ideal and nonideal surfaces, are close to each other, however the deviation is systematic. Moreover, these two models give qualitatively different behavior at boundary values of partial pressures. For instance, according to an ideal surface model, at high partial pressures, the reaction rate obeys zero order kinetics, meanwhile at low pressures, the reaction
240 order is equal to unity, in drastic contrast to the mechanism for nonideal surface, which do not show zero order dependence in a wider range of partial pressures. 7. 2. 2 I n d u c e d n o n u n i f o r m i t y
The derivation above considered a two-step sequence on biographically nonuniform surfaces and followed the treatment first developed by Temkin. For induced nonuniform surfaces the reaction rates of elementary reactions are described by eq. 3.103-3.105. These general equations which take into account all the possible lateral interactions between all the surface adsorbed species on the surface can be applied to treat the same two-step sequence. Z+A<--->ZI +F ZI+B<--->Z +D A+B <--->F+D
(7.87)
It can be assumed that the effective charges of the transition states are equal in both steps and are proportional to the effective charge of I. As in the case of ideal surfaces, the steady state approximation could be then applied ( k , P A + k 2P~,)(1 - O; )e ~&:o,,' _ ( k .P~, + k2PB)Oze (' ~),/co, r = 0
(7.88)
The reaction rate is . . . . . . -~,/co, r _ k P o e (1-a)ll2(7OIT r = r+t - r_t = KlrAI, -- v z ) e -1 FVl
(7.89)
For the region of medium coverage 0i/(1-0 i)~1, then 0i -
TlnU'
r/2C
(7.90)
U ' - k~R4 + k 2PD k iRa + k2PB
(7.91)
where
Thus the reaction rate for the region of medium coverage TlnU'
r =- -
q2C
k3P~k2PB - k
•
1P~ k ,PI,
-
(7.92)
( k , P A + k 2P~,)~(k2PB + k ,P~ )' ~
is similar to expression (7.81) for biographically nonuniform surfaces. However for induced nonuniform surfaces in the region of medium coverage, the concentration dependence is slightly more complicated, than according to derivation for biographical nonuniform surfaces, since parameter U' is dependent on partial pressures of reactants and products.
241 7. 2. 3 T h e r m o d y n a m i c consideration o f kinetics on non-ideal surfaces
The relationship between the rates of the forward and reverse reactions is usually expressed by the following equation similar to eq. (1.65) r -r+ [ O- e (P)/K)j
(7.93)
where r is the overall rate, r+ is the rate of the forward reaction, q~ (P) is a function of only reactant(s) concentrations, and K is the equilibrium constant, which is the ratio of rate constants of the forward and reverse reactions. This ratio is valid when the system operates close to thermodynamic equilibrium. It is, however, typical for heterogeneous catalysis to occur far from equilibrium in an open, non-linear, dissipative, distributed, and nmltiparametric medium. Thus heterogeneous catalytic reactions exhibit diverse non-linear phenomena: the multiplicity of steady states (stable and unstable); hysteresis phenomena; the ignition and extinction of the process; critical phenomena; phase transitions; a high sensitivity of the process to changes in the parameters; oscillations and wave phenomena; chaotic regimes; the formation of dissipative structures and self-organization phenomena. The dynamic properties of reaction systems ultimately depend on the nature of interactions between molecules. The non-linearity of macroscopic rate laws is due to the participation of more than one species in an elementary act and the complex cooperative interaction of adsorbed atoms and molecules with each other and with the catalyst sm'face. The non-linearity of macroscopic rate laws is also due to phase transitions in the adsorption layer, surface restructuring during the reaction, catalyst surface (energetic) non-uniformity, and the influence of mass, heat, and pulse transfer processes on the reaction rate due to the delay in the feedback. Thus, the mechanism of catalytic processes near and far from the equilibrium of the reaction can differ. In general, linear models are valid only within a narrow range of (boundary) conditions near equilibrium. The rate constants, as functions of the concentration of the reactants and temperature, found near the equilibrium may be unsuitable for the description of the reaction far from equilibrium. The coverage of adsorbed species substantially affects the properties of a catalytic surfhce. The multiplicity of steady states, their stability, the ordering of adsorbed species, and catalyst surface reconstruction under the influence of adsorbed species also depend on the surface coverage. Non-linear phenomena at the atomic-molecular level strongly affect the rate and selectivity of a heterogeneous catalytic reaction. For the two-step sequence (eq.7.87) when step 1 is considered to be reversible and step 2 is in quasi-equilibria, it can be demonstrated for ideal surfaces that (7.94)
where r' corresponds to the rate on ideal surfaces, 00 is the fraction of vacant sites, 0~ is the coverage of I, PA, etc. are partial pressures (in case of gas-phase reactions) or concentrations (for liquid-phase reactions), K[ ---- k~ , Ki
k_1
~ ~
= K I K2
When non-linear phenomena are taken into account the reaction rate is expressed as
(7.95)
242
r" =klP~Ooe~°q(1 k-'~2 ~P~) ~' 1 P~Pj) ktk2 p--~) = r+(1 K' ~ )
(7.96)
where r" corresponds to the rate on non-ideal surfaces, rate constants kl, etc., obey the conventional Arrhenius temperature dependence, o~ is the Polanyi parameter, o) is the interaction parameter r and K' - kl k2 k q k_ 2
Thus, in a system that is far from equilibrium, despite the non-linearity, the rate for the overall reaction can be expressed with equation (7•93). The equilibrium constant, which is determined as the ratio of constants of the forward and reverse reactions, does not include the lateral interactions in the adsorbed layer.
Z 2. 4. Ammonia synthesis The most famous example of the application of the concept of ninuniformity in catalytic kinetics is the mechanism of ammonia synthesis• Schematically, in the simplest treatment, the ammonia synthesis mechanism near equilibrium was pictured in the following way 1.]72+ Z¢::>ZN2
(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
(7.107)
The equilibrium constant of the overall heterogeneous catalytic reaction can be expressed by the multiplication of the equilibrium constants of the elementary steps, which constitute the overall reaction. In the particular case of ammonia synthesis (eq.7.97), the equilibrium constant is given as p2
S=
(7.108)
"P~Px~"
k l
it follows then that Ov
kl n
Oo
k,
~'~ -
ve,,e
-v2o~c'/r
~
(7.109)
The reaction rate is expressed as n
,~
r = r 1 - - F 1 = Klt'Zv-2Uoe
aq2Ov2C/T
-k
ION2
e( 1 a)q20~2c/T
(7.110)
where r/~ = a t /
(7.111)
The balance equation for surface coverage is given by 0iv2 + 0 o =1
(7.112)
245 leading
to
[- kl -~20'?'~7' 1 1 0v~(l+/--P~v e ] )=1 '
(7.113)
/k_,
The estimation of surface coverage (eq.7.113) and reaction rate (eq.7.110) should be performed numerically. For instance, the value of surface coverage, could be solved using the Newton-Raphson procedure. An analytical expression (although not strictly correct) can be obtained for the region of medium coverage when it is usually assumed, that 0N2/0o~,l, hence similar to eq. (7.90) TlnU* Ov2 - - /]2C -
(7.114)
with
p2 kl ( - NH,5 PN2-klK,p~ "
g*= -
-
(7.115)
2
From the balance equation (7.1 12) one arrives at 00 =1
TlnU* q2C
(7.116)
Equation (7.110) can be transformed (not strictly, we should emphasize) into TlnU* r = k~P,~, (1 - - ) ( U * ) /72C
TlnU* ~ - k, - - ( U * )
-
/.]2C
1~
(7.117)
Then we finally arrive at an equation for ammonia synthesis in the region of medium coverage
r=k~(1
TlnU
Pn~ ~
-k~TlnU
i
' ~
(7.118)
which is in fact very similar to eq.(7.98). Practically, in the region of medium coverage, the difference is so small that it can be neglected. In certain conditions the rate of the ammonia synthesis reaction in the reverse direction is very small and the overall reaction rate is determined by the ammonia synthesis rate in the tbrward direction. According to the conventional derivation based on the model of biographically nonuniform surfaces this rate in the forward direction r+h is given by
p3 ( ze2 ,V r÷b = k+p~.2,~?.. A~
(7.119)
246 Comparing (7.118) and (7.119) one would expect intuitively certain deviations t}om the first order kinetics towards nitrogen if eq. (7.118) is valid. Such experimentally observed deviations within the framework of the model of biographically nonuniform surfaces were attributed to changes in molar volume of the activated complex at high pressures. Note that the values of molar volumes of activated complexes cannot be determined experimentally, therefore the pressure correction factor was an adjusted parameter, during parameter estimation. It can be numerically shown that if the interaction parameter is sufficently high, then devitaions from the first order dependence become pronounced in the case of induced nonumiformity. It is interesting to compare the mechanism on biographical surfaces, (7.105 with corresponding equation 7.106), which is sucessfully applied in the design of ammonia catalytic convertors with the one recently presented in the textbook on kinetics and catalysis by Chorkendorff and Niemantsverdriet, based on catalytic kinetics over for ideal surfaces 1.N2(g) + *=N2* 2. N2* + *~2N* 3. N* + H * - N H * + * 4. NH* -- H*-NH2* + * 5. NH2* + H * - N H 3 * -- * 6. NH3*=NH3(g) + * 7. Ha(g) + 2 ' =2H* N2+3H2=2NH3
(7.120)
Hans Niemantsverdriet and lb Chorkendorff Quasi-equilibrium assumption for all steps, but the second one gives expressions for surface coverage of dinitrogen, hydrogen and ammonia
ON~ = K1Pv20,
(7.121)
OH = K ~ T P ~ 0,
(7.122)
O~H = 3
1 K6
(7.123)
P~H & ,
3
Equilibrium of step 5 gives
0A/H2 -- ~
--
KsK64K7P&
(7.124)
247 in a similar way for step 4 ONH ~O, ONH
--
-
-
K40H
PNH~
K4KsK6( K~7P~2)2
O,
(7.125)
and step 3
O~v.O,
PN,,~
(7.126)
OV - K30-----~- K3K4K~K6 " . ( ~ K7P;% )3 O, ' From the balance equation the coverage of vacant sites is O, =
(7.127)
1
PvH~
K~_
I + K1PN~ + ~
+
1
1
P~-H~+ KsK6 K~Tp~ (1+ K4 K~7P~ ( 1+ K3 K~TP~ ))
The reaction rate is given by the rate of the slowest (not quasi-equilibrium) step 2
.
- K~K4K, K6 ( _
.
K7*PH2)3
(7.128)
with the coverage of vacant sites expressed by eq. (7.127). Derivation of eq. (7.128) was essentially simplified by assuming that all steps but one are quasi-equilibria, otherwise due to the nonlinear character of these steps an explicit expression could not be obtained in a closed form and kinetic modeling should require numerical treatment. In general it can be concluded, that for ideal surfaces models for ammonia synthesis could be obtained that are fairly consistent with the rates of nitrogen chemisorption as well as ammonia synthesis over a wide range of experimental conditions. On the other hand detailed analysis of models based on the Langmuir approximation revealed (B. Fastrup, Microkinetic analysis of ammonia synthesis based on surface reaction studies of iron catalysts as compared to single-crystal studies, Journal of Catalysis, 168 (1997) 235) that it is necessary to take coverage dependences arising from intrinsic surface heterogeneity or from adsorbateadsorbate interactions into account, meaning that the extended Temkin models are still superior.
7.2.5. Linear sequences on nonuniform surJaces The kinetic models considered above for intrinsic nonuniform surfaces with esentially two non-equilibrium steps were extended by Snagovskii and Avetisov to several types of reaction mechanisms. Only linear steps were considred which occur either with or without changes in the number of adsorbed species leading respectively to AI+Z <--->ZIl+A2
(7.129)
248 (7.130)
B2+Z1 <--->ZI2+B2
It was assumed that the rate constants of elementary steps do not depend on the parameter (see eq.2.30). The rate constants of the steps in the forward direction (e.g. adsorption type) which correspond to eq. (7.129) contain the term e <e, while the desorption type constants have a term ef¢ . The values of the transfer coeffcients o~ and 13 are considered to be the same and distribution functions are similar for all surface species. The kinetic equation for a uniform patch is expressed by
(7.131)
P = L De ~ +He w
where L is the number of surface sites, C, D and H are some functions of the reactants partial pressures. In a similar fashion as for the two-step sequence on non-ideal surfaces, the reaction rate is obtained by multiplication of eq. (7.131) by the distribution function with further integration in the region of medium coverage. Here we present only the result of such treatment, which is an extension of eq. (7.80). r =
Jr7 C (J-1)sinner D"H"
(7.132)
with m = o~-7 and n=[3+7. Several examples were considered by Snagovskii and Avetisov and will be given here without derivation. For the two steps sequence the values of C, D and H are correspondingly C = o 0 2 D - o 0 2/g , D = o 0 ] + o 0 2, H = o 0 1+o02
(7.133)
which after insertion in (7.132) will lead to eq. (7.80). For a two-route reaction sequence with two surface intermediates (Z and ZI)
Z+AI<--->ZI +B1 ZI+A2<--->Z --B2 ZI+A,~<--->Z + B3
N ~)
N (2)
1 1 0
1 0 1
(7.134)
N(I): Al+ A2 <-->BI+ B2 N (2) :A]+ A3 ~ Bl+ B3
the values of parameters C, D and H are C I =O02D-O0 2H , C 2 = O03D-O0 3H , D=o0~ +o0 2 +o0 3, H = o 0 ~+o02 +o03
(7.135)
where Cl and C2 correspond to the rates along the first and the second route. For instance, imagine the hydrogenation of a mixture of organic compounds with very weak adsorption of organic compounds, then in scheme (7.134) A1 stands for hydrogen, A2 and A3 for different organic compounds and Bi is absent.
249
1 .Z+A~<-->ZI 2.ZI+A2<-+Z +B2 3.ZI+A3<-->Z + Ba N(l): A~+ A2 <--->B2 N (2) :A~+ A3 <--->B3
N (~)
N(2)
1 1 0
1 0 1
(7.136)
Then the reaction rates are given by F NI =
¢2-y 0)2 ([7°1-}-°0-2 q- 0)-3 ) - 0)-2 (0)-1 -t- 0)2 -}-0)3 ) (e J -1)sinner (C°l +0) 2 +co 3)'(0) , +o02 +°)3)"
(7.13 7)
rN~ =
3r7 0)3(0)1 -}- 0)-2 -}- 0)-3) - 0)-3 (0)-1 -}-0)2 -}-(03) (e ~ -1)sinner (0), +co 2 +co 3 ) ' ( ( ° , +o)2 + o0~)"
(7.138)
where the frequences of steps CO1 = klPA,, 0)2 = k2PA2 , 0)3 = k3PA3
(7.139)
co_, =k_l, co ~ = k ~P< co ~ = k ~G~
(7.140)
The same scheme (7.134) can be applied to hydrogenation of one component A2 with two functional groups, then the mechanism takes a form (A1-H2, A2-S, B2-P1 and B3-P2)
1 .Z+ H2<--->ZH2 2.ZH2+S<---~Z +P~ o.ZH 2+ S<--~Z+ Pp N(~): H2+S <--->P~ N (2)" H2+S <--->P2
No) 1 1 0
N (2) 1 0 1
(7.141)
leading to the reaction rates FN1 =
rN2 =
2"g~" k2Ps(klPH2 q-k 3.Pf½)-k 2Pf] (k 1-]-k3Ps) (e > - 1)sin nTr (k 1PH,_+ k-2Pe~ + k-3Pr. )" (k-1 + (k2 + k3 )1~s)"
~7"
k3Ps'(klP'2 +k-RP~)-k-3Pv2(k-1 +kRP~') (e ~ - 1)sinner (kiP,2 +k zPr; +k 3Pr,)"(k, + ( k 2 + k3)P, )"
(7.142)
(7.143)
Contrary to example (7.136), if two reactants adsorb on the catalyst surface and further react with another reactant(s) coming directly from the fluid phase then there are three adsorbed intermediates (Z, ZI1 and ZI2) and the mechanism can be described in the general form
250 N (I)
N (2)
1.Z+A j<-->ZI~+B~ 1 2.ZII+A2<-->Z +B2 1 3.Z+A3<-->ZI2 +B3 0 4.ZI:+A4<-->Z +B4 0 N(~): A~+ A2 <--->BI+ B2 N (2) :A3+ A4 <--->B3+ B4
0 0 1 1
(7.144)
Such a mechanism allows for accounting of adsorption of mixtures. The general eq. (7.132) is valid with 3( °0 1 '}-0-)2)(003 -}-co 4)
(7.145)
D = ((-°1+ ( 0 2)(093 +(-°4)+((0 t +(02)((03 +(0 4), H = ((-° 1-}-(02)((-03 + ( 0 4 )
(7.146)
CI = ( 0 1 H - ( 0
1((01 --I-(0 2)(00 3 '}-0)4), C2 =(03 H - O )
In a particular case of say hydrogenation reactions, A2 and A4 are hydrogen molecules, Bl and B3 are absent and AI-SI, A3- $2, B2-P1, B4-P2 • The mechanism is then simplified
1.Z+S~<-+ZS~ 2.ZSt+ H2<-+Z +Pt 3.Z+S2<--->ZS2 4.ZSz+_ ~<-->Z +Pz N(I): S1 + H2 <--->P1 N (2) :$2+ H2 <-+ P2
N(1) 1 1 0 0
N (2) 0 0 1 1
(7.147)
and the kinetic equation can be easily constructed from eq. (7.132), (7.145) and (7.146) with 2 , (04 = k4PH,
(7.148)
(0-1 = k-l, (0-2 = k-2Pr~ , 0-)-3 = k-3, 0)-4 = k-4PP,
(7.149)
(0, = k,P,
, (02 =
2PH2 , 0,3 = k3e
For a mechanism with two surface intermediates, produced in subsequent linear steps of the Christiansen type 1.Z+Al<-->ZI~+B~ 2.ZII+A2+->ZI2 +B2 3.ZIa+A3~+Z +B~ NO): A~+A2 +A3++ B~+B2+B3
(7.150)
the parameters in the eq. (7.132) are C = 0.)3 (0) 1 + CO_3 ) -- CO_3 H , D =
(1 +
(0-2 / (02)((0-3 + (01), H = co 3 + (0-10")-2 / 0)2
(7.151)
For the mechanisms with hanging verteces, e.g. with adsorption equilibrium steps where either the free form of the catalyst
251
1.Z+AI<-->ZII +BI 2.ZII+A2+-->Z +B2 3.Z+A!-ZI: AI+A2 <--->BI+ B2
1 1 0
(7.152)
or the substrate-catalyst complex are reversibly blocked 1.Z+AI<--->ZII +Bl 2.ZI I+A2<-->Z +B2 3.ZII+A3-ZIp. Al+A2 4-->BI+ B2
1 1 0
(7.153)
application of the biographical nonuniform surfaces concept leads to the general equation (7.132) with
C=0)l(0)1q-0)2)-0) l(0)l q-0) 2),D=(0) l-}-0)2)0)3/0) 3-}-(0)l-}-0) 2),H=0)1q-0) 2
(7.154)
C = 0), (0)_, +002)-00_1(01 +0)_2),D = (0), +0)_2)0 3/00_3 +(0)1 +0)-2) 'H =0)-1 +0)2
(7.155)
in case of mechanisms (7.152) and (7.153) respectively. Mechanism (7.152) corresponds for instance to ammonia synthesis on iron catalysts in the presence of water vapour.
7. 2. 6. Kinetic considerataions ofloptimum catalyst for two-step sequence
The relationship between thermodynamics and kinetics in chemical reactions is usually expressed by the Bronsted equation (eq. 3.52 in chapter 3.4) k = g K ~ , where k is the rate constant, K is the equilibrium constant of the elementary stage, and g and c~ (Polanyi parameter) are constant values for a serious of reactions. These constants are determined by parameters characterizing the elementary mechanism (composition and structure of the activated complexes, etc.) thus allowing for the existence of an optimum catalyst, on which the rate of catalytic reaction per unit of surface has a maximum value. Equations of the type (3.52) were used for the explanation of "volcano-curves", when catalytic activity as a function of thermodynamic characteristics follows a curve with a maximum. An example for a volcano curve in methanation of CO is given in Figure 7.6. It can be demonstrated that tbr an optimum catalyst an optimum surface coverage exists. We will consider below the two-step reaction mechanism 1.A +Z+--yZI+C 2.B+ZIg--yZ+D A+Bg~C+D
(7.156)
where A and B are reactants, C and D are products, Z is the surface site and I is an adsorbed intermediate. For unitbrm surfaces in the conditions when the overall rate is at a maximum and values of c~ are equal for both steps the surface coverage 0 on the optimum catalyst is expressed by the following equation 0-~z.
252
Ru CO
F
NI
£ g
+
ir
@
N 0~0t ~.3
-3,3 -2.3 ~t.3 ~.3 0,7 Ofs~oNa'~teCO ad~r~Jon e n e ~ [eV]
Figure 7.6. Volcano curve for CO methanation (T. Bligaard, J.K. Norskov, S. DaN, J. Matthiesen, C.H. Christensen, J. Sehested, The Bronsted Evans@olanyi relation and the volcano curve in heterogeneous catalysis, Journal of Catalysis 224 (2004) 206). If we consider the same mechanism and take into account lateral interactions the rate o f the elementary reactions o f step 1) in mechanism (7.156) can be written ~l~0 ]
(7.157)
C, = k_,P~Oexp[(1 - oc,),gO]
(7.158)
r I = k, PA( 1 - 0 ) e x p [ and
where )v is a constant, accounting for the degree o f nonuniformity. Constants o f elementary reactions k4 and k~ are connected with the equilibrium constant/£1 by
K1_
kI _ ~,0 eXO k, PA(1-0)
(7.159)
Similarly
K2 -- k2 -- PD(1 -- 0) e -,;0 k 2 PB0
(7.160)
Although the equilibrium constants o f the elementary reactions are dependent on coverage, due to the nonuniform behavior o f adsorbed species, the equilibrium constant o f the overall reaction is independent o f the coverage as well as on the nature o f the catalyst. K = K 1K 2 -- P c P D
p4pB
(7.161)
Constants o f elementary reaction kl and k_l can be expressed using the Bronsted equation k, = g, K~*
(7.162)
253 (7.163)
k ~ = g ~ K ] '-I
If the values of the Polanyi parameters are equal for both steps (z1--(x2--(xin the steady-state the reaction rate is given by r = r 1 - r t = kl/° 4
(1 - O)e ~;.o _ k l P ~ O e ~, ~>o
(7.164)
Equation (7.164) can be rearranged taldng into account prevoius thermodynamic consideration (eq. 7.96)
t" =
k i p A (1 -- O)e ~o (1
= ~-(1 -
1
Oa
O) 1 a
[
K PcPD) C P , = g' p ~ ( 1 - 0 ) e ~°
1
P,(1-o)e
~;.°(1 K1
) -
(7.165)
where a ~
~y:gl~,Pj
l
1 Pc,P1, -
a
(1
K )rArB ~
(7.166)
The rate of the overall equilibrium reaction is a maximum over a catalyst when Or/00 -- 0, leading to 0 : ~ [ ~ o ~-' ( 1 - o ) '-~ - (1- ~ ) o ~ ( 1 - o ) -~ ]
(7.167)
and therefore 0=a for ideal surfaces as well as for a surface where there is a mutual interaction of adsorbed species. Ii is interesting to compare rates on optimum catalyst with or without mutual interactions of adsorbed species. To simplify the matter, we will consider a case that is often observed (e.g. 0=a=0.5). Further on we will consider that both steps are irreversible and that frequencies of steps are the same for uniform and nonuniform (with interactions) surfaces. Then the ratio of optimum reaction rates on a surface with ( r i ) and without (r~) mutual interactions between adsorbed species is given by / G = e[(t-~)~;~] = e[(°25;~]
(7.168)
If )~ ~10, which is in good correspondence with electrochemical data on adsorption of hydrogen on Pt in the form of H +, then the activity of the best catalyst could be an order of magnitude higher on the catalytic surfaces with mutual interactions between adsorbed species in comparison with the uniform surface without the interactions.
7.3. Selectivity In chapter 4 the selectivity in consecutive and parallel reactions was considered tbr the case of ideal surfaces without taking into account interactions in the adsorbed layer. Here we will
254 consider the catalytic surface with mutual interactions in the adsorbed layer. Let us assume that for the consecutive reactions the following sequence of steps holds
1. A+Z<=>AZ 2. A Z + B ~ CZ 3. C Z + B ~ DZ 4. CZ <=>C+Z 5. DZ ---~D+Z N(~): A+B=C; N(2): C+B=D
N (l)
N (2)
1 1 0 1 0
0 0 1 -1 1
(7.169)
For the sake of clarity the treatment below is simplified by neglecting the adsorption of B and assuming that the number of surface sites for the adsorption of all reactants is equal to unity. We also assume that steps 2 and 3 are irreversible and step 5 is very fast. The rate of step 1 in the forwaxd direction is (7.170)
rI =tqPAOo e-~v-Je
with
v / = ('IAOA+ rlco
:)u'/r
(7.171)
Here ot is the constant of proportionality between the effective charge in the transition state of step 1 and adsorbed species and has the meaning of Polanyi parameter. In a similar way the adsorption of C is given by 1"4 = k 4P(70oe-w4rhW
(7.172)
The rates of the elementary desorption or reaction steps are given as r2 = k2P~O4e (' ~)'~'#
(7.173)
r3 = k3PR4:e ° ~>~"~
(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 <)v#') / klPAOo e <,l.,w
(7.177)
If one assumes that the values of Polanyi parameters are equal ~z~ = a 2 = a 3 = a 4 = a
(7.178)
255
the fraction of vacant sites is (7.179) From steady-state conditions the relationship between coverage of C and the other species can be established. The selectivity to C is defined as
S --
-F3 r2
1 Pck3k4(k-1-[-k2PB) P, k, k2(k4 + k3P )
N3PB exp[c~(qA -rlc)W ]
(7.180)
k4 +k3P,
The explicit form of eq.(7.180) cannot be obtained without further simplifications and therefore the rigorous treatment requires numerical simulations when eq.(7.180) should be solved together with the balance equation 0 A + 0 C +00=1
(7.181)
However, it follows from eq. (7.180) that the nonuniform character of the catalytic surface is not manifested when the effective charges of adsorbed species are equal, for exanaple in hydrogenation of organic compounds with two symmetrical nonconjugated functional groups. In the instance of hydrogenation reactions, a nonuniform surl?ace is assumed for the gas phase kinetics, while in the case of liquid-phase the surface is totally covered by organic compounds, and therefore the kinetics obeys the corresponding laws for uniform surfaces. The analysis, presented here, indicates that in the case of slow adsorption of reactants and equality of the effective charges, the selectivity pattern would be similar in gas phase and liquid phase reactions. However, when the adsorption/desorption of C is fast in comparison with the elementary reaction steps, the third term in the right side of eq. 7.180 can be neglected. In that instance, the selectivity does not depend on the pressure of B and the selectivity pattern is the same for gas and liquid phase reactions, even if the adsorption properties (and therefore effective charges) differ substantially. Similar to consecutive reactions in the case when the reaction network has a parallel character the ldnetic equations can be derived for the instance of nonuniform surface, assuming that the reaction proceeds via the following sequence of elementary steps
1'. A+Z <::5AZ 2'. AZ+B--> CZ 3'. AZ+B---~ DZ 4'. CZ <=>C+Z 5'. DZ <=>D+Z N(W: A+B=C; N(2)': A+B=D
N (1)
N (2)
1 1 0 1 0
1 0 1 0 1
(7.182)
The selectivity to product C can be expressed in the following way S-
r2'
-
k2 ' exp[(1 - ~z2)rL42W']
(7.183)
256 where w'=
+
+ .,,o.)g'/:r
(7.184)
It clearly follows from eq.(7.183), that when the values of the Polanyi parameters and effective charges are equal for both elementary reactions 2' and 3', the selectivity would be exactly the same for uniform and nonuniform surfaces. However, if molecule A has two functional groups, which are different in their abilities of react with B (for example, olefinic and carbonyl groups in unsaturated aldehydes), the charge transfer toward catalyst surface is different for two adsorbed configurations (e.g.rlA2mlA3) and the selectivity will differ tbr an energetically nonuniform surface in comparison with a uniform surface.
7.4. Polyatomic nature of reactants and coverage dependent adsorption mode
The multicentered nature of adsorption on catalysts surfaces was discussed in section 2.5.6 and it was indicated that the number of adsorption sites on which a molecule (especially in organic catalysis) is adsorbed is not equal to unity as is frequently (and tacitly) assumed. Moreover, in some reactions (for example hydrogenation) there could be a different number of adsorption sites for different substances (for hydrogen and say benzene). This results in a competition tbr sites between participating molecules. Let us consider as an example: a hydrogenolysis reaction utilizing the concept of multiple sites for adsorption and competition between hydrogen and an organic compound for surface sites. The model presented below considers an uniform surface and does not include all the features related to possible detects present at the surthce of real catalysts. 1) HC + ZS ~ HCzs 2) H2+ 2S ~ 2Hs 3) HCzs + Hs + Products
(7.185)
Here steps 1 and 2 are adsorption steps, step 3 is the fracture step followed by hydrogenation and desorption steps. The surface coverages of the adsorbed products are negligible since the hydrogenative desorption of these species is assumed to be fast. In (7.185) HC denotes the hydrocarbon used, while Z is the number of free neighbor potential sites <<S >> 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)
<~ = k3OEH O/z
(7.195)
and the quasi-equilibria for the competitive adsorption of hydrogen and ethylbenzene
o; Kn
pnOJ
(7.196)
258
0E K v - PE0; r
(7.197)
the rate expression can be described as k,k2k3(K,p,)30~:O j /(k r = I+k3KHp, /k 2+k2kB(K,p,)
,k 2) 2 /k
ik
2
(7.198)
The balance equation for competitive adsorption is given by (7.199) and should be solved iteratively unless X is equal to unity. Another example worth mentioning is catalytic enantioselective hydrogenation of ketones. This reaction over non-chiral catalysts when a ketone contains a prochiral center produces racemic mixtures of optical isomers. The kinetics of 1-phenyl-l,2-propanedione hydrogenation was studied in the presence of a chiral modifier -natural alkaloid cinchonidine (Figure 7.7)
IIO~ ~ OH (~)
OX~oH~ (B)
H().~OH (F)
o5o (H) (C) (G) Figure 7.7. Scheme of 1-phenyl-l,2-propanedione hydrogenation (E.Toukoniitty, B. Sevcikova, P. M~iki-Arvela, J. W~-ngt, T.Sahni, D.Yu. Murzin, Kinetics and modeling of 1-phenyl-1,2-propanedionehydrogenation, Journal of Catalysis, 213 (2003) 7).
The most pronounced effects on enantioselectivity (es=B/C) and regioselectivity (rs (B+C)/(E+D)) were observed when the modifier (cinchinidine alkaloid) concentration was varied. The reaction rate also depended on the modifier concentrations used. The reaction rate exhibited a maximum as the modifier concentration was increased. Both of the maxima in es and rs appear relatively close to the 1:1 molar ratio of modifier-to-surface Pt. Cinchonidine, being a bulky molecule, reduces the accessible active platinum surthce as it adsorbs and should causes some deactivation with respect to racemic hydrogenation. The decrease in formation rate of the main product after the maximum can be a result of poisoning by adsorbed spectator species, which inhibit enantiodifferentiating substrate-modifier interaction. Adsorbed cinchonidine in parallel mode (active form) provides an enantioselective site (Figure 7.8) and when the reactant is adsorbed in the vicinity, interaction between reactant and modifier leads to such orientation that hydrogenation towards the main product (e.g. B or 1-R enantiomer) is preferred. However, when the tilted form (Figure 7.8) of
259
modifier (spectator) is adsorbed in the vicinity of the actor species, the site becomes poisoned and the overall activity decreases.
M
3
Figure 7.8.Different forms of adsorption of cinchonidine of the surface (D. Ferri, T. Bargi, A. Balker, in situ ATR-IR study of the adsorption of cinchonidine on Pd/A1203: differences and similarities with adsorption on Pt/A1203, Journal of Catalysis, 210 (2002) 210).
The following mechanistic proposals could be applied. At low coverage, cinchonidine adsorbs mainly via re-bonding of aromatic quinoline rings, the ring system being almost parallel to the metal surface. The adsorbed parallel form of the modifier would be involved in the formation of enantiodifferentiation substrate-modifier complexes on the catalyst surface. At higher coverage, two additional tilted adsorption modes of cinchonidine were observed. The parallel and tilted adsorption modes of cinchonidine require different amounts of primary Pt sites for adsorption, the former one occupying more active metal sites than the latter one and being active in the enantiodifferentiating interaction. The coverage dependent adsorption mode of the modifier is coupled with the coverage dependent adsorption of the reactant as well. Over a non-modified catalyst, the reactant can readily adsorb in a planar mode, where both of the carbonyl groups and the phenyl ring are in the same catalyst plane. This assumption is supported by the experiments, which revealed that the phenyl ring of A is hydrogenated in the absence of the modifier. The formed cyclohexyl products can be taken as an indication of the planar adsorption mode of A, i.e. in order to hydrogenate the phenyl ring, it has to be adsorbed parallel to the catalyst surface via rebonding. If small amounts of the modifier are added, the phenyl ring hydrogenation is no longer prefered. This is indicated by the disappearance of the cyclohexyl products. The adsorbed modifier occupies the surface in such a way that the planar adsorption of the reactant is no longer facile. This imply that the carbonyl groups are adsorbed on a somewhat tilted orientation. Therefore, also the adsorption mode of the reactant depends on the coverage of the modifier on the catalyst surface. The regioselectivity aspects would require two different adsorption modes for the reactant, because both carbonyl groups cannot adsorb simultaneously. The same type of reasoning regarding the number of occupied sites could be applied also for substrate- modifier complexes. The proposed mechanistic assumption could be now translated in the rate expressions. The quantitative treatment of the hydrogenation kinetics presented below is based on the reaction scheme displayed above. Two adsorption modes of the modifier are selected for modeling. One mode is the parallelly adsorbed species of cinchonidine involved in the enantiodifferentiation and in the other second the adsorbed in tilted, and are only spectators on the catalyst surface. Two adsorption modes were used for the reactant as well, namely the adsorption of carbonyl group 1 and 2, leading to regioselectivity. There is a zero order dependence on hydrogen. The adsorption steps of the organic compounds are assumed to be rapid compared to the hydrogenation steps, which implies that quasi-equilibria are applied to
260 the adsorption steps of the reactant (A) as well as the modifier (M). The adsorption of the product molecules is neglected. Only the first hydrogenation steps to B, C, D and E are considered here, since the amount of diols formed is minor during the first stage of the experiments. For similar reasons the phenyl ring hydrogenation is disregarded. The hydrogenation steps are presumed to be irreversible and determine the rate of product formation. The reaction mechanism can be summarized as follows (vacant surface sites are denoted by*): [I] [II] [III] [IV] [V] [VI] [VII] [VIII] [IX] [X] [XI] [XII] [XIII] [XIV]
A + m * - - Alto* A+n* - - A2n* M + p * - - Mp* M + q * - - Mq* Alm* +Mp* +f* - - A1M(mpf)* A2n* +Mp* +1" - - A2M(npl)* A l m * + H(ads) ~ A1Hm* A1Hm*+ H(ads) ~ 0.5B +0.5 C+m* A2n*+ H(ads) ~ A2Hn* A2Hn*+ H(ads) ~ 0.5D +0.5 E+n* A1M (mpf)*+ H(ads) ~ A1MH (mpf) A I M (mpf)*+ H(ads) ~ B+ m*+p*+f'*+M A2M (npl)*+ H(ads) ~ A2MH (npl) A2MH (npl)*+ H(ads) ~ D +n*+p*+l*+M
(7.200)
Steps [I] and [II] correspond to adsorption of the reactant (A), while steps [III] and [IV] denote adsorption of the modifier. In scheme (7.200) Alm* and A2n* denote the adsorption modes of 1- and 2-carbonyls, respectively; and n, m, denote the number of sites required for adsorption of them; Mp* denotes the parallel adsorption mode of the cinchonidine involved in the enantiodifferentiating interaction with the adsorbed reactant An* or Am*; and Mq* denotes the tilted adsorption mode of the cinchonidine, which adsorbs on the catalyst surface as an spectator. Reactant - modifier interactions are supposed to be essential for transferring chirality to a pro-chiral reactant and it is reasonable to suppose that the site requirement for the substrate-modifier complex might be higher than the sum of the sites occupied by the reactant and the modifier separately. In the kinetic treatment, the formation of substrate modifier complexes is considered and they are supposed to be formed on the Pt sites in the steps [V] and [VI], where f and 1 are extra sites, which are required for the substrate-modifier complex to be formed compared to the site requirement for them separately. Rigorous kinetic modeling in principle should include formation of half-hydrogenated species, which for racemic hydrogenation of the reactant on unmodified sites is described as steps [VII], [VIII], [IX] and [X]. The elementary reactions, which are only summarized in equations [VII]-[X] could be more complicated and they are considered here as schematic representations, giving the correct stoichiometry of racemic hydrogenation. Enantioselective hydrogenation of the reactant to (R)-enantiomer proceeds via reactions of the substrate-modifier complex in reactions [XI], [XI1] and [XIII], [X1V]. Although some hydrogenation to the (S)-enantiomer may take place on the modified sites, the contribution of these steps is easily obscured by the contributions of the racemic hydrogenation steps and can be disregarded for the sake of clarity here.
261 The use of the quasi-equilibrium hypothesis for the adsorption steps i-IV implies that the concentrations o r A l m*, A2n*, Mp* and Mq* are expressed by CA|m. =
|CAC.
C 42., = K2CAC,
C~;~. = K.~c~c. p, cMq, = K4c,wc. q
(7.201)
If one applies also the quasi-equilibrium hypothesis for the steps V and VI, the concentrations of the complexes A1M (mpf)* and A2M (npl)* are expressed as CAIMm[?/,
=
K5CAIm,CMp,
C,f
r: r: r: = IklI~31k5CACMC
,
m+p+f
CA2Mnpl* = K6C A2n*CMP*c*I = K2K3K6C ACalC*n+F+I
(7.202) (7.203)
The surface coverage of half-hydrogenated species could be computed applying quasi state assumptions, e.g. r7 = r8, r9 = rl0, ri1 = q2, q3 = ri4
(7.204)
leading respectively to Cram"* =
(7.205)
kTC.4|°'* /ks
CA2H~'* = k9c A2"* / k|o
(7.206)
CAIMHmPt* = kI~CA~A~"'~/* / k~2
(7.207)
C A2MttnP l* = k l 3 C A2M~'P l* /
(7.208)
k14
All these expressions are inserted in the total balance of metal sites, +n(l+kg/klo)C~2..
m(l+k
7/ks)CAl,,,,
npl(1 +
k|3 / kl4 )C AeMnpl, -[- C ,
+pcA4p. +qC~lq, + mpJ'(1 + k~/kl2)CAL~j,,,t~/, +
= CO
(7.209)
where Co and c* denote the total concentrations of accessible sites and the concentration of vacant sites, respectively. After inserting the quasi-equilibria into the total balance of sites, coverage of all species can be expressed via the fraction of vacant sites, O-c*/co: c A ( i n k 1(1 + k 7 /. K . . ,.). c. .o 1u. .+ n.K 2. ( l +. k 9. / k l. o ) C 0 10 j~ )+cM(pK3coP-lOP +qK4coq-loq)+ + (m + p + f ) K I K 3 K 5 (1 + kjl/kl2)CACkfCo m+p+l lore+P+'/ +
(7.210)
+ (n + p + 1 ) K 2 K 4 K 6 (1 + kt3 / kl4)CAGsc0n+m/-lo n+p+t + 0 = 1
After introducing t
1
\
K 1' = m K 1(1 + k 7 / Ks ) c
nl I
o
(7.211)
262
K 2 ' = r t K 2 (1 + k 9 / k , o ) C o " 1
K3'=
(7.212) (7.213)
pK3co ;'<
K4, = qK4co q t
(7.214)
K 5 '= ( m + p + f ) K 1 K 3 K 5 (1 + kit / k12)Co "'+p+t-I
(7.215)
K6'= (n + p + I ) K 2 K 4 K 6 (1 + k,3 / k,4)Co"+~'+~ I
(7.216)
the site balance obtains a more compact form f ( O ) = c A ( K t ' O " + K 2 ' 0 " ) + c M ( K 3 ' 0 p + K 4 ' 0 q) +
(7.217)
+ c~cv r ( K 5 ' 0 ''+p+-/ + K 6' 0 "+p+t) + 0 - 1 = 0
The rates of the hydrogenation steps [VII, IX] and as follows F7 = NTCAI*, ?'9 = kgcA2* , rll = klICAIM*
[XI, XIII] in scheme (7.200) are expressed (7.218)
, /"13 = kl3CA2M*
For practical purposes, the rate expressions could be written as r7 = kT' c A O ' , r9 = kg'C AOn, ~1 = ktt' c Ac,vrO'+p+y, r13
=
kll'C,4ca101~+p+!
(7.219)
where the combined parameters (ki') contain the rate constants (k), the equilibrium constants (K) as well as the total concentration of sites (co). The consumption rate of the reactants is (7.220)
FA = - - ( F 7 q-F9 q-Fii-1- FI3 )
and the generation rates of all other compounds J% = 0 . 5 F 7 -'}-Fii , r C = 0 . 5 F 7 ,
FD =0.5F9"l-FI3 , Fh, = 0 . 5 F 9
(7.221)
The next step is to combine the reaction rates to the mass balances of the components. The fraction of vacant sites (0) should be calculated from the balance equation during the estimation of kinetic parameters. The non-linear equation could be solved using, for instance, Newton's method, i.e. O(k+lj--O(k) -f(O(k~)/J"(O(~), w h e r e f ' is the derivative o f f while k denotes the iteration index. Parameters m, n, p, q, l and f as well as the rate constants should be estimated by data fitting. A certain initial estimate of coverage is used at the very beginning of the parameter estimation. Later on, a previously converged value of 0 is utilized to start a new iteration cycle. During the parameter estimation the ordinary differential equations are solved for all of the components. A stiff ODE-solver could be used to guarantee a rapid and
263 stable solution. The residual sum of squares in the parameter estimation could be minimized applying different algorithms (for instance simplex or Levenberg-Marquardt methods). More information about numerical treatment is given in Chapter 10.
7.5. Ionic species Intermediates which are involved in heterogeneous catalysis could have ionic character, which require an extention of the general treatment of complex reactions. As an example we can consider the catalytic hydrogenation over oxides and sulphides, where intermediates of cationic character were proposed. Ionic intermediates are also possible in catalysis over metals, for instance in the case of neopentane transformations over electron deficient palladium, which occur via formation of carbocations. If we consider olefin hydrogenation over oxides or sulphide with a heterolytic dissociation of hydrogen, the mechanismn of this reaction can be presented in the following form -
t
+
1.H2+Z + Z' > ZH +Z H 2.ZH- +R => Z R H 3.ZRH- +Z'H+ = > 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<, and PR denote partial pressures of hydrogen and olefin respectively. Then from (7.223)
[Z]+kfH2r kfH2 2 =1 k - ~ [ Z ] "2 +k3--d_--[z)[Z]
(7.230)
Three real values of surface coverage can be obtained as a solution of this equation depending on the values of constants and partial pressures. A second example also treats hydrogenation of olefins (R) on oxides occuring by the following mechanism M+-ORH M+-ORH +M+H - --) M+-RH2 +M +- 02. M+-RH2 "~M + + RH2 H2+ M+-OH 7XH20+M+HH20+M + + M +- 02- 7a 2M+-OH -
1. R + M + - O H - E
2. 3. 4. 5.
(7.231)
Steps 1, 3, 4 and 5 are considered to be in quasi-equilibria and step 2 is rate determining. The symbol M + corresponds to the cation and M +- 0 2- is surface oxygen. Expressions for the coverage of surface species are computed from quasi-equilibria
[M+-ORH-] KtR [M--OH]
(7.232)
[ ~ f f ]-K4 He[~-OH]/HzO
(7.233)
[M+-RH2] Ks [M+] RH2
(7.234)
Ks-[M+-Off]2/ fl20[M+][M+- o27
(7.235)
The are six intermediates in scheme 7.231, five steps and only one reaction route. According to the rule of Horiuti-Temkin there should be two balance equations. One is the balance for coverage of surface species
[M+-OH7 +[M+-ORHT+[M+HT+[M+-RHU+[M - U7+[MT-1
(7.236)
and another one corresponds to the electroneutrality of the catalyst
[M+-R~] + [M+] - [ m +_o~7
(7.237)
Hence
[ ~ - U - ] - [ ~ ] [1 + 1£3RH2]
(7.238)
265 Replacing in (7.235) [M+-O2-] and making use of (7.238) one gets an expression for [M + ] as a function of [M+-OH]. [M+]:
[M+-Ott ] 4Ks[H20](1+K3[RI-I2b
(7.239)
[M+_OH ](I+KI[RI4K4[H2]
1
[H20] + 4Ks[H20](.I+K3[RH2b
(2K3[RH2]+2))= 1
(7.240)
or [H 2O] [M + -OH-]: [H20]+
K,[RI[H20]+K4 [H2 ] + 2K5 o.s4[H20~.1+K3[RH2b
(7.241)
The coverages can be determined from the balance equation explicitly. The reaction rate is proportional to the rate of step 2 (eq. 7.231) leading to
(7.242)
k2KI[RIK4[H2I[H20] ([H20]+K,[R][H20]+K4[H2]+2Ks
0.5
4[H2OI(I+K3[RH2b)-
The theory of complex reactions with additional balance equation also gives the possibility of computing the rate law of catalytic reactions with ionic intermediates where it is important to take into account participation of the bulk of the catalyst. In this case, from the viewpoint of the theory of complex reactions, it means that besides the balance equation which corresponds to the total electroneutrality of the surface species, there will be another one which describes electroneutrality of the whole catalyst.
7.6. Transfer of labelled atoms in heterogeneous catalytic reactions
The general principles of catalytic reactions can be applied tbr the description of the kinetics of the transfer of labelled atoms by reaction. Usually identification of molecules by labelling is done by substitution of an atom with an isotope. It is assumed that the content of labelled atoms is determined only in the participants of the reaction, i.e. in the reactants and the products - but not in the intermediates and that only the total content of isotopes in equivalent atoms of a given substance is determined. The significance of the last restriction is that no distinction is made between H2+D2 and 2HD. It is also assumed that a complex reaction proceeds in a stationary manner. As an example of the transfer of labelled atoms we will discuss the ammonia synthesis mechanism presented by the following scheme
266 1 .N2+Z¢::> 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#<=::7ZO # 02 3.ZO*O2<=>ZO*+ 02 ZO+ 02 *= ZO*+ 02
1 1 1
(7.262)
where ZO denotes a surface oxygen atom which has a special position in the three atom complex ZOO2. i f we assume that the exchange mechanism consists in the label being trasnsfered from the molecule 02* to ZO and that ZOO2* and ZO*02 are intermediates along this rome, then eq. (7.260) can be directly applied (~t=l)
p =
~o2rlr2r3 - ~ z o r l r 2 r 3
(7.263)
rzr 3 + rlr+3 + r l r 2
If we neglect the kinetic isotope effect, then the rates of the chemically similar steps are the same /~ = 123,/~-I = r3, r2 = r-2
(7.264)
and eq. (7.263) can be simplified to (OeF1F2F3 -- (ZOI~IF2F3 P =
F1F2((O) -- ( Z O )
~ r 3 + r 2 + rzr 3
2 ~ + r3
(7.265)
From the steady steady-state approximation fi - KI = r3 - K3
(7.266)
and (7.264), we get that • = r 3 and thus
(<,n r3 -4"z,,r, r2r3 P =
r2t ~ + r [ + t ) r 3
=
2t) +r~
(7.267)
When the rate of the second step is much faster than the rate of the first one we have the adsorption-desorption mechanism which governs the rate of oxygen exchange. When q >> r z the rate is defined by the rate of the redistribution of oxygen atoms within the three-atom complex P = rz((o~ - ( z o ) = kzO((o~ - ( z o )
where 0 corresponds to the surface coverage of the three-atom complexes.
(7.268)
270 7.7. Electrocatalytic kinetics A special case of heterogeneous catalytic kinetics, that we will briefly discuss, is electrocatalysis, which is defined as the acceleration of an electrodic reaction by a substance that which is not consumed in the overall reaction. For a catalytic reaction with the following step Sads+Had~SHads this is only one of the possible mechanisms for the involvement of hydrogen in the catalytic reduction. Another mechanism is the electrochemical or ionic mechanism where the adsorbed hydrogen serves only as an electron source for the reduction process. This type of reaction is formulated by the following reaction steps: Hads~H++e S+e-~SS-+H+~SH From an electrochemical point of view this type of catalytic reduction can be conceived as two coupled electrode processes. In the case of this hydrogenation reaction, the catalytic and electrochemical reduction differ from each other only in the means for the achievement of the reduction, e.g. molecular hydrogen, redox systems, or pure electrochemical means. Similar to heterogeneous catalytic processes, electrochemical reactions tend to occur as a sequence of very simple steps. For example, hydrogen evolution occurs as two steps, with two alternatives for the second step, corresponding to two reaction routes: N~
2 1. H20 + e - ~ Hads+OH 1 2a. 2 Hads~H2 2b. Had + HaO+e-~ H2+OH- 0 N ~, Nr." 2 I~20+2e-fi2+ OH-
N2 1 0 1
(7.269)
Scheme (7.269) can be presented in a slightly modified way
1. H + + e + * ~ nads 2.2H~d~H2+2* 3. Had+H++e-~ H~+* N 1, )~2:2 H + +2e- --H2-
N~
N2
2 1 0
1 0 1
(7.270)
There are 3 steps in scheme 7.270, two intermediates (adsorbed hydrogen and vacant sites), one balance equation which relates these two intermediates, and then respectively two independent routes according to the Horiuti-Temkin rule. Steps 1, 2 and 3 are usually referred as Volmer, Tafel and Heirovskf reactions respectively acknowledging the names of researches who emphasized the importance of these processes.
Max Volmer
Julius Tafel
Jaroslav Heyrovsk~
271 The relevance and the need to addreess electrocatalytic kinetics is closely associated with the recent developments of fuel cells. The process for fuels cells, with hydrogen as fuel can be represented in a simple way 2H2
+ 02
=:~ 2 H20 + energy (electricity)
One fuel cell type, called a proton exchange membrane fuel cell, carries out this reaction in the following way. The hydrogen fuel (H2) enters one side of the fuel cell, where it encounters a catalyst, for example platinum, which splits the hydrogen atoms into a proton (H +) and electron (e-). The proton then travels through a membrane (the proton exchange membrane) to the other side of the fuel cell. But the electron cannot easily permeate through the membrane. Instead, its travels through an electrical wire to get to the other side. The buildup of negative charge follows the path of least resistance via the external circuit to the other electrode (the cathode). It is this flow of electrons through a circuit that creates electricity. To complete a electrochemical cycle as the electrical current begins to flow, hydrogen protons pass through the membrane from the anode to the cathode. After transferring energy, the electrons react with oxygen and the hydrogen protons at the cathode to form water. Heat emanates from this union (an exothermic reaction), as well as from the frictional resistance of ion transfer through the membrane. This thermal energy can be utilized outside the fuel cell. To summarize there is an anode reaction (reverse of the one in scheme 7.270) H2 - - > 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 <
PotentialN P ~
~ ¢0
~0 0.1 02 0.3 t1~ 115 0,6 1).7 0,8 0.9 1.0 PotentiaW Pd/H
0.0 0.1 0.2. 0.3 0.4 0.5 (x8 0.7 0.8 o.g 1.0 F~te~
Pd/H
Figure 7.12. Cyclic voltammograms for different platinum surfaces (G.A. Attar& A. Ahmadi, D.J. Jenkins, O.A. Hazzazi, P.B. Wells, K.G. Griffin, P. Johnston, J.E. Gillies. The characterization of supported platinum nanoparticles on carbon used for enantioselective hydrogenation: A combined electrochemica|-STM approach. ChemPhysChem, 4 (2003) 123).
The rates of the overall electrochemical reaction can be treated according to the general principles of the theory of complex reactions. It is clear that besides the rates of the catalytic reaction (2) in scheme 7.270, it is necessary to describe the rates of electrochemical process, e.g. electron transfer reactions. Some basic information on a single electron transfer reaction
274 will be presented below. If we consider the following reaction between two surface species O and R with the participation of an electron (coming from an electrode) O(s)+e-(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++e<=>A 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* <-+H20 + O* + *
4 2
0 0
(7.308)
N0*)4CH4 + 02 -+4CH3" + 2H20 The catalytic radical production cycle provides a heterogeneous initiation, which is much faster than the homogeneous initiation. The produced methyl radicals react further in the gas phase where they can either react to form formaldehyde through the following propagation steps N (1)
N (2)
4. CH3" + 02 -+CH20 + OH"
0
1
5. CH4 + OH" -+ CH3" + H 2 0 N (2) CH4+ 02 -+CH20+H20
0
1
(7.309)
or couple to form ethane in the presence of a third body
6 . 2 C H ( +M --)'C2I-I 6 +M N0)4CH4 + 02 -+2 C2H6+ 2H20
N (1)
N (2)
2
0
(7.31o)
Some other routes involving branched chains are possible N (3) CH3" + HO2" --+CH30" + OH CH30" + M .+CH20 + H'+ M
(7.311)
CH20 + CH3".+CHO'+ CH4 CHO" + M -+CO + H'+ M 02 + H'+ M -+HO2 + M CH4 + HO2"-+CH3"+ H202 H20~ -+2OH" N (3) CH3 + 202 --+CO + 3OH If we limit our considerations to the mechanism (7.308)-(7.310), and consider step 1 as being in equilibrium, it leads to 0o = ~ 0
V
(7.312)
where 0 o is the coverage of adsorbed oxygen and 0 V is the fraction of vacant sites. From the quasi-equilibria of step 3
283
K 3 -
(7.313)
OOH
and eq. (7.312), an expression for the surface coverage of hydroxyls is O°H = (K~, P''2° ~
)0.5
(7.314)
From the balance equation for the surface sites one can get an explicit expression for the oxygen coverage
0o =
~
1+ ~
1
(7.315)
)05
The rate of initiation is proportional to the rate of step 2 (scheme 7.308)
r 2 = k2Pc< 1 + ~
+(1 P H , o ~ ) ° 5
St.(,,
(7.316)
Here Scat is the catalyst surface area. The termination is described by the rate of step 6 (eq.7.310)
r6 = k6C~2,,(S,.,,;;a + S.,oH.~)
(7.317)
with its rate proportional to the surface area of the catalyst (and any other solid species) and the walls of the reactor. The steady-state conditions require that these rates are equal and thus (stoichiometric numbers of steps should be taken into account) 4r2 = 2r6
(7.318)
which results in an explicit expression for the concentration of methyl radicals ~0.5
=
k2Pc;'~ ~
2(1+
K3
S~a' 2
~
•
,.2
]
(7.319)
/
The generation rate of formaldehyde is expressed by the rate of step 4 (or 5 as they are equal to each other when quasi-stationary conditions are valid), therefore
284
0.5 (7.320)
rCH20 : k4CcH~ ])02 : k4~)2
1
) o s ) S~ol, a + SwalL~
Methane is consumed in both routes N (l) and N (2), which are given by steps 2 (eq.7.308) and 4 (eq.7.309). Its consumption rate (per mole) is
- r,.H4 = k2PcH ~
S~.,, +
# ,
/ 05 v ...2
K3
_, ~ , .J2
(7.321)
)
For more realistic reaction mechanisms, even more surface catalytic reactions and gasphase radical chain and branched reactions should be included, coupled with carefull consideration of mass-transfer, which obviously requires a rather sophisticated numerical approach. In conclusion, a few comments about the practical importance of heterogeneoushomogeneous reactions from kinetic viewpoint. Usually industrial reactors have a relatively large free volume and the packing material is also present in addition to the catalyst. If homogeneous reactions are beneficial for the overall process, then they will be retarded due to deactivation of radicals and the rate will be lower compared to laboratory reactors with more advantageous reactor geometries.
285
Chapter 8. Dynamic catalysis 8.1. Transient kinetics
For the development of more efficient catalysts, an improved understanding of reaction mechanisms and kinetics is of crucial importance. Under steady-state conditions each individual elementary step of the composite reaction proceeds at the same velocity, and one can only obtain the relationship between the reactant and product concentrations. It is, thus, difficult to improve the insight into the knowledge of a dynamic system. Typical experimental parameters are reactant and product concentrations, total pressure and reaction temperature. Steady-state experiments give valuable information about the overall rates of catalytic reactions, but they cannot reveal the underlying reaction mechanisms in an unequivocal way, because different reaction mechanisms give steady-state rate equations, which do not differ very much from each other. Thus, the experimentally recorded kinetic data can typically be fitted to several kinetic equations and the true nature of the underlying catalytic surface reaction mechanism remains unknown. Interaction of the reaction medium with the catalyst results in a reconstruction of the catalyst surface. An example of such a reconstruction is the well-known process of reconstruction of the surfaces of metal monocrystals such as in the microfaceting of Pt(1 0 0) under the influence of CO, or the creation of oxide surface phases on Ni(1 1 1) under the influence of 02. The invention of scanning probe microscopy with nanometer scale resolution has made direct observation of surface reconstructions readily possible on various surfaces, providing opportunities to study the role of local reconstructions and surface defects in catalytic reactions. Changes in the catalyst activity could be not only due to changes in the surfaces, but also changes in the phase composition of the catalyst and changes in the bulk composition (tbr instance palladium is known to absorb hydrogen, possessing then properties different from the pure catalyst). Such nonstationary operation could result in formation of criticalities and even oscillations.
8. 1.1. Relaxation to steady-state
The stationary rate is not established instantaneously, but after some relaxation time. Under nonstationary conditions, one should understand the conditions when the characteristic time of changes in reaction parameters is the same order as relaxation time. If the relaxation time, e.g. time to reach a steady-state state is longer than the duration of a catalytic experiment, than the reaction occurs under nonstationary conditions. Sometimes it could be even beneficial to perform a reaction under such conditions, periodically changing initial parameters of the reaction system., e.g. temperature, pressure, concentrations, or flow velocity. Under nonstationary conditions two types of processes could operate. In the first one, the concentrations of intermediates in the catalytic cycles are at non steady and such changes are
286 due to the intrinsic reaction mechanism. The non stationary behavior is associated with the rate of the chemical reaction, in the second case, changes in the rate constants of some elementary steps is associated with the side reactions, not catalytic cycles. The relaxation time of these side processes could be higher than the relaxation time of the reaction itself. Let us consider as an example, the self-relaxation for a two-step sequence on a uniform surface. Z+AI<-->ZI +B1 ZI+Az<-->Z +B z AI+A2 <--->BI+B2 After defining the rate as the number of catalytic acts per unit of time per unit of surface we have for the rate of formation of intermediate I
r =
L dO dt
(8.1)
where L is the number of catalytic acts per unit of surface. This intermediate is generated in the steps one in the forward direction and step two in the reverse direction and consumed in the reverse direction of step 1 and forward direction of step 2 leading to the following expression dO L7
= rl +
- (,'-1 +
(8.2)
Defining the rates through the frequencies of the steps 0)1 = klPA, ,0)-1 = k-l P~, ,0)2 = k2PA, ,0)-2
= k-2P~ 2
(8.3)
we have for the rates of the steps r I = 0)1(1- O),/'_ I = 0)_10,//'2 = 0)20,F_2 = 0)_2 O
(8.4)
and consequently
L
dO =0)1 -t-0)-2 --(0)1 +0)-1 -t-0)2 +0)-2) 0 dt
(8.5)
Defining the steady-state coverage as 0~, which is achieved at the steady-state dO / dt = 0 900
0)1+0)2 0)1+O)1+0)2+O) 2
and introducing "c (relaxation time)
(8.6)
287
L
Z'=
(8.7)
0)1 + 0 ) I + 0 9 2 + 0 ) 2
eq. (8.5) after diving both sides b y 0)1 + 0), + o)2 + 0) 2 takes the f o r m
r
dO =G -0 dt
(8.8)
After integration, using the limit t = 0 a t 0 = 0 0 one arrives f r o m (8.8) at (8.9)
0 : O~ +(00 _O~)e "~ where O~ = O at t = co. The overall rate is equal to the rate o f the second step
r
= r 2 - C 2 = k 2 P & O - k _ 2 G 2 (1 -
(8.1o)
O) = (0)2 + 0)-2 )0 - o9_2
w h i c h at t>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 ~o<1 and r < 0~(1-O~)U~. Equality holds for irreversibility of both steps (~o = 1 ). The maximum of 0~ (1- 0~) is achieved at 0~ = 1/2. Then we arrive at an expression which relates the relaxation time with the time of one turnover at steady-state 1
v <_-~U~
(8.19)
For industrially relevant process the relaxation time is ca. 1-100 s. For construction of a kinetic model for nonstationary conditions, knowledge about the evolution of the concentrations of adsorbed species on the catalyst surface is needed. Under nonstationary conditions the changes of concentration fields in time, reactor space and catalyst surface (for heterogeneous catalysis) are interrelated by complex dependencies. Therefore, for experimental investigation under nonstationary conditions, lmowledge about the gas and surface composition is required. In chemical industry, the majority of processes are conducted in stationary conditions. Hence, it is not surprising that the steady-state kinetics is mainly studied, when the surface concentration of reactants is time independent. Investigation of reaction kinetics in gradientless reactors further simplifies the experimental routine. The main drawback of kinetic models, based only on steady-state data, is associated with the fact, that start-up and transient regimes cannot be reliably modeled. Kinetic models for nonstationary conditions should be applied also for the processes in fluidized beds, reactions in riser (reactor) - regenerator units with catalyst circulation, as well as for various environmental applications of heterogeneous catalysis, when the composition of the treated gas changes continuously. 8.2 R e l a x a t i o n
methods
The conventional steady-state flow methods that are used widely to investigate the kinetics of heterogeneous catalytic reactions are limited because under steady-state conditions, the
289 elementary steps proceeding in series take place at the same rate, and the overall kinetics provides somewhat limited insight into the mechanism. The relaxation experiments could be arranged in the following way. A system at the steadystate (or equilibrium) is perturbed by an external influence. The rate of relaxation to the steady-state (equilibrium) is measured. The Nobel Prize in chemistry was awarded to M. Eigen for the development of the group of methods to study fast reactions by monitoring the system responses to controlled disturbances.
Manfred Eigen In the flow technique the reactants are injected in a mixing chamber at a steady rate. The location of the movable spectrometer con'esponds to different times after reaction initiation. In the stopped-flow technique, reagents are driven to the mixing chamber and the time dependence of concentrations is monitored (Figure 8.1).
ONi~ Path
A
Dri~ Single M~x~ngStopped-FIc~+
Figure 8.1. Single mixingstopped flow (www.hi-techsci.com_) Temperature jump is another relaxation methods used for rapid reactions. The procedure for T-jump includes the following steps: let the system equilibrate at TI, increase the temperature instantaneously to T2 and then follow the change in the concentrations of the reactants and products (Figure 8.2). T~tu~
Figure 8.2. Temperature-jump(M.Eigen, Immeasurably fast reactions, Nobel Lecture, December 1 l, 1967)
29O For the reaction A ¢>Z we can write
(8.20)
d x / d t = k+ ( a o - x ) - k x
where ao is the initial concentration ofA. In equilibrium
(8.21)
dG,; / d t = k+ ( a o - G,; ) - k x ,.q = 0
Substraction gives d ( x - x~,;) / d t = - k + ( x - x~,,) - k_ ( x - X,q)
(8.22)
After defining x - x~q = A:c we have (8.23)
d A x / d t = - ( k + + k_ )zk~c
Integration of (8.23) leads to
(8.24)
ln(Ax / Ax0) = -(k+ + k ) t where Ax0 = a 0 - Ax. Defining the relaxation time as r
t, at which Ax
Axo/e, gives
r = (k+ + k_) '
(8.25)
Similar analysis of the reaction A+B <:vC gives Z~
1 =
k + ( G q +b,,q) + k _
(8.26)
and for the system in which both the forward and the reverse reactions are second order A+B ~=>C+D r
1
=
k+(ae, 1 +b,~l)+k(c~,~ l +d,~l )
(8.27)
The basic ideas of the transient method applied to heterogeneous catalysis were set forth by K. Tamaru. In general, the term "transient" refers to changing one or more of the system parameters. In transient kinetic studies, a dynamic change is introduced into a reactor system, and the response of a reaction quantity is observed. A typical transient response experiment is sketched in Figure 8.3. Transient state methods have applications in reactor modeling, optimization and control. By measuring the concentrations as a function of time at the catalytic reactor outlet qualitative conclusions concerning the surface reaction mechanisms can be drawn and the kinetic parameters included in the rate equations for the surface reaction steps can be estimated from the data by using non-linear regression analysis. Transient techniques have been recognized as useful tools for studying the mechanisms of heterogeneous catalysis. They give more information about the surface reaction mechanism than is deducible from an equivalent
291 number o f stationary measurements and they are useful, especially for preliminary screening to definitely limit the number o f possible mechanisms that are selected for a further evaluation. J transier~ J~
.~
steady state
~ J~ transient state
concentration change
o"
8 concentration change Time
Figure 8.3. A transient response experiment. In order to obtain accurate predictions o f reactant's conversions, a dynamic reactor model is required, i.e. a model, which allows the calculation o f the accumulation o f gas and the surface components. Solving the dynamic reaction model equations requires a very detailed knowledge o f the reaction kinetics involved. The transient methods can be classified into two main groups: the first one is when, due to the perturbation, the system has been transformed into another thermodynamic state. For example, the classical temperature program desorption (TPD) and concentration jump methods belong to this group. The second group is when the system remains in the same thermodynamic state during the transient experiment. This means different kinds o f labelling techniques. Several transient techniques are available, for example, step response experiments, pulse experiments, cycling feeds, isotope exchange experiments and non-isothermal experiments, such as TPD, temperature programmed reduction (TPR) and oxidation (TPO) as well as temperature programmed surface reaction (TPRS). The simplest way o f changing the concentration is to can T out a step-response experiment. When using the concentration step-response method, a deduction about the surface reaction mechanism can be made from the shape o f the response curves. The characterization o f some methods is provided in Table 8.1 Table 8.1. Characteristic features of transient techniques Transient techniques
Instruments
Characteristic features
Step response experiment
MS / rapid GC
Inlet paralneters are suddenly changed, such as c, T and P
Pulse experiment
MS / GC
Inlet is regulated by several groups of pulse inputs with the same time interval
Temperature programmed desorption
MS
Desorption is proceeded after adsorption at room temperature
Temperature programmed surface reaction Isotope exchange experiments
MS
T is changed after each steady-state
MS
A molecule is replaced by its isotope
Transient infrared studies
FT-IR
Analysis of the gas composition not only in the bulk phase but also on catalyst surface
MS = mass spectrometer, GC = gas chromatography, c = concentration, T = temperature, P = pressure, FT-IR = Fourier transform infrared spectroscopy
292
Isotopes are elements that have the same atomic number but different mass number divided in two categories: radioactive and stable isotopes. Due to their similarities in their behavior, isotopes are considered practically identical in their chemical behavior. It is possible to label particular molecules by isotopes and follow them through a sequence of physical or chemical changes. This method is called the isotope tracer technique and it has earlier been carried out with radioactive isotopes, but nowadays, thanks to the development of high resolution mass spectrometry, stable isotopes are used offering more facilities for investigation. The isotopic mass of an atom in the reactant molecule influences the reaction rate as well as the equilibrium. The change in the reaction rate with the substitution of an isotope is called the kinetic isotope effect and is distinguished from changes of equilibrium constants, called equilibrium isotope effects. In chemical reaction kinetics, isotope-labelled reactants are frequently employed to follow a reaction pathway and to determine the reaction mechanism (see Chapter 7.6). The isotopic tracer technique is a useful tool in catalyst surface analysis, because it enables determination of whether the adsorbed species present on the surface during the reaction are by-products or reaction intermediates. One of the adsorbed species is labelled by an isotope atom and its rate of disappearance is followed by surthce spectroscopy. Simultaneously, its rate of appearance in the product molecule is followed by mass spectrometry. When both rates are identical, it can be concluded that the observed adsorbed species is the reaction intermediate. Steady-state isotopic transient kinetic analysis (SSITKA) involves the replacement of a reactant by its isotopically labelled counterpart, typically in the form of a step or pulse input function. Producing an input function with isotope-labelled reactants permits the monitoring of isotopic transient responses, while maintaining the total concentration of labelled plus nonlabelled reactants, adsorbates, and products at steady-state conditions. It is assumed that there are no effects due to differences in kinetic behavior of the isotopic species from unmarked atomic species. However, for instance, deuterium substitution exhibits isotopic effects that can not be neglected. Chemical relaxation techniques have been employed to study the rates of elementary reaction steps. The two most useful variables for the system control are the concentrations of the reactants and the reactor temperature. The dynamic responses from the system after the changes of these variables are related to the elementary steps of the catalytic processes. Chemical relaxation techniques can be divided into two general groups, which are single cycle transient analysis (SCTA) and multiple cycle transient analysis (MCTA). In SCTA, the reaction system relaxes to a new steady-state and analysis of this transition furnishes information about intermediate species. In MCTA, the system is periodically switched between two steady-states, e.g. by periodically changing the reactant concentration. The apparatus for the step-response experiments can consist of gas inlet, reactor and gas analysis sections (Figure 8.4). The catalyst is placed in a cylindrical glass tube, through which the reactant gases flow-. A thermoelement could be installed at the exit of the catalyst bed. The gas flows are regulated with mass flow controllers. A small portion of the product flow can be taken through a capillary into a quadrupole mass spectrometer, and the mass numbers of interest can be monitored. The system can react to changes in the flow composition or temperature jump. A pulse method has also been introduced in heterogeneous catalysis and is used extensively. The transient flow reactor techniques that is frequently employed include the introduction of reactant as pulses, which provides unsteady state information, and step changes in reactant concentration, which provide intbrmation about the transient process fi'om one steady state to another. The pulse method is generally less informative than the step change method in
293 identifying the rate-determining step of a catalytic reaction. The pulse transient starts and ends with the same stationary state, whereas the step transient ends at a new stationary state, thus providing additional information. However, in the conventional step-change transient mode, with a continuous reactant flow system, the step change of a reactant concentration (usually from a high value to zero) may result in a significant change in the catalyst temperature. This is most pronounced for highly exo- or endothermic reactions. Therefore, it may be difficult to hold the catalyst temperature constant during a step change. In contrast, in the pulse mode, the amount of reactant employed in each pulse may be so small that only a negligible change in the catalyst temperature occurs. Furthermore, in the common pulse techniques, the pulse width is in the range of 1-100 s. It is about 1 ms in the temporal analysis of products (TAP) technique. Because the pulses are so small, it is difficult to extract additional information by making a step change in the reactant concentrations during the regular pulse or by introducing a pulse of an isotope. The combination of transient and isotope methods has provided much useful insight into catalytic reaction mechanisms.
T
T
Figure 8.4. Experimental set-up for the step-response experiments (l=oven, 2=reactor, 3=catalyst).
N20 decomposition As an example of an application of transient analysis we will discuss self-decomposition of N20 to N2 and 02 over Rh/A1203. The decomposition of N20 as a function of temperature on Rh/A1203 is presented in Figure 8.5.
0.012 N20
0.01
N2
.~ 0.008 0.006
02
0,004 0.002 0 100
150
200
250
300
350
400
450
Temperature, ° C
Figure 8.5. Decompositionof N20 as a function of temperature on Rh/A1203
294
Transient responses over the same catalyst at two temperatures are given in Figure 8.6 and 8.7. Mole
0.010
fraction
-
0.009 "
N20 0.000
0.007 "
0.006 -
0.005 "
0.004 -
0.003
0.002 -
0.001 -
0.000
1
1
1
200
400
600
Time,
s
Figure 8.6. 1%N20/Ar step-response experimenton Rh/AI203 at 250°C.
O.OLO O.OO9
/
o.oos o.oov .~
o.oo6
O.OO4 0.003
-
0.002
-
~2 O
o.OOl o.ooo
-
-
1 2011
7
7
4DII
Time,
6[HI s
Figure 8.7.1% N20/Ar step-responseexperimento n Rh/A1203 at 350°C. Based on the transient experiments, the following surface reaction mechanism was proposed. N20 is adsorbed on a single empty site: N20 +* ~ N20*. The mechanism assumes splitting of the N-O bond in N20* to form N2 and O*: N20*--->N2+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)<0, 0 < z < l
(8.39)
The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation sot~ware used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over Rh/A1203 (Figure 8.8) demonstrates how adequate the mechanistic model is. OOl u.o~
0.007
~" ~ "~
o o~ o .o~ o.m~
....... o o
'--~
o.m2
Figure 8.8. Comparison between experimental data and model simulations for N20/Ar step responses over Rh/AI203 (K. Rahkamaa-Tolonen,Investigationof catalytic NOx reduction with transient techniques, isotopic exchange and FT-IR spectroscopy,2001, PhD Thesis, Abo Akademi University).
tt2/D: exchange As an example, in isotope exchange we will consider the dynamics of the isotope exchange between hydrogen and deuterium over a silver catalyst, supported on alumina. In general, hydrogen dissociates easily over metallic surface, but reports on nondissociative adsorption of hydrogen on Ag systems m'e available in the literature. Hydrogen is
297 initially pre-adsorbed on the surface. Once deuterium is introduced to the reactor and hydrogen taken to the by-pass, the isotopic exchange takes place with the hydrogen atoms already adsorbed, and formation of HD is immediately observed (Figure 8.9). 0.012 0.01 .~ 0.008 "5 0.006 ,,Re o 0.004 0.002 0 1365
1415
1465
1515 Time (s)
1565
1615
Figure 8.9. H2-D 2 isotope exchange over Ag/alumina catalyst (H. Backman, J. Jensen, F. Klingstedt, J. W~rn& T.Salmi, D.Yu. Murzin, Kinetics and modeling of Hz/D2exchange over Ag/A1203,Applied Catalysis A. General, 273 (2004) 303) At this point the formation of HD starts to grow until a maximum, after which it decreases in a short period of time. The production of water could be noticed in the first and third step of the experiment, which can be attributed to the reaction of chemisorbed hydrogen with OHgroups present in the alumina support and oxide precursor. This can, however, be considered as negligible because the molar fraction value of water is extremely small compared to the values of H2 and D2. Different reaction mechanisms for the H2/D2 exchange have been reported in the literature. Bonhoeffer and Farkas proposed a mechanism involving dissociation of hydrogen on the metal surthce and chemisorbed atoms link up randomly in pairs and evaporate as molecules. An alternative mechanism was proposed by Rideal where he stated that interaction takes place between a chemisorbed atom and physisorbed molecules, according to a exchange reaction between H* and D2 to fo1Tn HD and D*. The fast response of HD obtained in the experimental result indicates that Rideal is pathway is probably the one followed. It can be thought that at the beginning of the experiment, when the surface was covered by H*, the exchange reaction was enhanced due to the lack of free active sites for deuterium to dissociate. Once there were more vacant sites, it was possible to form D* to react with the H* atoms. 1. H, +D2 ~ D , + H D 2. D , + H 2 ~ H , + H D 3. H2 + 2,
-~ 2H,
4. D2 + 2,
~ 2D,
5. H , + D ,
--,HD+2,
(8.40)
To clarify the reaction mechanism and to reveal the importance of the steps included in the proposed mechanism, parameter estimation and modeling should be performed. The proposed reaction mechanism includes adsorption, dissociation, surface reaction and desorption steps. The adsorbed species are denoted by H,, D, and HD, where * is an active empty site on the surface. The rates of the elementary steps are given by the following equations tbr the gas phase species and surface species respectively
298
RH2
= - r 2 + I" 2 -
Rm)
= rI -
F 3 4- F 3
r I +r 2-r
(8.41)
2 +r 5
RD2 = - r 1 + F 1 - - t~ 4- F 4
and
R, =-2.r~
+2.r 3 -2.r 4 +2.r 4 +2.r 5
RD, =/~1 - r-t - r2 +r_2 +2. r 4 - 2. r_ 4 - G R~. = - r ~ -}-F 1 + r 2 - r 2 +2.15 - 2 . r
(8.42)
3 -r 5
where the single steps are expressed as:
r j = k t.cz) 2.c.,
r~ =kl.C~),.cHj ~
t5 = k? • c.~ • el),
F_ 2 = k _ 2 . C H * • C H D
= k 3 • CH2 • c , 2
r_3 = k_~ • c11,2
r 4 = k 4 • cr) 2 • c. 2
t'_ 4 = k_ 4 • CD, 2
(8.43)
t~ = k 5 . c H , .cz~,
In these equations the reaction rate is denoted by ri, concentrations by ci, the rate constant by k~ and adsorbed species by *. The concentration o f vacant sites is expressed as c,. The kinetic analysis o f a reaction can be performed with the use o f model fitting and non-linear regression analysis. The transient step-responses could be described quantitatively with a dynamic plug flow model as discussed above. A comparison between the experimental data and the simulations is given in Figure 8.10. C ^^*e O O
o O
0 0
e O
I~me Is) Figure 8.10. Comparison between the experimental data and the shnulations tbr H2-D2 isotope exchange over Ag/alumina catalyst (H. Baclcman, J. Jensen, F. Klingstedt, J. W~irn~, T.Salmi, D.Yu. Murzin, Kinetics and modeling of H2/D~ exchange over Ag/A1203, A p p l i e d Catalysis A. General, 273 (2004) 303) Typical responses
Some typical responses were analyzed by Kobayashi and are given in Table 8.2.
299 Table 8.2. Typical responses (M. Kobayashi, Characterization of transient response curves in heterogeneous catalysis--I. Classification of the curves. ChemicalEngineering Science, 37 (1982) 393). Instantaneous
0
a)Surface reaction b)Adsorption
t
Monotonic
c)Desorption or combination of a) and c); b) and c)
0
t
Overshoot response
0
d)Regeneration of active surface species e)Competitive adsorption of reaction components
t
h)Presence of some stable intermediates
S-shape response /
/
/
Three-phase systems To summarize, transient kinetic experiments are an established and valuable tool in the investigation o f heterogeneously catalysed gas phase reactions. For liquid-phase systems, transient studies are m u c h more rare than for gas-phase systems. It is probably related to slower dynamics and the fact that the intrinsic kinetic p h e n o m e n a can be obscured by mass transfer effects and catalyst deactivation. As an illustration (Figure 8.1 1) we will consider three-phase continuous hydrogenation o f an organic c o m p o u n d leading to two products over a metal catalyst on a structured support (knitted silica). H2 H~
/ Cata,yst boa
Figure 8.11. Three-~hase continuous hydrogenation of an organic compound over a metal catalyst on a structured support E. Toukoniitty, J. Warngt, T. Salmi, P. M~ki-Arvela, D.Yu. Murzin, Application of transient methods in three-phase catalysis: hydrogenation of a dione in a catalytic plate column, Catalysis Today, 79-80 (2003) 383).
300 The tube reactor containing the catalyst layers was described with a dynamic axial dispersion model. The hydrogenation kinetics was independent of hydrogen partial pressure, i.e. the concentration of dissolved hydrogen, therefore the mass balance of hydrogen in the gas phase can be neglected. The reaction temperatures are low (25°C), which implies that the volatilities of the organic compounds are negligible. Thus, just the mass balances for the liquid phase remain. According to the concept of the dynamic axial dispersion model, the mass balance equation ibr an organic compound in an infinitesimal volume element (AV) can be written as
dl
) i.
nLi'°"
dl
) o,,~
After introducing the notations =
w,
=
Q/A, r ,
= L/w,,,
A V e = A A 1 , V~ = A L , z = l / L ,
AQ
= e
AG,
(8.45)
P e L = w L / D , m<~,~ = p ~ A V L
where A is the cross section area, D is the dispersion coefficient, 1 is the reactor length coordinate, L is the reactor length, L = (c0*S)-1, h is the molar flow rate, Pe is the Peclet number, S is the specific surface area, /) is the volumetric flow rate, VR reactor volume, ttime, w-flow velocity, z-dimensionless reactor length coordinate, s is the liquid hold-up, "c is the residence time, and p is the catalyst bulk density. Carrying out the differentiations and letting the volume element shrink, the mass balance becomes dcLi dt - riPn +
(pei~rvt~), d2CLi dz 2
(~Irr~
)1 d(~'[d dz
(8.46)
m:i:::::::~:::::~:~:~:~:~:~:~i~i~i~:~¸¸:~i~:i~:~:~:::
Jean Claude Eugene Peclet For the adsorbed compounds, the mass balance equations are simply given by the expression dc~
_ S
(8.47)
IK/
dt
where ci* is the concentration of the surface species and S is the specific surihce area of the catalyst. By introducing the fractional coverages (~= cj/Co*), the balance becomes
do, dt
- Lri
'
L =
(8.48;
301
The balance contains just two adjustable hydrodynamic parameters, rL*CL and Pe~. The Peclet number is estimated from the separate impulse experiments carried out with the inert tracer (NaCl), while the quantity rL*~L is estimated from the kinetic experiments in order to ensure a correct description of the reactor dynamics. The flow pattern of the reactor is characterised by separate impulse experiments with an inert tracer component injecting the tracer at the reactor inlet and measuring in this case the conductivity response at the outlet of the reactor with a conductivity cell operated at atmospheric pressure. In order to get a proper conductivity response, water was employed as the liquid phase. The liquid and hydrogen flow rates should be the same as in the hydrogenation experiments and the liquid hold up was evaluated by weighing the reactor. Some results from the tracer experiments are given in Figure 8.12.
E ( t ) "~
0
05
1
15
2
25
3
35
4
t/'~
Figure 8.12. Tracer experiments in a three phase tubular reactor with knitted silica catalyst layers (E. Toukoniitty, J. Wfirnfi,T. Salmi, P. Mfiki-Arvela,D.Yu. Murzin, Application of transient methods in three-phase catalysis: hydrogenation of a dione in a catalytic plate column, Catalysis Today, 79-80 (2003) 383). As the Figure 8.12 reveals, the flow pattern deviates from plug flow. The residence time distribution function E(I) is calculated from the experimentally recorded responses, after which the F(t) function was obtained from integration of E(t). The experimental functions are compared to the theoretical ones. The expressions of E(t) and F(t) obtained from the analytical solution of the dynamic, non-reactive axial dispersion model with closed Danckwerts' boundary conditions were used in comparison. A comparison of the results shown in Figure 8.12 suggests that a reasonable value for the Peclet number is Pe=3. A fit of the model based on adsorption quasi-equilibria which describes very well the batch reactor data is not sufficient to describe the transient behaviour of the system (Figure 8.13).
0014& .
0 0
0
~
0
0
0
n
u
o
o
o
o
o
so
60
0 008 C o
0 004
°0°0°020 ............... 0
10
20
30
o
o
40
70
810
time
Figure 8.13. A fit of the model based on adsorption quasi-equilibria to transient data (E. Toukoniitty, J. W~irnfi, T. Sahni, P. Mfiki-Arvela, D.Yu. Murzin, Application of transient methods in three-phase catalysis: hydrogenation ofa dione in a catalytic plate column, Catalysis Today, 79-80 (2003) 383)
302 The assumption of adsorption quasi-equilibrium predicts responses which are too rapid, indicating that the sorption dynamics of the compounds is of crucial importance. When the kinetics of adsorption is included in the model, the fit of the model is significantly improved (Figure 8.14).
/ / 4,
¢.-9c
10
20
30 tim40
50
60
70
80
Figure 8.14. A fit of the kinetic model which includes sorption dynamics (E. Toukoniitty, J. W~rnfi, T. Salmi, P. M~ki-Arvela, D.Yu. Murzin, Application of transient methods in three-phase catalysis: hydrogenation of a dione in a catalytic plate column, Catalysis Today, 79-80 (2003) 383) SSITKA- compartment model In SSITKA (steady-state isotopic transient kinetic analysis) developed and actively applied by Happel, Biloen and Goodwin, it is common to consider the catalyst surface to be composed of a system of interconnected pools, also termed compartments, where each pool represents a homogeneous or well-mixed subsystem within the reaction pathway.
James G. Goodwin A separate pool is assumed to exist for each unique adsorbed reaction- intermediate species or type of catalytically active site. it is assumed that there is essentially no mixing or holdup time associated with each pool or within the reaction pathway except for the residence time of a reaction-intermediate species adsorbed on the catalyst surface.
R
P
Figure 8.15. Concept of sepm'ate pools in series. (S. L.Shannon, J. G. Goodwin, Characterization of catalytic surfaces by isotopic-transient kinetics during steady-statereaction, ChemicalReviews, 95 (1995) 677). Figure 8.15 presents this concept where n number of intermediate pools in series represents the steady-state reaction pathway. Here N,~ is the steady-state abundance of the number of atoms of the isotope in each individual i-th pool (i = 1,2 ..... n) where the overbar, "-", I
indicates a steady-state condition and r f species R due to adsorption.
is the steady-state adsorption rates of reactant
303 Now consider (Figure 8.16) that an instantaneous and complete switch occurs at time, / = 0, in the isotopic labeling of species R, R ~ * R , at the inlet of the reaction volume as indicated by *R.
*R
4
7/'
p
Figure 8.16. Introduction of a isotopic label (S. L.Shannon, J. G. Goodwin, Characterization of catalytic surfaces by isotopic-transient kinetics during steady-statereaction, ChemicalReviews, 95 (1995) 677). The new isotopic label in reactant species R progresses along the reaction pathway via reaction intermediates A, resulting in a transient condition in the isotopic labelling for product species P, P ~ * P . Ultimately, all of the old isotopic label is displaced by the new isotopic label as indicated in Figure 8.17.
*R
*P
Figure 8.17. Final displacement of the old isotopic label by the new one. (S. L.Shannon, J. G. Goodwin, Characterization of catalytic surthces by isotopic-transient kinetics during steady-state reaction, Chemical Reviews, 95 (1995) 677). The distribution of the isotopic labels is dependent upon the steady-state transfer rates, between pools. Analysis of SSITKA is rather complicated and is outside of the scope of the present text. it can be summarized, that SSITKA is a powerful technique to determine in situ kinetic information about the catalyst-surface reaction intermediates and mechanisms at steady state without substantial disturbance of the catalyst-surface behavior, contrary to some other transient techniques.
Temporal analysis of products The temporal analysis of products reactor system (TAP) developed recently by J. Gleaves is another technique to study fast responses.
John T. Gleaves The key feature which distinguishes it from other pulse experiments is that no carrier gas is used and gas transport is the result of a pressure gradient. A TAP pulse response experiment is a special type of transient response experiment that involves injecting a small gas pulse of very short duration into an evacuated microreactor containing a packed bed of particles. When the number of molecules in the pulse is sufficiently small (~ 10 ~0 mol) convective flow disappears, gas transport occurs by Knudsen diffusion, and the molecules move through the
304 bed independently. The kinetic regime of a TAP Knudsen pulse response experiment is located at the boundary of the surface science regime. Theoretical analysis of TAP-pulse response data is based on the reactor model that corresponds to the particular reactor configuration that is used. Either the one-zone reactor model, which represents a reactor uniformly packed with catalyst particles, and uniformly heated over its entire length, or a model corresponding to two packed zones, the first of which contains an inert material and the second an active catalyst sample, could be used. More complicated is the three-zone reactor configuration that contains three adjacent packed zones, with the center zone containing the active material, and the outside zones containing inert material (Figure 8.18).
T
~nel
~n~2
he3
Vacuum I 0 ~ 8 t~r
Figure 8.18. Three-zone TAP reactor (D. Constales, G. S. Yablonsky, G. B. Marin and J. T. Gleaves, Multi-zone TAP-reactors theory and application, i11 Multi-response theory and criteria of instantaneousness, Chemical Engineering Science, 59 (2004) 3725). The pulse residence time under vacuum conditions is much shorter than in conventional pulse experiments. Thus, a high time resolution is achievable. TAP pulse transients are capable of providing information about short-lived reaction intermediates, which are not detectable with a conventional step-transient technique. Therefore, transient experiments using TAP pulsing are valuable mostly for the detection and identification of reaction intermediates to ~brmulate a reaction mechanism.
8.3. Temperature programmed desorption In temperature programmed desorption, the preadsorbed species are heated up at a certain rate. Desorption is performed in a vacuum. Once a molecule has desorbed it is rapidly removed from the reactor by the pumps and is analyzed by mass spectrometry (Figure 8.19) Vacuum Chamber C~al
T(t)
= To
Figure 8.19. Schematic of temperature programmed desorption (TPD) The pressure is always very low and re-adsorption can be neglected for this reason. Increasing T causes the rate constant for desorption to increase smoothly but rapidly tbllowing the Arrhenius exponential dependence When the rate constant is sufficiently high, the desorption rate becomes significant and the coverage drops to zero as all molecules have left the surface. The area under a TPD peal, is proportional to the coverage. The desorption rate goes through a maximum (Figure 8.20), as
305 at low T the coverage is high, but the rate constant is low and the desorption rate is low. As the temperature increases the rate constant and thus desorption rate increase. At intermediate temperatures, the desorption rate is high as both the coverage and the rate constant are high. At higher T, the coverage is virtually zero and although the rate constant is enormous, the desorption rate is virually zero
Temperature (T) / K Figure 8.20. Dependence of the desorbed amount on temperature
Temperature programmed desorption can be used in order to calculate the activation energy of desorption. The desorption rate is r~.~ -
dO ~j~/Rr -- kCrca~tat,,e dt
(8.49)
where surface reactant concentration is in fact coverage. For a n-th order reaction (8.50)
d O _ kO,,e_A~.~, ~e7 dt
Often, first order desorption kinetics is assumed. For a linear heating rate, the temperature is given by (Figure 8.21) (8.51)
T =T O +fit
where 13is the heating rate [3 - dT/dt
(8.52)
gradient
T
t Time (t) / s Figure 8.21. Illustration of linear heating rate in TPD
306 From(8.50) and (8.52) kO -~,,~,/RT
dO -
-
--
dT
(8.53)
e
fl
It holds at the peak corresponding to the temperature d20 dT 2
Tp(Fig. 8.20) that (8.54)
-0
Differentiating - d O / d T with respect to T results in
-- d2~O : O :2F ~ d T
~
.....
+ dT
)
(8.55)
From (8.55) we arrive at dO _ 0 AEd..... dT RT 9
(8.56)
Now replacing - d O / d T in eq. (8.56) with (8.53) /~kEddS __ k
RL 2
P
e - ~des
/ R]'p
(8.57)
Thus from the measured values of temperature at maximum, it is possible to calculate the energy of desorption after rearranging (8.57) and taking logarithms (8.58)
e A~'~'''e'q'- R T g k
AEt~, ' / R T
= ln( RT~'2k )
(8.59)
Further modifications results in 1/T,-
R AE d~,,
ln(
)+
ln(
)
(8.60)
AE j~,,
Plotting the reciprocal values of temperature at maxima vs ln(T~,2 /fl) gives a straight line with the slope R / AE~o, (Figure 8.22).
307
25
24 2S
22 7
s
9
lO
11
12
hKTpS/beta)
Figure 8.22. illustration of activation energy calculations.
8.4. Oscillations
8. 4.1. Homogeneous catalysis The most studied and cited example of complex dynamic behavior in homogeneous catalysis is the Belousov - Zhabotinsky (BZ) reaction named after B. P. Belousov who discovered the reaction and A. M. Zhabotinsky who continued Belousov's early work. The reaction is theoretically important in that it shows that chemical reactions do not have to be dominated by equilibrium. These reactions are far from equilibrium and remain so for a length of time. In this sense, they provide an interesting chemical model of nonequilibrium biological phenomena, and the mathematical model of the BZ reactions themselves are of theoretical interest.
Boris Belousov
Anatol Zhabotinsky
Belousov did not succeed in publishing his results as they were extremely unusual at that time (1950s), displaying changes in color (oscillations) in a system containing a mixture of potassium bromate, cerium(IV) sulfate, and citric acid in dilute sulfuric acid, due to changes in the ratio of concentration of the cerium(W) and cerium(Ill) ions (Figure 8.23).
Figure 8.23. Color oscillations in Belousov - Zhabotinsky (BZ) reaction
308 In its classical form the BZ reaction consists of a one-electron redox catalyst, an organic substrate that can be easily brominated and oxidized, and a bromate ion in form of NaBrO3 or KBrO3 all dissolved in sulfuric or nitric acid. As catalysts, mostly Ce(III)/Ce(IV) salts, Mn(II) salts and ferroin are used. As organic substrate normally malonic acid (HOOC-CH2-COOH, MA) is used, instead of Belousov's original citric acid; he was looking for an inorganic analog of Krebbs cycle. In fact there are no violations of the second law of thermodynamics as was incorrectly suspected by the reviewers of the original contributions of Belousov and the reaction is not oscillating between the reactants and final products, but rather there are fluctuations in the concentrations of the intermediates. The mechanism of the Belousov-Zhabotinskii reaction can be considered using a simple Oregonator model. For the reaction A+B m P+Q
(8.61)
the following sequence of steps is proposed 1.A+Y ~
X
2.X+Y ~ P .B X ~ 2 X +Z 4.2X~Q 5.Z~Y
(8.62)
where X,Y and Z are intermediates. The concentrations of these intermediates are interdependent. For the intermediate X the steady-state approximation results in dx
7
= klay - k2xY + k3bx - 2k4x 2
=0
(8.63)
where x, y and z denote concentrations of respective intermediates. Solving this equation one can get two real solutions
(8.64) 4k 4
x -
(k2Y- k3b) 4k4
- 1+ 1+ _
(8.65) (k2y _ k3b) 2
When y is large, the concentration of x is determined by reactions 1 and 2, thus xvkla/k2. Then the Oregonator mechanism can take a simplified lbrm 1.A+Y ~
X
2.X+Y ~ P A+2Y ~ P
(8.66)
309 For this sequence of steps, X in a steady state consumes Y, therefore the concentration of Y falls reaching a certain limiting value (Y)lim, alter which x is determined by reactions 3 and 4 (5) in (8.62), thus x~k3b/lq and instead of (8.62) the following submechanism is operative 3.2B+2X ~ 4 X +2Z 4.2X~Q 5.2Z~2Y 2B ~ Q+2Y
(8.67)
In (8.67), X is generated, giving a rise to the concentration of X and gradually leading to a new steady state tbr X. The link between mechanisms (8.66) and (8.67) is reaction 3, where Z is formed and further regenerates Y in step 5. As the concentration of Y increases there is a switch to mechanism (8.66). It can be thus concluded, that x, y, z oscillate between maximum and minimum values, however, the reaction is not oscillating between the reactants and final products, but rather there are fluctuations in the concentrations of the intermediates. When there is no reactant available anymore the reaction stops completely, indicating that there are no violations of the second law of thermodynamics. Summarizing, the essential features of the Oregonator (Field-K6r6s-Noyes) mechanism are
Richard J. Field, Endre K6r6s and Richard M. Noyes • competition between reactions 2 and 3 for X and its dependence on the concentration of Y • consumption of Y in mechanism I • auto-catalytic reaction of X - reaction 3 • second-order consumption of X in reaction 4 • regeneration of Y in mechanism II Going from more general consideration to the chemistry of the Belousov-Zhabotinskii reaction we can split the overall reaction 3 CH2(CO2H)2 +- 4 BrO3---+4 Br- + 9 CO2 "1-6 H20
(8.68)
into several processes. In the first one, bromate ions oxidize bromide ions (Br ) to produce bromine. 1. 2. 3. 4.
BrO3- + Br- +2H + ~ HBrO2 + HOBr HBrO2+ B r + H + m 2 HOBr 3HOBr- +3Br- +3 H + ~ 3Br2 +3 H20 3B1~2+3 CH2(COp_H)p_ =:>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+ E2<---~Y E2 YE2-+Z+ E2 Y+ EZ<-->E~
(8.79)
Step 1 corresponds to conversion of glucose to F-6-P. Steps 2 and 3 describe the enzymatic conversion of F-6-P (X) to F-1,6-bP (Y) catalyzed by PFK (El) and steps 4 and 5 describe further reactions to glyceraldehyde-3-phosphate (Z) catalyzed by aldolase (E2). Step 6
314 represents the activation of PFK by F-1,6-bP. The time dependence of the concentrations is given by a set of differential equations
d[x]_ ~,Co - k~[xl[E, ] + ~_~[xE, ] dt
(8.80)
d[Y]_ k3[XEl]_ k4[Y][E2] + k_4 [YE2 ] - k6[E' Iy]+ k_6 [El ]
(8.81)
d[El ] - k2 [X ][Ell -[-k ~[XE1 l-l- N3[XEI]-[-N6[E 111/1- k_6 [El ] dt ---
(8.82)
d[Xe']-k2[xl[E,l-~ 2[xE, I- ~3[xE,]
(8.83)
dt
dt
dt
k6[g_..~][Y]-}l k 6[Eli
d[rE~]_ diE, l_ t,~[r][E~]- k ~[rE~]- k~[r~] dt
(8.84)
(8.85)
dt
Using the steady-state approximation for all of the enzymes a minimal two variable model is obtained, which shows stable steady states and stable oscillations, i.e. limit cycles. The richness of this model stems from the mechanism, which includes an enzymatic MichaelisMenten step, and is connected to the autocatalytic step (activation of PFK by its product). 8.4. 3. Heterogeneous catalysis Rate oscillations, spatiotemporal patterns and chaos, e.g. "dissipative structures" were also observed in heterogeneous catalytic reactions. If compared with pattern formation in homogeneous systems, the surface studies introduced new aspects, like anisotropic diffusion, and the possibility of global synchronization via the gas phase. Application of field electron and field ion microscopy to the study of oscillatory surface reactions provided the capability of obtaining images with near-atomic resolution. The most extensively studied reaction is CO oxidation, which is catalyzed by group VIII noble metals. Pattern formation during the catalytic CO oxidation reaction was observed on the Pt(110) catalyst.
Figure 8.26. Pattern tbrmation during the catalytic CO oxidation on the Pt(110) catalyst.(A, von Oertzen, A.S. Mikhailov, H.H. Rotermund, G. Ertl, Subsurface oxygen in the CO oxidation reaction on Pt(110): Experhnents and modeling of pattern formation, Journal ofPhysical Chemistry B, 102 (1998) 4966).
315 An example of FEM-images of the oscillating CO-oxidation reaction is given in Figure 8.27. / ,caN
C'*~1
ql' ~ .
I Tll ',
1001 ~'7-'~ ° ~ ,,,/ ~ 133 \ /
*~
- " &~01
111
o,,313 171
a)
d)
-~
b)
8s
e)
0s
19s
c)
f)
0,15 s
3Is
Figure 8.27. FEM-images of the oscillating CO-oxidation reaction on Pt ( (1 0 0) (E.I. Latkin, V. I. Elokhin, V. V. Gorodetskii, Spiral concentration waves in the Monte Carlo model of CO oxidation over Pd(l I 0) caused by synchronisation via COads diffusion between separate parts of catalytic surface. Chemical Engineering Journal, 91 (2003) 123). Oscillations were also found in a number of other oxidation reactions where H2, NH3, hydrocarbons, methanol and ethanol are oxidized by molecular oxygen. Another large class of oscillating reactions includes catalytic reduction of NO with CO, hydrogen or ammonia as reducing agents. Rate oscillations may also occur in hydrogenation reactions. Heat and mass transfer limitations might play an important role and these factors have to be taken into account properly in an analysis of the experimental data. Isothermal models used to explain oscillations in reaction rate involved: a) participation of a slow buffer step, which is not in equilibrium with the fast steps of the mechanism; b) schemes with coverage dependent activation energies associated with surface heterogeneity or lateral interactions.; c) requirement of a vacant site in a surface reaction, if either a decomposition step with more than a single product is involved or if a structural rearrangement of an adsorbed molecule has to precede the reaction step; d) island formation. Below we will discuss, as an example, only a reconstruction model for CO oxidation developed by Ertl and co-workers.
Gerhard Ertl For single crystals, the driving tbrce tbr reconstruction of clean surfaces is minimization of the surface energy. The Pt (110) surface is reconstructed from a 1 x 1 to 1 x 2 missing row (Figure 8.28).
316 pt {1tol CO covered
¢teoa
tO011
1.1
~
I
*2
Figure 8.28. Reconstruction ofPt(110) surface.
Relative stability can be switched if an adsorbate is more strongly bound to the "clean" surface than to the "reconstructed". The sticking coefficient of oxygen depends on the surface. Thus for a surface covered by CO the 1 x 1 adsorption rate of oxygen is high leading to high catalytic activity. When more CO is consumed than supplied, coverage decreases below a certain critical value 0CO,crit-0.2, which leads to surface reconstruction from 1 x 1 to 1 x 2. For the latter surface the sticking coefficient of oxygen is lower and the CO coverage is increased, increase of CO coverage occurs up to a certain critical value 0co<~it -0.2, after which the surface reconstruction back from 1 x 2 to 1 x 1 takes place. Translating these mechanistic considerations into mathematical expressions one gets
dOc° - k l p ( ' ° I I - ( ~ )c3o]s-
E
doo -g -
1
G,,,o,
k20( - ksO( °O°
oo
12
- k o oo.
(8.86)
(8.87)
with ,-~
Ix2
S02 • O'lxlSO2 Jr- 01, 2
(8.88)
The amount of 1 x 1 surface is calculated from dOix ] - - dt
ksO, x ~
dOl~1 dt •
s - ks(~
q, 3 0~: i o - 0,,,)
for 0co < 0.2
(8.89)
for 0.2 ___Oco ___0.5
(8.90)
tbr 0co _ 0.5
(8.91)
i=0
dO'x1 dt
k~(1- 01~,.1)
A polynomial expression is used to reproduce experimentally observed behavior. In (8.86)(8.91) kl, k2, k4 describe adsorption, desorption of CO and adsorption of oxygen respectively.
317 The kinetic constant, k3, corresponds to the reaction between CO and oxygen, while the phase transition of the 1 x 1 to 1 x 2 is expressed by eqns. (8.89-8.91).
8.5. Activity changes (deactivation) 8. 5.1. Heterogeneous catalysis The majority of catalysts are subject to deactivation, e.g. to changes (deterioration) of activity with operation time. The time scale of deactivation depends on the type of process and can vary from a few seconds, as in fluid catalytic cracking (FCC), to several years, as in, for instance, ammonia synthesis. Due to the industrial importance, the modelling of deactivation was mainly developed for heterogeneous catalysis. Although the reasons for deactivation (inactivation) of homogeneous and enzymes could differ from solid catalysts, the mathematical approach can sometimes be very similar. The main causes of deactivation in heterogeneous catalysis are poisoning, fouling, thermal degradation (sintering, evaporation) initiated by the often high temperature, mechanical damage and con'osion/leaching by the reaction mixture (Figure 8.29).
iii, S
, ~,~,,,~, ,
~,,
:;:,: :~:;:,,:;: ,;:,: :z, :~:; :~;;,,:z,,,:,:; ,z: :z:~i:i:
0 ~
t~ ~ s
~
Figure 8.29. Deactivationmechanisms.(J.A. Moulijn, A.E. van Diepen, F. Kapteijn, Catalyst deactivation: is it predictable?: What to do? Applied Catalysis A: General, 212 (2001) 3). Poisoning is defined as deactivation by strong adsorption of, usually, impurities in the feed. Poisoning depends upon adsorption strength of such poisons relative to the other species competing for catalytic sites. Adsorbed poisons may not only block active sites, but change the electronic or geometric structure of the surfaces as well. Surface restructuring by poisons can also occur. The poison may effect the nearest neighbor metal atoms and possibly its next nearest neighbor atoms and modifying their abilities to adsorb and/or dissociate reactant molecules. Fouling is associated with surface covering by a certain deposit, which is quite often hydrogen deficient carbonaceous material (i.e. coke). Such coking makes the active sites inaccessible. Catalysis leaching is extremely important particularly for liquid-phase reactions. Often leached metals are responsible for the high catalytic activity in some reactions, which are considered to be heterogeneous catalytic, but are in fact homogeneous. Leaching is particularly a problem in oxidation catalysis owing to the strong complexation and solvolytic properties of oxidants (H202, RO2H) and/or the products (H20, ROH, RCO2H, etc.). Mechanical deactivation is due to strong stresses of packed catalysts beds during start ups, shut-downs and catalyst regeneration.
318 Thermal degradation is a physical process leading to catalyst deactivation because of sintering, chemical transformations, evaporation, etc. Sintering is the loss of catalyst active surface due to crystallite growth of either the support material or the active phase. Due to the industrial importance of deactivation, various kinetic models that account for deactivation have been advanced in the literature. Probably the most frequently used approach, is based on different empirical and semi-empirical equations. However, increasingly, the nature of deactivation is considered as a constituent part of the reaction scheme. This approach provides more possibilities for elucidating deactivation mechanisms, which should be an essential part of any new catalyst development. In the simplest case, the catalytic activity is proportional to the number of active sites NT, intrinsic rate constant and the effectiveness factor. Catalyst deactivation can be caused by a decrease in the number of active sites, changes in the intrinsic rate constant, e.g. changes in the ability of surface sites to promote catalysis and by degradation in accessibility of the pore space. When the reaction and deactivation rates are of different magnitudes, the reactions proceed in seconds while the deactivation can require hours, days or months, and moreover the deactivation does not affect the selectivity, the concept of separability is applied. The reaction rates and deactivation are treated by different equations. A quantity called activity, (a) is introduced to account for changes during the reaction. In the general principles of kinetic modelling, the reaction rate is defined as a function of the intensive variables in the system, typically the concentrations of the components involved and the temperature. In a deactivating system, the initial intrinsic rate on the fresh nondeactivated catalyst is I"o = ft (ci ,T)
(8.92)
The rate over a deactivated catalyst is r t = f2 (c,,T)
(8.93)
The activity is defined as the current reaction rate divided by the initial intrinsic rate, a = r~ / r o
(8.94)
By carrying out experiments in a fixed bed reactor activity changes are observed with timeon-stream. The initial rate ro remains unknown, and one has to extrapolate to zero time to obtain this value. This is not always easy, particularly tbr rapid deactivation, because of the influence of the starting up procedure of the experiment. Assuming that the loss of catalytic activity is due to the decrease of active sites, we can define the fraction of active sites, (a) as a = N, / N O
(8.95)
where Nt is the number of active sites on the deactivated catalyst at time (t) and No is the number of active sites on the fresh catalyst. More generally, the structure of kinetic equations for activity was proposed by Levenspiel to have the following form
319
da _ f ( c , , T)q)(a) dt
(8.96)
where ~o(a) is a function of catalyst activity, e.g. deactivation function. Different equations have been proposed in the literature for the deactivation function, for instance, a power -law one q~(a) = a"
(8.97)
The deactivation functions of the various reactions depend upon their mechanisms, e.g. on the number of sites involved in the rate-determining steps of the main reaction (m) and deactivation (h):
da - f(c,,T)ad; dt
h-1 d = 1+-m
(8.98)
For the linear mechanisms eq. (8.96) could be transformed to
da _ .fo ( c , , T ) f ( A , T ) f , ( a ) dt where Jo(ci,T)
(8.99)
is a function of only the main reaction, f ( c i , T ) is a function which is
determined by deactivation kinetics and f , (a)is the catalyst activity function. Equation (8.99) is an example of separable kinetics, e.g. the deactivation function can be separated from the activity function. Often deactivation is expressed in terms of time. This is not the true variable, as it could lead to incomplete predictions. More correctly, the deactivation function has to be expressed in terms of the deactivating agent: the coke precursor or the poison, which means that the amount of coke (or poison) on the catalyst site should be known. The determination of a rate equation for the formation of the coke precursor is thus an integral part of the kinetic study of the process. For the formal treatment of deactivation in the case of separable kinetics we will define the fraction of non-coked or non-poisoned catalyst ( f ) in terms of the concentration of the poison or coke on the catalyst surface (ce) and the catalyst capacity for coke or poison (cp, o).
f _ ce, o - cp Cp,o
(8.100)
Accumulation of coke on the catalyst is given by
de> _ R~(c~, p , T , W H S V ) dt
(8.101)
where WHSV stands for weight hour space velocity for fixed bed operation in continuous mode. Separating activity and deactivation we get
320
ac. R.(c.)O(p,T,WHSV)
(8.102)
_
dt
Assuming a power law for the dependence of Rp on coke concentration, equation (8.102) takes the following form
at> _ &,,.o -c,,)" o(p, r, w m v )
(8.1o3)
dt
which means that the rate of coke formation is proportional to some power (n) of unused capacity for adsorbing coke. Combining equations (8.100) and (8.102) the time derivative o f f is obtained. df _ dt
(8.104)
k c,",-~f"~b(p, T, W H S V )
Integrating for constant values of Cp.o and f gives linear exponential and hyperbolic relationships between fraction of coke and time l
f:
i,
l+(n
f = exp(-
- "~" U K c , ," '0 t
1
k'c,",,-~It)
n~l
(8.1o5)
n=1
where k' = k ~b(p, T, W H S V ) . Observe that k' is only an apparent constant, as it depends on temperature, WHSV and the reactant pressure p. Combining (8.105) with (8.97) one gets
.=[1 11'-'1 -J- (r/ - .~1' ~7 1 U K cp, o t
'n-It)
a = exp(- mk cp, o
n~l (8.106) n=l
Introducing Y = m / ( n - 1)
(8.107)
the following forms of the activity functions are obtained
.
n>l,m>O
(8.1o8) .2 = O - k 2 ; ) "
OO
321 8.5.1.1. Coking
Knowledge of deactivation can be more explicitly taken into account. We first consider a scheme for the reaction A + * <--->A* <--->B* <--->B + *
(8.109)
$1" S* which can be expressed by a mechanism depicted in Figure 8.30.
Figure 8.30. Mechanism of deactivation. Here 0s denotes the fraction of catalyst surface covered by coke.
In general, the coke (or carbon deposit) could be produced from any surface species, not only A*. Deactivation rs and self-regeneration r_s proceed simultaneously. In order for a chemical reaction to occur, deactivation steps rs and r_s must be essentially slower than the reaction steps. This means in turn, that quasi steady state (Bodenstein) conditions could be applied for the main reaction, but not for deactivation. In terms of surface coverage it follows then that dO 4
dO °
dt
dt
do.
do °
dt
dt
" ~--=0
(8.110)
=0
(8.111)
while dO t,
a 0
(8.112)
dt
Here 0 ° denotes surface coverage of A at deactivation free conditions, in the more general case when there are i species present on the surface dO,
dO,°
dt
dt
~ O
(8.113)
The total coverage is therefore expressed as (8.114) i
322 leading to 0 A = 0 ° ( 1 - 0~,)
(8.115)
For simplification it can be supposed that adsorbed A is the most abundant surface intermediate, hence the surface coverage of species other then A can be neglected. The changes in coke coverage as a thnction of time-on-stream are given by d(1 - On) _ ksOA _ k s(1 - OA)
(8.116)
dt
Here ks' and k_s denotes deactivation and self-regeneration constants. The rate of reaction (A --->B) 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
dO<:, all
(8.133)
and
'
k+4KAcA 1 + K 4c A + K~c~
/
(8.134)
If deactivation is much slower than the main reaction the solution procedure can be decoupled. If pseudo-steady state is assumed to prevail in the fluid phase and hence, d a / d t is considered with constant temperature and concentrations. Denoting K ' = k +4K Ac A
1
(8.135)
1 + KAC A + K ~ c B
one obtains do~
dt
- K'~
(8.136)
326 which has the analytical solution similar to eq. 8.120 a=a
= e K'~
(8.137)
This result is dependent on the initial assumptions on the deactivation kinetics. By making different initial assumptions about deactivation steps, several empirical forms of deactivation: linear, exponential and hyperbolic decay can be derived (Table 8.4). Table 8.4. Typical empirical deactivation functions. Order n=0
Type Linear
n=l
Exponential
Differential form da _ k' dt da _
Integral Ibrm a(t)= l-k't
k'a
a ( t ) = e I~,,
k,a2
a(t) =
dt
n=2
Hyperbolic
da _ dt
1 - k't
The mechanism of the main reaction can be linear, while the deactivation is nonlinear rs - ~tdOs' _ kdOjO '
(8.138)
where Oj,O~ denote coverage of intermediates taking part in the reaction, and i can also be equal to/. Then instead of eq. (8.115) it holds 0j = 0~(1 - 0s); c~ = 1 - 0~
(8.139)
leading to oc ( 1 - 0s)-
(8.140)
dt
As an example we consider a mechanism similar to (8.122), but with the rapid desorption of B. 1. A + * " A* 2. A* ~ C* 3. A * -
(8.141)
~ B+*
The rates of the steps 1) -3) are written as t3 = k+,cAO ~ - k
,0 A
(8.142)
r2 = k20 ~
(8.143)
t:~ = k+30~ - k 3c1~0,,
(8.144)
327 The total balance o f the sites is given by 0 A + 0 c +0,, =1
(8.145)
The mass balance equation for surface compounds gives dO A
= c~>
(8.146)
= a (r, - vA r~ - ~ )
dt
dt
(8.147)
= at2:, = d v c r 2
We now differentiate eq (8.146) once and insert dOe~dr from (8.147) into the second-order differential equation. The result becomes d204 + d B dOA + d 2 A v c k 2 0 A = 0 dt 2 dt
The notations dB = d, and
o£2Av c k 2 = d o
(8.148)
are introduced. The differential equation has the
general solution (8.149)
0 A = C1 er~t + C 2 e ~t
where rl and r2 are the roots o f the characteristic equation r 2 + d~r + do
VI'2 =
-all +4d~ -4d° 2
O,
(8.150)
The integration constants Cl and C2 are obtained from the boundary conditions 04 = 0 at t = 0 and 0 c = Oc,~ at t = o0. The solutions for 0 A and 0 c become in closed form as follows:
\r 2
r,)
r1 1
1
)/ (8.152)
The parameters A and B in eq (8.148) are dependent on the reaction rate constants and the concentrations. With these parameters, d0and % can be calculated and inserted into eq. (8.150) giving ri and r2. The coverages are solved from eqns. (8.151) and (8.152) and are
328 inserted into rate expressions (8.142) - (8.144). Finally, the stoichiometry gives the generation rates for substrates A and B rA = - rl; rB =
(8.153)
r3
For a continuous fixed bed reactor the pseudo-steady state mass balances for the fluidphase species A and B are solved numerically with respect to the space time (r) at different times. If the main reaction contains nonlinear steps (Figure 8.34)
oi, oj
Os
Figure 8.34. Deactivation mechanism with nonlinear steps. then the activity is a - r - - 0j0, r o 0°10°i
(8.154)
Similarly to (8.115) we express the coverage o f i a n d j
o, =oo(1-o,),
= oo(1-0,)
(8.155)
leading to a = (1-0,,) 2
(8.156)
The generation rate o f S is
d G - k~O,O, dt
(8.157)
Differentiation o f a from 8.156 gives da = 2 ( 1 - O s ) ( - d O s ) , which in combination with (8.155) and (8.157) results in
da- k O,0 O,0 dl
(8.158)
In Table 8.5, several linear and nonlinear models are collected, including mono- and bimolecular deactivation mechanisms for reactant and product molecules as well as a reversible deactivation reaction. In the Table 8.5 the models are presented in a general form, e.g. with surface species leading to coke denoted as X* or Z*. They could be reactant A or product B, giving rise to slightly different generation and consumption rates (right hand column in Table 8.5).
329 Table 8.5. Deactivation models with corresponding rate equations (F. Sandelin, Reaction kinetics and catalyst deactivation in dynamic two-phase fixed bed reactor models, Doctoral thesis, Abo Akademi University, 2 0 0 4 ) . Model Rate equations Stoichiometry 1 . A + * - " A* 1-0c, :--~--IVA['.4 2. A * --+B*
3. B* 4. X * ~
0v -
" g+*
rA
1 + Kc A + Kc B
r~ = , ~ - [ v ~ l r 4
C* r 2 = k'+2cAO V
r4 = k'4cx®
l~.. = r 4
V
~A-~ I~AI=1, Iv~l =o ~-~
1.A+* 2. A * ~
"
A*
B*
1-0c,
FA -
0v - 1 + Kc"A + Kcu
3. B * " B+* 4. X * + Z * ~ 2 C *
-r~
- I~AIF4
~ --'~ - I ~ 1 ~
r2 = k'+2CAOv t
I~1 =o, I,~1 =,
rc, = r4
2
1"4 = k + 4 c x C z O V I 2, I,,BI 0
XA, ZA~I~
I~< 1, Iv~l 1 X-B, Z 8-~ I~1-0, I'<-2
XA, Z B-~
1. A + * 2. A * ---~ B *
"
A*
1-0c, 0v -
3. B * " B+* 4. X * + Z --+ C *
1 + Kc A + Kc B ¢
r2 = k+2cAOv
rc, = r 4 r 4 = k'+4CxCzO v
~A, Z=A--~ IrA] =2, ]vs] =0
Xe, Z=B~
1 . 1A+*, --+ B * 2.
"a*
1-oc .
0v
.
.
.
rA
--r2
IvAI=o, I~1 =2 Ilea I1.4
l + K c A + Kc"e
3B,
-
- u+,
4. X * + * --+ C * *
,
/'2 = k+2cAtgv
, 2 r4 = k + a c x O v
r~ = , ~ - I v ~ l r 4 rc,=r4
~A-~ IvAI=,, I,~BI=o
1. A + * 2. A * +
" A*
' - Oc
,~ : - r2 - I v A Ire
0v - l + K c A + K c e
B*
3. B *
-
"
g+*
4. X *
-
"
C*
F2 =
k'+2CA[gV
r 4 = k'+4CxO v - k _ 4 0 c ,
FB = F 2 - - I I / B B F 4
re* = r 4
XA--~ [VA[=1, [VB[=0 ~-~
i~i :0, i"~i : '
Derivation of a rate equation is based once again on the theory of complex reactions, considering deactivation as an independent route leading to coke on the catalyst surface. Equation (8.117) took into account self- regeneration. The rate constant of this step is difficult to extract from catalytic experiments, therefore a notion of stationary activity is often used. This activity corresponds to a situation when the deactivation and self-regeneration are
330 equal to each other, which means that the stationary activity is activity at infinite time (e.g. as in eq. (8.117) or a~ in eq. 8.159 and Figure 8.35). The activity could be then expressed as F
a -
- ia
F0
-
a~o
(8.159)
a 0 -- a~
02 O~ 0? O~ 02
a~
Ox 02 00
02
~4
Og
08
1O
Figure 8.35. Activity profile according to eq. 8.159.
8.5.1.2. Batch reactors
In the case o f batch operation reactions similarly to (8.159), it can be proposed that the decrease o f activity is proportional to the amount o f formed product in the following way r=r(l
N° N N O- N.~-)
(8.160)
where N is the molar fraction o f substrate in the mixture with the product, No is the initial value o f N at t - 0 , N~ is the value o f N at total catalyst deactivation, and ra is the rate in deactivation free conditions. Considering, as an example, a reaction which follows the kinetics -
kN i
(8.161)
l+k'N
and taking into account this intrinsic kinetics determined in deactivation tree conditions and the mass balance for a batch reactor, r -
n dN mdt
(8.162)
where m is the catalyst mass, and n is the initial molar amount o f substrate the following equation is obtained
331
(8.163)
n2t=(No_N~o)I( 1 +k')lnNo-N~__+ 1 In N 1 n L kNo, k N-N~o kN ~o
Contrary to experiments in fixed bed continuous reactors, deactivation o f catalytic reactions performed in batch modes can be hidden, as the observed behavior could be similar to first order kinetics. Then it is advisable to perform experiments under identical conditions but recycling the catalyst. Hypothetical experimental results are presented in Figure 8.36, where the numbers 1 and 10 correspond to the first and the last (the 10th) experiments in a series o f repeated experiments at the same conditions. It follows from the data in this example, that at around 20% o f the catalyst activity is lost after 100 hours. Reactant concentration, %
loo£ 9o-" 80-" 70-"
I I
60"
i|
m
so40-
• 10
| 1
;
" s'0 " 1;0 " 1;0 " 2;0 ' 2;o
3~)0
time, min
Figure 8.36. Reactant concentration as a function of time in consecutive experiments in a batch mode. For the modeling o f deactivation, a semi-empirical equation can be used
Cn = C O _ alle k,~
(8.164)
where al is a complex parameter (could include reactant concentration), CR° is the initial concentration o f reactant, k d is the deactivation constant, t is the time, and r is the total time starting from the beginning o f the series. Deactivation can have a profound influence on selectivity. Let us consider a molecule with two reactive functional groups Ar and Br, which react to products ArBp and ApBr in a parallel fashion (Figure 8.37), and are different in their deactivation behaviour.
k ~ . , ¢ ApBr ArBr
~"
0*s~
Figure 8.37. Deactivation mechanism for a reactant with two reactive functional groups As there are two functional groups it can be assumed that there could be two modes o f adsorption, one by the A and another by the B group.
332
(8.165)
0 A = KAOoc4,.B, O~ = KBOoc4~ ~
where Oois the fraction of vacant sites. It is feasible to assume that the decrease in activity along different routes is associated with the decrease in the number of sites where the reactant is respectively adsorbed either by one or another functional group. Assuming that deactivation is essentially slower than the reaction steps and applying the quasi steady state (Bodenstein) conditions only to the main reaction, but not for deactivation, one arrives at the following expression for the coverage of OA (8.166)
0 A = KAOoc4e ~e k~ O , = K ~OoCA,.e,.e
(8.167)
l~r
where k~J, and k~/ are deactivation constants for the A and B functional groups respectively and r is the overall reaction time. For a series of consecutive experiments in a batch reactor using the same catalyst batch, the value of z-is obtained by the summation of the reaction time in the previous experiments and time in a given experiment. The rate of reaction is then given by (8.168)
dC4,~ _ klKxOoC~,~, e k:;~+ K2tk yoC4~, e dt
For the sake of clarity the coverage of species other than the reactant is neglected. The selectivity is clearly influenced by deactivation, since dC+,~ /dCz,"
. . . B)e ---- (k~K A /. .g21~
.
Integrating eq. (8.166) for the boundary conditions t-0, N~B ~ = 1 the expression for reaction time is obtained o
--t = /7
N~e, - N,B" ) +
in NA~B~
1
(8.169)
JVA. B~
with A~ =K~l~xC4~Be
+K2~XBCA~e
'
and
A2 = K A+ K s
(8.170)
8. 5.1.3. Multilayer coking
Often when organic compounds react on catalyst surfaces with adsorbed molecules undergoing and leading to species which are retained on the catalyst, these species stay on the surface and build up several carbonaceous layers. The model for multilayer coking was proposed by N. Ostrovskii (Figure 8.38) and in the treatment below we will follow his considerations.
.~3.~
I I
a
o0:Zo,
o:~o.
1
.,4
Figure 8.38. The model for multilayercoking The deactivation reaction can be presented in the following way 1. R+0o---> 0j 2. R+0~---> 02
(8.171)
n. R+0._I-+ On The rate of the first, second and n-th reaction steps in the eq. (8.17 l) are given by r~ = k,,OoC~, ~ = k~,Oic~, r n = kj, O,, ICR
(8.172)
Here R denotes the reactant and 0~, is the surface covered by coke. The rate of coke generation is composed from the rate of monolayer and multilayer formation N
1 dc~, _ k°,Oo~ + k ~ O , , , 4
dt
(8.173)
~,=z
where ~ is the quantity of coke, derived from 1 mole of reactant. Eq. (8.173) takes the form 1 dC(, _ kmc~ (1 - 0~,) + k~,c~ (Op - O~ )
¢Jt
(8.174)
Activity changes in the main reaction are only due to coke formation in the first layer, which blocks the active sites. Therefore, an equation for the coverage of coke 0t~ is C,, dOt, _ kmcR(1 _Or, ) ¢ dt
(8.175)
where Cm is the capacity of a monolayer, or the maximum amount of coke (gram of coke /gram of catalyst), which could be adsorbed on the catalyst in case of monolayer coking. Equations (8.174) and (8.175) should be solved simultaneously, which is a very demanding task. A reasonable simplification is to assume that 0~ = 0, leading to
334
1 dC c _ kmcR (1 - Or,) + kpc~@ 4dt
(8.176)
From (8.175) and (8.176) one gets ____
1 dC c _ l +
C,. dot,
kpO,,
k,. (1- Op)
(8.177)
if the mechanism of the main reaction is linear, then a = 1 -Op and dC 1-a ---c____2- 1 + c p - -
1
C,,
da
(8.178)
a
where (p = kp / k,,. After integration we arrive at an equation that relates the activity to the coke concentration Cc
- (1 - q~)(1 - a ) - q~ I n a
(8.179)
Cm This equation includes linear and exponential dependences as special cases. When multilayer coking is not profound, then q~ = k~, / k,, -~ 0 and C(, / C,,, = (1 - a). When the deactivation constants for monolayer and multilayer coking are the same and q~ = k~, / k,, = 1, then C c / C , , = - l n a and a = 1 - C c / C , , , . Similar treatment could be extended to cases of nonlinear deactivation and nonlinear mechanisms for the main reaction.
8. 5. 1.4. P o s o n i n g
When a poison molecule (P) deposits on a vacant surface site, P+*
-
" P*
(8.180)
the rate of poisoning is given by r r, = k+cpc v - k _ c p ,
(8.181)
where cp. and c,, denote the surface concentrations of the deposited poison and vacant sites, respectively. By considering the mass balance for the deposited poison in the reactor volume, assuming that the reacting compounds are in rapid adsorption equilibria, replacing the coverage of vacant sites with the more general term activity (a) and introducing an empirical exponent (n) it is possible to obtain the following equation for activity da -
dt
.k'(a
- a * ) '~
(8.182)
335 Integration gives the following results:
(8.183)
,n=l
a ( t ) = a * + ( a o - a * ) e -k't
1
a(t)=a* + [(ao-a*)'n+k'(n-1),] ' '
,n;el
(8.184)
A general equation (8.183 and 8.184) is thus obtained from which the many empirical forms used in describing deactivation can be derived. Often the deactivation is considered to be irreversible, a* 0 and the initial activity is set to unity, ao 1. Then by varying the order (n) of deactivation the different forms of the empirical functions can be obtained as special cases. Comparing eq. (8.183) and (8.117) it can be concluded that the activity profiles are very similar in the cases of coking and poisoning. Let us consider as an example, hydrogenation of sitosterol,
HO ~ which is a product of the pulp and paper industry and contains impurities such as sulphur, chloride and phosphorus. The catalyst deactivates after working tbr a few hours. Interestingly, higher amounts of catalyst exhibit higher hydrogenation rates (Figure 8.39) when plotted against the normalized abscissa (mass of the catalyst x time).
90
concentration, %
8o 70 60
0.5 g - exp 1.0 g -exp
I
5o 40
3o 2o 10 o~
;
'
)o
'
~o
'
;o
'
;o
'
1'oo
mxt, gmin
Figure 8.39. Hydrogenation of sitosterol in a batch mode with different catalyst amounts. Comparison between experimental and calculated (eq. 8.186) data.
The possible reason tbr the deactivation of the Pd surface could be poisoning by S, C1 and P which are present in the feed, blockage, and coke formation. XPS results unequivocally demonstrated the presence of C1 in the spent catalyst. For the first order kinetics in a batch reactor the following equation holds 1 dC - -
mdt
-
kC
(8.185)
336 However, when a poison is present in the reactant feed, a certain part of the catalytically active sites can be blocked. The number of these sites is proportional to the amount of the poisons. Therefore, instead of m (catalyst mass) an effective catalyst mass ( m * = m - f ) , should be used, where f is the amount of catalyst irreversibly blocked by the poisons. Assuming that f is independent on reactant concentration, one arrives at C = Coe k(., / ,
(8.186)
where C = C o at t-0. Reasonably good agreement between experimental data and calculated values can be achieved (Figure 8.39) when applying eq. (8.186).
8. 5. 1.5. Enhancement o f activity while deactivating
So far we have considered deactivation mechanisms where active sites due to blocking, posoning, etc, are withdrawn from the catalytic cycles, diminishing the number of active sites while not changing their activity. However, one can imagine a situation when the modified ("deactivated") sites are different in their collective properties than nondeactivated ones. Imagine a situation where deactivation occurs according to the linear mechanism 0A--~0S Then the reaction rate is defined as r = r ° ( 1 - Os). For the sake of simplicity we consider a zero order reaction, thus r = k(1 -Os)
(8.187)
If the activation energy of the main reaction depends linearly on the coverage of coke AE = AE ° +P0~,
(8.188)
then the rate constant is k = k°e -p°~/Rr
(8.189)
and the rate equation takes the form r = k°(1 - O~)e ~0,/R1.
(8.190)
leading to a non-linear dependence of activity on the coverage of coke a = (1 - Os)e -~°"
(8.191)
with d, = f l / R T . It follows from the numerical analysis of the activity data (Figure 8.40), that if deactivation leads to an increase of activation energy, then 2 > 0. The activity decrease is rather steep compared to the case when the activation energy does not change with coverage. Interestingly, if the decrease in activation energy is significant enough 2 < 1, even a temporary increase of catalytic activity higher than initial one is possible.
337
3.0-
2.5
..............................
~ ...............................
..............................................................
•-
2.0-
.............................
~.~
..............................
1.0-
...:~
.........
...... .
0.5-
............................
~ :
.""
.......................
...............................
......................
.................... i............................... ...............................
d.~::~:::.
• ....... :
...............................
....... .........
0
0.2
...............................
1
..........,
5 0.4
i
0.6
...............
.....................
:
-i .............
o.o 0.0
.............. . . . . . . . . . . . . . . . .
-3 " , ) ,
,
',
i......... '[ ....... ; ?;::""
0.8
t
",
1.0
0 Figure 8.40. Numerical analysis of activity dependence on coverage (eq. 8.191).
8.5. 2. Enzyme deactivation Similarly to heterogeneous catalysis, the activity of enzymes can decline with time. As an example we can refer to catalytic reactions over immobilized enzymes in continuous reactors. In general it is known that proteins are prone to denaturation, the reasons being some perturbations in the native protein geometrical and chemical structure. Denaturation results in the loss of structure due to protein unfolding, which in tulTl leads to loss of the catalytic function (Figure 8.41).
ed
%
, ?,
U
nlMded
Figure 8.41. Deactivation of enzymes by protein unfolding.
Protein denaturation can be due to physical reasons, e.g. heat (Figure 8.42), cold, mechanical forces, radiation or influence of chemical factors (acids, alkali, salts, solvents, surfactants, oxidants, heavy metals, chelating agents).
rate~
den~urafion T---~
Figure 8.42. Influence of temperature on the rates of enzymatic reactions.
338 Considering the irreversible deactivation of an enzyme following the pathway E,,~v ~ ~ E~.~,,~.... if the activity decline is proportional to the concentration of the active form of enzyme one gets ~, =k~[E~]
(8.192)
then the concentration of the active form in a closed system diminishes with time
d[E.]_ .kd[E~]
(8.193)
dt
finally giving after integration an explicit expression [ E a] t
[E~] ° exp(-kdt)
(8.194)
which demonstrates, that after a certain period of time an enzyme will become totally inactive (Figure 8.43). Activity
08 \\ 06" 04 e2.
time
Figure 8.43. Activity profile in enzymatic reactions corresponding to eq. 8.194. Such a type of deactivation behavior was observed in the case of chemotrypsin deactivation under ultrasound. As can be noticed, the same mathematical expressions as given by eq. (8.194) can be used to describe the deactivation behavior in the case of heterogeneous and enzymatic catalysis. Let us now consider deactivation for the reaction mechanism which obeys MichaelisMenten kinetics. kl
k2
E + S ,(zz~ E S ~ E + P
(8.195)
kl
If deactivation of only the free form of enzyme is occurring, then the decrease of the total concentration of enzyme ( [E,]tot ~ E,,,,,i~, ) is proportional to the deactivation constant and the concentration of the tree tbrm of enzyme. d [ E , ]tot _ .kj [E o ] dt
(8.196)
339 The latter one is expressed via the total enzyme concentration, substrate concentration and Michaelis -Menten constant [ E a ] - Km[E~]'°' [E~]'°' K,,, + [S] 1 + [SI/K,,,
(8.197)
Combining (8.196) and (8.197) we arrive at d[Ea],o ~ _ dt
kd
[E~],,,
(8.198)
1 + [S]/K m
Analysis of eq. (8.198) shows that the rate of deactivation depends on substrate concentration. However, if both the complexed and free form of enzyme deactivates with the same deactivation constant, then the deactivation rate does not depend on the concentration of substrate.
341
Chapter 9. Mass transfer and catalytic reactions 9.1. Catalytic multi-phase systems In any catalytic system not only chemical reactions p e r s e but mass and heat transfer effects should be considered. For example, mass and heat transfer effects are present inside the porous catalyst particles as well as at the surrounding fluid films. In addition, heat transfer from and to the catalytic reactor gives an essential contribution to the energy balance. The core of modelling a two-phase catalytic reactor is the catalyst particle, namely simultaneous reaction and diffusion in the pores of the particle should be accounted for. These effects are completely analogous to reaction-diffusion effects in liquid films appearing in gas-liquid systems. Thus, the formulae presented in the next section are valid for both catalytic reactions and gas-liquid processes. Gas-liquid diffusion can be essential in homogeneous and enzymatic reactions if the catalyst is dissolved in the fluid phase, while one of the reactants has to be first dissolved. As an example we can refer to alkylation of aromatic compounds by olefins in the presence of A1C13, where diflhion in the fluid film is of importance. Such a type of diffusion is also present in heterogeneous catalysis for three-phase systems (three-phase catalytic hydrogenations or oxidations) as illustrated in Figure 9.1. Ca
Gas
Gas-liquid diffusion
Liquid absorption equilibrium
Catalyst
CA
Figure 9.1. Mass transfer processes in three-phase systems
Physical transport processes can play an especially important role in heterogeneous catalysis. Besides film diffusion on the gas/liquid boundary there can also be diffison of the reactants (products) through a boundary layer to (from) the external surface of the solid material and additionally diffusion of them through the porous interior to from the active catalyst sites. Heat and mass transfer processes influence the observed catalytic rates. For instance, as discussed previously the intrinsic rates of catalytic processes follow the Arrhenius
342 law, while mass transfer hinders such pronounced dependence, decreasing the apparent activation energy. The intraparticle and interphase mass transfer coefficients display a lower temperature dependence as visualized in Figure 9.2 and discussed later. In k
film diffusion
pore
slope = 0
slope=
kine2
EA 2R
.. transition region
transition region
\
slope=
\
gA R
1/T
Figure 9.2. Temperature dependence of catalytic reactions. Moreover, the mechanism for the transport of mass and heat is different from the one for chemical reaction, therefore a different dependence on concentration can also be expected.
9.2. Simultaneous reaction and diffusion in fluid films and in porous materials
The mathematical desription of the influence of mass transfer will be based on the general conservation laws. Balancing the amount of mass for a given volume element we have IN + GENERATION =OUT + ACCUMULATION
A general continuity equation (mass balance) for a component in a layer, where diffusion and chemical reactions take place simultaneously, is written for an infinitesimal volume element (N~A)m +r,c~AV = (N~A) ..... ~
dn i
(It
(9.1)
where Ni and ri denote the flux and the generation rate, respectively. A is the cross-section area of the flux, AV is the element volume and n is the amount of substance in the volume element. The factor a equals 1 for homogeneous kinetics, but equals the catalyst density (a pp) tbr catalytic systems. The amount of substance in the volume element, ni, is expressed by the concentration (ci) and the volume of fluid phase in the volume element (eAV). The factor (e) is 1 for homogeneous systems but <1 for porous catalysts, where it indicates the catalyst porosity. By denoting the difference A( N , A ) = ( NiA),,,,, - ( N,A)m
(9.2)
343 we get - A(N,A) + q a A V = ~AV dc dt
(9.3)
Division by AV gives and letting AV ~ 0 gives dc i dt
-
d(NiA) - +r,a dV
(9.4)
This is a very general form of the transport equation. Further treatment depends on which models are used for the geometry and diffusion. The most common geometry is a slab with the volume element d V - A d r , where dr is the thickness element of the catalyst (Figure 9.3), leading to dc i dt
r
dN i - - +~o~ dr
(9.5)
r+dr
R Figure 9.3. Slab
For spherical geometry (Figure 9.4), on the other hand, we have A 4:rtr2 and dV-4zt-r2dr giving the relationship dc i c dt
d(Ni r2 ) r2dr + ~o:
(9.6)
Figure 9.4. Sphere.
Since d(yx2) x2 dx
x2 dy 2x x 2 dx +-£-5-Y
(9.7)
344 equation (9.6) becomes c
dc,_ dt
dN, dr
2 Ni + ~ a
(9.8)
r
Analogously, it can be shown for the third ideal geometry, an infinitely long cylinder, that the balance equation becomes
dc dt
d(Ny) rdr
c---
--+~a
(9.9)
suggesting that a general formulation is 6
clc, dl
d(N,r 2) ÷t~a r*'dr
- -
(9.10)
i.e. c
dc I
dt
~
-
dr
S
r
N~ + ~ a
(9.11)
where s=0 for slabs, s=l for infinitely long cylinders and s=2 for spheres. All real systems can be placed within s ~ [0, 2]. The diffusion flux of a component, Ni is related to the local concentration gradients of the compounds. Depending on the mathematical model used for diffusion different expressions are obtained. The pioneering work in diffusion was carried out by Stefan and Maxwell during the 19th century.
James Clerk Maxwell
Jozef Stefan
Adolf Eugen Fick
According to the general law of Stefan and Maxwell the diffusion flux of each component (xi) is related to every concentration gradient (dci/dr): dc
N=F~
dr
(9.12)
where F is the coefficient matrix, the structure of which depends on the system. Already for molecular diffusion, the theory of Stefan and Maxwell gives a complex relationship. For porous catalysts, additional effects appear: diffusion originated by molecular collisions with pore walls, and surface diffusion. Thus the vector F_has additional elements for these cases.
345 A commonly used approach to simplify the mathematical description of diffusion is to introduce some kind of effective diffusion coefficients (Dei) being formally related to one corresponding concentration gradient only, i.e. the law of Fick cd
N, = - D e ----:--
(9.13)
dx
This concept essentially simplifies the mathematical treatment of diffusion; tbr instance tbr the simplest geometry (the slab or fluid film), we get:
d2c,
dc, = De * c (_It ~dr +r*a
(9.14)
However, the simplification should be investigated case by case. The basic concepts of diffusion and reaction will be used as a basis for the treatment for heterogeneous catalytic processes as well as catalytic gas-liquid reactions.
9.3. Gas-liquid reactions The presence of two phases, namely gas and liquid, is characteristic for non-catalytic or homogeneously catalysed reaction systems. Components in the gas phase diffuse to the gasliquid interphase, dissolve in the liquid phase and react with components in the bulk liquid phase. The liquid phase may also contain a homogeneous catalyst. Some of the product molecules desorb from the liquid phase to the gas phase and some product molecules remain in the liquid. The processes taking place in a gas-liquid reactor are displayed in Figure 9.5.
d~
I c
Bulkliquid
A+CK-f-~B(~) ol
6~
Gas-liquid interface I
8L
Figure 9.5. Mass transfer on gas-liquid interface. Multiple reactor constructions for gas-liquid reactors are available, because of the large number of different application areas. Spray columns, wetted wall columns, packed columns and plate columns are mainly used for absorption processes. The gas concentrations are low in the case of absorption processes, hence a large interracial contact area between the gas and the liquid is important to enhance the absorption process. These column reactors usually operate in counter-current mode. Counter-current operation is the optimal operating mode, because at the gas outlet where the gaseous component concentration is lowest, the gas is in contact with a fresh absorption solution. The low- concentration of the gaseous component can then partly be compensated by the high concentration of the liquid component. Two reactor types dominate in the synthesis of chemicals in the case of gas-liquid reactions: the tank reactor and the bubble column. Both types can be operated in continuous or
346 semi-batch mode. In the semi-batch operation, the liquid phase is treated as a batch and the gas phase flows continuously through the liquid. A typical tank reactor used for gas-liquid reactions is displayed in Figure 9.6.
Ei~ia
Figure 9.6. Tank reactor.
For this kind of reactor it is very important to have a good gas dispersion in the liquid. The gas is fed through a sparger located under the impeller. The advantage of a tank reactor is its good mixing capabilities making it useful in the treatment of highly viscous fluids. The heat transfer capabilities are appreciated in the case of highly exothermic reactions, where to certain extent the reactor temperature can be regulated by the feed of cold reactants. In semibatch operation a similar problem with temperature control and product quality can occur as in the case of homogeneous liquid phase systems. The biggest disadvantage of tank reactors operating in continuous mode is the low reactant concentrations the reactors operate at. Another disadvantage is the complex mechanical structure that causes increased investment and operating costs. An often used gas-liquid reactor is the bubble column. The gas is usually fed from the bottom through a sparger and the liquid flows either cocurrently or counter-currently. Counter-current operation is more efficient than co-current, but for certain types of parallel reactions, cocurrent operation can give better selectivity. Bubble columns are often operated in semi-batch mode: the gas bubbles through the liquid. This mode of operation is attractive in the production of fine chemicals which are produced in small quantities - especially in the case of slow reactions. The flow patterns can vary a lot in a bubble column. Generally, as a rule of thumb, the liquid phase is more back-mixed than the gas phase. The plug flow model is suitable for the gas phase whereas the liquid phase can be modelled with the backmixed, dispersion, or plug flow model. The mathematical models for different kinds of gas-liquid reactors are based on the mass balances of the components in the gas and liquid phases. The bulk gas and liquid phases are divided by thin films where chemical reactions and molecular diffusion occur. The flux of component i from the gas bulk to the gas film is Nfli and the flux from the liquid film to the liquid bulk is NSI , . The fluxes are given with respect to the interfacial contact area (A) according to N i A = I m ° l m 2
Lm2 J"
The flows N~I, and N~ can have both positive and negative values, depending on the direction of the respective flow-. Different theories and methods are available for the calculation of these molar fluxes. According to the two-film theory, simultaneous molecular diffusion and chemical reaction occur in the liquid film and only molecular diffusion occurs in the gas film. The liquid and gas film thicknesses are denoted as 6L and ~c, respectively. Ficks
347 law describes the fluxes in the gas- and liquid films. If Ficks law is valid for both the gas- and the liquid film, the fluxes N~iiand N{'÷~ are defined as
N;,~i = +D~i ( dc~;, )
(9.15)
and
b
NLt z-
D Li ( -dc,~, ] -
(9.16)
where DGi and DLi are the diffusion coefficients in the gas and liquid phases, respectively. The different signs in equations (9.15) and (9.16) are due to the different coordinate system definitions for the gas- and liquid films. The following balance is valid for component i in the gas film
i~7
ollr
where 0in-0out=A(DGi dcoi/dz). Dividing eq. (9.17) with Az gives
A D,si dz ) _ 0
(9.18)
Az Assuming that the diffusion coefficient is approximately constant and allowing Az~0, we obtain the balance equation
D~i d2ccsi - 0 dz 2
(9.19)
This balance equation has an analytical solution. Integrating the equation once gives d C G i __ C I
(9.20)
dz Integrating it once more yields
(9.21)
C(÷i = C I z .-.I-C 2
The following boundary conditions are valid for eq. (9.19): surfixce s CGi = C G j =CGi,z=O~
bulk b Cck = C ( ; ~ = C ( ; t , Z = ( ~ ( ;
(9.22)
348 The integration constants, C~ and C2, can be determined by inserting the boundary conditions into eq. (9.21) b
b
C~ - c°i - cGi
(9.23)
C 2 =cclis
(9.24)
The expression for the concentration profile, c~i, can now be written as
ccsi (z) = ccjj -ccs i
+ccs j
(9.25)
The flux, N~, is obtained according to the definition
(9.26)
The ratio, Dgi/Az, is often referred to as the gas film coefficient 1,~i kcji -
D(7~
(9.27)
The flux, N~), can be expressed in the form
' )~ Ncb;i -- kc;, c(;i - cc~ The flux in the gas film is dependent on the gas film coefficient and the concentration difference between the gas bulk and the gas film interface. For the liquid film, the mass balance can be written as
dc,,~ A+r, A z = - ( D]+,dc~,] - ( D,+,~-) dz] in
A
(9.29)
out
where AAz is the volume element in the liquid film and ri is the generation rate of component i. The difference, 0in-()out, can be written as A(DLi dcLi/dz). Dividing eq. (9.29) with the volume element, AAz, gives A(
dc'I ~"~ Az
+ i-, = 0
(9.30)
if the diffusion coefficient, DLi, is assumed to be independent of the concentration and allowing Az--->0, eq. (9.30) is transformed into
349
d2 cLi D,+, d-~7-_~+ < : 0
(9.31)
Eq. (9.31) describes simultaneous diffusion and chemical reaction in the liquid film. The concentration profile o f component i, cLi(z), can in principle be solved from eq. (9.31) and the flux, N~,, is obtained from the derivative, dcLi/dz, in eq. (9.16). Eq. (9.31) has the boundary conditions, s
N ~ i = NGi ,b
ci, i = cr~,
z=0
(9.32)
z = 6r,
(9.33)
at at
The boundary condition, eq. (9.32), means that the component fluxes, defined according to the gas- and liquid film properties, must be equal to each other. At the interface chemical equilibrium is usually assumed. The concentration o f i in the gas and liquid phases are then related to each other according to K = c~i~ / cr.'
(9.34)
where Ki is the equilibrium state. For gases with low solubility, Ki is often called Henry's constant. An analytical solution o f the differential equation (9.31) with the boundary conditions, eq. (9.32) and (9.33), is only possible in special isothermal cases. These cases are characterized by the interdependence o f reaction and diffusion velocities and by the reaction kinetics. The following special cases can be distinguished according to the reaction kinetics: physical absorption, very slow reactions, slow reactions, normal reactions, fast reactions and infinitely fast reactions. A more detailed description o f the different reaction types is summarized in Table 9.1. Table 9.1. Summary o f different reaction types. Physical absorption
No chemical reaction in the liquid film and bulk. Linear concentrations in the films.
Very slow reaction
The same reaction velocity in the liquid film and in the liquid bulk. No concentration gradients in the liquid film.
Slow reaction
No reaction in the liquid film, chemical reaction in the liquid bulk. Linear concentration gradients in the fihns.
Finite speed reaction
Chemical reaction in the liquid film and in the liquid bulk. Non-linear concentration profiles in the liquid flm.
Fast reaction
Chemical reaction in the liquid film. No chemical reaction in the liquid bulk. Nonlinear concentration profiles in the liquid film. The gas phase component concentration is zero in the liquid phase.
Infinitely fast reaction
Chemical reaction in the reaction zone in the liquid film. The diffusion rates of the components determine the reaction velocity.
350 Analytical expressions will be derived tbr the fluxes in the case of different reaction types and reaction kinetics, which are of importance for homogeneous catalysis, e.g. slow and finite speed reactions.
9. 3.1. S l o w r e a c t i o n s
Slow reactions are characterized by the fact that the diffusion resistances in the gas- and liquid films suppress the absorption velocities. No chemical reactions are assumed to occur in the liquid film. For the diffusion flux and for the gas-liquid equilibrium the following equations are valid for the component A N (~A b = NLA l, K A = c "(;A
(9.35)
(9.36)
~ CL4
Equation (9.36) states that the concentrations at the gas-liquid interface are different from the concentrations in the bulk phases. The flux through the gas film is given by (L4
= k (iA c
- c~,
(9.37)
Because no reactions are assumed to occur in the liquid film (ri-0), the transport equation, eq. (9.31), for the liquid film can be solved analytically, in the save way the transport equation for the gas film was solved. The solution is analogous to the gas film reaction solution and the flux through the liquid film is obtained as ~"
b )
N~'L, = kLA CL, - CLA
(9.38)
where the liquid film coefficient, kLA, is defined in an analogous way as the gas film coefficient k~A (9.39)
kLA = DLA / CYL
Here DLA and 6L are the liquid phase diffusion coefficient and the liquid film thickness, respectively. Equations (9.37) and (9.38) can be set equal to each other, according to eq. (9.35), and the expression for c~,A is inserted i, - K ACL4 ~' k<jA C<JA
= k L4 czd - cL4
The unl~lown interface concentration, c ~ , can be solved from eq. (9.40). The result is as follows: b kxA b s
CGA -- ~(2A CLA
CLA =
kx A
KA+
k~A
(9.41)
351 Insertion of the surface concentration, CLA'"into , the expression for N2A eq. (9.38), gives b
N'
LA
-
b
cG.4 - K~ cL.4 KA 1 -
-
q_
kL~
-
(9.42)
-
k~j~
Eq. (9.42) shows that the film transport resistances can be added to each other and an overall transport coefficient can be defined as 1
K A 1 -- + -k,A - ki.A kc÷A
(9.43)
If the diffusion resistance in the gas film is negligible in comparison to the diffusion resistance in the liquid film - which is fi'equently the case for slow reactions - the term 1/kcA vanishes.
9. 3.2. Reactions with a finite liquid f i l m reaction velocity
For reactions in this regime, no general expressions can be derived for the diffusion flux, NA, through the gas and liquid films. Chemical reactions proceed in the liquid film and the following conditions are valid for the diffusion fluxes through the gas and liquid films: (9.44) The first condition states that the diffusion flux for component A in the liquid phase changes because of the reaction taking place in the liquid film. For the gas-liquid equilibrium, at the interface, the following ratio is valid K A = c<jA / CLA. The transport equation for the component A must be solved separately for each kind of reaction kinetics. An analytical solution for eq. (9.31) is possible in some special cases, which one can encounter in homogeneous catalysis, e.g. zero or first order kinetics. The liquid film must, however, be considered as isothermal, which is a quite reasonable assumption. Below we present an analytical solution for the mass balance equation of the liquid film,
d2CL4 DI.A = ~ dz+
r~ = 0
(9.45)
for zero and first order reactions.
9. 3.3. Zero-order reactions
For zero-order kinetics, the reaction rate, rA, is r A = K~R = E4k
(9.46)
352 where v A is the stoichiometric number. Inserting rA into eq. (9.45) gives
d2CLA dz 2
1/k
(9.47)
DLA
--
Integration of eq. (9.47) twice gives the concentration profiles for the component A in the liquid film vk
(9.48)
cl A ( z ) - - - z 2 + C l z + (72 2Dr, A
Insertion of the boundary conditions = CL4,
(9.49)
= CLA
makes it possible to determine the integration constants C~ and C2 in eq. (9.48). ~,
,
C, - cLA - cLA +
~31 vk k
C,
"'
(9.50)
insertion of eqs. (9.50) into eg. (9.48) gives the concentration profile, CLi(Z)
CIA (z) - - 2Dj,A
•
+ Cr,A - CrA + 2DIA, ) \ 61,) + Cr4
To obtain the flux, NI.s A , we calculate the derivative, dCkA/dz dc'i~A
~-z
vAk61
-
DL~
z
~, cS'L) + ~,
CLA
-
+
cI~A
dr.
)
"
•
(9.52)
2DLA
According to the definition, the flux N)' A is obtained as ( dc'jA)
DI~
+
vAkdl
(9.53)
and the term -VAkSL/2 can be written as v A k6~
v A kDL4
2
2kL~
By defining the dimensionless quantity, M, accordingly,
(9.54)
o
o
.~5.~
v4 kDt~A
2.2
M-
b
(9.55)
t(,LA CLA
the expression for the flux, N~A ' can be written as ( s b ) b N~A = kLA CLA - CLA + k~CLAM
(9.56)
The flux is also affected by diffusion through the gas film. For the gas film diffusion we have: N".A = N ~ = k.A, c
(9.57)
-c.A
By setting eqs. (9.56) and (9.57) equal to each other and inserting the equilibrium definition, eq. (9.36), the unknown interface concentration, c~,sA, can be solved. ,b
s
CLA =
.b
kz.Ac,.A(1 - M ) + k<÷,4c<÷ A
(9.58)
kl, A + kc÷AK4
Insertion of c)'A according to eq. (9.58) into eq. (9.56) gives the final form of the flux, N)~IA, C~A + KAC~.~( M s
NLA =
KA --+
1)
1 -
(9.59)
-
kL4 k,. where M is defined by eq. (9.55). If the gas film resistance is negligible, the term kaA vanishes in eq. (9.59). For gas-liquid reactions an enhancement factor is often defined. The enhancement factor is the ratio between the chemical absorption rate and the physical absorption rate. Eq. (9.42) is valid for component A when only physical absorption occurs in the liquid film. The enhancement factor, EA, is in this case defined as
N>IA 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, <,~ N)[A = - D , . A [ ~ )
k,.<~ M '/2
=o- tanhMV2
.,
Cb ~"<~
]
(9.76)
c,.A coshM,<2)
If diffusion through the gas film affects the flux, the expression N"L.4 :
N
: k<;A co;A-co; A
(9.77)
is combined with eq. (9.76) in order to eliminate the unknown surface concentration, c~A. The result is kLAM1/2 b CLA b GAc G,4 + tanh M 1<2 cosh M I/2 c:.,j = kL~ Mi/2 kc;~K4 + k
tanh M 1/2
(9.78)
356 Insertion of the concentration, eq. (9.78), into eq. (9.76) gives the final expression for the flux, N~A ' for first-order reactions b
KA
C(5~4
N ~ = tanh
;,
c o s h M v2 cI'A
kc~
M I/2
+
(9.79)
1
KA
M 1/2
kcja
where M is given by eq. (9.71). M 1/2 is often denoted as the Hatta number, H a and as follows from the analysis of eq. (9.72) is a complete analog of Thiele modulus. The enhancement factor, EA, is obtained by dividing eq. (9.79) with the expression for physical absorption
c~;A - cosh
kzA + k<JA E A =
K A tanh M v2 kr,A M 1/2 +
M 1/2
cL4
(9.80)
c<~ - K4cr, 4 • -~L k i÷~4
- - b - -- - ~
"
if gas film resistance is negligible eq. (9.80) can be simplified cb
E4 =
•
KA
b
(;A cosh Ha c lA
Ha
b c~A
tanh Ha
(9.81)
t KAC~A
Hu
In case of small Hatta numbers,
lim---1, tanh H a t la--+O
c Ha -Jr-e Ha
and
coshHa =
2
-1,
the
enhancement factor is approaching unity. For fast first order reactions, e.g. high values of Hatta number, and the liquid-phase concentration of the gaseous compound approaching zero (c<;';_~= 0 ), it can be demonstrated that tanh ~
1
(9.82)
leading to KA
EA
k;,, KA
+
[
k,;A
(9.83) 1
krA " f M + k<;A 9. 3.5. U t i l i z a t i o n f a c t o r
For practical purpose it is, however, important not only to estimate the enhancement factor, but also to calculate the impact of mass transfer on the reaction rate occuring on the liquid side of the interface. Defining now the effectiveness factor (liquid utilization factor) as the
357 ratio of reaction rate with transport limitations and the reaction rate at conditions prevaling on the liquid side of the interface ,2
~ll. - NrAa'-------~
(9.84)
kcL
where, for first order reactions, the flux N~A is defined according to a slightly modified eq. (9.76)
_ / LAH" (es
N)~A tanh Ha ( LA
cosh Ha
(9.85)
one obtains for irreversible first order reactions
av kL~Ha 1 CLA r/L - k tanh Ha c~A cosh Ha
(9.86)
As the mass transfer coefficient is defined through the film thickness according to eq. (9.39), at v = -1 from (9.71) we arrive at
(9.87)
Ha - ~ kLA
Taking into account (9.87), the liquid utilization factor can then be defined as
rlL
z
(
> /
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<5000. Typical values of the Hatta number for industrial reactions with homogeneous catalysts are 10-2
Of. =
rate" observed"
(9.90)
= qLHa
" diffusion" rate
dependence of the utlization factor as a function of • L at different Sherwood numbers can be constructed (Figure 9.6).
o,1-"~I Sh 0.01
10-~
1000
10,~
10-~
10 4
-~0 o
101
0~. = qLHa 2 Figure 9.7. Dependence of the utlization factor as a function of Ol~ at different Sherwood numbers (F. Kapteijn, G.B. Matin, J.A. Moulijn, Catalytic reaction engineering, in Catalysis: an integrated approach, Elsevier, 1999).
As can be seen, a utilization factor close to unity can be achieved if the reaction is mainly occuring in the bulk of the liquid phase. From a practical point of view, it means that homogeneous catalytic reactions are performed in reactors with continuous liquid phase and dispersed gas phase.
9.4. Liquid-liquid diffusion, phase transfer catalysis An interesting example of a system where mass transfer plays a crucial role is phase-transfer catalysis. Phase-transfer catalysis is used in reactions between anions (and also neutral molecules and transition metal complexes) and organic substrates. It is needed since many anions in the form of their salts and neutral compounds are soluble in water and not in organic solvents, whereas the organic reactants are not usually soluble in water. Phase-transfer catalysts transfer the inorganic anions from the aqueous solution or organic anions generated via deprotonation of the appropriate precursors in the organic phase. Two types of systems are used: liquid-liquid and liquid-solid. The concepts and practice of phase transfer catalysis were strongly promoted by the effors of M. Makosza, C. Starks and A. Br~indstr6m. Detailed account on phase transfer catalysis is given in the special issue of Catalysis Reviews, 46 (2003). The catalysts are often quaternary ammonium salts (e.g., tetrabutyl ammonium, [C4H914N+), symbolized by Q+. The ion pair Q+Nu- (Nu- being the anion to be reacted) is a much looser ion pair than for instance Na X , which is an additional factor in the increase in activity. At the +
-
359 end of the reaction, an anionic group is usually generated. This anionic leaving group is conveniently brought to the aqueous phase by the shuttling catalyst. This mechanism is called the extraction mechanism of phase-transfer catalysis first proposed by Starks, 1971. R-Y + Q +Nu-
~- R-Nu + Q+Y
1L organ,cphase
1L
aqueous phase M+Y - + Q+Nu- -
~
M+Nu - + Q+y-
According to this mechanism, the lipophilic ion pairs with nucleophile ions are formed via migration of QNu into the aqueous phase where the critical ion exchange takes place. The ion exchange between two phases is a combination of a few processes: diffusion of the components from the bulk of both phases into the interthcial region, ion exchange in this region, and diffusion of the ion pairs back to the bulk. The rates of these processes depend on the mobility of the ion pairs, size of the interfacial area, viscosity and temperature. Not surprisingly, the observed rate was found to be dependent of the agitation efficiency, as the higher stirring speed increases the interfacial area. This extraction mechanism can be schematically presented by the following reaction steps 1.(Q+Y) org q- (Nu-)aq ~ (Q+Nu-)org+ Y-aq 2.R-Y o~ + (Q+Nu-)o~_~ R-Nuo~_~ + (Q+Y-)o~ (Nu-)aq + R-Y org ~ R-Nu org -t- Y-,q
(9.91)
Step 1 is eq. (9.91) describes the competitive extraction of reactant and product anions between the aqueous and organic phases in the presence of a catalyst cation. The rate constants of this step includes the effects of mass transfer across the interfacial region and depends on the mass transfer. A kinetic equation for this mechanism can be easily derived from a general expression for a two step sequence (Chapter 4, eq. 4.93):
r = c~,
0)10)2 --(-O1(2) 2
0)~ + 0)2 + (0_~ + o)_2
=
k,C[~v, ]~,k2C[~yl,~~ - k ,C[y ],k 2C[~,,,]<~
k~C[_~;,~,, + k2C[~yG + k
(9.92)
where C[e, 1,~, is the total concentration of the salts ((Q+Y) org a n d (Q+Nu-)org) in the organic phase. For an in'eversible reaction (i.g. irreversibility of step 2 in eq. (9.91)) and if the chemical reaction is slow compared to the rates of mass transfer k~C[~. ],~ + k IC[~ ],~ >> k2C[~G, , then
klC[.~;, ],, k2C[~YLr~, r =
k,C[,v,'
+ k ,C[y
%' ]
....
(9.93)
360 Since the denominator is almost constant during the reaction, the reaction follows second order kinetics, as observed for several systems. If on the other hand, the organic phase reaction is very fast compared to the rate of mass transfer, and if k 2 ~ 0, than instead of (9.92) we arrive at
V = kiC[Nt, ]eqC[Q+l,rg
(9.94)
which is first order with respect to the reactant anion and zero order with respect to the organic reactant, which has also been observed experimentally. Some other limiting cases can be analyzed for this two -step mechanism as well. An alternative mechanism for carbanion formation, the Makosza interfacial mechanism, was proposed in 1973. Makosza concluded that the Starks extraction mechanism does not operate for the generation of carbanions based upon theoretical and observed effects. In theory, the extraction mechanism would require the transfer of anions into the organic phase as an ion pair with the quaternary ammonium salts (often referred to as quat). In the case when the anion is hydroxide, the quat hydroxide should act as a base in the organic phase. However, because the affinity of hydroxide anions to the organic phase is much smaller compared to chloride anions, the hydroxide anions do not enter the organic phase. The conversion of the quat chloride to the corresponding quat hydroxide proceeds only to a very small extent, thus it cannot be a step in the catalytic process, in addition to this equilibrium effect, observed effects such as stirring rate affecting reaction rate, no hydrolysis of active intermediates (indicates no O H or water present in the organic phase), and highly lipophilic quats, practically insoluble in water, pertbrming well as PTC catalysts, confirmed the need lbr another mechanistic picture: 1.RHorg +(OH-)aq<-->(R-H20)inteffacialregion 2. (R'n20)interfacial region+(Q+Nu") org<--->(Q+R H 2 0 ) org +(Nu-)aq 3. (Q+ R-H20)org + R'Nu org ---> (Q+Nu- H20)org + RR'o~g
(9.95)
The crucial point of the Makosza interfacial mechanism is that deprotonation of a CH acid by the hydroxide anion occurs at the interfacial region. The carbanions formed cannot enter the aqueous phase because of the strong salting-out effect nor can they enter the organic phase because the accompanying sodium cations cannot move with carbanions to the organic phase. In the interfacial region the carbanions are in low concentration and have low chemical activity. The quat salt forms lipophilic ion pairs with carbanions helping them enter the organic phase where further reactions proceed. The first step in the Makosza interfacial mechanism involves the reaction of OH- with the organic acid at the interthcial region to produce the corresponding solvated carbanion. The second step involves the transfer of the carbanion from the interfacial region into the bulk organic phase as an ion pair with the phasetransfer cation. The final step is the alkylation reaction within the organic phase to produce product. Apparently, for many types of reactions, there is no need for the quat catalyst to migrate into the aqueous phase to enter the ion exchange process. In a similar way as for the two-step mechanism, a kinetic equation for the three step sequence, where the last step is irreversible can be derived based on the general formula (eq. 4.114) r =
0)10)20) 3 0)20) 3 q- 0)30) I q- 0).10.)3 q- O.)ICO2 q- 0.).20) 1 q- 0).20.).1
(9.96)
361
C o n s i d e r i n g (R-H20)interfacial region and (Q+ R - H 2 0 ) org as intermediates, eq. (9.96) then takes
the folTn
r =
k,[RH]o,.g[OH ],qk2[Q+Nu l,,.gk3[R'Nu]o~g
(9.97)
D
with
D = k2[Q +Xu ]orgk3[R' Xu](,,.,.+ kl[RH]orff[OH ]aqk3[R' Xu](,,.,.+ k_lk3[R' Xulorv +
(9.98)
Charles Liotta modified the Makosza interfacial mechanism. In this modification deprotonation takes place at the interfacial region and is assisted by the quaternary cation.
Mieczyslaw Makosza
Charles Liotta
1. (Q+Nu-) org +(OH-)aq <--->(Q+OH-) if +(Nu-)aq 2. (Q+ OH-) if + RH org<-+(Q+ R-H20) org 3. (Q+ R-H20)org + R'Nu org-->(Q+Nu- U20)org + RR'org
(9.99)
Similarly considering ( Q + O H ) i f and (Q+RH20)org as intermediates the rate for the modified interfacial mechanism is
r=k,[Q+N.L [OH
(9.100)
D'
with
Dt=k2[RH]orgk3[S'NbllJrg +kl[Q+NIA ]~rg[OH ]~qk3[atNg,ll,rg @k_l[N~/]~qk3[R'Nbtl,rg+ +
+
L[oH
-+ + ++++,[o L[oH L +
L
(9.101)
Note that nominator in (9.100) is effectively the same as in (9.97). In general it can be stated, that although there are numerous industrially important applications of phase-transfer catalysis, only a few mathematical models involving explicitly mass transfer phenomena have been reported.
362
9.5. Catalytic two-phase systems 9.5.1. Chemical reaction and diffhsion inside a catalyst particle In a porous catalyst particle, the reacting molecules must first diffuse through the fluid film surrounding the particle surface. They then diffuse into the pores of the catalyst, where the chemical reactions take place on active sites. The formed product molecules, of course, need to follow the opposite diffusion path. The phase boundary area is illustrated in Figure 9.8. e.g, gas I
•,~# r0,)
c0~ f/ "
"J~"
i I
C(y) ]
I Fluid film
Catalyst particle
Figure 9.8. Phase boundary area. The diffusion resistances in the fluid film around the particle, as well as inside the particle, are the reason for the fact that the concentrations of the reactant molecules inside the particle are lower than those in the main bulk of the fluid. The result, in the case of most common reaction kinetics, is that the reaction rates inside the pores assume lower values than what would be expected with the concentration levels of the main bulk. Let us study a component i that diffuses into, or out from a catalyst particle. The species i has the flux Ni (mol m -2 s-b in an arbitrary position in the particle. Often the flux Ni can be described with sufficient accuracy with the aid of the concentration gradients of the components (dci/dr) and its effective diffusion coefficient (Dei): dc N, = - D ~ c/x
(9.102)
In the calculation of the diffusion in a porous media the effective diffusion coefficient is applied, because the diffusional cross section is smaller than the geometric cross section (porosity e) and the catalyst has irregular pore structure (expressed via tortuosity ~) as illustrated in Figure 9.9.
Figure 9.9. lllustartion of porosity and tortuosity concepts. D =D--
(9.103)
z"
The following general balance equations can be established for a volume element AK and component i, at steady-state:
363
[incoming i by means o f diffhsion] + [generated i]
[outgoing i by means o f diffusion]
Quantitatively, this implies (9.104)
(N,A),, + gain = (N,A)o,.
in which A denotes the diffusion area and Am the mass of catalyst inside the volume element. Various catalyst geometries can be considered, but let us start with a special particle. The diffusion area for a spherical element is A = 4~r 2 .The catalyst mass inside the volume element can be expressed with the density of the catalyst particle (pp) and the element volume:
Am = p p A V = pp43r 2Ar
(9.105)
Then the balance equation is d c i A 2~ (-D~, ~7-r ,+at ),, + G o 4ar2Ar = (-D~, 7dc,
A 2.)out ~-70"
(9.106)
The difference, (DeJdcjdr)out - (Deir2dcjdr)in, is denoted as A ( D e Y d c j d r ) and the equation transforms into a new form:
A(-D e ~ff~--42-r 2) +r~fl 4~2Ar = 0 ar P
(9.107)
By means of division of eq.(9.107) by Ar and letting Ar---~0, the tbllowing differential equation is obtained dZ
r~
dc~
2.,
t~et d r r ) r2
~
+ r,p, = 0
(9.108)
The derivation above was conducted for a spherical particle. It is easy to show- that the treatment can be extended to an arbitrary geometry, as the form (shape) factor s is applied. Equation (9.108) can then be written in the general form
1 d ( D , tic' r ' ) dr r" dr + r,pl, = 0
(9.109)
The form factor, s, obtains the values: s-2 for a sphere, s-1 for an infinitely long cylinder, and, s-0 tbr a disk tbrm of catalyst. A real catalyst geometry - such as a short cylinder - can be described by eq.(9.110), by choosing a suitable non-integer value for the form factor. The form factor, (s), for an arbitrary geometry can be estimated from the relationship Ae _ s + l Vp R
(9.110)
364
where Ap denotes the outer surface area, Vp the volume of the particle and R stands for the characteristic dimension of the particle. The balance equation (9.109) is usually solved with the tbllowing boundary conditions, which are valid for the centre point (r 0) and the outer surface of the particle (r R). s dq/dr=O,r=O; c~=c~,r=R
(9.111)
Here cj~' denotes the concentration at the outer surface of the particle. The first boundary condition, at r-0, tbllows from the symmetry, e.g. that the concentration profile through the pellet is symmetrical about the center line of the sphere. We can also consider the fluid film around the catalyst particle. The diffusion flux inside the fluid film is constant and equal to the flux on the outer surface of the particle:
( dc' l
(9.112) r=R
(c; -
(9.113)
The ratio Di / 6 is often denoted as k~i, k~i being the gas-film coefficient according to the film theory. Thus, we obtain (9.114) In the case of liquid-phase processes, k~i is replaced by the liquid-film coefficient kLi. The mass flow of component i at the surface of the particle is then given by dc i
: - D,
7A'
:
-
(9.115)
The two latter equalities in eq.(9.115) tell us that the fluxes calculated in the fluid film and at the outer surface of the particle should be equal. Let us now define the effectiveness factor for component i from a theoretical viewpoint
N,,(~=~)Ap
q~ - ~(C)ppVp
(9.116)
The general definition of the effectiveness factor states that the factor describes the ratio between the real molar flux (Ni) and the molar flux (Ni') that would be obtained if the reaction proceeded in the absence of diffusion resistance. This ratio is equal to the ratio of observed rate and the rate if the diffusion resistance does not have an influence on reaction rate. The effectiveness factor, qi, can even be expressed in another way. The differential equation (9.115) can be formally integrated as follows,
365
Iod
dq .~1 Tr j :-,,,
J~ r,r'dr
(9.117)
where the upper integration limit, y, is given by ( dci~ y= Dc,~r )
R'
(9.118)
r=R
The lower integration limit is zero because for symmetrical reasons dc,/dr = 0, at r = 0. Integration of eq. (9.117) gives
dc, I N,.(,+=~ = - De.,\ dr )
Pp ~ , - R" for, r d r
(9.119)
I*=R
Insertion of this into the definition of the effectiveness factor gives
I]gr'drA,
(9.120)
., r,(cb)e, v,, -
As the definition of the form factor, eq.(9.110), is also taken into account, we get (s+ 1) j0~r,r'dr r/= r/(C)R,+ 1
(9.121)
Equations (9.120) and (9.121) illustrate that the effectiveness factor represents the ratio between a weighted average rate and the rate that would be obtained, if the conditions were similar to those in the bulk phase. If no concentration gradients emerge in the particle, r, in eq.(9.120) can be replaced with a constant, ri (cb), and the integration of (9.120) gives the effectiveness factor, r/i =1. For arbitary reaction kinetics, the balance equation for a catalyst particle, eq.(9.109), must be solved numerically, with the boundary conditions (9.111). Thereafter, the effectiveness factors can be obtained from eqs.(9.120) or (9.121). For some limited cases of chemical kinetics it is, however, possible to solve the balance equation (9.109) analytically, thus, obtain an explicit expression for the effectiveness factor. We shall take a look at such few special cases.
9. 5.2. First-order reaction If the diffusion coefficient, Dei, is assumed to remain constant inside the catalyst particle, eq. (9.109) is transformed as follows:
d2c,
s dc,
tgvri
~dr
+ r --& -
D,,,
(9.122)
366
As the reaction is of first-order, with respect to the reactant i, the reaction rate, R, is given by (9.123)
R = kc i
and the generation velocity becomes (9.124)
= v,R = v~kc',
The substitution, y = ci r, is subsequently inserted into eq.(9.122), which gives dq
dy 1
y
drr
r2
(9.125)
m
dr and d2ci
d2y 1
dr 2 - dr 2 r
2 dy
2
r2 dr + 7 y
(9.126)
Insertion of the derivatives, eqs.(9.125) and (9.126), into the balance equation (9.122) gives
ld2y (s+2)@ 0-2) r dr 2 +
r2
dr
r3
p ,ky Y
Dei
(9.127)
r
which is simplified to r 2d2y (s 2)r dr-----Z- + -
+ (2-s)+
D~,i
r2
y =0
(9.128)
This is the transformed Bessels differential equation, which has a solution for arbitrary values of the form factor, s, that can be expressed with the aid of the Bessel functions.
9.5.2.1. S p h e r e a n d f i r s t order kinetics
For a spherical geometry the analytical solution becomes particularly simple: s=2 and the eq.(9.128) is reduced to d2y p~ v,k drY+ ~ y =0
(9.129)
which is the second-order differential equation v~ppk r '2 +
Dei
- 0
(9.130)
367
with the roots,
q'.2 = -+
(9.131)
Net
The solution of a second order differential equation ay"+by'+cy=O
is y=C~e ~'x +C2 e~'~"
where J,tl and g2 are roots of the characteristic equation a p 2 + b p + c = 0 . solution to eq. (9.128) can be written as
The analytical
y = Clef1"+ Cze-
(9.132)
Y = CleV /),,, r + C2e ~ 1),,, r
(9.133)
Let us implement a dimensionless parameter, p, defined as - v,p k ~b2 - - R2 De,
(9.134)
as well as the dimensionless coordinate, x - r/R. The parameter, ~0, is called the Thiele modulus, named after Ernest W. Thiele, who introduced this modulus in a seminal paper in 1939 independently from Ya. Zeldovich and G. Damk6hler, who heavily contributed at almost the same time to considerations of the reactions and diffusion in porous media.
This parameter can be presented in a slightly different form. The stoichiometric number of a reactant A in the reaction A---~Products is eqial to -1, therefore eq. (9.134) takes a form
¢ = R k~pp
(9.135)
~D, In the equation above the units for the rate constant are [m3/(s kg)]. i f the first order rate constant has the units s-1, then instead of (9.135) we have an expression which is very frequently encountered in catalytic literature
~b= R.] k'
V
(9.136)
368 Eq. (9.133) can now be simply rewritten: (9.137)
y = C~e ~" + C2e -¢"
The boundary conditions give u s y 0 a t x O ( r R) s i n c e y cirandy S, a t x 1 (r R). Insertion of these conditions into eq.(9.137) allows us to determine the constants Cl and C2. From y(o) = ( c y "
+ GC - )lx=o = c , ¢ + c ¢ , = o
(9.138)
one arives at C 1 z -(-~2 and utilizing another boundary condition y(1) = (C~e °~ + C2e °~)].,.=l = C S + C2e ~ = C I (e ~ - e ~) = y~
(9.139)
one gets the following dependences y'~
C1
y ~"
e0_e_¢,C 2
e- ~ _ e ¢
(9.140)
and consequently using hyperbolic functions we arrive at Y = y.~ (e ~ - e ¢~) _ s i n h ( # ) e ~ --- ~-£ sinh(~b)
(9.141)
where the hyperbolic functions are e ~r - e
sinh(grc) - - - , 2
~
e g +e
cosh(c,ar) - - -
~Y
2
tanh(cvc) - sinh(q~) _ e ¢~ - e -°~ _ 1 cosh(cpx) e ¢~+ e ¢~ coth(gvc)
(9.142)
(9.143)
The concentration profiles in a sphere can, consequently, be obtained form eq.(9.141), since y~ = c s R, and, y=cir, c, = c~' -
sinh(¢ ) xsinh( )
(9.144)
The concentration dependence at different values of the Thiele modulus inside a sphere can be calculated and is presented in Figure 9.10. It is clear that the higher the value of Thiele modulus is (e.g. more prominent are the effects of mass transfer), the steeper the concentration profile decrease is.
369
c/c s 1.0
0.8-
0.6-
0.4-
0.2-
OlO;
:
]
0.0
0.2
.................................. I
0.4
'l
I
I
0.6
I
0.8
1.0
r/R Figure 9.10. The concentration profile dependence inside a spherical particle.
For calculation of the flux, N~, at the outer surface of the catalyst, equation (9.144) needs to be differentiated, taking into account the coordinate x, dc i
dc i
- - R : sin~(~b) ~bco ( ~ )
dx
sinh(&)/ x~ -)
(9.145)
which on the surface of the particle (x 1) becomes dc i
dx
Cs
x=l--
(9.146)
sinh(~b) (~bc°sh(~b)- sinh(¢))
As the boundary condition at x 0 (r R) was y 0 (y ci 0, it is interesting to see if eq. (9.145) results in a concentration gradient at the center of the spherical particle equal to zero (dc
= 0 ).Modifying eq. (9.145) we arrive at
d X x=
d ( c , / c, ) 1 (.(beC~+ Oe ¢~ dx sinh(~b) x
OeC~- Oe ~ .) x2
(9.147)
At the boundary conditions ~ x=0the concentration gradient in the centre of the sphere is
d(c,/c,) dx
1 _~ x=o sinh(b) ( o -
0eO'x~e~x= ° -5
"
)
(9.148)
The second term on the left hand side can be rearranged using a Taylor series neglecting the higher terms •
370
(~r)2 - 1 + #v -
2!
#e ~" x~-#e ~'~ ~=o = ( 1 + # +
X2
2!
=
2#
(9.149)
(#)2) o Tx:o
which in combination with eq. (9.148) gives ~
= 0.
On the surface of the spherical particle, the flux is
( dc, ~ N, =-D~,~),.=~e-
DeiCi' / ~, - R~b)(cosh(~b)-sinh(¢))-
D,c;' - R
¢b
.)
tanh(~b)-i
(9.150)
This flux is equal to the expression of the flux through the fluid film
R
tan )
1
(9.151)
The unknown surface concentration, cis, is now solved from eq.(9.151): [} Ci
,3 ¢1 =
(9.152)
l+ Rk(;, ~'(tanh(~b)- l~bl The dimensionless quantity, R kGi/Dei, is called the Biot's number for mass transport: Bi~e
-
RkC~i
(9.153)
Oeg
Jean-Baptiste Biot The Biot's number gives the ratio between the diffusion resistance in the fluid film and in the catalyst particle. Usually BiM > > 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+ <
tanh(~b)
The final goal, the effectiveness factor, rh, is now obtained from expression (9.116). After 4~R 2 insertion of the ratio, A p / V p 42"R3 / 3 ' for the spherical geometry, as well as Ni for first-order kinetics we obtain
/7-
3Nih R vikci p ,
(9.156)
(1 :/ 3D,,,
¢ tanh(~b)
'=-vR2kp, m+ ¢ Bi,,
1
1]
(9.157)
tanh(~b)
The final result can be rewritten in the form
1] =
3
ta (~b)
(9.158)
Bi m tard~(~b) Equation (9.158) gives the effectiveness factor for first-order reactions, in a spherical catalyst particle. Certain limiting cases are of interest: if the diffusion resistance in the fluid film can be neglected - as often is the case, since BiM is large - qi becomes: 3 1 1 r/= ~ [ t a n h ( ~ b ) ~ b )
(9.159)
If the Thiele modulus (~0) has a large value - in other words, if the reaction is strongly diffusion resisted - the asymptotic value for the effectiveness factor is obtained:
3 q = --
¢
since lira (~o --~ o0 tanh ~o 1.
(9.160)
372 9. 5.2.2. Slab a n d f i r s t order kinetics
Another simple geometry of interest is a slab (flake) catalyst particle (s-0). Particularly in three-phase reactors, in which often only the outer surface of the catalyst is effectively used, the slab approximation (s 0) represents the catalyst particle well. The differential equation (9.122) is transformed, for a slab -formed catalyst particle and first-order kinetics, into the equation:
d2ci ppv, kc, dr2
- D
=0
(9.161)
Equation (9.161) is directly analogous to the transformed equation for a spherical catalyst particle, therefore the solution is c~ =C,e ~ +C2e -# , where x denotes the dimensionless coordinate, x r/R, and the Thiele modulus is (o2 = (-v~ ppk)/(DeO R 2. For the boundary dc = 0 leading to conditions in case ofeq. (9.161) we use ~x=0
t d ( 0 ) = (CI~}C oh" -- C2~e-ch')lx=o = C l ~ - C 2 # = 0
(9.162)
and thus one arives at C, = C 2 . Utilizing another boundary condition c ( 1 ) = ( C l e # +C2e
(It)Ix=1 =CleO +C2e ~ =Cl(e~ + e ~ ) = c ~'
(9.163)
one gets the following dependence CS
C, -
(9.164)
and consequently using hyperbolic functions the concentration dependence is
c i = c;'
(e e + e #) _ cosh(&) e~ + e ~ cosh(~b)
(9.165)
The concentration profiles inside a slab are given in Figure 9.11. Note the difference between concentration profiles in a slab (eq. 9.165) and the sphere (eq. 9.144). This difference is not very pronounced at higher values of the Thiele modulus (Figure 9.12). The concentration gradient for the slab can be calculated from eq. (9.165) dc, _ dC, R _ dx dr
c;~# sinh(~) cosh(q~)
At the outer surface of the particle (x=l), the concentration gradient is
(9.166)
070
1.0
~
c / c ~ .................... ~ . s o.a--
i
....... ~
"
0.0
0.2
i ..................
......... .................... .......... ' - ~ . 4 ; ~ ; i ] ~ .<,.'i;I i
~ ........
0.4
~
,/,=lO
0.6
o.s
I r/l~o
Figure 9.11. Concentration profiles in a slab 1.0~ 0.5, shpege
............... 52 ..._....................... ....................."'" "'"::"1
o/oS£ ................... o;:-:i:b
o.~
• '"".""/1
~+~, .....................""
t
/I
2, slab..-'"
0k
71
....:::77
't
............ ...............
0.0
0.2
,~ 10 sphere,"/
" 0.4
~J
i/
0.6
0.8
1.0
r/R
Figure 9.1ft. Comparison of concentration profiles in a slab and a sphere at different values of Thiele modulus. dci
= c;~btanh(~b)
(9.167)
d x .r=l
The flux on the surface of the particle is, thus, given by
Ni = -
D ei (dc' ~ ~.- &- )r=R
D<'ic;(btanh(~b) R
(9.168)
By setting the fluxes through the fluid-film and on the particle surface equal, we obtain
NI
z
k ()-t \(ct'-c;") i
D
~'ic'~btanh(~b) R
(9.169)
The surface concentration, c[, can now be solved. The result becomes Ci
s Ci
--
b
1 + Ctanh(~b)
Bi,,,
(9.170)
374 where Biot's number is defined according to the eq.(9.153). For the flux through the outer surface of the particle, we obtain N~ =
D~,e~ ~btanh(~b) R 1 _~ ~btanh(~b) Bi,,
(9.171)
The definition of the effectiveness factor gives for a slab geometry (Ap/Vp = 1/R): Ni ~1 - R v, kc~'pr,
(9.172)
The final result, the effectiveness factor, tbr a reaction of first-order and a catalyst slab is tanh(~b)
1 ~btanh(~b)
- -
~/=
~
1+
(9.173)
Bim
case that the diffusion resistance in the fluid film is negligible, BiM----)oQ, reduced to In
tanh(q})
tl - - -
eq.(9.173)
is
(9.174)
¢
If the Thiele modulus has a large value, an asymptotic value is again obtained for the effectiveness factor, J/i 1
q = --
(9.175)
since lim(q~---~m) tanh(q))=l. Let us recall the definition of the Thiele modulus 42 =-vikpFR2/D~.,. The asymptotic results, eqs.(9.175) and (9.160), indicate that an asymptotic value for q,, for an arbitrary geometry, might be
tl -
s+l
(9.176)
O
since the nominator is equal to unity at s=0 (slab) and 3 at s=2 (sphere). Let us now define the generalized Thiele modulus, ~0s,for a first-order reaction according to the following:
~2
OeTel (S~) 2
~ ~Ap)
aei
(9.177)
375 where L is the linear dimension (equal to the length of a slab, or R/2 in case of cylindrical particles, or R/3 for spheres). Now the asymptotic value of the effectiveness factor, for an arbitrary geometry, can be written as 1
r/= - -
(9.178)
A comparison of the expressions for spherical, cylindrical and slab geometries (Figure 9.13) shows that the effectiveness factor is not very sensitive to the particle geometry.
cylinder slab I I
I I
I I
I I I i~ IIIIIIb~\PI
,7 I I I IIIIIII I
I I IIIIIII
,g
I I I IIIII I IIIIIII
lllllll P~.IIIIIII
I I I IIIIIII I BJIIIII
Figure 9.13. A comparison of the expressions for spherical, cylindrical and slab geometries (F. Kapteijn, G.B. Marin, J.A. Moulijn, Catalytic reaction engineering, in Catalysis: an integrated approach, Elsevier, 1999).
9. 5. 3. Asymptotic effectiveness Jactors Jbr arbitrary kinetics The expressions in the previous section were derived for first order kinetics. Although analytical expressions cannot be derived for arbitrarily reaction kinetics, useful semianalytical expressions can be still obtained for the effectiveness factor. We shall hereby restrict ourselves to the slab geometry (s=0). Consequently, the balance equation, is reduced to
d2c,
---+ dr 2
tOv~ D ~i
=0
(9.179)
The dimensionless coordinate, x=r/R, and the dimensionless concentration, y=c/ci b, are introduced. Equation (9.179) is thus transformed to
d2y F IOvR2
---
dx 2
c,"D ~,,
r, = 0
(9.180)
Let us define the relationship ri' as follows
r,' = ~ / r~(c b) Now, a generalized Thiele modulus can be defined as
(9.181)
376
02 -
(9.182)
Pfli(cb) R 2
and eq.(9.180) can be re-written to d (~Yx)-~b2r,' = 0
(9.183)
dx
Multiplication of eq.(9.183) with dy, followed by integration leads to dv / dx 7
,/
"
f ~o ay d/dY) = #"ir jy dx k dx ) ,,o
(9.184)
The result becomes, = 2~ 2 fr'dy
(9.185)
)*
in which y* denotes the dimensionless concentration in the middle of the particle, y , _ ci,/r 0)
(9.186)
c;
and dy/dx gives the dimensionless concentration gradient on the outer surface of the particle. The concentration gradient becomes, consequently: - )?
]0.5
(9.187)
~ --
L
Y*
J
The flux of component i, on the particle surface, is N, =-Dci dci dr
D ,,,c it, dy R dx
(9.188)
and the ratio, Ap/Vp, for a slab particle is 1/R. After taking into account eqs. (9.187) and (9.188), together with the definition of the effectiveness factor, eq. (9.116), we obtain the expression for the effectiveness factor: F v,
rl - - 11,(c h)ppR 2
] °5
377 By taking into account the definition of the Thiele modulus, a new form of the equation describing the effectiveness factor is obtained: F y'
]0,
/ The surface concentration, yS, is affected by the diffusion resistance in the fluid film, around the catalyst particle. After setting the fluxes at the outer surface of the catalyst equal (x-l),
Nj =-k(7,(c I' - c ; ) -
D~i dci R dx
(9.191)
the following relation is obtained
D~,i dy _ kc;,(1 - y') R dx
(9.192)
This equation gives the dimensionless surface concentration, yS : y'=l
1 dy Bi,. dx
atx= 1
(9.193)
The Thiele modulus, according to the definition becomes ~b2 -
11'(~=')P~'R 2
(9.194)
The effectiveness factor, eq.(9.190), approaches a limiting value as the Thiele modulus increases in value. For large values of the Thiele modulus, the concentration in the centre of the particle approaches zero. In other words, lim(~0--->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<
dr
) ~ R : ( - a H , 9 , o , , =0
(9.203)
j
This expression is valid for spherical geometry only. it can be shown that for an arbitrary geometry, the energy balance can be written with the form factor, 09:
1
r"
dr
+ ~ R, (-AH,:,)pp = 0
j
(9.204)
The energy balance, eq.(9.204), has the following boundary conditions:
dT
= 0, r = 0
(9.205)
T = T', r = R
(9.206)
dr
The first boundary condition follows from symmetrical reasons. In practice, the effective heat conductivity of the catalyst, 2e, is often so high that the temperature gradients inside the particle are minor. On the other hand, there often emerges a temperature gradient in the fluid film around the catalyst particle, since the thermal conduction of the fluid is limited. The energy balance of the fluid film is reduced to
1 r"
dr ) 0
(9.207)
dr
since no reactions take place in the film itself. The heat conductivity, Zf, in eq. (9.207), denotes the conductivity of the fluid. Because the fluid film is extremely thin compared to the catalyst particle, and the heat conductivity of the fluid can be assumed as approximately constant, eq.(9.207) can be simplified to
381 d2T
0
-
dr 2
(9.208)
having the boundary conditions, T = T',r=
(9.209)
R , T = T~',r = R + 6
where ~ denotes the thickness of the fluid film. The equation system, eqs.(9.208) - (9.209), is analogous to that of mass balances for the film. Equation (9.208) implies that d T / d r = a , where a is a constant. Integration of it gives T = a T + b , i.e. the temperature profile in the fluid film is linear. Insertion of the boundary conditions, (9.209) offers the opportunity to determine the constants a and b: T D __ T s
a=
(9.210)
3
(9.21 1)
b - T~ ( R + c~) - R T h
g The temperature profiles in the film are, consequently, given by Tb-T' T(r) - - a~
r +
T~(R+6)-RTb S
=
(Tb-T')(r-R) c~
~T"
(9.212)
The temperature gradient in the film becomes dT
T t' - T"
(9.213)
8 The concentration profiles and gradient in the film can be written in a similar way ci(r )
dci
=
ci
dr
(c, b - c ,
)(r-R)
,
~-c i
g b
-- Ci
(9.214)
s
(9.215)
6
The heat flux through the film, M. becomes
Mr=~
"\ dr )r=R
Mr = R-
g
~-
(9.216)
~lb-T)'
(9.217)
382 The quantity 2f/~ is called the heat-transfer coefficient of the film, h. Thus, we obtain
Mr=,e=-h(Tb-T ~)
(9.218)
The energy balance for the catalyst particle, eq.(9.204), can be integrated as follows, (9.219)
where y denotes the upper integration limit, y=)ceRS(dT/dr),.=R.For the heat flux at the surface of the particle, we thus obtain R
M~=~ =
"t dr/,.=e
PP I~-, R/(-kHr/)r"dr
(9.220)
R' 0 j
The heat fluxes, eqs.(9.220) and (9.218), are set equal, and a new relationship, is obtained 19
R
(9.221) 0 .I
that gives an expression for the temperature of the surface, 7*: R
r
r" + hR' p" ! 2
(9.222)
If the dimensionless variable x, O R
r xR, is applied, eq. (9.222) is transformed into
I
V ~= V" +S~_ jy" R,(_AHr,)xdx
(9.223)
0 l
Equation (9.223) is global and general: it is valid for those cases when the temperature varies inside the particle, T(r)# 7~, as well as for the cases when the whole particle has the same temperature, T(r) = 7~. In the latter case eq.(9.223) can be applied through iterative calculation of the surface temperature: the mass balance equation for the particle is solved ~br an assumed temperature, the concentration profiles are obtained and, finally, the integration of eq.(9.223), can be conducted. Eq.(9.223) gives, thus, a better estimate for the surface temperature. If considerable temperature gradients emerge inside the particle, the original energy balance needs to solved together with the molar balance equation for the particle. The effective heat conductivity of the particle, )~e, is, however, constant in practice. Therefore eq. (9.204) can be simplified to
~'~ dr 2 +-r
+
Rj(-AH,.j)pp = 0
(9.224)
080
The boundary conditions, eqs.(9.205) - (9.206), are still valid. However, whether a temperature gradient does, indeed, exist in the fluid film or not, the boundary condition is applied accordingly. The boundary condition d T / d r = 0,r = 0 is always valid, whereas the boundary condition T = T b..r = R
(9.225)
is valid in the case that no temperature gradients emerge in the fluid film. The boundary condition, dT _ h (T b _ T),r = R dr A,,
(9.226)
is valid if a temperature gradient exists in the fluid film. The solution of this coupled system of molar mass balances and the energy balance always needs to be conducted numerically. Analytical solutions cannot be applied, since the energy and mass balances are coupled through concentrations in the reaction rate expressions and through the exponential temperature dependencies of the rate constants. The numerical solution leads to the following dependence of the catalyst effectiveness factor on the Thiele modulus, Arrhenius and Prater numbers qo = j {~b, ~,, fls]
(9.227)
where the Thiele modulus for instance of a first order reaction is defined as
_ v,, /k(r)p,,
(9.228)
and the Arrhenius and Prater numbers are respectively 7/-
Ea RE
fls- (-AH"A)D 'CA ZE
(9.229)
(9.230)
For exothermic processes the reactions cause a temperature rise inside the particle. This usually leads to increased values of the rate constants. This increase of the rate constants can sometimes overcompensate for the lower concentrations (compared to those in the bulk fluid) caused by the diffusion limitations in the particle. As a result, the reaction rate becomes higher than the reaction rate that is obtained with the concentrations in the bulk phase and temperature. Consequently, the effectiveness factor exceeds 1! This effect is particularly emphasized at small values of the Thiele modulus. The catalyst effectiveness thctors as a function of Thiele modulus at different values of the Prater numbers are illustrated in Figure 9.15.
384
t,0
0,1 N,II
l,O
1~,0
Yhiele modulus
Figure 9.15. The catalyst effectiveness factors as a function of Thiele modulus at different values of the Prater numbers. Another interesting phenomenon can emerge under non-isothermal conditions; for strongly exothermic reactions there will be multiple solutions to the coupled system o f energy and mass balances even fbr the simplest f~rst-order reaction. Such steady-state multiplicity results in the existance o f several possible solutions for the steady state overall effectiveness factor, usually up to three with the middle point usually unstable. One should, however, note that the phenomenon is, in practice, rather rarely encountered, as can be understood from a comparison o f real parameter values (Table 9.2).
Table 9.2. Values of the Prater and Arrhenius numbers and Thiele modulus for industrial processes. Reaction
13
Y
q)S
Ammonia synthesis
0
29.4
1.2
Synthesis of higher alcohols from CO and Hz
0.001
28.4
Oxidation o f CH3OH to CH20
0.0109
16
1.1
Synthesis ofvinylchloride from acetylene and HC1
0.25
6.5
0.27
Hydrogenation of ethylene
0.066
23-27
0.2-2.8
Oxidation of H2
0.1
6.75-7.52
0.8-2.0
Oxidation of ethylene to ethylene oxide
0.13
13.4
0.08
Dissociation of N20
0.64
22
1-5
Hydrogenation of benzene
0.12
14- | 6
0.05- | .9
Oxidation of SO2
0.012
14.8
0.9
385
9.6. Mass transfer and enzymatic kinetics In section 9.5.3 the arbitrary kinetics for the slab geometry was considered. In this section we will use Michaelis-Menten kinetics for describing immobilized enzymes and spherical particles following the nomenclature and approaches frequently encounted in the literature on bioreaction engineering; for instance in the treatment provided by J.E. Bailey and D.F.Ollis presented here with the details of the derivation.
James E. Bailey
David F. Ollis
The balance equation for a spherical particle is written in the following form for a volume element with a surface area A
dc dc A~+d~DcT;Ix+~ - A~Dc T;l~ : rAdflX
(9.231)
We are presuming that the intrinsic reaction kinetics of the immobilized enzyme catalyzed reaction is of the Michaelis-Menten type, therefore eq. (9.231) takes the form 4
(x
ds
ds
=4 x 2
Vm.~S
dx
(9.232)
K,,, + s
where Do is the effective diffusion coefficient, s, Kin, Vm~x, following the commonly accepted notation in biokinetics stand for substrate concentration, Michaelis-Menten constant and maximum rate. Note as dx is small x+dx x, the first term in eq. (9.232) can be simplified to
ds ds 4~:r2D GI~+,~-4~RD ~GI ' '
x=
47t3C 2
V,.,~xs dx K,,, +s
(9.233)
Dividing both sides by 47r gives X2
--IdS - X 2 dS dx Ix+dx dx x dx
=
Vmax S
D~(K m +s)
x2
(9.234)
Eq. (9.234) can be rearranged to d (x 2 __)ds_ V,naxS X2 dr dx" D e,(K m + s)
(9.235)
386 The expression on the left hand side is transformed to
(x2 d._.~_~),=2x_~_+ x 2 d2S ax ax dx 2
(9.236)
Taking into account (9.236) and dividing the left and right hand sides in (9.235) by x 2 we arrive at the differential equation
des 2 ds gmaxS 1 -- + dx 2 x dx K m + S D ~
(9.237)
which has the following boundary conditions. Since the concentration profile through the pellet is symmetrical about the center line of the sphere, then & 71,=0=0
(9.238)
The surface concentration at the external surface of the pellet is equal to the bulk substrate concentration So S]~ R=S0
(9.239)
In the following treatment we will neglet the effect of film mass transfer resistance since usually it is less important than the internal diffusion. Multiplying eq. (9.237)by R 2/S0we arrive at R 2 d2s
----
S o dx 2
+
2 ds R 2
gllnaxS R 2 1
- - xdxS 0 K,.... + s S 0 D
(9.240)
Eq. (9.240) can be transformed to a dimentionless form with the dimentionless concentrations and radii -£ = S / S o ; F = x / R
(9.241)
Since
d2s - d ( ~ )
s0
= d2s
(9.242)
60
then the first term in eq. (9.240) is written in the form R 2 d2s
d2-d
S o dx 2
dF 2
Also the second term on the left hand side of eq. (9.240) is rearranged to
(9.243)
387
2dsR
2
2R ds R
2 dg
x dx S O
x S O dx
FaY
(9.244)
The term on the right hand side of eq. (9.240) is transformed to Vmax S R2 Vln~xS R2
1
R2 Vmax t K., s
R2 Vm~x t K., s
D
De
K,, S O
~
K + s S OD
(I+K)D ~
, (l+Km)
SO
SO
s ,So) ( I + K SO
(9.245)
Introducing the Thiele modulus
~=3~
(9.246)
D~
and the saturation parameter/7
/7-
So
(9.247)
Km the right hand side of eq. (9.240) can be written in the following manner
R2 Ulnl~×/K., s S O = 9(~2
D,,
(1+ S° s ) K,, S o
"S 1+ ,6~-
(9.248)
Finally eq. (9.240) becomes d2s
--
ay 2
t
2 ds F ctF
vR 2
- --
s
- 9~b2 - -
D So
(9.249)
l + flY
This equation cannot be easily evaluated analytically and requires numerical treatment. We can however evaluate the catalyst effectiveness factor, which is defined as the ratio of the observed reaction rate to the rate in the absence of mass transfer: robserved
r/-
(9.250)
Since the amount of consumed substrate inside the pellet is equal to the flux through the outer surface then the observed rate is expressed by r.~........, = D ~ A x ( d s / d r ) r = R
(9.251)
388 The rate in the kinetic regime is equal to the rate at the outer surface and is defined as (9.252)
~(So + K,,,)]
r = vp
where Vp is the particle volume, leading to the following expression for the effectiveness factor D e A x ( d s / dr),.=e
q =
(9.253)
Vp[VmaxS0/(S 0 + Kin) ]
After rearranging eq. (9.253) D~A~.(d~/dr)~= R [
r] = Vp[<-~S0/(S0 +Kin)]
D~,4rcR2(ds/dr),.=~
=
4KR3{I/maxSo/(So+K.,)] 3
"
= 1
D~.(dr/dr)~= R Vm~xSO/Km
-R[--- ' " - - - ] 3
(9.254)
So/K,,+I
and muliplying by R/S0 R D ( cls / dr ),.=R So 71= 1 R[ V'axS° / K " ] R
S o / K , , +I
So
R D ( ds / dr ),. ,~ = So V,n~xSo/ K,, ] R
3
S o / K , , +I
So
=I
D,. (dY / dF)>l .... / K,,
3
fl+l
(9.255) ]
we arrive at
q=
3
(dY / d?-),.< R2 V.~ax 1
(9.256)
9 D,K,,, l + f l
which could be further rearranged to (ds/dg)>l q = 3~be[1/(1 + fl)]
(9.257)
Since (dr/d~)F_ 1depends only on the Thiele modulus and saturation parameter, the catalyst effectiveness factor for immobilized enzyme catalysis also depends only on these two parameters. A practical problem in applying the Thiele modulus is that the values of the kinetic constants V,,o~ and Km are frequently unlmown. Therefore an observable Thiele modulus qb was introduced which depends on the measurable overall rate, but not on kinetic parameters:
¢=
rob, (R_)2 D cSo 3
(9.258)
389 Let us analyze what is the relationship between the Thiele modulus ~band observable Thiele modulus ~ . For that we will set S = S O. At these conditions the observed rate is equal to the rate in the absence of internal diffusion, thus ~'o
D~-o ( ) 2 -
1 V,,..×So R2 DeSo (Kin+So)(3 -)
1
(I+So//Km)
V,,,~x/K,, D~ ( )2
(9.259)
and finally
1 02 qbs°-l+fl
(9.260)
The effectiveness factor can be also presented using the following ratio q-
v°~" -
v°~"
(9.261)
vk,,~,,~, v~So/(So + K,,) which after some manipulations ~ ( R /3)-vob, q = (R/3)2VmaxSo/(So +K,,)
(R/3)2(Vo~s/K,,)(S o +K,,,) So (R/3)2(v ..... /K,,)
R (/3)-(vob~/K,,,) 2 (R / 3) (v ..... / K,,)S O/(S O+ K,n)
(R/3)2(Vo~s/SoDc)(So +Kin) K,, (R/3)2(v ..... /K,,D )
(9.262)
leads to a dependence of the catalyst effectiveness factor on the observed Thiele modulus, the Thiele modulus and the saturation parameter cD(l+p)
(9.263)
Plots for q - q~ for different values of the saturation parameter are given in Figure 9.16, showing that the effectiveness factor is relatively insensitive to the values of ft. In the general case of immobilized enzymes not only the internal diflilsion addressed above, but also diffusion through the film should be taken into account. Similarly to heterogeneous catalysis the catalyst effectiveness factor for slab geometry and low substrate concentrations (first order kinetics) is decribed by eq. (9.173) in a somewhat different form q=
tanh ~b ~b[1+ (~btanh ~b) / Bil
Eq. (9.264) can be written in a form
(9.264)
390
0
#2
(9.265)
tanh ~b Bi 10
i
-
05
d ,o > u
005 --
Spherical geometry Slab geometry
0.01 0.010
•
I
I irlll 0.05 0.1
I
i
z l llJJ] 0.50 1.0
I
~
I
I Jill:
5.0
10.0
50.0 100.0
Figure 9.16. E f f e c t i v e n e s s factor for i m m o b i l i z e d e n z y m e catalysts with M i c h a e l i s - M e n t e n kinetics (J. E. Bailey, D.F. Ollis, Biochemical engineeringfundamentals, M c G r a w - H i l l , 1986).
In the case when external mass transfer is negligible, Bi is high and catalyst effectiveness factor is equal to the effectiveness factor for internal diffusion as expressed by eq. (9.174). Thus, similar to electric circuits where the overall resistance is equal to the sum of resistances in series, it holds that 1
-
q.,
1 q
~
~2 Bi
(9.266)
9.7. External mass transfer
If there is only transfer of mass from the bulk to the external surface of the catalyst and internal diffusion does not play a role, the mass balance can be expressed by the following equation N = kjA~,(c b - c ~)
where
An
(9.267)
is the external surface area of the catalyst particle and k/ is the mass transfer
coefficient in the film layer surrounding the catalyst particle. At steady state, this flux is equal to the reaction rate in the particle N
=
Vjv(c' )
(9.268)
where Vp is volume of catalyst particle, and r~.is the rate per particle volume, which is related to the rate per unit mass (r W) and particle density (p~,) in the following way t; = pf, r~. For a first order irreversible reaction eq. (9.267) and (9.268) give
391
k l(c ~-c") =~f]k,,c' = 1/c'C"a'
(9.269)
where a'= Ap / Vz, is volumetric external surface area. The concentration at the external surface is determined from eq. (9.269) by rearrangement c" -
k/a'
(9.270)
ch
k,, + k j a ' which in combination with an expression for the reaction rate gives
r,~'h' = k,,c" -
k,,kja' ch k,, + k f a '
(9.271)
This equation can be further transformed and expressed via the external effectiveness factor ~]ext
F ob.s _
'
1
k,,c ~ = rl<,,vk,.ct'
(9.272)
kv +1
ksa' where
q,,xt
.
1 1 . . . k,. +1' l + D a
(9.273)
k~a' In eq. (9.273) D a is the Damk6hler number k~./k/a'.
Large values of Da correspond to
strong mass transfer limitations, therefore the obselwed kinetics in the domain of mass transfer is of first order. Comparison with (9.266) shows that when Bi number is small, e.g. internal control is not significant, and the kinetics is fast (Da>>l), the overall effectiveness factor is expressed as 1 Bi 7/= rL,x, - D----a ~b2
(9.274)
For immobilized enzymes when only external mass transfer is essential, eq. (9.269) should be modified to account for the Michaelis-Menten kinetics kL(S, - S~) - 1 Vm,xS, a' K,,, + S,
(9.275)
where S~ and S~ are substrate concentrations in the bulk of the liquid and on the surface.
392 The Damk6hler number, the ratio of the maximum reaction rate to maximum mass transtbr rate, then takes the form D a - ~nax/Sl kLa'
(9.276)
The external effectiveness factor is expressed as the ratio of the observed rate to the reaction rate at bulk fluid conditions Vm,,:& q _ C,t,., _ ~ _ Km+ S_____________~, S, / S L K,, / S,. + 1 _ x (k + 1) r~,,,/~ rkin,.ti~~ fmaxSL K,, / S L + S, / S L 1 (k + x) K,, + S~.
(9.277)
where k=Km
SI.
, x = S, SI.
(9.278)
When mass transfer is substantial (high values of Da) the substrate concentration at the external surface is negligable compared to the concentration in the bulk and (Sr. - S, ) ~ Sr~. In this case, the observed rate is determined by the rate of mass transfer and the external effectiveness factor is expressed via Da and k (this parameter is analogous to the reciprocal value of the saturation parameter, eq. 9.247)
k,.(Sj.-S,.)_ -
V,,,a,SL
K,, + S L
k,.S~,
l+k
(9.279)
V,,,a,S--------~-D----~
K,, +S~.
9.8. Internal diffusion and selectivity Not only catalytic activity but also selectivity can be influenced by mass transfer phenomenon. Analytical treatment is available for simple reactions and as the complexity increases, numerical treatment becomes necessary. A few very basic reaction types will be analyzed following the interpretation of Ioffe et al. (I.I. Ioffe, V.A. Reshetov, A.M. Dobrotvorskii, Heterogeneous catalysis', Chimia, 1985). For a consecutive reaction sequence A ~ B ~ C when the rate equations are of first order and diffusional limitations can be neglected dc 4 _ kACA dc~ = kACA _ kvc~, dcc - kvc~ dt ' dt dt
the expression of selectivity in intermediate component B is given by
(9.280)
393 dcB kAcA--kBcB . . . de"A kAc A
s .
1
kBCB
(9.281)
kACA
Differential equation (9.281) is an equation o f type I+M y x
-
dx
(9.282)
where M = k ~ / k A , which can be solved by introducing u = y / x , gives dy = udx + xdu and the following expression dx
du
x
which subsequently
(9.283)
(M-Ou-1
Intergration o f (9.283) at the boundary conditions c B = 0 for c A = c,0A finally gives at M
C/3 --
1 I CAM : ~:/ 1
1 - M I tcA)
CA
I
:
1
(9.284)
which gives the amount o f intermediate B as a function o f unreacted substrate (or conversion). From (9.281) and (9.284) an expression for selectivity follows
(F
1M-1 /
J
(9.285)
Let us now consider the case o f pore diffusion in a slab geometry with a characteristic dimension L, then for dimentionless concentrations c' it holds (see 9.5 for details) 2
v
d c A _ ~b2c,A dx 2 d2c'B
J2
- - -
v
-~ C A + 7
dr 2
(9.286) 2J2
O CB
(9.287)
where ~b2 = L 2 kA
Oe~A
y2 _ k~D,,A kADcm
(9.288)
(9.289)
394
The
boundary
conditions
are
at
x TMI:
i
bulk
D
c' A = c A / c A = 1, c' B - D
C bulk
e,B B
h,Jk --(C'~)L
and
e,ACA
dc'_____ 2 _ dc'______~_ 0 at x=0. Solution of eq. (9.287) is given by eq. (9.165), as eq. (9.286) is dx dx
independent ofeq. (9.287), e.g. c' - c ~ _ (e ~" + e ~') _ c o s h ( ~ ) A CAt'"lk e ~ + e ~ cosh(¢)
(9.290)
where c A~'~'¢k is the concentration in the bulk.~ since external diffusion is not considered. Eq. (9.287) is then rearranged to d2c', cosh(qir) d x 2 - Y-O2c'~-~b2 cosh(¢)
(9.291)
Solution of eq. (9.291) gives dimentionless concentration of B inside the catalyst pore c°sh(~byx) + - - - ~ 1 I.c°sh(¢yx) c'B = (c'B)L cosh(y~b) 1 - 72 ~ cosh(y¢)
c°sh(&).l cosh(¢) J
(9.292)
The generation (consumption) rates for reactants A and B are given as - Fhulk
A
=
~
hulk
,.CA
~
b,,/k
r • ,=, ~!K t , ~z J .~ c ~
(9.293)
tanh0
tanh(~by)-~cA
butk tanh ~b- y tanh(~by) l_y2
(9.294)
Differential selectivity is defined as the ratio of rates 1 __ s - ryk bulk rA
_
-tanh(~by) - y tanh ¢ 1 -- y2
D
bulk
y tanh(¢y) c,~c~ bulk tanh ~b D c,.4c A
(9.295)
When both reactions in the consecutive sequence occur in the kinetic regime, e.g. ¢ <<1 and y << 1, then eq. (9.295) is transformed into eq. (9.281), since tanh~b ~ ¢ and tanh(~by) ~ ~by. Differential selectivity in consecutive reactions depends on ~b and y . The higher the value of the parameter y = ~/k~De, A / k 4 D e , ~ is, the more pronounced the influence of diffusion is and the less the selectivity of the intermediate B is (Figure 9.17).
395 S
~0~ 2
0,e o,6
¢ Figure 9.17. Dependence of selectivity for consecutive reactions on Thiele modulus at different values of 2" ( 1.1. Ioffe, V.A. Reshetov, A.M. Dobrotvorskii, Heterogeneouscatalysis,Chimia, 1985).
The boundary conditions are c'
A,
= c'
.4~
= 1 (at x=l);
dc'A~ - dc'A~ - 0 (at x=0) dx dx
(9.296)
Differential selectivity could be analytically calculated if the competing reactions are independent leading to c'
- c°sh(~blx) c' - c°sh(¢2x) A, cosh(~bl) ' ~2 cosh(~b2)
(9.297)
and the effectiveness factors tanh ~b1
ql---,
tanh ~b2
;72 - - -
(9.298)
The ratio o f observed rates is then
k c bulk
rl
rh t ".4,
s - t'2
t 12 k c l2, , l.42 k
tanhqtl
~bl
q/2 tanh~b2
1
bzdk
tqcA,
~ k2
kl
D~A,_, tanh~bl cb';kA,
kc2 A2 b'lk ( k ~
k2
tanh(}2 Cb"lk A2
(9.299)
AI
and =
s
tanh ~b~ cA~
~k2D~A, 2 tanh~b~_ C;'";kA2
In the diffusion region (tanh ~b~ = tanh ~b2 = 1 ) eq. (9.297) is transformed to
(9.300)
396
Ik~D~A, tanh~b1 c A1b'lk
=
~
~
c bUlk
A,
(9.301)
which gives for equal diffusivities of A1 and A2 ~1
bulk C A~
sa~/'~'i°~ = 1l~-~ ~ ~2
(9.302)
A2
In the kinetic regime the differential selectivity of independent reactions is, in the case of first order reactions, bulk
ski~,i~., - kl cA'
(9.303)
(.,.bulk
k9 -
A2
Comparsion between eq. (9.302) and (9.303) show-s, that S
Jit~,.,,o~ _ ~ 1 / k2 _ ~ s~m~,,i~.~ kl / k 2
/ k,
(9.304)
Therefore, if the first reaction A ~ B ~ is the desired one and it is faster (e.g. kl>k2), the selectivity in the diffusion regime is lower than under pure kinetic control. For parallel reactions A ~ B ~ and A ~ B 2 , the differential equation describing the concentration profile of component A is
d2c'A +1 c' dx z - ¢~t2 ( f (c'A) y f2- ( A ))
(9.305)
where [~
,~
bulk., i
~b~= L I~IK1FltCA--)-/ C
bulk
(9.306)
and ~
bulk
7 ' - KIrltcA
)
k~r~(c ~I''lk)
(9.307)
If the reaction orders are different, for instance the reactions above are respectively n-th and m-th order ( f (c' A ) = (c' A )n; j~(c, A) = (c'~)", then the differential equation describing the concentration profile of component A is
397
d2c' dx 2
2 , ). = bl ((CA +
Y'
)
(9.308)
with
I ( bulkin-1
y,= ~(cAbUlk)..... ;01= L kl,C;, 4J
(9.309)
The solution of an equation for the ratio of diffusional to kinetic selectivity in product Bi when 7' << 1 given by Ioffe e t a l . ,
Sd~[/i~sio, Sk,~e,c ,
m +]
(9.310)
2n + 1 - m
implies, that when the reactions are of the same order, the differential selectivity in diffusion and kinetics is the same, regardless of the regime. If the desired reaction is of lower order (e.g. n<m), then according to (9.310) it is preferential to conduct the reaction in the diffusion region, since the highest penalty in the case of pore diffusion limitations is on the highest order reaction.
9.9. Internal diffusion and deactivation
For the analysis of the impact of deactivation in the case of internal diffusion limitations let us consider slab geometry. The diffusion model is then d2c , d x 2 - ~b2r ( c ' ) a
(9.311)
where a is the relative activity (or fraction of unpoisoned sites). Eq. (9.311) for uniform poisoning and the first order kinetics is simplified to d2c , dr 2 - (~2a)c'
(9.312)
Analogous to previous considerations, we can directly write an expression tbr the catalyst effectiveness factor in the case of particle deactivation r/,Oi~on~J _ tanh(~b~a) ~b~a
(9.313)
Apparent activity is defined as the ratio effectiveness factors for the poisoned and unpoisoned catalyst under conditions of pore diffusion limitations
398
tanh(gi~a) q ~o,,o,,~J _ Z - rl.,,poi,o.~,j
¢~aa
~ a tanh(O~a)
tanh(~b)
-
(9.314)
tanh(~b)
When diffusion limitations are negligible (e.g. tanh~b~a ~ O~a; tanh(~b) ~ ~b), eq. (9.314) is simplified as expected to Z = a. When, however, the Thiele modulus is large, Z = ~ a (Figure 9.18). 1.0
0.8
./ f
large Thie
Z" °6 T ¢0.4-
O ¢O
o i/
LI_ 0.0
I 0.0
i
0.2
,
i
0.4
,
i
,
0.6
i
,
0.8
1.0
Activity Figure 9.18. Dependence of the initial activity fraction on activity.
9.10. Elucidation
of the impact of mass transfer
9.10.1. Criteria for the absence of mass and heat transport limitations (fixed bed) In a general case (Figure 9.19) temperature and concentrtaion gradients are possible in the catalyst particles and in the gas film.
i
,,,,~/ /-j
c
T C
Ik gas Exothermic
Endothermic
Figure 9.19. Temperature and concentration gradients in the gas film and catalyst particle (after J.A. Moulijn).
399 The absence o f mass and heat transfer transfer is often verified by applying several criteria, which are usually derived in a way that deviations o f the observed rates from the ideal rates are not more, than 5%. q=
G~,
r(Tb, G )
-1+0.05
(9.315)
The advantage o f applying different criteria is that observed values o f reaction rates are used, but not kinetic constants such as in the Thiele modulus, which are often unknown. The criterion tbr external heat transfer was proposed by D.E. Mears (J. Catal. 20 (1970) 127).
[AH~[Gh'R •
<0.1
hTa
5 RTb
0.15 --EA Y
(9.316)
where h is the heat transfer coefficient, sometimes denoted as c~, Tb is the bulk temperature, AH. is the reaction enthalpy, EA is the activation energy. The criterion o f external mass trasnport is also due to Mears (Ind. Eng. Chem. Process Des. Develop. 10 (1971) 541) who considered the mass transfer rate equal to the observed rate fl(G - G ) 47cR2 = (4 / 3)r, bdrR 3
(9.317)
From (9.317) one gets
cl, - c, - r°b~'R 3fl
(9.318)
It can be assumed following eq. (9.315), that G - c , < 0.05 G , then the criterion for the n-th order reaction takes the form Gt,,R -
-
0.15 < --
fig
(9.319)
n
Experimentally a way to determine whether external mass transfer is present is to determine the dependence o f conversion on the flow rate.
CCA,1
t vvl
t
2
vv2
t
G Figure 9.20. Elucidation of mass transfer by changing the flow rate and catalyst amount
400 If the conversion at a fixed residence time in the reactor does not change with changing flow rate (and respectively catalyst amount), e.g. ~za,1= aA.2 , when W~/F°, = W 2/F22 (Figure 9.20) or conversion is proportional to the residence time while chaning the flow rate and keeping the catalyst amount constant (Figure 9.21) then external mass transfer limitations can be neglected. ~A, 1
Figure 9.21. Elucidation of mass transfer by changing the flow rate.
Internal heat transport limitations were considered by J.B. Anderson (Chemical Engineering Science 18 (1963) 147) and according to the fbllowing criterion such limitations do not occur if IAH~[r.~,~,R2
2~
RT~, 0.75 < 0.75 - - - - -
Ej~
7
(9.320)
where 2 is the thermal conductivity. Probably the most widely applied criterion is the one for internal mass transfer limitations in an isothermal catalyst particle, e.g. for pore diffusion. Due to Weisz and Prater (Advances in Catalysis 6 (1954) 143) no pore diffusion limitation occurs, if the Weisz modulus c~ = r'~t'~'R2 _
~b2q
(9.321)
c~,D, for the first order reaction is below unity (qb < 1 ), for zero-th rder reaction qb < 6 and for the second order reaction the Weisz modulus is below 0.3.
Paul B. Weisz For a non-isothermal catalyst particle according to the Weisz-Hicks criterion no internal heat and mass transport limitations occur if CI)exp( Yfl--@~/ < 1 ~l+pp,)
(9.322)
401 where PPr is the Prater number and y is the Arrhenius number. As follows from the considerations above for the calculation of the influence of mass transport phenomena mass transfer coefficients as well as diffusion coefficients have to be determined.
9.10. 2. Mass' transfer coefficients
Various correlations were proposed in the literature for heat and mass transfer utilizing dimentionless numbers. The Sherwood number, discussed above (eq. 9.89), is quite often also presented in the following form (9.323)
Sh - f l d D
where d is the characteristic length, and fl is the mass transfer coefficient, representing the ratio of total mass transfer to diffusive mass transfer. The Nusselt number is an analog of the Sherwood number and corresponds to the ratio of the total heat transfer to conductive heat transfer hd Nu = - -
(9.324)
2
Wilhelm Nusselt
Osborne Reynolds
Ludwig Prandtl
Ernst Schmidt
The Reynolds number is defined in the ~bllowing way Re-
Vd _ Vdp
v
~t
where V is velocity, p density, ¢t dynamic viscosity and v kinetic viscosity.
(9.325)
402 Mass and heat transfer correlations in packed beds reported in the literature are summarized in Table 9.3. Table 9.3. Mass and heat transfer correlations in packed beds. Mass transfer
Heat transfer
Gases Sh -
0.357 Re0641 Scl/3
oep
Nbl --
0.428 Re0641 prt,3
Cp
Sp
./, = 0.357 Re 0359 3
= 0.25 Re0.69 SCI/3
Sh = 1.09
Range
Ep
Liquids Sh
Mass transfer factor 0.416<e <0.788
Rel/3 ScVS
Nb/ = 0.30 ReO69pr~/3
j , = 0 . 2 5 R e 031
55
~'p Nu = 1.31
cq'p
Re//3 pr~/3
Jl) = 1.09Re 0.67
in Table 9.3. the Schmidt and Prandtl numbers are respectively
0.0016
and P r - Cp~ /I and e is the interparticle void fraction of the bed of particles. If the values of Re and Sc numbers are known, then the Sherwood number can be calculated and consequently the mass transfer coefficient can be determined. Analogously, heat transfer coefficients can be computed. Sc = v/D
Another approach is the use of massjz) and heat transferjH factors for packed bed reactors: Sh
Jz~(~s) - Re(Sc),/3
(9.326)
Nu
J" -
KtV)ne'-r"/3
(9.327)
Experimentally established correlations between the mass and heat transfer factors and Reynolds number j , = f ( R e ) are presented in Table 9.3 and are graphically illustrated in Figure 9.22. Obtaining Jz) from (9.326) or experimentall correlations and inserting it into the following equation,
p(,/2 3 (9.328)
where G is the mass flow ( G = Vp ), the mass transfer coefficient fl (kg/s) can be calculated.
403 1.0 O.g ~
0,4
~-
°3
i'N \ "X,
0.2
0.1 O.08 J
0.06 0.04
0.02
0,01 10
20
30 411
60
gO 100
200
300 400
~
800 1000
) Re = dpGtl.~
Figure 9.22. Correlationbetweenmass and heat transfer factors and Reynoldsnumbers. For calculation of heat transfer coefficients, the Chilton-Colburn analogy is applied. From Figure 9.22 it can be concluded that j J j ~ 0 . 7 and consequently from eq. (9.327) the heat transfer coefficient is obtained 9/3
J"
_
h C, G
(9.329)
For slurry reactors several correlations were proposed in the literature, which relate the dimensionless Sherwood number to the Reynolds and Schmidt numbers. Data collected by Temkin for various slurry reactors (Kinetika i kataliz, 28 (1977) 493) show that the Sherwood number can be described by the following equation Sh = 1.0Re 1~2Sc 1/3
(9.330)
In the case of low Reynolds numbers, eq. (9.330) should be modified to contain a correction term, which accounts for mass transfer in the absence of stirring, as in the correlation of Sano Sh = 2 + 0.4 R e f 5 Sc 1'3
(9.331)
The dependence of the Reynolds number on the local velocity can be established in terms of the Kolmogorov theory of turbulence.
Andrey Kolmogorov
404 According to this theory, the only parameter needed to describe the probability distribution of the relative velocity field V~ in a homogeneous isotropic turbulent fluid is the average energy dissipation ~, i.e. V2 oc (g)[)l/3
(9.332)
which leads to R e d 0(2 ( w ) 1/3
(9.333)
v 3
Equation (9.333) provides a possibility to estimate the gas/liquid and liquid/solid mass transfer coefficients flcL, ills according to:
(9.334)
(9.335)
where c denotes the specific mixing power, D°Ae is the mutual diffusion coefficient of solute A in solvent B, p is solvent density, r/solvent viscosity, db,, is the diameter of gas bubbles, and dp is the diameter of the catalyst particles. Theoretical calculations of the maximum specific mixing power are based on the assumption that all of the energy of the impeller is dissipated in the liquid. Since the power of the motor is known, E = m~lJgrc, r
PLVL
(9.336)
The specific mixing power can also be determined from (9.335) by experimentally determining the rate of a mass transfer or the heat released in the process. For instance, the dissolution of solid materials is limited by mass transfer and could be used for calculation of mass transfer coefficients and specific mixing power. An example illustrating how the calculations could be performed for a three-phase system will be presented in the following section.
9.10. 3. Calculation of dif]hsion coefficients' As became apparently clear from the treatment above, a reliable calculation/estimation of diffusion coefficients is a must for correct evaluation of the influence of mass transfer. In the gas-phase molecular diffusion prevails (Figure 9.23), however in small pores, not only molecular, but also Knudsen diffision (Figure 9.24) could play an important role. In the latter case, the mean free path becomes comparable to the size of the pore and molecules are just as likely to hit the walls as they are to hit each other.
405
Q
O
O
o
o
°°
o
o Figure 9.23. Molecular diffusion
O/
"AO O~.....~O
Figure 9.24. Knudsen diffusion
The theory of diffusion in the gas phase is well developed. In general the diffusion flux of a component i (Nj) depends on all of the components. According to Stefan-Maxwell theory, the diffusion flux and the concentration gradient are governed by the matrix relation FN
dc
(.~37) 9 ~,
_
dr
where F is a coefficient matrix, numerical inversion of which is required for a rigorous computation of the diffusion fluxes. This matrix contains the contribution of moleculm" and Knudsen difthsion coefficients. An approximate calculation is based on the Fick law, which gives a simple relation between the diffusion flux and the concentration gradient. In porous media, the effective diffusion coefficients are utilized. In case both molecular D ~ and Knudsen Dtc~ diffusion are present, the Bosanquet approximation is used 1 -
D
-
1
1
DAB
DKA
-
(9.338)
The first part of this equation originates from intermolecular collisions (molecular diffusion) and the second from collisions between the molecules and the pore walls (e.g. Knudsen diffusion). Equation (9.338) is valid, strictly speaking, only for equimolar diffusion in a binary solution ( N A = - N n ) . in porous media, the diffusion coefficient calculated from (9.338) should be corrected fbr porosity and tortuosity according to eq. (9.103). For Knudsen diffusion molecular gas theory gives that the Knudsen diffusion coefficient is proportional to the pore radius and the mean molecular velocity 2 D x = ~-r~u
(9.339)
.5
where re is the pore radius, and u is the average velocity
u=
8 RT
erM
(9.340)
406 hence the Knudsen diffusion coefficient is proportional to the square root of temparature
2 r ~__ RT
(9.341)
Dr = 3 "~]~r M
The pore radius in the case of cylindrical pores can be calculated from the total pore volume Vp (cm3/g) and the total surface area S (cm2/g), since their ratio is Vp S
70e2L -= 0.5r~ 2~L
(9.342)
Hence eq. (9.341) takes the form .
_~ S
~c M
3Sp.
(9.343)
M
where @ is porosity and p:, is the particle density. Wilke has developed an approximate method to calculate the molecular diffusion coefficients starting from the Stefan-Maxwell theory 1 -
x I
(9.344)
Dln~ X 1
]=2~ Ol: whereDt, , is the difthsion coefficient of a certain component, xl is its mole fraction, and
D~/
are binary diffusion coefficients. Different methods were proposed in the literature to calculate the binary diffusion coefficients. One of the most common is the equation of Chapman and Enskog, which is based on kinetic gas theory and uses Lennard-Jones paremeters
1 / 05 DAB(Cm2/S) = 1.8829"10 3
P( AdaA
(9.345)
r'l Sydney Chapman
David Enskog
In equation (9.345) M,A and MrB are relative molecular masses (dimentionless), p is total pressure (kPa), o-As (nm) is the characteristic length (Le~mard-Jones parameter) for a pair of
407 molecules, £2AB is a collision integral and is a function o f I'BT/CAB, where eAB (J) another Lennard-Jones parameter and kB (1.38 • 10 -23 J/K ) is the Bolzmann constant. Often in the literature, values Of~aB are given in A (10 1° m), then total pressure should be in 105 Pa, if the units o f the diffusion coefficient are cm2/s.
Charles Satterfield
Charles R. Wilke
The procedure to calculate diffision coeffcients in binary mixtures, as presented in the literature by Satterfield, will be given here. The Lennard-Jones parameter, C~AB,is calculated by the following equation,
crab
1 ~(crA +crB)
(9.346)
while the collision integral depends on the constant ~A~, which is calculated for a binary system according to CAn = ~
~B
(9.347)
The Lennard-Jones force constants for some molecules are given in Table 9.4.
Table 9.4. The Lennard-Jones force constants for some molecules Substrate g/mol H2 He Ne Ar Kr Xe air N2 02 CzH 2
Cell4 C2H6 C3H8 i-C4H10 n-CsH12 n-C6Hi4 n-CsHIs cyclohexane
Molar mass 6 , A 2.016 4.003 20.183 39.944 83.80 131.3 28.97 28.02 32.0 26.04 28.05 30.07 44.09 58.12 72.15 86.17 114.22 84.16
2.915 2.576 2.789 3.418 3.61 4.055 3.617 3.681 3.433 4.221 4.232 4.418 5.061 5.341 5.769 5.909 7.451 6.093
e/kB,K 38.0 10.2 35.7 124.0 190.0 229.0 97.0 91.5 113.0 185.0 205.0 230.0 254.0 313.0 345.0 413.0 320.0 324.0
Substrate g/mol CO CO2 NO NeO SO 2
C12 Br2 12 CH4 CHzC12 CHC13 CC14 C2N2 COS CS2 Cc,H6 CHBC1
Molar mass ~Y, ~ 28.01 44.01 30.01 44.02 64.07 70.91 159.83 253.82 16.04 84.94 119.39 153.84 52.04 60.08 76.14 78.11 50.49
3.590 3.996 3.470 3.879 4.290 4.115 4.268 4.982 3.822 4.759 5.430 5.881 4.38 4.13 4.438 5.270 3.375
e/kB,K 110.0 190.0 119.0 220.0 252.0 357.0 520.0 550.0 137.0 406.0 327.0 327.0 339.0 335.0 488.0 440.0 855.0
408
V a l u e s for the c o l l i s i o n integral are collected in the T a b l e 9.5. T a b l e 9.5. V a l u e s for the c o l l i s i o n integral kBT/~AB 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30 1.40 1.50 1.75
~AB 2.662 2.476 2.318 2.184 2.066 1.966 1.877 1.798 1.729 1.667 1.612 1.562 1.517 1.476 1.439 1.375 1.320 1.273 1.233 1.198 1.128
f2AB
kBT/~AB 2.0 2.5 3.0 3.5 4.0 5.0 7.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 200.0 300.00 400.00
1.075 1.000 0.949 0.912 0.884 0.842 0.790 0.742 0.664 0.623 0.596 0.576 0.560 0.546 0.535 0.526 0.513 0.464 0.436 0.417
Diffusion coefficients which are calculated using the Chapman-Enskog equation and the procedure outlined above in general agree quite well with experimental data for diffusion coefficients. However, in practice a semi-empirical modification of the Chapman-Enskog equation developed by Fuller-Schettlet-Giddins gives better agreement ,,75( 1 DA,(m 2 /s)=l.O,lO 7
TJ
(MrA + M r B j
P /[ atm
1 / 05
I'vA
~,'3
(9.348)
~/3¥ + vB
}
where VA and VB are the diffusion volumes of the gases. Diffusion volumes of some common molecules are given in Table 9.6. Table 9.6. Diffusion volumes of some common molecules Substrate Diffusion volume Substrate Diffusion volume H2 He Ne Ar Kr Xe Nz 02 air
6.12 2.67 5.98 16.2 24.5 32.7 18.5 16.3 19.7
CO CO2 NH3 N20 SO2 CI 2
SF6 Br2 H20
18.0 26.9 20.7 35.9 41.8 38.4 71.3 69 13.1
409
vB c a n b e a l s o e s t i m a t e d compound
from
volume
increments
associated
with
each
element
in the
(Table 9.7).
T a b l e 9.7. A t o m i c a n d s t r u c t u r a l d i f f u s i o n v o l u m e i n c r e m e n t s Element
increment
Element
Increment
C H O N Aromatic ring
15.9 2.31 6.11 4.54
F C1 Br 1
14.7 21.0 21.9 29.8
-18.3
S
22.9
T y p i c a l v a l u e s o f g a s p h a s e d i f f i a s i o n c o e f f i c i e n t s a r e ca. D A - 1 0 4 m2/s. S o m e v a l u e s o f b i n a r y d i f f u s i o n c o e f f i c i e n t s a r e p r e s e n t e d i n T a b l e 9.8. T a b l e 9.8. B i n a r y d i f f u s i o n c o e f f i c i e n t s Gas pair T DABp (K) (cm2s "1 atm) Argon-neon 293 0.329 Dichloro difluoromethane ethanol 298 0.047 H20 298 0.105 -
-
Gas pair Air-ammonia - benzene - benzene - diphenyl - ethanol 12 - C12 - CO2 CO2 - Hg - methanol - naphthalene - O2 - SO~ - toluene - toluene - H20 - H20 -
Ethane-methane - propane
293 293
0.163 0.085
Helium-argon benzene ethanol - H2
273 298 298 293
0.641 0.384 0.494 1.64
CO2-benzene ethanol -H2 Nz - methane - methanol - propane
318 273 273 298 273 299 298
0.071 0.069 0.550 0.167 0.153 0.105 0.086
CO-ethene - N2 - 02 - He
273 288 273 273
0.151 0.192 0.185 0.651
Nz-ammonia - ethene - 12 - 02
293 298 273 273
0.241 0.163 0.070 0.181
-
-
-
-
-
- HzO - H 2
O2-alnmonia - benzene - ethene - CC14 Water-ethene - H2 - helium - CO - N2
- methane - 02
T (K) 273 273 298 491 298 298 273 273 1000 614 298 298 273 273 273 298 273 298 1273 273 293 296 293 298 307.7 307.2 307 307.4 307.5 307.6 308
DaBp (cm2s "l atm) 0.198 0.077 0.096 0.160 0.132 0.834 0.124 0.136 1.32 0.473 0.162 0.061 0.175 0.122 0.071 0.084 0.220 0.260 3.252 0.612 0.253 0.094 0.182 0.071 0.204 1.020 0.902 0.202 0.256 0.292 0.282
T h e t h e o r y f o r d i f f u s i o n c o e f f i c i e n t s i n l i q u i d s is n o t v e r y w e l l d e v e l o p e d . S i n c e t h e g e n e r a l t h e o r y f o r t h e c a l c u l a t i o n o f b i n a r y d i f f u s i o n c o e f f i c i e n t s i n t h e t h e l i q u i d p h a s e is m i s s i n g , semi-empirical equations are often used. These equations describe the diffusion of a dissolved
410 component in a solvent, ibr the conditions, when the concentration of solute is much lower than the solvent concentration. The treatment of the diffusion of a molecule A in a solvent B is based on the Stokes-Einstein equation, which correctly describes the trends in liquid-phase diffusion.
George Stokes
Albert Einstein
The ditlhsion coefficient tbr a sperical particle is given by
RT Dx. - - 67r¢l~R A
(9.349)
where ~t B is the solvent viscosity and RA is the radius of the solute molecule. Estimation of this radius could be difficult, especially tbr non-spherical molecules, moreover polar solvents can form dimers or trimers. Therefore, by considering these factors, Wilke and Chang proposed an equation for the estimation of liquid diffusivities D°AB, which gives accurate results for the prediction of diffusion coefficients of gases in liquids
DoA~ = 7.4x 10 s(~b M~)V2T [cm2/s ]
(9.350)
,.b cP L The dimensionless association factor ¢ is equal to 2.6 for water, 1.9 for methanol, 1.5 for etahnol and 1 for benzene, heptane, and other non-associated solvents, MR is the molecular mass of solvent, ¢t B is solvent viscosity in cP at temperature T [K], Vb(Aj is the liquid molar volume at the solute's normal boiling point in cm3/mol. Another method used in the literature was introduced by Hayduk and Laudie
DoAB _ 13.26x10 5 [cm2/s] xlA4 Tz0.589 /d:,) v:,(x )
(9.351)
where /.t B is the viscosity of the solvent (cP), and VA is the molar volume of the solute (cm3/mol). Molar volumes of some molecules are given in Table 9.9. Table 9.9. Molar volumes of some molecules Methane 37.7 Methanol 42.5 Propane 74.5 Dimethylether 63.8 Heptane 162 Acetone 77.5 Cyclohexane 117 Acetic acid 64.1 Ethylene 49.4 Methyl formate 62.8 Benzene 96.5 Ethyl acetate 106
Acetonitrile Methyl chloride Phosgene Ammonia Water SO2
57.4 50.6 69.5 25 18.7 43.8
411 For a general compound the molar volume can be estimated from the atomic increments following Le Bas (Table 9.10) Table 9.10. Atomic increments Atom Increment, cm3/mol C 14.8 H 3.7 O (except as noted below) 7.4 -in methyl esters and ethers 9.1 -in ethyl esters and ethers 9.9 -in higher esters and ethers 11.0 -in acids 12.0 -joined to S,P,N 8.3 N - double bonded 15.6 - in primary amines 10.5 - in secondary amines 12.0
Atom Br C1 F I S Ring -3 membered -4-membered -5-membered -6-membered -naphthalene -anthracene
Increment, cmB/mol 27.0 24.6 8.7 37 25.6 -6 -8.5 - 11.5 - 15 -30 -47.5
9.10. 4. Heat conductivity Analogously to mass transfer, for heat transfer there is a necessity to calculate the heat conductivity A in the pores, which appears in the Fourier's law relating the heat flux through a sur~hce with the temperature difference along the path of heat flow
dT q =-2,, dr
(9.352)
The heat conductivity/t can be computed according to a following equation (9.353) where Z,otij,Zg are respectively heat conductivity of the solid and gas in the pores, and x is the porosity. Since heat conductivity of solid materials is several orders of madnitude higher than the conductivity of gases, the second term in eq. (9.353) can be neglected. The values of heat conductivities for industrial catalysts are ca. 10-I- 10-3 Watt/m K.
9.11. Three
phase
systems
in the case of three phase heterogeneous catalytic reactions, the rate of the process and its selectivity can be determined either by intrinsic reaction kinetics or by external diffusion (on the gas-liquid and gas-solid interface) as well as by internal diffusion through the catalyst pores. Careful analysis of mass transfer is important for the elucidation of intrinsic catalytic properties, for the design of catalysts, and for the scale up of processes.
412 Let us as analyse now a process, when all the mass transtbr limitations are present in a threephase system, e.g. mass transfer at the gas-liquid interface, transfer in the liquid to the external surface and internal diffusion. Heterogeneous catalytic hydrogenation reactions is a typical example of such a process. Quite often the reaction rates are defined through hydrogen partial pressure, but not hydrogen concentration in a solution. Therefore the following treatment will include hydrogen partial pressure. Then the rate of the gas-liquid mass transfer is given by ~ , i = G , , ( cg"',,, - c " % , ,
) = G,,s(P
- p)
(9.354)
where flg/~ is the mass transfer coefficient across the gas-liquid interface, y is the Henry's constant, P is the hydrogen partial pressure, and p is the fugacity of the dissolved hydrogen. The rate of hydrogen transfer from the bulk of the liquid to the external catalyst surface is defined as rH,, = flt/,,Y(P - P , )
(9.355)
where /7~/.~ is the external mass transfer coefficient, and p, is the fugacity of hydrogen at the external catalyst surface. The fugacity of the dissolved hydrogen is calculated from eq. (9.354) and (9.355) and r~,/l = rn, to be
p =
fl:4/~P + fll/~,P,,
(9.356)
leading to an expression for the mass transfer rate
r
_
A.,,P,,,z
fig,', +fl,/,, (P-p,)=B(P-p,)
(9.357)
where B is the overall mass transfer coefficient, which incorporates gas/liquid and liquid/solid mass transfer.
B-
fl~/,Pl/,, Y At, + P,<,
(9.358)
As an example we will consider the following reaction mechanism 1. Z+A-=ZA 2. ZA +Hg---~Z+B A +H2~B
(9.359)
where step 1 is in quasi-equilibrium, and step 2 is irreversible. The kinetic equation for this simplified mechanism is
413
F =
k2K~CA(ps)& 1+ K1Cx(ps)H ~
(9.360)
From eq. (9.357) and (9.360)
.k/(B - K1CABPH~ + k2K~CA) 2 + 4K~C ~B2Pue PS
2KICAB
(9.361)
B - K1CABPH~+ k2K1CA _ ~ + f - I 2K~C ~B 2K~CxB where 9
I = B - KtCABPH~ + keKtCA, f = 4KICAB~PH~
(9.362)
This value should be inserted in (9.360) to get the rate expression
k2K1C4(~l 5 + f - l)
r =
•
(9.363)
2K~CAB + KIC4(~/F ÷ f - l ) When
1.545/2 > f
a substitution can be made
~ff7+f=l+f/2l,
hence the term
@T + f _ 1 can be replaced by f / 2l giving
k>KIC ~PH2B
r -
-
"
(9.364)
B + k 2K 1CA
I f we now take into account internal diffusion limitations eq. (9.364) is written as
qkzK~C'AP"~ flg/I + l~i,,, r =
(9.365) flg/lfll/,'Y t- Ilk2KICA
Aj,
The first order in hydrogen is observed at all pressures, while the reaction order in the substrate can vary from first to zero order. Thus the observed kinetic regularities could be the same in the kinetic and mass transfer regimes under certain conditions, demonstrating the necessity to verify experimentally and by calculations if the kinetic regime is achieved. The rate o f the gas-to-liquid mass transfer can be calculated i?om the ibllowing equation: , =
-c, )
(9.366)
414 where flcJL is the coefficient of mass transfer through the gas film into the bulk liquid phase (often denoted as kD and given by eq. (9.334); F is the gas-liquid interface area; ceq is the equilibrium hydrogen concentration, obtained from hydrogen solubility data; and cL is the actual concentration of hydrogen in the bulk of the liquid. The maximum rate of dissolution corresponds to the case when cL is essentially smaller than Ceq, i.e. assuming Ceq >> cL one arrives at rg/, = f i ~ F G x = fi~,FyP.~
(9.367)
As discussed in 9.10.2, the calculation of the mass transfer coefficient fiG1 requires knowledge of the value of the dissipation energy c. It can be calculated at diflbrent stirring rates and liquid loads by measuring the dissolution rates of some solid materials, for instance, benzoic acid. An example is the recent calculations of the influence of gas-liquid mass transfer were performed for the case of cinnamaldehyde hydrogenation in propanol (J. Hfijek, D.Yu. Murzin, Liquid-phase hydrogenation of cinnamaldehyde over Ru-Sn sol-gel catalyst. Part I. Evaluation of mass transfer via combined experimental/theoretical approach, Industrial & E n g i n e e r i n g C h e m i s t r y Research, 2004, 43, 2030-2038). The experimentally obtained dissipation energy was ~/6=0.77 (W/kg) 1/6 (Autoclave Engineers EZE Seal 0.5 1 reactor stirring speed 1500 rpm, liquid volume 200 ml). The density of 2-propanol is p - 705 kg/m 3 and its viscosity is 17 - 0.427 cP - 4.27x10 -4 N.s/m 2 (at reaction conditions 373 K, 70 bar). The diffusion coefficient of hydrogen was calculated according to the Wilke-Chang equation D°H2 - 1.5x10 -s m2/s (373 K, 70 bar). The diameter of the hydrogen bubbles was taken dbu0.15 cm - 1.5x10 -3 m. The calculation gave the value of the gas-liquid mass transfer coefficient flGL = 5.44x 10 .4 m/s. The gas-liquid interface was calculated to be 91 cm 2 taking into account the gas-hold up ~0c (< 0.4) and liquid-phase volume VL F = Vr 6~;
(9.368)
• db u
The equilibrium concentration of hydrogen in 2-propanol at reaction conditions (373 K, 70 bar) is ceq ~ 3.27x10 -4 mol/cm 3. Consequently the calculated value of the maximum dissolution rate is 6.6xl 0-4 mol/s. The measured initial hydrogenation rate at 373 K and 70 bar was 3.25x10 -6 mol/s (catalyst mass 0.5 g), thus it can be concluded that the hydrogen dissolution was several orders of magnitude higher than the rate of hydrogen consumption. Apparently, the reaction was not mass transfer limited on the gas-liquid interface. An alternative experimental method that can be used to estimate the influence of film diffusion on the gas/liquid interface is to perform experiments with different catalyst masses. If the reactor productivity is proportional to the catalyst amount then film diffusion is not a limiting factor. At high catalyst mass, the rate becomes independent of the catalyst mass and is only limited by gas-liquid mass transfer which is independent on the catalyst amount (Figure 9.25). Plot of substrate concentration vs normalised catalyst mass in cinnamaldehyde hydrogenation (Figure 9.26) indicates that gas-to-liquid mass transfer resistance does not play a significant role at the tested experimental conditions.
415
Reactor productivity, a.u.
Gas-liquid mass transfer
Catalyst mass
Figure 9.25. Reactor productivity as a function o f catalyst mass.
o t
7ff
5 /
y3
to
@
/
1
02
04
06
08
I
12
Catalysts mass [g]
Figure 9.26. Substrateconcentrationvs normalised catalyst mass in cinnamaldehydehydrogenation. The external diffusion limitation (mass transfer through a liquid-solid interface) is determined by the diffusion rate of the reactant to the external surface or the product out from catalyst particles surface. The flux to the external surface is defined by:
J = p,, s,o,(c, - e , )
(9.369)
where i l l s is coefficient of mass transfer through the liquid film around the catalyst particle, S~,~ is the external surface area of the catalyst particles, cL is the hydrogen concentration in the bulk liquid and Cs is its concentration on the catalyst surface. In the kinetic region ct, ~ Cs thus cL - Cs ~ O. For calculation of the mass transfer coefficient in case of 200 cm 3 solvent load the same values of ~, D ° H 2 , p and 7"/were used as for the calculation of gas-liquid mass transfer. Considering spherical particles (d~-4.5x 10 -3 cm) of density pc,t-0.75 g/cm 3, a mass transfer coefficient through the liquid-solid interface fliF1.53x104m/s, the external surface area of catalyst S - 889 cm 2 (0.5gcat) and the experimentally obtained reaction rate of 3.25 x 10 -6 mol/s, the calculations indicate a negligible hydrogen concentration difference between the liquid bulk and particle surface cl, - Cs - 2.39x 10 -8 mol/cm 3. Comparing this value with the value of hydrogen solubility one arrives at the conclusion that liquid film diffusion does not limit the reaction rate.
416 A common test to varify if mass transport controls a catalytic reaction is to vary the rate of agitation. Agitation of a reaction mixture is usually done by stirring, although reactors in the form of shakers are also used for laboratory purposes. The mass transfer coefficient depends on the energy dissipation according to eq. (9.334) and (9.335). The latter is a function of mixing power W, often related to the stirring speed according to 3
5
(9.370)
W = KlOliq,,idrl d U,,'rer)
where n is the stirring speed, K is a constant and dstirrer is the stirrer diameter. Utilizing eq. (9.370) the dependence of the mass transfer coefficient on stirring speed can be established (9.371)
fl oc n 1/2
If the rate is then independent on the agitation efficiency at sufficiently high stirring speed (Figure 9.27), then it is conventionally assumed, that mass trasnport effects are minimized.
m _ _ - - m
•
._>
~0-
/
/
g2-
1-
500
/
1000
1500
200O
2500
3000
3500
4000
stirring speed (rpm)
Figure 9.27. Reactor productivity as a function of stirring speed.
However, one should bear in mind, that energy dissipation can have a complex behavior depending on the selection of solvent, stirring rate, liquid volume and design of the reactor internals. Recent experiments with an Autoclave Engineers EZE Seal 0.5 1 reactor (J. Hfijek, D.Yu. Murzin, Liquid-phase hydrogenation of cinnamaldehyde over Ru-Sn sol-gel catalyst. Part I. Evaluation of mass transfer via combined experimental/theoretical approach, Industrial and Engineering Chemistry Research, 43 (2004) 2030-2038) demonstrated that the dissipation power at high stirring rates (Figure 9.28) in fact does not follow eq. (9.371).
14-
1210O8O60402O0 '
ICCO'
'
I~
'
~
stirring speed
Figure 9.28. Dissipation energy as a function of stirring speed.
417 The energy of dissipation can also depend on the liquid load of an autoclave or a shaking reactor (D.Yu.Murzin, V. Yu.Konyukhov, N.V.Kul'kova, M.I.Temkin, Diffusion from surfaces of suspended particles and specific mixing power in shacked reactors. Kinet.katal. 33 (1992) 728). At higher liquid load, increasing the shaking frequency does not necesserly lead to improved energy dissipation (Figure 9.29), therefore the rate of catalytic reactions could be indepenent on the agitation simply due to the fact that the mass transfer coefficient is the same. 1.81.6-
y=1/3
~ . ~ "
1.41.2O
1.0-
",~ "~ 0 . 8 .o
y=1/2 -
0.6CO 0.4-
0.2
i 120
i 140
i 160 Number
i 180 of d o u b l e
i 200
i 220
i 240
movements
Figure 9.29. Specific mixing power as a function of double movements frequency (left) in a shaked reactor
(right). This shows the apparent danger of establishing the kinetic region based exclusively on the results of catalytic experiments at different stirring speeds, since the independence of catalytic activity on the agitation could be explained simply by the fact that higher stirring speed or shaldng frequency do not necessarily influence the specific mixing power. Therefore, experimental verification of the absence of mass transfer should be combined with calculations. Treatment of the internal diffusion in the case of three-phase systems is the same as for twophase heterogeneous catalytic reactions, discussed in Chapter 9.5. In the case of cinnamaldehyde hydrogenation the mean radius of the catalyst particle was R = 2.3x 10 .5 m, the rate constant k - 7.0x 10 -5 s -1 and the effective diffusion coefficient of hydrogen D d in 2propanol was calculated using the values of catalyst porosity ( and tortuosity Z respectively 0.5 and 3, resulting in the effective diffusion coefficient of hydrogen Dd.= 2.5x 10 -9 m2/s (373 K, 70 bar). Calculation of the effectiveness factor according to eq. (9.136) and (9.159) gave qd---~ 1, indicating that hydrogen diffusion inside the catalyst pores does not affect the reaction rate. An experimental approach to verify the impact of internal diffusion is to perform experiments with catalyst of different particle sizes. The experimental results on the impact of particle size in cinnamaldehyde hydrogenation are presented in Table 9.11, demonstrating that a decrease in mean particle size increases both the catalyst activity and selectivity and that the smallest catalyst fraction (_< 45 ~tm) can be considered to be sufficient to safely eliminate the influence of internal diffusion, as confirmed by the calculations.
418 Table 9.11. Dependence of activity and selectivity on particle size in cinnamaldehyde hydrogenation Mean Particle Size [gm]
Initial Activity
Selectivity
Mean Particle lnitial Activity Size [gm] [molx 104/min.gcat]
[mol x I 0-4/min.go.t]
[%]
23
12.3
63
95
8.0
61
50
11.4
61
165
7.5
55
54
11.0
61
750
3.7
53
Selectivity [%]
However, if there is no change in activity with pellet size one cannot say for sure that there are negligible intraparticle concentration gradients. For instance, the influence of internal diffusion in the small pores of a bimodal pore distribution may still be important, and the test may just indicate that there is no diffusional influence in the larger set of pores. Therefore, another experimental criterion advanced by Koros and Nowak and advocated by Madon and Boudart could be applied. This criterion is based on making measurements on catalysts samples in which the concentration of the catalytically active material has been changed. Since the reaction rate in the kinetic regime is directly proportional to the concentration of active material, then say a two fold increase of activity will be obtained if the amount of the active material is doubled. The limitations of this method are assosiated with the practical difficulties of preparing catalysts samples with different loading without simultaneously altering some other important characteristics, such as dispersion.
419
Chapter 10. Kinetic modelling 10.1. Basic principles Procedures for obtaining kinetic parameters involve several steps, such as selection and construction of experimental equipment, planning of experiments, conducting them, checking the consistency of the experimental data, developing kinetic models and evaluation of the kinetic data. The latter task can be done by classical methods, which are mainly based on graphical procedures, or by modern approaches, which rely on statistical methods. Statistic based methods of evaluation of kinetics require implementation of a particular kinetic model on computer and subsequent parameter estimation. The process does not end here, since the physical and statistical consistency of the parameters has to be determined. If the values of the parameters are for some reasons unacceptable, then the parameter estimation procedure is repeated, sometimes with additional experiments. A block diagram of the procedure is given in Figure 10.1.
q
Reaction mechanism
-stoichiomctry, thermodynamics ~l Model generation
-derivation of rate equations -derivation of mass and heat transfer equations for phase boundaries and catalyst particles -derivation of mass, heat and momentum balance equations tor the reactor -flow modelling Design of experiments
I
I I i
Data generation with experiments
I I I
Implementation of the model
I I
Parameter estimation and evaluation
-identifiably ofparamelers, errors, sensilivily, -correlation between parameters, physical meaning
Yes Simpli~"model
@ m a t i c
Figure 10.1. Kinetic modelling procedure.
deviati~
420 The m o d e l structure is presented in Figure 10.2. Reaction rates, rJ = L (_k, K,_c)
Generation rates g = Z vorJ ]
Mass balances Figure 10.2. The model structure R e a c t i o n rates c o r r e s p o n d to intrinsic kinetics, while generation rates are specific for a particular reaction network. Let us consider as an e x a m p l e (Figure 10.3) the reaction n e t w o r k for a h e t e r o g e n e o u s catalytic reaction
A
D
Figure 10.3. Reaction network of a heterogeneous catalytic reaction. T a k i n g into account possible adsorption o f reagents the reaction rates are r, =
kl K Ac'J~ 1 + KAc A + KBCB + KcC c + KDCD k~KAc 4
r2 =
-
"
(10.1)
(10.2)
1 + K4c~ + K e c ~ + K c c c + KDc o
1; =
k3KBcB 1 + KAc A + KRc ~ + K c c c + K~c~
k 4K(,c c,
r4 =
(10.3)
(10.4)
1 + KACA + Knc B + K c c c + K~)c~)
where k~ is the rate constant o f reaction 1, K a is adsorption constant and CA is the concentration o f c o m p o n e n t (A) etc. The generation rates for particular c o m p o n e n t s are r 4 =-~-r2;
r~ = r l - r ~ ; r c = r z - r 4 ; r D =r~ + r 4 ;
(10.5)
421 As clearly seen from eq. (10.1)-(10.5) they are independent on the reactor type. The characteristic features o f different reactors were discussed in Chapter 1. In Table 10.1 we summarize design equations for batch, fixed bed and continuous stirred tank reactors Table 10.1. Design equations for batch, fixed bed and continuous stirred tank reactors Batch Plug flow reactor CSTR reactor Design equation dc~ dc, c~ - cot = c~r/~ = c~r/~ - - co?~ dO dO r ® in design equation t - time r -residence time r residence time r=v/V
r=v/V
For homogeneous systems a = 1, while for heterogeneous catalytic reactions c~ = m,, t / Vr~ = p~. For simple cases o f kinetics analytical solutions are possible, while numerical solution provides a general approach to model simulation and parameter estimation. Several examples o f analytical solution for first order kinetics were discussed in Chapter 1. Here we consider analytical solutions ibr a batch (plug flow reactor) and CSTR in the case o f enzymatic Michaelis-Menten kinetics, written in the following way r - mVii/lax C ~,
(10.6)
K m+ Q
The generation rate o f the substrate in a reaction S---~P is r; = G r , v, = -1
(10.7)
where v, is the stoichiometric number, i f the catalyst effectiveness factor is equal to unity, eq. (10.6) and (10.7) together with the mass balance give ibr a batch reactor des
~nax Cs
dl
K m + Cs
(10.8)
which can be then integrated using the boundary conditions t=0, c, = co, /
(10.9) V,.axe,
0
After some manipulations
422
~'i K,,dc, + co,f dc = /id t ..... Enax C; c~ Vlnax o
(10.10)
a design equation for Michaelis-Menten kinetics in a batch reactor is obtained Km
lnC°' +
1
(Co.,-G)=t
(10.11)
Deft ning conversion as 6 = (Co, - G) / Co, eq. (10.11 ) is transformed to K,,
1
ln(1 - c~) + T@-- (Co~Y) = t
Vmox
(10.12)
Gax
For a heterogeneous catalytic reaction A--->P the catalyst density should be included in the model
1 de"x -pdt
-r A
(10.13)
For a reaction mechanism 1. A+Z=AZ 2. AZ--->P+Z A--->P
(10.14)
where the first step is in quasi- equilibrium, the reaction rate is
kiK 4c~ kc . . . . 1 +KAC4 l+k'c
r .
(10.15)
which is in lhct similar to eq. (10.8), since the latter can be written in exactly the same tbrm as eq. (10.15) &
dt
_
_
I + K q # T e s.
kc
l+k'c
(10.16)
The design equation for a heterogeneous catalytic reaction for which the kinetics is described by eq. (10.15) is almost the same as for an enzymatic kinetics (10.11), with the only exception being that the catalyst density should be taken into account klnCc]. + ~ (c°~'-G)= pt
(10.17)
423
The design equation for a plug flow reactor with Michaelis - Menten kinetics is the same as eq. (10.11) - K . , ln(1 - 8 ) + (Co,,.8)
= Z'Vma ×
(10.18)
with r being the residence time, e.g. the ratio of the reactor volume to the volumetric flow rate r = VII)0 . For a CSTR, in the case of Michaelis-Menten kinetics and catalyst effectiveness factor equal to one, we arrive at Cs -- Cos i
~naxCs
r
(10.19)
K,, + c~
from the general design equation, leading to
I)V- --1` I K.,
C~
~-(Co., - c, ) 1
(10.20)
or using conversion 8
r-~--
V~n.x K,,
+c0~,8
(10.21)
Similarly, design equations can be derived for some simple reaction kinetics. Summarizing, we can state that two kinds of models dominate in catalytic kinetics: algebraic and differential models. Algebraic models can be represented by the equation
2) = f(x,p)
(10.22)
where 5' is the dependent variable representing, for instance, the rates measured in CSTRs. Equally, these can represent the measured rates in differential reactors or analytically integrated balances of batch reactors. Examples of algebraic models were given above. Differential models can generally be described by ordinary differential equations (ODEs), such as d~ _ f ( y , p) dO
(10.23)
where the symbol @ represents time, length, volume etc., depending on the particular case. Partial differential equations, particularly the hyperbolic and parabolic ones, can be transformed into the ~brm presented in eq.(10.23) by discretization. The dependent variable (50 corresponds to
424 concentrations or molar amounts measured in batch and semi-batch reactors, or, concentrations or molar flows at the outlet o f plug flow or, alternatively, fixed bed reactors. Here we should introduce the notion o f linearity and nonlinearity o f the model with respect to the parameters, since they are the unknown quantities. The following notations will be used here: y-estimated dependent (measured) variable; x-independent variable, p-parameter. The concept o f linearity-nonlinearity is illustrated with a simple example: assume that batch reactor experiments for a first-order reaction, A~P, r A = -kc~, have been carried out. The concentration o f component A , CA, has been measured as a function o f the reaction time. The differential model for this system is according to the mass balance for a batch reactor dc A
-
(10.24)
kc A
dt
Translation o f this model equation to the nomenclature used in parameter estimation implies that: ® = t, y = c A, p = p~ = k. Thus, we get from the standard form o f the differential model for this case dY _
(10.25)
PlY
dO
where the reaction time, ® , is the independent variable, the concentration, y, is the dependent variable and the rate constant, D, is the unknown parameter to be estimated by regression analysis. The simple problem presented above can be solved as an algebraic model, too. Integration o f the differential equation gives - ln2-~- = kt
(10.26)
C4
and
we
can
define
the
variables
and
parameters
for regression
analysis
as
follows
Y = - I n ( c A / c ° ) = Y~, Pl = k, x = x I = l leading to an expression o f the algebraic model: Yl p l xl . This equation represents a linear algebraic model, since it is linear with respect to the
parameter (Pl - k). On the other hand, if we use the form ln~- = -kt
(10.27)
CA
and solve it analytically the result is c x / c ° = e k, and the independent and dependent variables are defined along with the parameter y -- c A/ c~, p~ -- k, x -- Xl -- t. The model becomes yl -- e ~,m, which is still an algebraic model, but a non-linear one with respect to the parameter (P0. The use o f algebraic and differential models will be illustrated in section 10.4, as the concepts o f nonlinear regression analysis are presented.
425
10.2. Heuristic design of experiments A heuristic approach to experimental design for kinetic data generation will be briefly discussed here. The following road map to help plan an experimental program is after S. Fogler (Figure 10.4).
Define objectives
[
+ Choose responses
[
÷ Identify variables
]
Design experiment
I
+ Perform experiment
+ Analyze results
[~ i TM
I i
Act on results
] I
Figure 10.4. Planning of an experimental program. The definition of the objectives is extremely important in kinetic studies, especially how comprehensive the experimental program will be. Two diflbrent types (independent and dependent) variables were briefly discussed in section 10.1. The responses are the dependent variables. As the experimental program is designed, the important dependent variables to be measured must be identified. While constructing the experimental program it should also be decided which independent variables have the greatest influence on the dependent variable and what are the ranges or levels of these variables to be examined. In some cases, instead of changing each independent variable separately, a group can be formed. As an example we can refer to hydroformylation reactions, where such a group could be the hydrogen to carbon monoxide ratio. On a practical level, the heuristic approach includes first collecting all the possible data during the experiments as a function of the parameters which are deemed to be important, i.e. concentrations, temperature, pressures, pH, catalyst concentration, volume, etc. Then the rate constants are estimated by regression analysis and the adequacy of the model is judged based on some criteria (like residual sums and parameter significance, which will be discussed further). If a researcher is not satisfied, then additional experiments are performed, followed by parameter estimation and sometimes simulations outside the studied parameter domain. The latter procedure provides the possibility to test the predictive power of a kinetic model. The kinetic model is then gradually improved and the experimental plan is modified, if needed. This process continues until the researcher is satisfied with the kinetic model.
426 When designing experiments one should also take care o f possible experimental errors, decide what is the minimum number o f experiments and define the necessity o f repeated experiments. The minimum number o f experiments that must be performed is related to the number o f important independent variables that can affect the experiment and to the experimental precision. The frequently used strategy is to carry out first experiments at the extremes (maximum and minimum setting) o f the range o f the controlled variables. Moreover in a classical factorial design, in the case o f linear relationships, only two conditions for each independent variable should be tested: at the lowest and highest possible setting. If the dependent variable is a function o f three independent variables (A, B, and C), each o f which can take on two possible values or l e v e l s , then to test all possible variable combinations the number o f experiments is equal to the number o f levels, N, raised to the power o f the number o f independent variables, n, e.g. N" = 23 = 8 (Figure 10.5 and Table 10.2). A+ A,tB+
¢+
f+
,I+ B¢+
¢+
A-
R
Q)
¢-
Figure 10.5. Example of a classical factorial design. Table 10.2. Experimental program tbr classical ~hctorial design. Variable/Number 1 2 3 4 5 A + + B + + C + +
6 + +
7 +
8 + +
+
If nonlinearity is involved, then at least three conditions for each independent variable should be tested, leading to a 3" design. Intermediate designs can be used, introducing for example a centre point, through which nonlinearities can be determined. As an example data on liquid-phase hydrogenation o f thymol are presented in Figure 10.6, demonstrating quite strong nonlinear dependence. NA 1 ~ - , , , . . . . 0.4 _ _ 0.8
~
0.6
0.4 1.4 _ _
~
1.4 t 2.4 _ _ 2.4
,=
04 0.2 0 0
5
10
15
20
25
30
35
40
Figure 10.6. Liquid-phase hydrogenation of thymol (2-isopropyl-5-methylphenol) over 50%Ni/Cr203 at 160°C in cyclohexane at different hydrogen pressures: 0.4, 1.4 and 2.4 MPa (A.I. Allakhverdiev, N.V.Kul'kova, D.Yu.Murzin, Kinetics of thymol hydrogenation over a Ni-Cr203 catalyst, Ind. Eng. Chem. Res. 34 (1995) 1539).
427 If there is some error associated with measuring the outcome of an experiment, repetition of some trials should be considered. Obviously, the less precise the measurements are, the more data must be collect. Repetition of experiments at the same conditions with averaging results is the typical approach. Traditionally, in designing kinetic experiments, it is believed that kinetic data should be obtained under isothermal conditions. The reactants are preheated separately and brought together for the desired reaction to proceed. However, as solid catalysts are involved, in practice the procedure becomes more complicated. With which component to combine the catalyst during preheating and how to pretreat the catalyst; does the pretreatment procedure contribute to catalyst deactivation; are typical questions raised when handling with solid catalysts together with gases and liquids. In practice, many experiments are carried out in such a way that all of the reactants are simply brought together with the catalyst, the mixture is heated up to the reaction temperature, and while heated, the mixture undergoes, at least, some reaction under the rising temperature gradient to the set temperature. Nowadays, this is not a technical problem since the temperature profile can be followed on-line and stored on a computer (Figure 10.7).
T/ i
i
~
i
i
i
i
q
time
Figure 10.7. Temperature measurements during a catalytic experiment. The temperature dependence of the kinetic model is incorporated in the temperature dependencies of the rate and equilibrium constants, e.g. according to the laws of van't Hoff and Arrhenius. All data sets are merged together and the parameters are determined by non-linear regression, which will be addressed in the following sections. The mutual correlation between the parameters can be suppressed. A further challenge appears, if the reactor equipment itself is non-isothermal. A typical example of this kind is a catalytic fixed bed. If the reaction has considerable heat effects, for instance highly exothermal reactions, it is difficult to maintain completely isothermal conditions in a tubular fixed bed. Instead, a hot spot appears (Figure 10.8).
Figure 10.8. Appearance of a hot spot in a multitubular reactor.
428 A way to surmount this dilemma is to place thermoelements along the reactor tube and to screen the exact temperature field (Figure 10.9).
N7
-
-
t;1~
Figure 10.9. Temperature ~rofiles in an industrial multitubular reactor, showing that hot spot temperature is 100°C higher than the temperature close to the reactor inlet (real plant data).
Later on, the temperature profiles can be incorporated in the reactor model. Principally, two possibilities exist: to describe the reactor with a complete non-isothermal model including the energy balance, or to utilize the experimental temperature profiles more or less directly. The first approach is theoretically correct, but it has the drawback that the heat transfer parameters of the reactor tube need to be determined. This introduces an additional uncertainty factor to the model, and bottom line, it is not relevant since the kinetic and thermodynamic parameters are of primary interest. Thus, it is better to interpolate in the experimental temperature field, for example, by fitting an empirical model for each temperature profile, T-J(reactor length), using for instance a polynomial model. The laws of van't Hoff and Arrhenius are incorporated in the thermodynamic and kinetic parameters, all data sets are merged together with the empirical temperature profiles and non-linear regression is used to obtain kinetic parameters. As an example, fitting a kinetic model to a water-gas shift reactor in a non-isothermal fixed bed reactor was discussed (T. Salmi, J. W~irnS,, Modelling of catalyst packed-bed reactors-comparison of different diffusion models, Computers Chemical Engineering, 15 (1991) 715). The example demonstrated clearly that nonisothermal data, although rather seldom used in kinetic experiments, indeed, can be used for kinetic analysis. The crucially important issue is to obtain primary data with a large enough concentration and temperature domain, and to record the temperature profiles frequently and precisely. If these requirements are fulfilled, the parameter estimation exercise can be safely carried out under non-isothermal conditions. 10.3. P a r a m e t e r estimation: classical methods
Classical methods are used for two different approaches. In case of the differential method, the rate expression is given as
429
(lO.28)
1. = f ( c , r )
while the integrated form of the rate equation is (10.29)
ci = g ( c , T, t)
For differential methods, moderate mathematical efforts are required. However, the accuracy in calculations of the rate is low since numerical or graphical differentiation is used to determine the rate values from concentration vs time diagrams (Figure 10.10).
,.|
time
Figure 10.10. illustration of graphical differentiation used to determine the rate values In such a way after obtaining CA as a function o f t , c4(t) experimentally, dc4/dt is estimated at each point (t, CA(t)) by differentiation. In the case of a power law kinetics ?n
(10.30)
r = kc i
after taking the logarithm l n r = Ink + m lnc i
(10.31)
the slope of the straight line gives the reaction order and the y-intercept is the logarithm of the rate constant (Figure 10.11).
Ink Figure 10.11. Estimation of reaction order.
430 In practice such analysis is done by regression, using numerical methods. If the reaction order is known apriori it is possible to plot the rate (- dc~l/dt) as a function of c;~'. Deviations from the straight line would indicate that probably, an incorrect value of the reaction order has been used. Otherwise, the slope of the curve gives the rate constant. Another possibility is to use hyperbolic approaches, similar to those discussed in Chapter 6, like the double reciprocal plot of Lineweaver -Burk, which has the form 1
K~,± 1
---
r
V, ..... c
+--
1
(10.32)
V, ....
Thus, plotting the reciprocal value of the rate vs l/c, the values of the constants can be obtained (see 6.1). Several other methods are often used in enzymatic kinetics, like the Eadie-Hoftsee and Hanes-Woolf methods. Similarly, in the case of heterogeneous catalytic reactions, if the rate is given tbr instance by eq. (10.15) after rearrangement c r
-
1
k' +--c k k
(10.33)
C
a linear plot in the coordinates - vs c gives k'/k as the slope and 1 / k as the y-intercept. r In the integral method, the concentrations of the reactants and products are plotted as a function of the residence time and the integrated forms are used in graphical evaluation. For example, in the case of power law kinetics (10.30) with the reaction order different from unity one arrives at
(10.34) \Co ) The value of the reaction order is determined from the y-intercept, while k can be determined from the slope of the dependence of (1/c)" 'w' l , e.g. from (m-1)k (Figure 10.12).
t Figure 10.12. Determination of kinetic parameters from eq. 10.34. For first order kinetics a plot ofln c vs time gives the rate constant as a slope (Figure 10.13).
431 ,lnc
t
Figure 10.13. Determination of kinetic parameters for first order kinetics.
In some cases, for instance in reactions involving transformations of solid substances, which are described by Avrami-Erofeev crystal growth kinetics, where a is conversion a = 1 - e -~f~"
(10.35)
Boris Erofeev double logarithmic plots (Figure 10.! 4) can be used to deduce the values of kinetic constants. J
•
in t
In (-ln(1-c~)) Figure 10.14. Application of double logarithmic plots for determination of kinetic parameters
10.4. Parameter estimation: regression The models in chemical kinetics usually contain a number of unknown parameters, whose values should be determined from experimental data. Regression analysis is a powerful and objective tool in the estimation of parameter values. The task in regression analysis can be stated as follows: the value of the dependent variable (y) is predicted by the model; a function 09, contains independent variables (x) and parameters (p). The independent variable is measured experimentally, at different conditions, i.e. at different values of the independent variables (x). The goal is to find such numerical values of the parameters (p) that the model gives the best possible agreement with the experimental data. Typical independent variables are reaction times, concentrations, pressures and temperatures, while molar amounts, concentrations, molar flows
432 and reaction rates are dependent variables. The parameters to be estimated are usually rate and equilibrium constants, sometimes even mass and heat transfer coefficients. Since most models in chemical kinetics and more general chemical reaction engineering are nonlinear with respect to the parameters, the treatment here is limited to non-linear regression analysis. The model equation is written according to eq. (10.22)~ = f ( x , p ) , thus non-linear regression strictly speaking is valid for algebraic models, but it can be applied to differential models, as well. For differential models, the solution (y) is obtained numerically from the model equation. The objective function to be minimized by regression is defined by:
O(P)=ZZZb,..%t--f(x,.s4,P)2~'i.s.I
(10.36)
i=1 s'=l t=l
where i denotes each component in the reaction mixture, s denotes the different data sets and t refers to the data points, y,..,., is the experimental value and J(x,.,., ) is the corresponding model prediction, w,.¢, is the weight matrix which is used, if experimental scattering varies between different data. In the case that the weight factors are equal, the objective function is simplified to:
Z ZZ b,,,,- s(x,,.,,,,4'] i=l
s=l
(10.37)
t=1
The necessary condition for obtaining a minimum value of the objective function, Q, is that all of the partial derivatives with respect to the parameters are zero.
Op!
,..,., - J'(x,+,.,, p)2
Op!
= 0, j = 1..ns,
(lO.38)
The total number of parameters is n~, and i=l..n,,,. Differentiation gives a system of np equations,
(10.39)
or in a simplified form
2(y, - f )
= 2gj = O,
(10.40)
=
the roots of which are found numerically. The algebraic equations can be solved by the NewtonRaphson method, which gives the following algorithm for multiple equations (numerical methods will be discussed in more detail in the following section)
433
1
(10.41)
P(,,,+I) = P(,,,) - G m g(,,,) where m denotes the iteration index. The Jacobian matrix consists of the elements
Og, ag,
Og,
~Pl C~P2"" ~PnF ~g2
Op, (10.42)
G= Og. ~
Og,, ,,
~1)1 ......... ~Pllp
The elements of matrix (G) are further elaborated
@---7 @---]
(Y' - f )
lzE =
----
+(y,-.D
c3Pk OP j
°~f
(10.43)
Since (y~ - f ) ~ O, eq. (10.43) can be modified to
Op~
Opk Opj j
(10.44)
By using the following matrix
A=
(10.45)
the Jacobian matrix is written in a compressed form
- ArA = G
(10.46)
434 The vector f is defined by y,
_f ya-f2
(10.47)
I*,,,,,-L,,,I Recalling that ~(Yi-.~) ~@ = g j i=L
(10.48)
j
and the definition of the objective function, we have - A r f
= g. Since the Newton-Raphson
solution is given by eq. (10.41), it can now be written as T
I
7
p(,,+,) = p(,,) - (A(,,) A(,,)) A (,,)f(,,,)
(10.49)
which provides important information: the iteration of the parameter vector (p) is dependent on the previous function values (/) and on the partial derivatives of the model equations. The algorithm above it suffers from a serious disadvantage: the convergence is guaranteed only in cases that the initial guess (p) is close enough to the true solution. Therefore, a further development of the algorithm has taken place. Levenberg proposed a compromise algorithm between the Newton-Raphson and steepest descent algorithms:
=
P(w,+l)
P(m) -- (2--/-~-
7' (A(,,)A_(,,))
1
A__7 (,,)J(,,)
(10.50)
where the identity matrix _/is the simplest nontrivial diagonal matrix -1 0...0 0 1...0 I=
(10.51) :
:
".:
0 0 ...1 and the Marquardt parameter 2 is a scalar. If the scaling parameter, Z, is large, the steepest gradient method is obtained, while for the case that ~. = 0 eq. (10.50) gives the Newton-Raphson method. Typically, the value of the scaling parameter, )~, is high in the beginning of the iteration,
435 but decreases as the minimum is approached. Marquardt in 1963 proposed a strategy for the selection of)~; therefore the method is today called the Levenberg-Marquardt method.
10.5.
Numerical strategies
As became apparently clear from the discussion above, during parameter estimation, analytical derivatives cannot be calculated, theretbre, there is a need to solve algebraic and differential equations by numerical methods. In this section we briefly present strategies for numerical solution of algebraic and differential equations.
10.5. 1. Solution of algebraic equation systems The algebraic equation system of unknown variables (y)
f(x,y) = 0
(10.52)
can be solved by the method of Newton-Raphson. In eq. (10.52), x denotes a continuity variable (for instance time or residence time). The Newton-Raphson algorithm for the solution of the equation system (10.52) is -I
Yk = Yk-, -_J_ fk-1
(10.53)
where k denotes the iteration index, f~j, is the function vector at iteration cycle k and J-J is the inverse of the Jacobian matrix containing the partial derivatives
(10.54)
J=
m
@N
Isaac Newton
Carl Gauss
Carl Runge
436 The inversion of d ~ is typically carried out by Gaussian elimination. The derivatives 8fi/Oyj are obtained either from analytical expressions or by a numerical approximation. The approximation of 8fi/Syi can be done with forward differences accordingly,
df ~ Af _ f(y/÷ Ay,)-f(y,)
4yj ayj
(10.55)
Ay,
The choice of the difference Ayj is critical: the smaller Ayj is, the more accurate the approximation of the derivative is, but if the accuracy of the computer is exceeded, the denominator of eq. (10.55) becomes zero. A criterion for the selection of Ayj, is Ay, = max(xc, e~ Y.I )
(10.56)
where ER is the relative difference given by the program user and ec is the round-off criterion of the computer. The initial estimate, E(0), should be carefully chosen for a successful iteration according to the Newton-Raphson algorithm. In many cases a solution is expected as a function of the parameter x, which is then selected as the continuity parameter. The solution of the equation system (10.52) obtained for one parameter value of x (y(=)) can used as an initial estimate for the solution of the problem for the subsequent parameter value x + Ax:
_y(o)(X + 6x) = y_(~)(x)
(10.57)
The convergence of the Newton-Raphson algorithm is usually quadratic, which guarantees rather rapid computations. The Newton-Raphson iteration is interrupted when a sufficient accuracy is achieved. A conservative criterion is to evaluate the relative change of each variable, i.e. to require that Y"k~-zYi(k-1) < ~, i=l..N I Yi(k)
(10.58)
where g is the relative error tolerance defined by the user. For a system with only one unknown variable, the general algorithm is reduced to
,f((k 1)
Y(k) = Y(k ~) f'(k-~)
(10.59)
Several computer codes exist where the solution of nonlinear equations is implemented as a part of a general solver for nonlinear equations and differential equations.
437
10. 5.2. Solution oJ'differential equations 10. 5.2.1. Ordinary differential equations First-order ordinary differential equations (ODE)
dy = f(y,x) dx -
(10.60)
for which the following initial conditions are valid, __y= Y0 at x = x 0 can be successfully solved by semi-implicit Runge-Kutta methods and linear multistep methods. The algorithms will be presented briefly in the following section.
Semi-implicit Runge-Kutta methods" Runge-Kutta methods have the general form q
Y. = Y,,1 + ~-~bk,
(10.61)
i=l
where ~ and ~_z give the solution of the differential equations at x and are obtained from
x-dx. The coefficients k~
where h=Ax. If the coefficient ail= 0 for l - i the method is explicit; if ail ;~0 for l - i the method is implicit. For systems of stiff differential equations an explicit method cannot be used. In order to achieve a stable solution for such kinds of systems, an impractically short step length (h) would be required. On the other hand, the implementation of an implicit method would require the solution of a non-linear algebraic equation system to obtain k,. After developing ki in a Taylor series around y~ + ~-'i=lla,k l and truncating after the first term, a semi-implicit Runge-Kutta method is obtained, according to which ki is calculated from
(I_- Ja, h)ki = f(Y, + '~ a,/kl)h
(10.63)
where i denotes the identify matrix, eq. (10.51) and J is the Jacobian matrix eq. (10.54). Application of the semi-implicit Runge-Kutta method requires computation of the Jacobian matrix. The Jacobian directly influences the numerical values of ki and thus even the final solution.~. Based on this reasoning, it might be favourable to use analytical expressions for J. For cases when it is cumbersome to obtain an analytical expression for the Jacobian, an
438 approximation with forward differences can be used, eq. (10.55), where the increment AZi is selected according to the criterion in eq. (10.56). Eq. (10.63) implies that a linear equation system is solved every time in the calculation of _k~. The matrix !-Ja~lh is inverted by Gaussian elimination. After obtaining ~ according to eq. (10.63), v, is easily calculated from (10.61). An extension of semi-implicit Runge-Kutta methods is the Rosenbrock-Wamler method. In this method eq. (10.63) is replaced by the expression
/I
/
/I
(I-Ja,zh)k,= f y +~_a,zk, h+~_c,,k_, --
--
--
_ n
--
l=1
(10.64)
l=1
Coefficients for a tburth-order Rosenbrock-Wamler method are given below: a21-0.4~8, a31=0.9389486785, a32=0.0730795421, b1=0.7290448800, b2=0.0541069773, b3=0.2815993624,b4=0.25, c21=-1.943744189, c3~=0.4169575310, c32-1.3239678207, c41-1.5195132578, c42-1.3537081503, c43- -0.8541514953.
all=0.~ 5,
Linear multistep methodf The following general algorithm applies for the solution of the ordinary differential equations: K1
_y = Z a , _ y
K2
+hZfl,.f ,
i=L
(10.65)
i=0
Dependent on the values of Kl and K2, Adams-Moulton q I
a/=I,K t =I,K 2 =q-l,
__y,,=Ynl
+h~-,fl, fnl
(10.66)
1=0
and backward difference-methods are obtained: q
K~ =q,K z = q - l ,
Y_n= ~ a ' - Y n 1 +hflo.f~,
(10.67)
i=l
Both methods are implicit by their nature because _in usually is a non-linear function of Z.. Eqns (10.66) and (10.67) imply a solution of a non-linear algebraic equation system of the type (10.68)
x =L, where an is given for Adams-Moulton method by q-1
a n = y ~ , + h~]p~f~ 1 J=O
(10.69)
439 or for the backward difference method by a~, = ~ azy_~,_,
(10.70)
1=1
For the solution of~,, a~ is known from the previous solutions. The problem thus consists of an iterative solution of the equation system (10.68). This can be achieved by the Newton-Raphson method. The coefficients ai and 13i for the Adams-Moulton and backward difference networks of various orders are listed in Tables 10.3 and 10.4 respectively. Table 10.3. Coefficients in the Adams-Moulton method. i /~li
0 l
1
2
3
4
2/~2 i
l
l
12/~3i
5
8
-1
24/~4i
9
19
-5
1
720/~5i
251
646
-264
106
-19
1440/~6,
475
1427
-798
482
-173
5
27
Table 10.4. Coefficients in the backward method. i ~1,
0 1
1 1
2
3
4
5
3 ~2,
2
4
-1
11 ~3i
6
18
-9
2
25 ~4,
12
48
-36
16
-3
137 ~5,
60
300
-300
200
-75
12
147 a6,
60
360
-450
400
-225
72
6
-10
For stiff differential equations, the backward difference algorithm should be preferred to the Adams-Moulton method. The well-known code LSODE with different options was published in 1980's by Hindmarsh for the solution of stiff differential equations with linear multistep methods. The code is very efficient, and different variations of it have been developed, for instance, a version for sparse systems (LSODEs). In the international mathematical and statistical library, the code of Hindmarsh is called IVPAG and DIVPAG.
Alan C. Hindmarsh
440 10. 5.2. 2. Partial difJerential equations In some cases, for instance in transient kinetics, there is a need to solve a set of partial differential equations. Such equations for two variables can be presented in the general form ~2u , 82u -o-~+c a di2 +
8u 8u di dj
+dSU+eOU+ fu+g=O di d~
(10.71)
Eq. (10.71) cannot be solved directly and requires discretization of the other variable. In difference methods, the area between the boundaries is divided into grid points, where the values of derivatives are approximated. For instance, for the first derivative, the five-point backward difference formula can be used, 1 3 10 18 --~Zz=zl:"~E'-~f(fl)-]-"~f(Zo)--'-~f( f l)-]-6.f(Z
)]
l f, z
2)--'~" t 3
(10.72)
For the second derivative, the same approach gives
d2f dz2 z:~ -
1 16 I E_ .i_.~ j.(z_2 ) q_.i_.~ j.(Z_l ) _ 3 0-j(Zo) + 16
Az2
.
1
.
iU(z+2)]
(10.73)
In collocation methods applied to chemical engineering problems by J. Villadsen, the solution is approximated with a set of basic functions, in particular for orthogonal collocation the areas between collocation points are approximated by orthogonal polynomials.
John Villadsen Commercially available software, like ModEst 6.0, which will be discussed in section 10.8, have implemented routines to solve partial differential equations.
10.6.Analysis
of parameters
10. 6.1. Statistical analysis An indicative and preliminary statistical analysis of the parameters obtained by regression analysis can be done by regarding two important quantities; the variances of the parameters and the correlation coefficients between them. These quantities are calculated from the objective function (Q) and the matrix of parameter derivatives A, given by eq. (10.45).
441 For the sake of convenience, a matrix L_is introduced L = (ArA)
(10.74)
1
The variances of the parameters (PD, ~rx2, can be estimated from the objective function (Q) and from the diagonal elements of L_ crk2 ~
Q
Lkk
(10.75)
17m - - n p
where nm and np denote the number of experiments and the number of parameters, respectively. The covariance of the parameters p~ and pl is defined by pkt o-~ o-1, where phi is the correlation coefficient between pk and pt. The fundamental statistical relationship is pklcrk or1 ~
Q
Lkl
(10.76)
F/m - - F/p
On the other hand, the variances are given by
,/0K
4z7,,
O-k_ ~ - - g / p " O" l
~--Rp
(10.77)
We get a simple expression for the correlation coefficient:
o-ko-, -
LY, 2 , / - / 7
(10.78)
The equation shows that the correlation coefficients are obtained from the elements of matrix A. Generally speaking, one of the goals in parameter estimation is to minimize the variances as well as the correlation coefficients of the parameter, i.e. ak should be as small as possible for each pk. At the same time, Pkl should be small for each pair ofparameters, pk -Pl.
t)kl =
~
4
Lk~r ' - - - - ,
4 L,,
Pkl
E
[0,1]
(10.79)
Small variances guarantee that the parameter is accurately estimated and small correlation coefficients indicate that the parameters cannot be mutually compensated. The variances are very much dependent on the precision of the experiments since they are directly proportional to the weighted sum of residual squares (Q), while the correlation coefficient depends heavily on the model structure as such. Special tricks to suppress the correlation between parameters exist, and should always be used.
442 The standard statistical analysis of parameters is based on linearization and does not always give a realistic picture of the accuracy of the parameters. For example, the variance expresses the confidence interval of the parameter in a symmetric way, which can sometimes lead to misunderstandings. For example, small parameter, e.g. a rate constant has a big variance, which means that even negative values of the parameter could be accepted. This is of course not true. A very simple and illustrative approach is to check the objective function (Q) as a function of each parameter value. All of the parameters except one is given the value which corresponds to the minimum value of the objective function, Q, and the influence on the numerical value of the objective function, Q, is investigated as a function of the parameter under consideration. Examples of these kinds of plots are shown in Figure 10.15 for two variables. Sensitivity
Sensitivity
900
~"~'~""#
85~
i
950 ~ 9OO 85O 8OO
16
750 0,02 0,03 0,04 0,05 0,06
$. = ...~::=::
800 750 700
: 10
12
14 kl
18
1 /
V k2
Figure 10.15. Sensitivity plots Sensitivity analysis gives additional information, for instance, concerning the symmetry: often the parameter is accurately defined in one direction, but less accurately in the opposite direction. In Figure 10.15 the first parameter (k0 is not very well identified, whereas the latter @2) has a prominent minimum and can thus be regarded as well identified. A standard way to describe the results of parameter estimation is a contour plot. The value of the objective function (Q) is investigated as a function of a parameter pair, and values, where the objective function gets the same, specified value are depicted in a single plot. An example of a contour plot is provided in Figure 10.16.
Figure 10.16 Contour plot.
443 If the correlation between the parameters is low, the contour plot consists of circles. In the case of a strong mutual correlation between parameters, contour plots become elongated. These kinds of plots bear a clear message: the numerical value of one parameter can be easily compensated by another parameter, either by increasing or decreasing its value. In any case, we get an equally good fit to the experimental data. From statistical, as well as a physico-chemical viewpoint, this is not desired: the exact values of the parameters remain uncertain, and the regression analysis has just performed a data fitting exercise. The model can be used for interpolation within the experimentally screened domain, but not for extrapolation outside it. Furthermore, the physicochemical significance of the parameters remains questionable. This discussion shows that suppression of the correlation between parameters is a crucially important issue.
10. 6. 2. Suppression of the correlation between parameters Suppression of the mutual correlation between parameters will be illustrated by a case study. Rates of catalytic reactions are very frequently measured in gradientless and differential reactors, and the rate expressions of the Langmuir-Hinshelwood type are frequently used in the interpretation of experimental data. The rate expression has the general form
r
=
N~
N+
~\~
i=l
N
i=~
(10.80)
(1 + Z K,c;' + Z K,c';' )~ i=1
J=l
or
N+
N
-I-I r =
,=1
N
t=i
(10 81)
(1 + ~ Kjc.; i )'~ j=l
where N-N++N , i.e. the number of components. N and N +, denote the number of reactants and products, respectively. The values of the exponents depend on the reaction stoichiometry and detailed reaction mechanism, as discussed in Chapter 7. The equilibrium constant (K) can be determined separately from the equilibrium composition and is not usually a parameter. The forward rate constant (k+) and the adsorption parameters (Ki) are usually obtained from kinetic experiments, by using regression analysis. A serious drawback is the strong correlation between the rate constant (k+) and the adsorption equilibrium parameters (Ki). If the value of the rate constant in the nominator of the rate expression is increased, its increase is compensated by the increase of the adsorption parameter, and one gets an equally good fit to the experimental data. A general rule of thumb is that the correlation between the parameters can be diminished by transforming rational functions to sums. Consequently, by taking the root of the rate equation and dividing it by the rate constant we get the transformed rate expression
444
_
1
1 _ ~ -N,
r '/'--w
k"
K
: c x,
(10.82)
++.~_--£;7 :
with N+
K+
N
i l l c,
_l--[c~,)t:'~
I=l
(10.83)
i=1
Thus, a new set of parameters are defined by 1
-N'
N K : c x' / ,j=I..N
(10.84)
ao =~;;;a:-+....~7? ./=l
Taking an additional advantage from the fact that the initial estimates of the parameters can be obtained from the initial rates by taking the reciprocal value of the equation we arrive at N
(10.85)
= ao + ~ a : x j ./=1
which de facto is a linear model with respect to the parameters ao ..... ax. The dependent variable is defined as 1 /)-
(10.86)
r~ .8
while the independent variables are: x/
x / = c/
(10.87)
Thus we arrive at a model that is linear with respect to the parameters and linear regression can be applied to it. A simple illustrative example can be considered: an irreversible, bimolecular, catalytic reaction, A + B - C, obeys the rate expression
kc,4 C ,4
r =
(10.88)
(1 + K ~ c A + KBc B + K(Cc) 2
The reciprocal square root leads to the following formula
.
.
.
.
~- - - -
+----
4
(10.89)
445
The dependent variable is ~ = 1/~]12r, while the independent variables are x I = 1/c~f~Ac% ,
X2=X/-~A/X/-~B, X~=X/-~B/X/-~4, X 4 = C c / C ~ : B and the parameters to be estimated by regression analysis are a, = 1/x/k,a, = K A/ x/k,a 3 = Ke / x/}-, a4 = K c / ~/k. A very simple linear model is obtained 3~= ~ a:x:, which can be used in linear regression to obtain reasonable initial j-1
estimates for the final parameter estimation, which is carried out by nonlinear regression. Correlation between the parameters often affects the results of parameter estimation in a severe way. Imagine, that the process model follows the dependence, y-po+p:x:, which is in principle a very simple model. In practice, the experiments are limited to a nan'ow interval o f x and, thus, the value ofpo is easily compensated by p: and vice versa. This is the case when the frequency factor (A) and the activation energy (Ea) of the rate constant are determined from the Arrhenius law. The values of the rate constant, k, are obtained within a certain temperature interval, and some experimental scattering typically appears in the k-values. In a logarithmic plot, in k versus 1/T gives lnk=lnA
E, 1 RT
(10.90)
The intercept (ln A) can be compensated by the slope (-EjR), and both parameters remain uncertain. In general, it is possible to obtain a reasonably good fit to the k-values, but the individual parameter values (A and E,) can remain highly uncertain (Figure 10.17)
00
Figure 10.17. Compensation effect. A way to improve the parameter estimation procedure is to orthogonalize (centralize) the experiments according to the following procedure. The abscissa axis is shifted to the average value of the experiments (x), x' x-x. After applying this to the Arrhenius law we obtain 1
g
1 -
T
1 --
T
(10.91)
446
The problem is reformulated to k = Ae ~ o, where ~ = Aexp(-E~ ~R-T), i.e. the rate constant at the average temperature. The advantage of the restructuring is that the experiments are centralized, which suppresses the correlation between the parameters. Furthermore, the frequency factor is replaced by the rate constant at the average temperature. It is much easier to make an initial guess of the rate constant, than for the frequency factor, in nonlinear regression. The lesson taken from this particular example, the Arrhenius law, is general. Whenever the mathematical model has this kind of a structure, the correlation between the parameters is suppressed by orthogonalization of the experimental domain, i.e. by taking averages and defining new coordinates x ' = x - x . Insertion into the original model j ) = a o+a~x results into)5 = (a 0 +alx ) +alx'. A new parameter, ao'=ao-al x - y , is defined and the model becomes a o +alx . 10. 6.3. Systematic deviations and normalization o f experimental data
The error model used in minimization assumes that the residuals have zero mean value and are normally distributed. The latter assumption could be violated, since not all of the deviations between the model and the data are of a stochastic nature. Analytical techniques, particularly automation of off-line analysis, such as gas and liquid chromatography and development of online analytical techniques (UV, FTIR, flow and sequential injection analysis) suppress the random scattering in the data to a minimum, and beautiful experimental curves can be plotted. Still, a lot of deviations appear between experimental and predicted data. The main reason originates from systematic deviations, which are easily recognized by graphical consideration of data sets, e.g. plotting the residuals as a function of dependent or independent variables. Cases in the Figure 10.18 correspond to a) adequate model with random distribution, b) inappropriate weighting, c) systematic deviations due to an incorrect model, d) poor experimental design incapable of supplying the information expected.
Figure.10.18. Residuals as a function of variables (A. Cornish-Bowden, Detection of errors of interpretation in experiments in enzymekinetics, Methods, 24 (2001) 181). Systematic deviations are caused by several reasons; the most common ones being inadequate stoichiometry, inappropriate kinetic model and poor calibration of the analytical equipment. That the kinetic model is 'wrong' is easily recognized from graphical plots (see Figure 10.18). The further measures are clear: an improved kinetic model is tried to the data, and a new comparison is made. The iteration is continued until an adequate fit is obtained.
447 Inadequate stoichiometry and poor calibration of the analytical device are interconnected problems. The kinetic model itself follows the stoichiometric rules, but an inadequate calibration of the analytical instrument causes systematic deviations. This can be illustrated with a simple example. Assume that a bimolecular reaction, A + B ~ P, is carried out in a liquid-phase batch reactor. The density of the reaction mixture is assumed to be constant. The reaction is started with A and B, and no P is present in the initial mixture. The concentrations are related by Ce--CoACA=Co~ -CB, i.e. produced product, P, equals with consumed reactant. If the concentration of the component B has a calibration error, we get instead of the correct concentration cB an erroneous one, c'~ act, which does not fulfil the stoichiometric relation. If the error is large for a single component, it is easy to recognize, but the situation can be much worse: calibration errors are present in several components and all of their effects are spread during nonlinear regression, in the estimation of the model parameters. This is reflected by the fact that the total mass balance is not fulfilled by the experimental data. A way to check the analytical data is to use some forms of total balances, e.g. atom balances or total molar amounts or concentrations. For example, for the model reaction, A + B ~-> P, we have the relation CA+C~+Cp -CoA+CoB-constant (again Cop-O). The sum of concentrations should remain constant in space and time, if not, something is wrong. Either the reaction does not follow this stoichiometry or more probably the analysis is not calibrated very exactly. A more advanced way to consider potential calibration errors is to look at the components individually. Let us consider a single reaction. The extent of reaction is defined by ~ = (y~-yo,)/v~, where 4 is the extent of reaction, yi denotes the molar quantity after some reaction or residence time in the reactor and y0i is the initial or inlet quantity. For a continuous reactor, yi is the molar flow, for batch reactors it is the molar amount of substance or concentration. For systems with a constant density, the molar quantities can simply be replaced by concentrations. Thus, any model predicts the stoichiometric relation fit =Y0~ +vj~. On the other hand, experimentally recorded concentrations exist for the components in the system. The task is now to find an optimal extent of reaction, which would minimize the difference for the entire data set. The objective function is defined as Q = ~-]wi@~- y i ) 2, where ,'i is the weight factor for component i. The stoichiometric relationship is inserted into the expression for the objective function leading to Q = ~] w, (Y0, + v ~ - y y . An optimal value for the extent of the reaction, 4, is found by differentiation of it with respect to 4, dQ / d~ = ~ 2,:, (Y0, + v,~ - y,)v, = 0, from which 4 is conveniently solved analytically ~ = ~[w,v~(y~-yoi)/~w~v2~.
Thus, stoichiometrically
consistent values of the molar quantities are obtained by inserting the expression for the optimized extent of reaction, 4, into the basic stoichiometric relation ~, = Y0, +v,~. For an arbitrary
component
(k),
we
therefore
obtain
the
stoichiometrically corrected
value
Yk = Y0, + v,~. The consistency of the data can now be checked by plotting the initial value (c3 and the corrected value (ck) in the same graph. Different weight factors are used if it is a priori known that some of the components are better calibrated than other ones. Furthermore, the weight factors can be used tbr testing purposes. The procedure presented above tbr a single-reaction case is easily extended to several simultaneous reactions with y~ = Yo~+ Zi): ~vij ~j, giving the objective
448 function Q = ZY ( y ,
- Yoi -
ZJ:/vi/f/)
2
wi,
differentiation of which with respect to each extent of
reaction gives after some rearrangements y,, w, Z/v,~ vi/g; y=
yo + V ~ =
yo + v A
Z, ( y ,
- Yo,)
v, w~, finally leading to
(10.92)
'B
where, v is the stoichiometric matrix and the elements of the array, A and B, are obtained from V/ VIU2....VIV S 9 V2V l V£....V2V S
A =
(10.93)
FsV 1VSV2....V~
(10.94)
2w, Cy,-yo,)V,+I Naturally, a computer implementation of this procedure is required. An example of the application of the algorithm is provided for the case of hydrogenation of D-xylose to xylitol, which was carried out in the laboratory of the authors in a batch wise operating slurry reactor. The analytical data were obtained by means of high-performance liquid chromatography (HPLC). The original data did not obey the stoichiometry exactly due to calibration problems typically connected to this analytical technique. This implies that systematic deviations remain in the estimation of kinetic parameters, and the fit of the model parameters is not very good. At the next stage, the original data were exposed to the normalization algorithm, and the parameter estimation was repeated with the stoichiometrically consistent data. The systematic deviations disappeared, and a much better fit was obtained (Figure 10.19).
60 50
40
4O
wl*%
.... ~
~'~0 20
10
S,~'" i0
~ iO0
150
200
It should be emphasized that the final goal should be the improvement of the primary data: to remove the shortcomings of calibration that distort the mass balance. The presented algorithm is
449 just a diagnostic tool, with which the deviations can be localized and their effect on parameter estimation can be investigated. Another example for the check of the stoichiometric consistency is the catalytic decomposition of nitrogen oxide (NO) with hydrogen (H2) over alumina supported Rh monolith. Besides the main reaction, the formation of nitrous oxide (N20) and self-decomposition also take place. The overall reactions are given below: 2NO+2H2~N2+H20, 2NO+H2~N20+H20; 2N20~2N2+O2. The system consists of three linearly independent reactions. The vector of chemical symbols is a T = [ N 2 N O O~ ? ( 2 0 H 2] and consequently, the stoichiometric matrix becomes A =
1
-2 0
0
-2
2
(10.95)
0
0
1
0
1
-2
-2
-1
0
Primary data were obtained from a laboratory scale tube reactor operating at atmospheric pressures and elevated temperatures. A typical experiment is shown in Figure 10.20. .....
00040-
00030-
0 O02(Y
./,, /h~. ~,v' t , , . , , + ~ . . . ~ W , ~ i , ~ j /f'f
: ~
500
1000 Time, s
~
i
1500
Figure 10.20. Primary experimental data in the catalytic decomposition of nitrogen oxide (NO) with hydrogen. As can be seen from the data, the calibrated primary signals have not only a stoichiometric deviation, but they also contain some noise, randomly distributed experimental scattering. To smooth the data, a moving linear regression procedure was applied: polynomials were regressed through neighbouring points, and a new, smoothed data point was obtained. Typically the smoothing was applied over 5-10 points. The smoothed data set is shown in Figure 10.21. 0.0050
0.0040
0.0030 H2
0.0020
__
/. ~:'/'no
-
-
-
j
_
A
--,
7
0 0010
IF /"~.'/ "
.._ N20
.
0.0000
500
1000 Time, s
1500
2000
Figure 10.21. Smoothed data for primary experimental data the catalytic decomposition of nitrogen oxide (NO) with hydrogen in Figure 10.20.
450 The smoothing procedure considerably improves the quality of the data and simultaneously preserves its original characteristics, e.g. the periodic self-oscillations, which where observed for this system. The next step involved the normalization of the data with the proposed algorithm. Again, the by-products were given rather high weights, in order to preserve their identity, whereas the main components were handled with lower weights. After some numerical experiments, a sufficient heuristics emerged, and a stoichiometrically consistent data set was obtained. The steady state part of the data set was used in the normalization: the averages of the successive steady state mole fractions were calculated, and the averages were then subjected to the normalization. A comparison of the normalized mole fraction and the original steady state mole fraction gave a correction factor, with which the mole fractions of the transient part of the experiment were multiplied. A new, normalized data set was then obtained. This procedure is necessary, since the transient data cannot be normalized directly: the overall stoichiometry is not valid during the transient period because of the non-steady state adsorption processes. For cases with stable periodic oscillations, an average was calculated over several periods, and it was used as a pseudo-steady state mole fraction in the normalization. Examples of normalized step responses are depicted in Figure 10.21. As revealed by this figure, the proposed procedure gives transient responses that obey the well-known overall stoichiometry of the process.
10. 6. 4. Physico-chemical testing Since parameters in a mechanistic model have a physico-chemical meaning, there are certain constraints. The most obvious is that the kinetic parameters, e.g. rate and equilibrium constants should all have a positive value. The values of activation energies should be positive, unless a combination of parameters is fitted and the obtained parameter for activation energy is an apparent one. Binding of the substrate to the catalyst results in the loss of entropy, this means that adsorption entropies are negative. In the case of adsorption on solid surfaces from the gas phase, the following constraints can be then formulated0 <-AS,,°, < S ° . Since the Gibbs energy of adsorption is negative the adsorption enthalpy should be negative -AH~°, unless adsorption is a more complex process and for instance also includes spillover. In the latter case, positive values of adsorption enthalpies indicating endothermic adsorption could be obtained during parameter estimation. For a kinetic model based on a detailed reaction mechanism sometimes the system is overparametrised in the sense that it is difficult to get statistically significant values of the kinetic parameters. An approach which is sometimes used is then to theoretically calculate at least some of these values. Such estimation for preexponential factors and activation energies of elementary process could be based on the transition state and collision theories, although for heterogeneous catalytic reactions this approach is somewhat limited due to possible surface heterogeneity and lateral interactions. Although the necessity to use theoretically sound values of parameters in kinetic modelling was recognized several decades ago, a useful framework for so-called microkinetic analysis was more recently advanced and strongly advocated by J.Dumesic in connection to heterogeneous catalysis.
451
James Dumesic
Charles T. Campbell
Also, quantum mechanical approaches, such as DFT, can be used to estimate thermochemical parameters, like enthalpies and entropies, as well as kinetic parameters like activation energies and frequency factors of chemical reactions. The microkinetic analysis is then a way to incorporate the basic surface chemistry in the kinetic modelling.
10. 6. 5. Significance of kinetic parameters It is instructive to remember that not all of the parameters are kinetically significant, e.g. not all of them are controlling the overall rate. The rate controlling step in a series of elementary steps is that step which has the maximum effect on the overall rate of the reaction. C.T. Campbell has suggested that the kinetic importance of a particular step in a reaction scheme may be ascertained by computing the relative change in the overall reaction rate upon changing the forward and reverse rate constants for that step, while maintaining the same value of the equilibrium constant for the step. Thus, according to Campbell, the "degree of rate control" for step i, X ac.; is given as
k; (6r.
(10.96)
xR,,,= 7
where the equilibrium constant for step / ( ~ ) and all other rate constants (k_~ are held constant. The derivative of the rate is 6=
(er] &;+ (at)5k, -~.c3k;)<
~
< K;,,,
(10.97)
This derivative corresponds to a change in the overall rate resulting from a change in k, by OTqand a change in c~7c; by 6k / K,.,,q maintaining the same equilibrium constant. The degree of rate control is then given by
<<,':
Defining dimensionless sensitivies in the following way
(10.98)
452
k ~ J kJ F equation (10.98) results in XR(,, = ~bi + ~b_,
(10.100)
The value of XR(., approaches zero as step i becomes quasi-equilibrated, while this value is close to unity for the step with the maximum sensitivity.
10.7. Model discrimination Often there may be more than one plausible model available which is able to fit the experimental data. Then the problem is how to select between plausible kinetic models. There are two possible approaches in the model discrimination. The first approach involves collecting experimental data, estimating the model parameters and then comparing the models based on such criteria such as residual sum, lack-of-fit and significance of the parameters. When such a method is applied, in several cases it could be difficult to discriminate between different models, especially when experiments are conducted at experimental conditions, when no discrimination is possible (Figure 10.22).
/
Figure 10.22. Illustration of difficulties in models discrimination.
For systems that are more complex, the second technique, true model discrimination is applied. This involves the use of a statistical criterion to select settings of the independent variables, e.g. experimental conditions, such that differences between models are emphasized. It should be noted that statistical techniques only provide guidelines. The judgment and experience of the researcher should not be underestimated, since they are crucial in determining the type and number of models as well as the level at which the discrimination is considered to be sufficient. Moreover, no discrimination technique can identify the best model if it is not included in the discrimination set and if none of the used models are adequate.
453 It has been recognized by researchers in chemical kinetics that while the kinetics of the system having a given structure, can be solved, the reverse question, what should be a structure of the model having the given kinetics, cannot be adequately answered. For this reason the interest in applying advanced experimental design tools, like sequential experimental design, somewhat declined in the last two or three decades. A simple approach for sequential experimental design is to determine the divergence of the predictions between two rival models at certain experiment settings D(Xk) =
[y(I) (Xk) -- ;(2)(Xk)]2
(lo.lol)
For m-models the criterion is written in the following way m
1
m
= Z
(10.102)
-
i=1 j=i+l Here the double summation is used to avoid mislocations of optimal discrimination conditions taldng each of the models as references. Since the divergence is also determined by statistical uncertainties, the variances should also be taken into account, for instance following the Box-Hill method
[ D(xk)=}--',~--~co, rc./,, (0_2 ----7,-7---7-
2 >(
2O-2 +O-~j+O-2
]
(10.103)
where 3r~,.is the degree of accuracy achieved after n experiments. The algorithm is then as follows. After n experiments the models are analysed and the following (n+l) experiment is conducted at conditions which maximize divergence. At the beginning the degree of accuracy of probability is the same for all models, however after n+l experiments, such probability is updated using the posterior probabilities proposed by T. Bayes
"¢Z'i,n+l
--
m
~;.,P;
Z ./=1
,i = 1..m
(10.104)
2 " / , ~ p j
where p; is the probability density of the (n÷l) observation y,,÷~ for model i. Discrimination is achieved when the value of the model posterior probability reaches unity. Other design criteria could be found in the special literature and mentioned in the recommended literature for this chapter. A few software programs for kinetic modelling like ModEst 6.0/6.1, which will be discussed in the following section, incorporate experimental design routines.
454 10.8. Software There are several software packages for modeling chemical and biochemical systems. They simulate the kinetics of systems of chemical and biochemical reactions providing tools to fit models to data, estimate parameters, perform simulations and optimizations among other options. In the general purpose software the task of the user is to develop a mathematical model and rewrite a model in the format expected by the software. For instance Athena Visual Studio offers an integrated environment for modeling, optimization, estimation, optimal experimental design, and model discrimination of chemically reactive and non-reactive process systems. A user has the possibility develop models, while the software takes care of the tasks associated with solving the underlying equations and estimating the model adjustable parameters. A convenient graphical user interface is combined with powerful parameter estimation and optimization software, which allows for the analysis of single, and multi-response experiments, model discrimination and optimal experimental design. Moreover, several examples are provided illustrating typical cases relevant to chemical reaction engineering (Figure 10.23).
AthenaVisualStud~ ~ l,,Lroduc!!or, :~ ~ proce~:Modeling l * ~ Modeiing,/~ithini~ialv'aiuePioblerns [ { ~ ModellngI niiial;Bounda[yValueProblelrs E~tinabon ~lh puie,Si~ebraicEqua[i,:n~ ~ E#Jmation~th O[dinar~.~Dif!elertiaiEqu~lion,', [~] Estir0ationwithDAE~~,iih~bgonalE Matrix. ~ Es~ma~onw~thDAEs,MthNon-Dla~onalE Matrix I~ ~ Eslirnaiionwi(h!niliai:BoundaryVal~eProblems EslrnmlonwithExpliclModel~ N'::n!healOpti~iZation ~ lrait)ingSamples /4::~ Gehe[alMathernali~s l : ~ DhernicaiR~actimEngii-eering ~'~]Bawh bOtherpnaiRe~to~ Se~:Bamh IsothermalReactol ~ IqmnisothetmalPlug FiowReactor PlugFloe Reacto~withS~deStream ~ Steaay-Sta~ReactoHwith,&xia]Dispei~ion ~ Oynamlctubukr~ eactorwithA:.JaIDispersion [)ynamic[)i{[usJonwiih EquililbiiumAdssipi;m, ~ DynamicR e~.tionwithDiffusionand Adsorption FixedBe~Reacio(',,,/ilhIntrapa~ticieOi~!usi~n Thetmodynamic~and Separaliom /:i ~N,::niinea i ParameterE~tima(ior, Sir,file Re~:pon~eA naly:~i:~ elilnE ~liciiModeb ~ l,.h_ndel DisGiminatior ,.'.i(h E~,~!icit ~,Iode!~ SirsleResponse/Anal~,~i,~ wi~hImplicitNodels ~ 14odeiDkciimination,.qthImplicit~odeb ~ Mul{!:Re~pon~eAnay~i~,~,ilhImNiciiModeb ~ ~.4ul~Re~on~e i N odeiDisermina!ionwiihImplicitModel Noniineal0 ptimizal!on 0 piimaJOon!rolPmblern
;¢4~ =or
[C , j
ZI ~ ~1 ~ Ka * fUJ
uh
~u¢lt
#,entre
Figure 10.23. An example of a steady-state isothermalCSTR in Athena Visual Studio.
455
In Athena Visual Studio, the selection of tasks (modelling, estimation or optimization) is combined with an option to select a variety of mathematical models (Figure 10.24).
Figure. 10.24. Selection of tasks in Athena Visual Studio. It is possible further to specify the details of a particular model (Figure 10.25)
Figure. 10.25. Specification of the model details in Athena VisualStudio. Specific programs devoted to biokinetics have the most often applied biokinetic models already implemented. As an illustration we can refer to Gepasi package (Figure 10.26) or Lucenz III
456 (Figure 10.27). In the latter case the visualization of the data includes the possibility to use a variety of diagnostic plots familiar to biochemists.
Figure. 10.26. Implementation ofbiokinetic models in Gepasi. N ......................e............ Weighted . . . . . . . . . 4.90e 30 R M S F,actionahesidual= 1 24eq5
1 Vm
5 OOe0O
÷/-
8 71e.15
4.oo~oo
+~
~o3o,5
3 Kb
2.00eO0
+/
5 73e 15
Figure. 10.27. Lucenz 111: a) implementation of biokinetic models, b) diagnostic plots. ModEst (from Model Estimation) is another software (Figure 10.28) that has been designed for parameter estimation of mechanistic mathematical models as well as for experimental design. It gradually evolved as a by-product of industrial modelling work with the participation of the authors. During that period, a great number of modelling tasks of various types has been solved.
457
Figure 10.28. ModEst 6.1- softwarefor parameter estimation. A few examples will be demonstrated in the tbllowing section. Each feature was incorporated in the software because it has been found necessary or useful in some modelling project. All the modelling tasks that previously required tailor made solutions in each project can now be solved in a unified manner in the ModEst environment. ModEst is able to deal with explicit algebraic, implicit algebraic (systems of nonlinear equations) and ordinary differential equations. As the standard way to handle PDE systems, the Numerical Method of Lines, which transforms a PDE system to a number of ODE components is used. In addition, any model with a solver provided may be dealt with as an algebraic system. The basic numerical tools are contained in the well tested public domain software (Bias, Linpack, Eispack, LSODE). The model are defined as ODE (Figure 10.29) or algebraic models (Figure 10.30)
Active proiect: Atiloclaue
Figure 10.29. Models definitionin ModEst ~ODE iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii:
Figure 10.30. Models definitionin ModEst algebraic.
~
458
The user is defining the model equations, (Figure 10.31).
i l i , krl
RR=8-31~3 z=i.O/(1.DO/Temp-l.
Ea2 kr3
DO/480.15)
EaS hr4
kl=krl*EXP( Eal/(RR*z)) k2=kr2.EXP(_Ea2/(RR.z) ) k3=kr3~EXP( Ea3/(RR~z)) kq=krq~EXP(_Ea4/(RR~z))
Ea4 kr5 me5 kr6 Ea6 krml Eaml krm2
kS=krS~EXP(-EaS/(RR~z)) k6=kr6~EXP(-Ea6/(RR~z)) ~r,l=krml~EXP (-E~nl/ (RRWz)) k~2=krm2 *EXP ( E~m2/ (RR*z)) KH=EXP (-d~I/RR) *EXP (dhl/ (RR~temp)) KA=EXP( d~2/RR)*EXP(dh2/(RR~temp))
!temp !temp
Eam2 dsl
KO=EXP(dsS/RR)~EXP(-dhS/(RR~emp)) !~emp Eroll=EXP(ds4/RR)*EXP(-dh4/(RR*temp)) !temp
!production oZ !roll-over
dhl ds2
Global Global Global Global Global Global Global Global Global Global Global Global Global Global Global Global Global Global Global Global
Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar Scalar
Real Real Real Real Real Real Real Real Real Real Real Real Real Real Real Real Real Real Real Real
Figure 10.31. Equations deffmition in ModEst.
Initial values and observable variable (experimental data) are also defined by the user (Figure 10.32) .
Figure 10.32. Experimental data definition in ModEst.
459 There is a possibility to define a task (simulation, parameter estimation, etc.) as well as the targets (Figure 10.33) and calculation tools (Figure 10.34).
Figure 10.33. Task and targets definition in ModEst.
Figure 10.34. Calculation tools definition in ModEst. After compiling and computing (Figure 10.35) results are then displayed graphically along with the basic statistics (Figure 10.36)
460
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S:\P~o{Math\,~et60>pauee P r e s s an
ke
to ¢ o , t i , u e
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Figure 10.36. Presentation of results in ModEst.
If needed more advanced statistical information (Figure 10.37), including sensitivity data
(Figure 10.38) and contour plots (Figure 10.16), is available. Residual S S S~d. ~rror of es~ma~:e
0~8S13~+~ 0.163~#99
Th~ Hess~an~
8.3S2~02
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• h~ ~ o r r e ~ a t & ~ matl:~x of the pa~amete=a: l,OOO
-'0,308 ~0,0~2
[,000 O.OZ4
&.808
Figure 10.37. Presentation of statistical results in ModEst.
461
i i i i i i i i i i i i i i!i i i!i i i i i i i i i i i i i iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii i i iii
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%
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Figure 10.38. Presentation of sensitivity data in M o d E s t .
In comparing how well competing models fit the experimental data, a convenient way defined in ModEst 6.1 is the calculation of a coefficient of determination, the R2-value,
R 2 =1
z=l j=, ~=l
ZZZ[(eij i=1
j=l
(10.105)
t=l
where ~,.~j., is the average of all the data points. The estimated sum of squares is divided by the sum of squares of the simplest possible model, the average of the values. This implies that R 2 values approaching 100% are desired. Typically values exceeding 95% represent a good fit. Unfortunately, the R2-value does not distinguish between random errors due to scattered data and systematic deviations caused by an inadequate model. Additional features, which are provided by existing software packages besides parameter estimation, are numerical simulation and optimization. The first option gives an opportunity to calculate, for instance, reaction rates for conditions (temperature, pressures, concentrations, etc) at which no experiments were performed. Besides interesting simulations, optimization tasks can also be performed, if, for instance there is a need to achieve an optimal result with minimal negative factors - harmful side products, costs, etc. The various goals are often contradictory and lead to a multitarget optimization task. Typically, the objective function has to be specified by the user, which means that while the objective functions for parameter estimation and optimal design are usually built-in routines in software like ModEst 6.1, the optimization task requires a user written subroutine. The desirability function technique is especially easy to implement and in many cases sufficient for multicriteria optimization.
462
10.9. Case study We will discuss here an example which concerns the pathway from intrinsic kinetics to diffusion-affected kinetics in catalytic hydrogenation (T. Salmi, T.-K. Rantakyl~i, J. Wfirn& P. Mfiki-Arvela, J. Kuusisto, I. Martinez, Modelling of kinetic and transport effects in aldol hydrogenation, C h e m i c a l E n g i n e e r i n g S c i e n c e 57 (2002) 1793). Modelling of catalytic liquidphase hydrogenation is a challenging topic, since complex surface processes on the catalyst surface interact with diffusion effects inside the catalyst particles as well as at the gas-liquid and liqui&solid catalyst interfaces. Experiments revealing the intrinsic catalytic kinetics are typically carried out with finely dispersed catalyst particles (@<100 btm) in batchwise operating slurry reactors in order to remove the diffusional resistance inside the catalyst. External mass-transfer limitations are suppressed by vigorous agitation of the reaction mixture and by having a sufficiently high liquid volume-tocatalyst mass ratio. Industrial practice, however, generally implies the use of continuous reactors, e.g. fixed beds and large catalyst particles. The amount of catalyst in fixed bed reactors calculated per reactor volume is clearly higher than the amount of catalyst in a slurry reactor. However, to minimize the pressure drop in a fixed bed it is necessary to use large catalyst particles (>1 mm), which implies that diffusional effects inevitably interfere with reaction kinetics. The scale-up from laboratory experiments carried out with small catalyst particles (<0.1 mm) to conditions where large particles (>1.0 mm) are used, is the key issue in successful process development. Kinetics and mass transfer effects in the liquid-phase hydrogenation process of an aldol (2,2dimethylol-l-butanal) to the corresponding triol (trimethylolpropane) were studied in a semibatch-wise operating autoclave, where finely dispersed and large catalyst particles of nickel chromium catalysts were used ]
CH20H I <~0
CH~OH [
CHzOH
CH2OH
°
Since the feed to the process inevitably contains formaldehyde as an impurity from a previous aldolization step R--( H 2 - - ( ~ I I
2
H H \ / C O
CH~{}H OH [ ~¢O ------------------------R - - ( - - C . H C~{2OH -
the hydrogenation of formaldehyde to methanol CH20+H2 ~CH3OH should be involved in the process description. The intrinsic hydrogenation kinetics was determined with the crushed catalyst particles at 40 80 bar H2 and 50 90°C in isobaric experiments. A kinetic model based on competitive adsorption and surface reaction between the aldol and hydrogen was successfully fitted to the experimental data. Physical measurements of the density, viscosity, as well as hydrogen solubility in the reaction mixture, were carried out. The density measurements of aldol and triol solutions in methanol water and water solutions showed that the density is linearly dependent on temperature t9 = A - B ( t ( ° C ) , but changes very little as a fimction of the actual composition, i.e. the reaction time. The viscosity data fbllowed Andrade's equation for the temperature dependence. Since the viscosity changed very little as a
463 function of the mixture composition under actual conditions, the ideal mixing rule was applied on the viscosities of aldol-triol mixtures. The derivation of rate equations for simultaneous hydrogenation of TMP-aldol and formaldehyde was based on a plausible surface reaction mechanism. According to the mechanism, the aldol, formaldehyde and hydrogen undergo competitive adsorption on the nickelalumina surface. Adsorbed hydrogen atoms add to the carbonyl bonds of the aldol and formaldehyde. These irreversible reaction steps are presumed to be rate-determining, whereas the product desorption is regarded as rapid. Consequently, the reaction mechanism is written as follows: H2+2*,~:>2H* AI+*<=>AI* A2+*<=>A2* 1. Al*+2H*----~A1H2+3* 2. A2*+2H*---~A2H2+3*
(10.106)
where Al-aldol, A2-formaldehyde, AiH2-triol and A2H2-methanol. Previously carried out kinetic experiments had shown that formaldehyde has a strong retarding effect on the adsorption and hydrogenation of the aldol (A1). This retardation cannot be explained by the simple competitive adsorption model, i.e. by a high value of the adsorption constant of formaldehyde as such. It was necessary to assume that the presence of lbrmaldehyde diminishes the adsorption affinity of the aldol. For the sake of simplicity, a model was applied according to which the adsorption enthalpy decreases linearly as a function of the coverage of the adsorbed species (10.107)
-AH 4 = -AH°A,-aOA2
In the rate equations applied in practice, the fractional coverage was replaced by the mole fraction of formaldehyde, CAZ/CTot,since the work was carried out in dilute solutions. The rates of the rate-determining steps were written with the surface coverage of the reactants as follows:
rl=kl0A10#,r2=k20A20#
(10.108)
The quasi-equilibrium hypothesis is applied on the adsorption steps. The surface coverage of hydrogen, formaldehyde and the aldol from the adsorption quasi-equilibria are inserted into the rate equations (10.108), after which the following expressions are obtained for the hydrogenation rates rj: _
f.
n
3
r l - k l c.41 A IcH/D , r2=k2'CA2CH/D 3
(10.109)
where .
,i
.
T~-
.
,i
D=ltlkAICAItlkA2CA2tlkH
1/2
1/2
CH
(10.110)
464 and (10.111)
k2'=N2KA2KH, kl'=klKA IKH
The exponent hA1 in eq. (10.109) was introduced somewhat empirically to obtain a better fit to experimental data. From the hydrogenation rates rl and r2, the generation rates of formaldehyde, methanol, the aldol, the triol and hydrogen are obtained in a straightforward manner by using the overall stoichiometry: Cdl =
rl,FA1H2=/'I,/~A2H2 =
r2,/~A2 =
r2,rH2=
(rl+r2)
(10.112)
In the absence of internal and external mass transfer limitations, the kinetic model is easily connected to the component mass balances in the pressurized autoclave. The mass balances for the liquid phase components can be written with the mass-based concentrations (in mol/kg): dci
m tOB
d/
,oL
(10.113)
where t is the reaction time, p~ is the catalyst bulk density and ri is the generation rate of the component i. The mass balance of hydrogen was discarded, since the hydrogen pressure was maintained constant during the experiments. The liquid-phase concentration of hydrogen was obtained from the correlation equation based on measured hydrogen solubility. The kinetic parameters in the rate equations were determined with non-linear regression analysis. The rate equations were inserted into the mass balances, which were solved numerically with the backward difference method during the parameter estimation. In the parameter estimation, the following objective function was minimized: (10.114) where Ci,t,exp is the experimentally recorded concentration and ci,t,~al~ is the concentration calculated from the model. Equal weight factors (wi) were used for all compounds. The aldol (Aj), formaldehyde (A2) and triol (A2H2) concentrations were included in the regression analysis, but methanol was left out because it was difficult to determine its concentration increase with a high precision since the mixture contained a lot of methanol as a solvent from the very beginning. The objective function was minimized with a combined Simplex-Levenberg- Marquardt method. The parameter estimation was commenced with the robust Simplex method, but was switched to the more rapid Levenberg-Marquardt method as the optimum was approached. The LSODE computer code implemented in ModEst 6.0 software was used in the solution of the differential equations. The fit of the model to the experimental data is compared in Figure 10.39, showing that the simulated curves follow the experimental trends. The parameter estimation results revealed that the governing kinetic parameters are quite well identified: typically the standard deviations of the parameters are <22%, and the correlation
465 coefficients of the parameters are below 0.54. The degree of explanation (eq. 10.105) was typically within the range 94 96%. 1
~ 0,,tool q,
.
~ ' C Ond SO bar . .
.
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L e 01~Do[ g)
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0
20
10
30
40
50
60
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l i,,,~ (',li'l)
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I
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so
40
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so
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-
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~ J ' c ~lid 40 b,l
1
1
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og
o
os
.
.
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.
as
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o
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07
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o
os
°
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o
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S0
ioa
12o
14o
o
2o
40
so
B0
i~,,,~ (,,m,) Ti
me (mill)
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12o
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50
t ime (,~,m)
Figure 10.39. Fit of the kinetic model compared to experimental data in aldol hydrogenation. Inhibition effect of formaldehyde on the reaction kinetics included. Abbreviations, Aldol=TMP-aldol (A), FA=formaldehyde, TMP=trimethylpropane (AIH2).
It should be noted that the inhibition factor a (see eq. (10.107)) is extremely important for the kinetic model, since a model without the inhibition factor is not sufficient for predicting a monotonically decreasing hydrogenation rate of aldol as the hydrogenation progressed, whereas the aldol hydrogenation rate in reality went through a maximum in most cases. The kinetic and solubility parameters obtained for the model were used together with the diffusional and textural parameters of the catalyst in a reaction-diffusion model in order to predict the performance of catalyst particles of industrial size. An agitated vessel operating in laboratory scale was considered, where the catalyst particles are placed in a basket and the gas phase is dispersed in the liquid phase, which is vigorously agitated. Under these circumstances, the interparticle transport resistances can be neglected. What are left are the intraparticle resistances, i.e. the heat and mass transfer effects inside the catalyst particles. Since the current case reflects the situation that few reactant and product molecules exist in an environment of solvent molecules, the simplest Fick's law- approach with effective diffusion coefficients can be considered as sufficient for the description of molecular diffusion.
466 The effective diffusion coefficient is obtained from the molecular diffusion coefficient (Di), the catalyst particle porosity E~, and tortuosity. The approximate formula of Wilke and Chang was used to calculate the molecular diffusion coefficients for a component dissolved in a solvent. The viscosity of the liquid was determined experimentally and the molar volumes of the components were estimated from the atomic contributions of LeBas, as discussed in Chapter 9. Applying the general expression for the mass balance the expression took the form
d(ciPL) dt
'[
I
~
r,Pt,
d(Nr2)] r2dr J
(10.115)
where r is the radius and N is the component flux. Eq. (10.115) is generalized for an arbitrary particle geometry
d(
c•l
'I l~) _ ~,
qp.
d ( Nr~') r"dr
(10.116)
where the shape factor (s) obtains the values 0, 1 and 2 for a slab, an infinitely long cylinder and a sphere, respectively. For non-ideal particles, the value of the shape factor is calculated from s+l-(Ap/Vp)R, where R is the characteristic dimension of the particle. When the model of the diffusion flux is inserted into the balance equation and a dimensionless coordinate (x-r/R) is introduced, if the effective diffusion coefficient can be considered to be constant inside the particle, we obtain
d(c'PL) -~F -11~Pp q_Jei ff d2(cflOL) - + sd(cflOL).I1
dt
R2~ dx 2
dx )~
(10.117)
For the heat transfer inside the particle, the simple law of Fourier is applied, which results in a differential equation for the temperature profile inside a pellet
dT dt
( d~-T s dT )
2,
(-all)rj /-1
Cp
p,c ,R
(10.118)
Flat concentration and temperature profiles corresponding to the bulk phase conditions were assumed in the beginning of the reaction (t=0). The boundary conditions at the centre of the catalyst particle (r=0) are dc/dr=0 and dT/dr=O because of symmetry reasons. After some rearrangements the mass balance took the form I
dci - m'c'~ I(s + 1)x'rflx dt I~lliquid0
(10.119)
467 The generation rates (ri) are obtained for each x-value by utilizing the concentration profiles which are solved numerically from the reaction-diffusion equation (10.117). The model equations were solved numerically by discretizing the partial differential equations (PDEs) with respect to the spatial coordinate (x). Central finite difference formulae were used to approximate the first and second derivatives (e.g. dcjdx, dT/dx). Thus the PDEs were transformed to ODEs with respect to the reaction time and the finite difference method was used in the numerical solution. The recently developed software of Buzzi Ferraris and Manca was used, since it turned out to be more rapid than the classical code of Hindmarsh. For the sake of comparison hydrogenation experiments with large cylindrical catalyst particles were carried out. The increase of the particle size diminished the velocity of catalytic hydrogenation. These experimental results provide a path for the process scale-up, i.e. a prediction of the hydrogenation rate on large catalyst particles starting from crushed particles. The values of the kinetic constants obtained for crushed particles were utilized and the ratio of porosity to tortuosity from the reactio~diffusion model was adjusted (0.167) to fit successfully the experimental data (Figure 10.40). c (mol/kg)
2__ o
20
40
60 80 Time, rain
100
120
140
Figure 10.40. Fit of the kinetic model compared to experimental data in aldol hydrogenation for catalyst pellets.
The temperature gradients inside the catalyst pellet turned out to be negligible, clearly less than 1 K for realistic approximations for AH and heat conductivity. Thus, the final calculations were made by the isothermal model by simply assuming the same temperature inside the catalyst particle as in the liquid phase. The concentration profiles of aldol and hydrogen during the experiment are displayed in Figures 10.41 and 10.42. 07[ C(mol/kg)
0
01
centre
o2
o3
o4
os
o~
07
oa
os
x
Figure 10.41. The concentrationprofilesofaldol.
468
0~25
0-2
g
O X
Figure 10.42. The concentrationprofiles of hydrogen. Figures 10.41 and 10.42 indicate that the process is limited by hydrogen and aldol diffusion in the beginning of the experiment, but the concentration gradient of hydrogen diminishes very much during the experiment. The diffusional limitation of aldol prevails throughout the whole experiment. Taking into account the fact that the description of the pore structure of the catalyst particle is very much simplified, we can conclude that the predictions are good and that they provide an example of a satisfactory scale-up approach based on the physico-chemical properties and kinetic behaviour of the system.
469
Recommended literature Chapter 1
I. Chorkendorff, J.W.Niemanstverdriet, Concepts of modern catalysis and kinetics, Wiley-VCH, Weinheim, 2003. Handbook of heterogeneous catalysis, 5 Volume Set, Editors: G. Ertl, H.Km6zinger; J. Weitkamp, Wiley-VCH, Weinheim, 1997 S.R. Logan, Fundamentals of chemical kinetics, Longman, Edinburgh Gate, 1996. R.I.Masel, Chemical kinetics and catalysis, Wiley-Interscience, New York, 2001. M.J. Pilling, P.W.Seakins, Reaction kinetics, Oxtbrd University Press, Oxtbrd, 1995. J.M.Thomas, W.J.Thomas, Principles and practice of heterogeneous catalysis, VCH, Weinheim, 1997. R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill, Catalysis: An integrated approach, Elsevier, Amsterdam, 2002.
Chapter 2
R.L. Augustine, Heterogeneous catalysis for the synthetic chemist, Marcel Dekker Inc., New York, 1996. A.S.Bommarius, B.R.Riebel, Biocatalysis, Weinheim, 2004.
Fundamentals and applications.
Wiley-VCH.
M. Boudart, Turnover rates in heterogeneous catalysis. Chemical Reviews, 95 (1995) 661. I. Chorkendorfl; J.W.Niemanstverdriet, Concepts of modern catalysis and kinetics, Wiley-VCH, Weinheim, 2003. A.Clark, The theory of adsorption and catalysis, Academic Press, New- York, 1970.
470 Handbook of heterogeneous catalysis, 5 Volume Set, Editors: G. Ertl, H.KnOzinger; J. Weitkamp, Wiley-VCH, Weinheim, 1997. K.W.Kolasinski, Surface science. Foundations of catalysis and nanoscience, Wiley, Chichester, 2002. K.J. Laidler, Chemical kinetics, Harper and Row, New York, 1987. R.i.Masel, Chemical kinetics and catalysis, Wiley-lnterscience, New York, 2001. Yu. S. Snagovskii, G.M. Ostrovskii, Modelling of kinetics of heterogeneous catalytic reactions, Chimia, Moscow, 1976. G.A. Somorjai, Introduction to surface chemistry and catalysis, Wiley-Interscience, New York, 1994. M.I. Temkin, The kinetics of some industrial heterogeneous catalytic reactions, Advances in Catalysis, 28 (1979) 173. P.W. van Leeuwen, Homogeneous catalysis: understanding the art, Kluwer Academic, 1999. R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill. Catalysis: An integrated approach, Elsevier, Amsterdam, 2002.
Chapter 3 M.Boudart, G. Djega-Mariadassou, Kinetics of heterogeneous catalytic reactions, Princeton University Press, Princeton, 1984. I. ChorkendorfL J.W.Niemanstverdriet, Concepts of modern catalysis and kinetics, Wiley-VCH, Weinheim, 2003. E.T. Denisov, O.M. Sarkisov, G.I. Likhtenshtein, Chemical kinetics. Fundamentals and new developments, Elsevier, Amsterdam, 2003. J.A. Dumesic, D.F.Rudd, L. Aparicio, The microkinetics of heterogeneous catalysis, American Chemical Society, Washington, 1993. B. Hammer, J.K. Norskov, Theoretical surface science and catalysis - calculations and concepts. Advances in Catalysis, 45 (2000) 71. L.P.Hammett, Physical organic chemistry, McGraw-Hill, New York, 1970. K.J. Laidler, Chemical kinetics, Harper and Row, New York, 1987.
471
S.J. Lombardo, A.T. Bell, A review of theoretical models for adsorption, diffusion, desorption and reaction of gases on metals, Surface Science Reports, 13 (1991) 1. R.I.Masel, Chemical kinetics and catalysis, Wiley-Interscience, New York, 2001. P.Stolze, Microkinetic simulation of catalytic reactions, Progress in SulJhce Science, 65 (2000) 65. M.I. Temkin, The kinetics of some industrial heterogeneous catalytic reactions, Advances in Catalysis 28 (1979) 173. Yu. K. Tovbin, Lattice-gas model in kinetic theory of gas-solid interface processes, Progress in Surface Science, 34 (1990) 1. R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill. Catalysis: An integrated approach, Elsevier, Amsterdam, 2002. F. Zaera, Kinetics of chemical reactions on solid surfaces. Deviations from conventional theory, Accounts" of Chemistry Research, 35 (2002) 129.
Chapter 4 J. Horiuti, Theory of reaction rates as based on the stoichiometric number concept. Annals of the New YorkAcademy of Sciences, 213 (1973) 5. J. Horiuti, T. Nakamura, Theory of heterogeneous catalysis, Advances in Catalysis, 17 (1967) 1. S.L.Kiperman, Foundations of chemical kinetics in heterogeneous catalysis, Khimia, Moscow, 1979. E.L. King, C. Altman, A schematic method of deriving the rate laws for enzyme-catalyzed reactions, Journal of Physical Chemistry, 60 (1956) 1375. L.A. Petrov, Application of graph theory to study of the kinetics of heterogeneous catalytic reactions, Mathematical Chemistry, 2 (1992) 1. Yu. S. Snagovskii, G.M. Ostrovskii, Modelling of kinetics of heterogeneous catalytic reactions, Chimia, Moscow, 1976. M.I. Temkin, The kinetics of some industrial heterogeneous catalytic reactions, Advances in Catalysis, 28 (1979) 173.
472 R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill. Catalysis: An integrated approach, Elsevier, Amsterdam, 2002.
Chapter 5 D.G. Blackmond, Reflections on asymmetric catalysis. A kinetic view of enantioselectivity, CATTECH 2 (1998) 17. E.T. Denisov, O.M. Sarkisov, G.I. Lildatenshtein, Chemical kinetics. Fundamentals and new developments, Elsevier, Amsterdam, 2003. F.G. Helfferich, Kinetics of homogeneous multistep reactions, in Comprehensive chemical kinetics, vol. 38. ed. R.G. Compton, G. Hancock, Elsevier, Amsterdam, 2001. K.J. Laidler, Chemical kinetics, Harper and Row, New York, 1987. R.I.Masel, Chemical kinetics and catalysis, Wiley-Interscience, New York, 2001. R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill. Catalysis: An integrated approach, Elsevier, Amsterdam, 2002.
Chapter 6 J.E.Bailey, D.F.Ollis, Biochemical engineering fundamentals, McGraw-Hill Book Company, NY. 1986. H.Bisswanger, Enzyme kinetics, principles and methods. Wiley-VCH. Weinheim, 2002. A.S.Bommarius, B.R.Riebel, Biocatalysis, Fundamentals and applications. Wiley-VCH. Weinheim, 2004. A. Cornish-Bowden, Foundamentals of enzyme kinetics (3d edition), Portland Press, London, 2004. E.T. Denisov, O.M. Sarkisov, G.I. Likhtenshtein, Chemical kinetics. Fundamentals and new developments, Elsevier, Amsterdam, 2003. V.Leskovac, Comprehensive enzyme kinetics, Kluwer Academic/Plenum publishers, New- York, 2003. S.D.Varfolomeev, K.G. Gurevich, Biokinetics, Grand, Moscow, 1999.
473
Chapter 7
I. Chorkendorff; J.W.Niemanstverdriet, Concepts of modern catalysis and kinetics, Wiley-VCH, Weinheim, 2003. A. Clark, The theory of adsorption and catalysis, Academic Press, New York, 1970. G. Djega-Mariadassou, M. Boudart, Classical kinetics of catalytic reactions, Journal of Catalysis, 216 (2003) 89.
H. S. Fogler, Elements of chemical reaction engineering (3d ed), Prentice Hall, N J, 1998. A.N.Frumkin, V.N.Andreev, L.I.Boguslavskii, B.B.Damaskin, R.R.Dogonadze, V.E.Kazarinov, L.I.Krishtalik, A.M.Kuznetsov, O.A.Petrii, Yu.V.Pleskov, Double layer and electrode kinetics, Nauka, Moscow, 1981. Handbook of heterogeneous catalysis, 5 Volume Set, Editors: G. Ertl, H.Kn6zinger; J. Weitkamp, Wiley-VCH, Weinheim, 1997. Yu. S. Snagovskii, G.M. Ostrovskii, Modelling of kinetics of heterogeneous catalytic reactions, Chimia, Moscow, 1976. M.I. Temkin, The kinetics of some industrial heterogeneous catalytic reactions, Advances in Catalysis, 28 (1979) 173. J.M.Thomas, W.J.Thomas, Principles and practice of heterogeneous catalysis, VCH, Weinheim, 1997. R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill. Catalysis: An integrated approach, Elsevier, Amsterdam, 2002.
Chapter 8
G. Ertl, Dynamics of reactions at surfaces, Advances in Catalysis 45 (2000) 1. K.J. Laidler, Chemical kinetics, Harper and Row, New York, 1987. N.M.Ostrovskii, Kinetics of catalyst deactivation, Nauka, Moscow, 2001. M.J. Pilling, P.W.Seakins, Reaction kinetics, Oxford University Press, Oxford, 1995. S.L. Shannon, J.G. Goodwin, Characterization of catalytic surfaces by isotopic-transient kinetics during steady-state reaction, Chemical Reviews, 95 (1995) 677.
474
K.Tamaru, Dynamic heterogeneous catalysis, Academic Press, New York, 1978.
Chapter 9 J.E.Bailey, D.F.Ollis, Biochemical engineering fundamentals, McGraw-Hill Book Company, NY. 1986. J.B. Butt, Reaction kinetics and reactor design, Marcel Dekker, New York, 2000. L. K. Doraiswamy, M. M. Sharma, Heterogeneous reactions: Analysis, examples and reactor design. Vol. 1, Gas-solid and solid-solid reactions, Vol. 2, Fluid-fluid and fluid-fluid-solid reactions, John Wiley, New York, 1984. H. S. Fogler, Elements of chemical reaction engineering (3d ed.), Prentice Hall, N J, 1998. G. Froment, K. Bischoff, Chemical reactor analysis and design, 2d ed., Wiley, New York, 1990. Handbook of heterogeneous catalysis, 5 Volume Set, Editors: G. Ertl, H.I~a6zinger; J. Weitkamp, Wiley-VCH, Weinheim, 1997. I.I.Ioffe, V.A.Reshetov, A.M.Dobrotvorskii, Heterogeneous catalysis, Khimia, Leningrad, 1985. O. Levenspiel, Chemical reaction engineering, 3rd Edition, Wiley, 1998. B. E. Poling, J. M. Prausnitz, J.P. O'Connell, The properties of gases and liquids, McGraw-Hill, 2000. T.Salmi, J.P.Mild~ola, J.W~irngt, Bridging chemical reaction engineering and reactor technology, Abo Akademi Press, Turku/Abo, 2004. Y. Sasson, R. Neumann, Handbook of phase transfer catalysis, Kluwer Academic Publishers, 1997. C.N.Satterfield, Mass transfer in heterogeneous catalysis, MIT press, Cambridge, 1970. J.M.Smith, Chemical engineering kinetics, McGraw-Hill, 1981. R. A. van Santen, P. W. N. M. van Leeuwen, A. A. Moulijn, B. A. Averill. Catalysis: An integrated approach, Elsevier, Amsterdam, 2002.
475 Chapter 10. K. A. Brownlee. Statistical theory and methodology in science and engineering. Wiley, New York, 1960. D. R. Cox, Planning of experiments. Wiley, New York, 1958. J.A. Dumesic, D.F.Rudd, L. Aparicio, The microkinetics of heterogeneous catalysis, American Chemical Society, Washington, 1993. G.F. Froment, L.Hosten, Catalytic kinetics: modeling, in. Catalysis. Science and technology (ed. J.A.Anderson, M.Boudart) Springer-Verlag, Berlin, 1981. C.W. Gear, Numerical initial value problems in ordinary differential equations, Prentice Hall, Englewood Cliffs, N J, 1971. Handbook of heterogeneous catalysis, 5 Volume Set, Editors: G. Ertl, H.Kn6zinger; J. Weitkamp, Wiley-VCH, Weinheim, 1997. P.Henrici, Discrete variable methods in ordinary differential equations, Wiley, NY, 1962. A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE Solvers," in Scientific Computing, R. S. Stepleman et al. (eds.), North-Holland, Amsterdam, 1983 (vol. 1 of IMACS Transactions on Scientific Computation), pp. 55-64 J. V. Villadsen, W. E. Stewart, Solution of boundary-value problems by orthogonal collocation, Chemical Engineerin,g Science, 22 (1967) 1483 B. J. Winer, Statistical principles in experimental design. McGraw-Hill, New York, 1962.
477
Subj ect index acid-base catalysis, 28 acidity function, 153 activation energy, 23 Adams-Moulton method, 439 adsorbate-adsorbate interactions, 57 adsorption, 47 adsorption isotherms, 46 adsorption modes, 71 affinity, 22 agitation speed, 416 allosteric enzyme, 205 ammonia synthesis, 242 Anderson, 400 apparent activity, 397 apparent activation energy, 25 Aris, 378 Arrhenius, 1 Arrhenius equation, 4, 23 Arrhenius number, 383,401 asymmetric amplification, 179 asymmetric catalysis, 179 asymmetric depletion, 179 Athena visual studio, 454 autocatalysis, 161 Avetisov, 248 Avralni-Erofeev kinetics, 431 backward difference method, 296, 439 Bailey, 385 Bangham equation, 98 basic routes, 121 batch reactors, 12 Bayes, 453 bell-shape, 220 Belousov, 307 Belousov-Zhabotinsky reaction, 307, 311 Berty reactor, 21 Berzelius, 1 Bessel, 366 Biloen, 302 biographical nonuniformity, 94 Blot, 370 Biot number, 370
bisubstrate reactions, 196 Bodenstein, 81, 83 Bodenstein conditions, 116 Boltzmann, 75 Boltzmann distribution, 73 Bond, 109 Bonhoeffer, 297 Bosanquet approximation, 405 Boudart, 95,418 Box-Hill method, 453 Bragg-Williams approximation, 58 Briggs, 190 Brusselator, 310 Brfindstr6m, 358 Bronsted, 88 Burk, 192 Buss loop reactor, 20 butadiene hydrogenation, 116 Butler-Volmer equation, 276 Buzzi-Ferraris, 467 Campbell, 451 carbonylation, 169, 175 catalyst market, 6 catalytic cycle, 2 chain polymerization, 183 Changeux, 205 Chapman, 406 Chapman-Enskog equation, 406, 408 chemisorption, 45 Chilton-Colburn analogy, 403 chiral, 30 Chorkendorff, 246 Christiansen, 172 Christiansen matrix, 172 cinnamaldehyde hydrogenation, 138, 414 citral hydrogenation, 231 cofactor, 201 collision integral, 408 collocation method, 440 compensation effect, 109, 445 competitive inhibition, 212 complex reactions, 111
478
concerted hypothesis, 205 conservation equation, 112 contour plot, 442 cooperative kinetics, 202 Cornish-Bowden, 215 correlation between parameters, 443 covalent catalysis, 38 coverage, 47 CSTR, 15 cyclic voltammetry, 272 Dalziel, 197 Damk6hler, 367 Damk6hler number, 391 deactivation, 317, 397 decomposition of N20, 293 de Donder equation, 22 degree of rate control, 451 deNOx removal, 278 design equation, 421 deterministic models, 104 dialkylbenzene hydrogenation, 129 diffusion, 342 diffusion coefficient, 345,404, 409 direct hydrogenation mechanism, 165 discrimination of models, 452 dissociative adsorption, 48 Dixon plot, 214 double logarithmic plots, 431 double reciprocal plot, 193, 198, 200, 214, 217, 219 Dumesic, 451 Eadie, 193 Eadie-Hofstee plot, 193,430 effective diffusion coefficient, 362 effectiveness factor, 356, 364, 375,384 Eigen, 289 Einstein, 410 electrocatalytic kinetics, 270 elementary reactions, 10 Eley-Rideal mechanism, 86 Elovich equation, 97 empirical deactivation functions, 326 enantioimpure catalysts, 179 enantiomers, 30
enantioselective hydrogenation, 258 energy balance, 379 energy dissipation, 404, 416 enhancement factor, 353 Enskog, 406 enzymatic catalysis 8, 35 enzyme deactivation, 337 enzymes, 35 epoxidation, 167 equilibrium, 4 equilibrium constant, 47 Erofeev, 431 Ertl, 315 Esson, 1, 9 esterification, 153 experimental design, 415 exponential nonuniformity, 55 extent of reaction, 9 external heat transfer criterion, 399 external mass transfer criterion, 399 extraction mechanism, 359 Eyring, 73, 91 factorial design, 426 Farkas, 297 femtosecond spectroscopy, 80 Fick, 344 Fick law, 345,405 Field, 309 film theory, 346 Fihner, 211 Flory, 184 Fogler, 425 formal kinetics, 8 Fourier, 379 Fowler-Guggenheim isotherm, 58 Freundlich isotherm, 56 fuel cell, 271 Fuller-Schettlet-Giddins equation, 408 gas-phase catalysis, 27 gas-phase reactions, 345 Gauss, 435 Gepasi, 456 Gibbs free energy, 3 Glasstone, 91
479
Gleaves, 303 Goodwin, 302 graphical methods, 429 graphs of catalytic mechanisms, 127 Guldberg, 1 Haldane, 190 Hammett, 88, 152 Hammett equation, 88 Hanes-Woolfplot, 194, 430 Happel, 302 Harcourt, 1, 9 Hatta number, 356 Hayduk-Laudie method, 410 heat conductivity, 411 heat of adsorption, 54 heat transfer coefficient, 382 heat transfer correlations, 402 heat transfer factor, 402 Henry constant, 349 Henry law, 52 heterogeneous catalysis, 6, 7 heterogeneous catalysts, 40 heterogeneous-homogeneous reactions, 278 Heyrovsk3), 270 Higgins, 313 Hill equation, 203 Hindmarsh, 439 Hinshelwood, 86 Hofstee, 193 homogeneous catalysis, 6, 7, 149 Horiuti, 107, 111,113,270 Hougen, 107 hydroamination, 166 hydroformylation, 170, 177 hydrogenolysis kinetics, 256 hyperbolic functions, 368 H2/O2 exchange, 297 ideal surfaces, 225 immobilized enzymes, 222, 385 increments, 409 indirect hydrogenation mechanism, 165 induced fit, 211 induced nonuniformity, 57, 240 inhibition, 39, 41,212
interfacial mechanism, 360 internal diffusion, 362, 392 internal heat transport criterion, 400 internal mass transfer criterion, 400 intrinsic nonuniformity, 235 Ioffe, 392 ionic species, 263 isotope exchange, 265 Jacobian matrix, 433 Kagan, 180 kinetic coupling, 143 kinetic modeling, 419 kinetic resolution, 177 King, 65 Kiperman, 109 Knowles, 30 Knudsen diffusion, 405 Kobayashi, 298 Kolmogorov, 403 Kooij, 3 Koros, 418 Koshland, 211 K6r6s, 309 labelled atoms, 265 Laidler, 91 Langmuir, 44 Langmuir- Hinshelwood-Hougen-Watson method, 107 Langmuir-Hinshelwood mechanism, 86 Langmuir isotherm, 52 lateral interactions, 58 lattice gas, 57 law of mass action, 1 Lennard-Jones parameter, 406 Levenberg, 434 Levenberg-Marquart method, 263 Levenspiel, 318 Liebig, 2 ligand-defficient catalysis, 172 linear free energy relationship, 88 Lineweaver, 192 Lineweaver-Burk plot, 192,430 Liotta, 361
480 liquid film, 350 lock-and-key concept, 37 logarithmic isotherm, 54 logarithmic plots, 11 Lucenz III, 456 Madon, 418 Makosza, 358, 360, 361 Manca, 467 Marquart, 434 Mars-van Krevelen mechanism, 227 mass balance, 13 mass transfer, 341 mass transfer correlations, 402 mass transfer factor, 402 Maxwell, 75, 344 Maxwell-Boltzmann distribution, 74 Mears, 399 Menten, 189 metal ions, 162 metallocene, 30, 182 metathesis, 169 Metropolis, 104 Michaelis, 189 Michaelis constant, 192 Michaelis-Menten equation, 192, 195, 421 microkinetics, 107 minimum energy path, 73 ModEst 6.0, 456 molecular diffusion, 405 Monod, 205 Monte-Carlo methods, 104 most abundant surface intermediate, 148 Moulijn, 398 multi-centred adsorption, 65 multicomponent adsorption, 49 multilayer coking, 332 multiple cycles, 176 multiplicity, 384 nanokinetics, 100 Nemethy, 211 Nernst equation, 276 Newton, 435 Newton method, 262 Newton-Raphson method, 432, 434, 435
Niemantsverdriet, 246 noncompetitive inhibition, 212, 215 non-ideal surfaces, 235 non-isothermal reactors, 427 nonlinear effects, 179 nonlinear regression, 432 nonuniform surfaces, 50 normalization of data, 446 Nowak, 418 Noyes, 309 Noyori, 30 nucleophillic catalysis, 158 Nusselt, 401 Nusselt number, 401 objective function, 432 observable Thiele modulus, 388 Ollis, 385 optimum catalyst, 251 ordinary differential equations (ODE), 262, 296,423,437 Oregonator, 308 organometallic catalysts, 32, 164 oscillations, 106, 307 Ostrovskii, 332 Ostwald, 1 oxidation, heterolytic, 164 parallel adsorption, 72 parallel-consecutive reactions, 138, 177 parameter estimation, 428 partition functions, 75 partial differential equations (PDE), 296, 440 pattern formation, 314 Peclet, 300 Peclet number, 300, 301 phase transfer catalysis, 358 physico-chemical testing, 450 physisorption, 45 pinene isomerization, 232 ping-pong mechanism, 196, 198 plug flow reactor, 12 poisoning, 334 Polanyi, 73 Polanyi parameter, 51, 88,275
481 polymer electrolite membrane, 271 polymerization, 34, 183 porosity, 362 potential energy diagram, 3, 5, 44 power law, 238,429 Prandtl, 401 Prandtl number, 402 Prater, 400 Prater number, 383,401 pre-exponential factor, 80 Prigogine, 310 promoter, 40 proximity mechanism, 38 quasi-chemical approximation, 65 quasi-equilibrium, 83 radicals, 280 rate-determining step, 83 rate equation, 10 rate law, 10 reaction orders, 10 reaction route, 112 reactors, 12, 421 recirculation reactor, 20 reconstruction of surfaces, 315 regression, 431 relaxation, 285 relaxation methods, 288 Reynolds, 401 Reynolds number, 401 Rideal, 86, 297 Roginskii, 50 Rosenbrock-Wanner method, 438 Runge, 435 Runge-Kutta methods, 437 Satterfield, 407 saturation parameter, 388 Schmidt, 401 Schmidt number, 402 Schneider, 378 screening parameter, 68 selectivity, 42, 135,253,392 -differential, 43,395 -integral, 43
Sernenov, 83 semi-competitive adsorption, 66 sensitivity plots, 442 separable kinetics, 319 sequential hypothesis, 211 shape selectivity, 135 Sharpless, 30 Sherwood, 357 Sherwood number, 357,401,403 shielding, 69 sitosterol hydrogenation, 335 Snagovskii, 67, 248 Starks, 358, 359 steady-state approximation, 81 steady state isotope transient kinetic analysis (SSITKA), 292, 302 steepest descent, 434 Stefan, 344 Stefan-Maxwell theory, 405 stiff differential equations, 439 stirred taN(, 12 stochastic models, 104 stoichiometric coefficient, 9 stoichiometric number, 111 Stokes, 410 Stokes-Einstein equation, 410 stopped flow, 289 structure insensitive reactions, 42 structure sensitive reactions, 42 support, 40 surface electronic gas, 60, 102 systematic deviations, 446 Tafel, 270 Tafel equation, 276 Tafel plot, 276 Taft equation, 90 Tamaru, 290 Taylor, 44 Temkin, 50, 91,113,235,240, 403 Temkin-Horiuti rule, 128 Temkin isotherm, 54, 56 Temkin-Pyzhev equation, 242 temperature jump, 289 temperature programmed desorption (TPD), 304
482
temporal analysis of products (TAP), 303 Th6nard, 8 Thiele, 367 Thiele modulus, 356, 367, 387 three-phase systems, 411 tilted adsorption, 72, 258 tortuosity, 362 tracer, 301 transient kinetics, 286 transient techniques, 291 transition metals, 29 transition state theory, 73 transmission coefficient, 79 tubular reactor, 16 turnover number, 42 two-step sequence, 227, 237 uncompetitive inhibition, 212, 218 utilization factor, 356 van't Hoff, 3
Villadsen, 440 virtual pressure, 95 Volmer, 270 Waage, 1 Weisz, 400 Weisz-Hicks criterion, 400 Weisz modulus, 400 Wenzel, 1 Whyman, 205 Wilhelmy, 8 Wilke, 407 Wilke-Chang equation, 410 Zeldovich, 50, 367 Zeldovich-Roginskii equation, 97 Zewail, 80 Zhabotinsky, 307 Ziff-Gullari-Barshad model, 105 Zucker-Hammett equation, 152
