Catalysis

CHAPTER 2

Catalysis 2.1. 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, acid-base, and by transition metals. The most common example of gas phase catalysis is the decomposition of ozone (O3) into oxygen (O2), which is catalyzed by CFCs (chloroflourocarbons), VOC, or nitric oxide (NO). Fig. 2.1 displays a catalytic cycle containing Cl and ClO; thus, formally ozone destruction has a chlorine catalytic cycle. 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.

2.1.1 Acid-Base Catalysis General Brønsted acid catalysis begins with the addition of a proton, whilst general Brønsted 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 H3O+ (H+) species and the reaction rate is given by: r ¼ kH + ½S½H + 

(2.1)

An example is the hydration of unsaturated aldehydes: O

O H + R -C -O

R -C -OR' +H3O+

+

H 2O

R'

O

O

H + R -C -O

+

2 H2O

R' Catalytic Kinetics http://dx.doi.org/10.1016/B978-0-444-63753-6.00002-6

R -C -OH +H3O+

+R'OH © 2016 Elsevier B.V. All rights reserved.

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36

Catalytic Kinetics

Fig. 2.1 Desruction of ozone layer by CFCs. (From http://icestories.exploratorium.edu/dispatches/ the-ozone-holeits-still-there/. © Earth System Research Laboratory, National Oceanic & Atmospheric Administration, http://www.esrl.noaa.gov/).

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

O

R -C -OR' + H OH

R -C -OH +OH– +R'OH

OH–

and the reaction rate is then expressed: r ¼ kOH ½S½OH 

(2.2)

For a reaction S ) P, which is catalyzed by both acid and bases: S + H + ) P + H + ; S + OH ) P + OH the rate follows the expressions: r ¼ kO ½S + kH + ½S½H +  + kOH ½S½OH  ¼ k0 ½S

(2.3)

Catalysis

log10k

lgk¢ = lgkO

pH

lgk¢ = lgkOH-+lgKW+pH

Fig. 2.2 Dependence of the rate constant on pH.

where for water solutions: k0 ¼ kO + kH + ½H +  + kOH KW =½H + 

(2.4)

as [OH] ffi KW/[H+] with KW ¼ 1014 mol2 dm6 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 (Fig. 2.2). Acid catalyzed reactions involve formation of π complexes: H+

+H

and carbonium ions: H +H

which react further and the proton is fully recovered. C

C

+ X–

+ H–R

C

C

X

C

H + R

+

C

C

+

+ O

R H

C O

C

C

R H

2.1.2 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

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Catalytic Kinetics

M(CH3)2

M(CH3)2

M(CH3)2

Fig. 2.3 Metallocene catalysts.

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: RhCl(PPh3)3 for olefins hydrogenation, Co2(CO)8 for carbonylation, and metallocenes for polymerization. Industrial applications are toluene and xylene oxidation to acids, oxidation of ethene to aldehyde, carbonylation of methanol and methyl acetate, polymerization over metallocenes (Fig. 2.3), hydroformylation of alkenes, etc. A fascinating 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.

Ryoji Noyori

William S. Knowles

K. Barry Sharpless

Catalysis

Fig. 2.4 Chiral enantiomers. (From https://upload.wikimedia.org/wikipedia/commons/thumb/e/e8/ Chirality_with_hands.svg/600px-Chirality_with_hands.svg.png).

Many of the compounds associated with living organisms are chiral (not superimposable on its mirror image); for example DNA, enzymes, antibodies, and hormones. Therefore, enantiomers, eg, pairs of optical isomers (Fig. 2.4) may have distinctly different biological activity. For many drugs, only one of these enantiomers has a beneficial effect, and the other enantiomer can be, in the best case, inactive, or even toxic. In 1968, Knowles at Monsanto Company 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. The aim of Knowles was to develop an industrial synthesis process for the rare amino acid L-DOPA, which had proved useful in the treatment of Parkinson’s disease. Knowles and co-workers at Monsanto discovered that a cationic rhodium complex containing DiPAMP (Fig. 2.5A), a chelating diphosphine with

(A)

(B) Fig. 2.5 (A) DiPAMP and (B) Synthesis of L-DOPA.

39

40

Catalytic Kinetics

two chiral phophorus atoms, catalyzes highly enantioselective hydrogenations of enamides, an important step in L-DOPA synthesis (Fig. 2.5B). Noyori discovered a chiral diphosphine complex, BINAP. Rh(I) complexes of the enantiomers of BINAP (Fig. 2.6) are remarkably effective in various kinds of hydrogenation reactions. Using titanium(IV) tetraisopropoxide, tert-butyl hydroperoxide and an enantiomerically pure dialkyl tartrate, the Sharpless reaction (Fig. 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 environment.

Fig. 2.6 Structure and application of BINAP complexes.

Fig. 2.7 Sharpless epoxidation. (From https://upload.wikimedia.org/wikipedia/commons/3/3c/Sharpless_ epoxidation_DE.svg).

Catalysis

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 (Fig. 2.8). Different types of elementary steps are possible with organometallic catalysts (Fig. 2.9). Catalytic cycles 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 Figs. 2.10 and 2.11, which could be presented in a general form for a reaction A ) P (Fig. 2.12).

Fig. 2.8 Formation of chemical bonds during catalysis.

Fig. 2.9 Elementary steps in organometallic catalysis.

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42

Catalytic Kinetics

Fig. 2.10 Mechanism for double-bond migration with an organometallic catalyst.

CH3–CH3 H2

H

L M

L

CH2 CH3

H

L M

L

H

C2H4

Fig. 2.11 Mechanism of a hydrogenation reaction with an organometallic catalyst.

A M P

Fig. 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: M

A

M’

P

and requires further transformations of palladium in the zero oxidation state to Pd2+.

Catalysis

PdCl 4 + C2 H4 ! PdCl3 ðC2 H4 Þ + Cl 2

PdCl3 ðC2 H4 Þ + H2 O ! PdCl2 ðC2 H4 ÞðH2 OÞ + Cl PdCl2 ðC2 H4 ÞðH2 OÞ ! PdCl2 ðC2 H4 ÞðOHÞ + H + PdCl2 ðC2 H4 ÞðOHÞ ! PdCl2 ðCH2 CH2 OHÞ Cl2 Pd  CH2 CH2 OH ! Pdo + 2Cl + CH3 CHO + H + C2 H4 + PdCl2 + H2 O ! CH3 CHO + Pdo + 2HCl This can be done, for example, by coupling ethylene hydrolysis with palladium reduction: C2 H4 + PdCl2 + H2 O ) CH3 CHO + 2HCl + Pd Pd + 2CuCl2 ) PdCl2 + 2CuCl 1 2Cu + + O2 + 2H + ) 2Cu2 + + H2 O 2 finally leading to formation of acetaldehyde from ethylene C2 H4 + 1=2O2 ) CH3 CHO, thus closing the catalytic cycle. M

A

M⬘

P

In industrial practice, the process is organized in such a way that reaction and reoxidation of Cu can be performed in one reactor or, alternatively, in two reactors. Several mechanisms for homogeneous catalytic reactions by transition metal complexes are rather complicated and consist of several steps. One example is polymerization reactions (Fig. 2.13). Another example is olefin metathesis, which allows the exchange of substituents between different olefins. R1

R2

R1

R2

+ R1

R2

cat.

R1

R1

R2

R2

R2

R2

+ R1

R2

R1

R1

This reaction was first used for the synthesis of higher olefins. The mechanism of metathesis proposed by Chauvin (Fig. 2.14) includes reaction of the metal methylene (metal alkylidene) with the olefin, forming a metallocyclobutane intermediate. This intermediate then cleaves, yielding ethylene and a new metal alkylidene. The ethylene formed contains one methylene from the catalyst and one from the starting olefin. The new metal alkylidene contains the metal with its ligands (indicated by the brackets around the metal) and an

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Catalytic Kinetics

Fig. 2.13 Catalytic cycles for polymerization reactions. M ¼ Ti, Zr, Cr, V; L ¼ PR3.

Fig. 2.14 Catalytic cycles for metathesis.

alkylidene from the substrate alkene. This metal alkylidene reacts with a new substrate alkene molecule to yield another metallocyclobutane intermediate. On decomposition in the forward direction, this intermediate yields the product internal alkene and metal methylene. Each step in the catalytic cycle involves exchange of alkylidenes. Schrock developed a whole family of molybdenum- and tungsten-alkylidene complexes of the general formula ½Mð¼ CHMe2 PhÞð¼ N  ArÞðOR2 Þ, with R being bulky groups, such as:

i-Pr

i-Pr

F3C F3C F3C F3C

N Mo

O O

Ph CH3 CH3

Catalysis

These compounds are at present the most active of the alkene metathesis catalysts known. Catalysts developed by Grubbs, such as the second-generation Grubbs catalysts, tolerate a wide variety of functional groups being very stable.

N Cl

..

N

Ru

Cl

H Ph

PCy3

In 2005, the Nobel Prize in chemistry was awarded jointly to Yves Chauvin, Robert H. Grubbs, and Richard R. Schrock for the development of the metathesis method.

Yves Chauvin

Robert H. Grubbs

Richard R. Schrock

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, and 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 compared with heterogeneous catalysis. One of the reasons could be that the generally adopted reaction schemes sometimes look too complicated for nonspecialists in kinetics, as analytical expressions could be very cumbersome to derive. A recent trend in application of organometallic catalysts is related to heterogenization of metal complexes onto inorganic supports (Fig. 2.15).

2.1.3 Biocatalysis Biocatalysis is related to catalysis by enzymes, which are proteins functioning as biological catalysts and mediating a vast array of biochemical reactions. Most reactions in a human body

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Catalytic Kinetics

M

O

M Silica

O O

Si

HOH2C

SO3–

O

O

HO O

OPh O

Ph2P PPh2 Rh (solvent)n

CO2Me NHCOCH3

cat. H2

*

CO2Me NHCOCH3

Fig. 2.15 Immobilization of organometallic catalysts. (From C. Li, Chiral synthesis on catalysts immobilized in microporous and mesoporous materials, Catal. Rev. 46 (2004) 419–492. Copyright Taylor & Francis, 2004).

are too slow to sustain life without a catalyst; for instance, the digesting (hydrolyzing) of food, the oxidation of glucose, making ATP (useable form of energy for cells), the synthesis cholesterol for membranes and copying DNA for cell division to name a few. Biological catalysts have been postulated since the early 1800s. 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 catalyses hydrolysis of urea, was crystallized in 1926. Comparison between chemical and enzymatic catalysis demonstrates the specificity of enzymatic catalysis:

Chemocatalysts

Enzymes

A variety of inorganic, organometallic substances Increase the rate of chemical reactions One catalyst can facilitate multiple reactions Could require high temperature/pressure

Mostly proteins, few are RNA Rate increases of 106–1012 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. In terms of elementary reactions, there are no substantial differences between enzymatic catalysis and other types of catalysis as the same type of chemical reactions such as breaking, forming, 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.

Catalysis

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 and hydrophilic/hydrophobic interactions at the same time, make enzymes 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 cosubstrates 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), as well as 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. The classification according to the Enzyme Commission emphasizes the ability of the enzyme to catalyze a process in one direction; however, as with other catalysts, enzymes also catalyze the reverse reactions. There are six major classes (with subclasses). Oxidoreductases catalyze oxidationreduction reactions, the transfer of hydrogen atoms and electrons, for example dehydrogenation of lactate: O

O–

O

Lactate dehydrogenase

C C O + NADH + H+

O– C CH + NAD+

HO

CH3

CH3

Pyruvate

L-lactate

This class of enzyme requires a cosubstrate. Transferases catalyze the transfer of functional groups from donors to acceptors: O C

O

C

C O

H2N-CH CH2 CH2 C

C

+ –

O O Glutamate

O–

C O CH3 Pyruvate

Alanine aminotransferase

O C

O–

H2N CH CH3 L-alanine

O–

C O

+

CH2 CH2 C

–

O O α-Ketoglutarate

Such group transfer reactions require two substrates with one of them being activated. Another class of two substrate reactions is catalyzed by hydrolases, which allow cleavage by the addition of water (hydrolysis). Nonhydrolytic cleavage of CdC or CdO, is catalyzed by lyases. According to their name, isomerases promote one-substrate isomerization reactions, eg, the transfer of functional groups within the same molecule.

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Catalytic Kinetics

O

H

CH2 OH

C HC OH HO -CH HC - OH HC - OH C

Phosphoglucose isomerase

C O HO - CH HC - OH HC - OH C

OPO32–

O Glucose 6-phosphate

O

OPO32–

Fructose 6-phosphate

Ligases use ATP to catalyze the formation of new covalent bonds, ie, CdC, CdS, CdO, and CdN bonds. Enzymes differ from ordinary catalysts, as the rates are typically 106 to 1012 times faster. Such higher rates are achieved at milder reaction conditions, eg, 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 (eg, modifiers and inhibitors). It should be noted that enzymes can be also used as catalysts for reactions with nonnatural substances in aqueous suspensions and in organic solvents as well as in ionic liquids. 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. Enzymes have a defined amino acid sequence, being typically 100–500 amino acids long, and a defined three-dimensional structure. The specific order of amino acids in the protein (the primary structure) is encoded by the DNA sequence of the corresponding gene (Fig. 2.16). Because of a possibility of hydrogen bond formation due to presence of the amide group and carboxyl groups, the protein chain can fold up on itself in two ways, resulting in two secondary structures: α-helix or β-sheet (Fig. 2.17). As a consequence of the folding-up of the 2D linear chain in the secondary structure, the protein can fold up further resulting in the tertiary structure (Fig. 2.18). 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. Due to the complex enzyme structure, the active site components are generally far apart in the linear amino acid sequence.

Catalysis

Primary protein structure is sequence of a chain of amino acids

Amino acids

Phe Leu Ser

Amino group NH2 H

C

Cys

COOH

Acidic R carboxyl R group group

Amino acid

Fig. 2.16 Primary structure of enzymes. (From http://en.wikiversity.org/wiki/Enzyme_structure_and_ function).

Fig. 2.17 Secondary structure of proteins (A) α-helix and (B) β-sheet. (From http://en.wikiversity.org/ wiki/Enzyme_structure_and_function).

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Catalytic Kinetics

Fig. 2.18 Tertiary structure of a mouse major urinary protein. (From http://en.wikipedia.org/wiki/Major_ urinary_proteins).

The substrate can bind to the enzyme at the active site via noncovalent interactions (eg, 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 of the size of the substrate. Complementarity in the structure and charges between the enzyme and the substrate is illustrated by the so-called lock-and-key concept (Fig. 2.19). Moreover, the active site can be pre-formed, but still undergoes some conformational changes upon substrate binding via so-called induced fit (Fig. 2.20).

Substrate

Enzyme-substrate complex

Active site

Enzyme

Fig. 2.19 Lock-and-key concept. (From https://upload.wikimedia.org/wikipedia/commons/9/9f/Lock_ and_key.png).

Catalysis

Substrate

Enzyme-substrate complex

Active site

Enzyme

Fig. 2.20 Induced fit concept. (From https://upload.wikimedia.org/wikipedia/commons/a/ab/Inducedfit080. png).

As with any other catalytic reaction, enzymes cannot change the overall thermodynamics of a reaction (ie, ΔG) and cannot make an unfavorable reaction favorable, which means that the 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 diminished. Binding energy can thus be used for the rate enhancement. Enzymes make both the forward and reverse reaction rates faster, meaning that either going forward or backward, the energy barriers are lower. Enzyme catalyzed mechanisms represent reactions fundamentally familiar from organic chemistry (Fig. 2.21). Acid-base catalysis is associated with the donation or subtraction of protons. Acid catalysis is a process in which a 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. Concerted acid-base catalysis, where both processes occur simultaneously, is a common enzymatic mechanism. 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 (Fe2+, Fe3+, Cu2+, Zn2+, Mn2+, or Co2+) 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 ΔG{ is the change in free energy between

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Catalytic Kinetics

Fig. 2.21 Illustration of the enzymatic catalysis. (From https://upload.wikimedia.org/wikipedia/commons/ 2/2d/Catserpcycle_v2.png).

the ground state and the transition state, then ΔG{ ¼ ΔH{  TΔS{. 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 (Fig. 2.22). However, if the energy is lowered too much, without a greater lowering of the corresponding activation energy, then catalysis would not take place. 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 transition state, the higher is the rate of the catalyzed reaction relative to the uncatalyzed reaction.

Free energy, (G)

Catalysis

+

ΔG+uncat

ΔGB +

ΔG+cat S

ES EP P

Reaction coordinate

Fig. 2.22 Energy profiles for a noncatalytic reaction and an enzymatic reaction. (From http://cbc.arizona. edu/classes/bioc462/462a/NOTES/ENZYMES/Fig8_6BindingEnergyCatalys.GIF. Department of Biochemistry and Molecular Biophysics, The University of Arizona. Copyright © 1998–2003).

Enzymes are chiral; therefore, their interactions with chiral molecules create diastereomeric relationships and allow them to distinguish between enantiomers. For example, if a racemic mixture of a certain molecule (drug, for example) is bound to an enzyme, a tight binding will be formed only in the case of one enantiomer (Fig. 2.23) Consequently, in a stereoselective reaction, one stereoisomer is formed preferentially over other possible stereoisomers, as illustrated in Fig. 2.24, or enantiomers can be separated as shown in Fig. 2.25 for acylation of 1-phenyl-ethanol with vinyl acetate using lipase as enzyme in the so-called kinetic resolution approach. Enantiomeric excess is often applied as a measure of reaction stereospecificity: ee ¼

R enantiomer

Binding site of the receptor

½R  ½S ½R + ½S

(2.5)

S enantiomer

Binding site of the receptor

Fig. 2.23 Binding of enantiomers to enzymes. (From http://www.kshitij-iitjee.com/Study/Chemistry/ Part2/Chapter3/109.jpg).

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Catalytic Kinetics

Fig. 2.24 Enantioselective synthesis of one stereoisomer with enzymes.

Fig. 2.25 Acylation of 1-phenyl-ethanol with vinyl acetate using lipase.

where R and S correspond to concsentration of enanatiomers. Enantiomeric ratio E ¼ R/S is another metric often applied, while in fact the ratio of rates for formation of enantiomers should be applied, rather than the ratio of concentrations: rS E¼ (2.6) rR E should be respectively above 100 or below 0.01 for S- and R-specific enzymes to ensure high steric purity. Kinetic resolution, even if allowing high stereospecificity, per se, can provide maximally the yield of 50%, thus to improve the yield dynamic kinetic resolution (Fig. 2.26) should be applied, which includes also racemization. Another 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

Fig. 2.26 Dynamic kinetic resolution of secondary alcohols.

Catalysis

Fig. 2.27 Competitive and noncompetitive inhibition. (From https://biochemanics.files.wordpress.com/ 2013/04/competitive_inhibit_c_la_784.jpg).

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 (Fig. 2.27). In the case of competitive inhibition, they bind to the active site and block access by the normal substrate. Some inhibitors can bind more strongly than the substrate, leading to irreversible inhibition. Inhibitors can also change the conformation of the enzyme in way that it does not bind to the substrate (Fig. 2.27). In order to reduce the costs of processes with enzymes, they should be used in a reusable form, which can be achieved by immobilization on porous supports. This approach results in introduction of mass transfer limitations and in this sense allows treating immobilized enzymes as conventional heterogeneous catalysts, which will be considered in the subsequent section.

2.2. HETEROGENEOUS CATALYSIS 2.2.1 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 (Table 2.1). 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 (Fig. 2.28). Components of the catalyst include the active agent itself and may also include a support, a promoter, and an inhibitor.

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Catalytic Kinetics

Table 2.1 Some catalytic processes for synthesis of basic chemicals Process or product Catalyst

Conditions

Hydrogenation

Methanol synthesis CO + 2H2 ! CH3OH Fat hardening Benzene to cyclohexane

ZnO-Cr2O3 CuO-ZnO-Cr2O3 Ni/Cu Raney Ni

Ni, Cu, Pt CuCr2O4 Co or Ni on Al2O3

250–400°C, 200–300 bar 230–280°C, 60 bar 150–200°C, 5–15 bar Liquid phase 200–225°C, 50 bar gas phase 400°C, 25– 30 bar 100–150°C, bis 30 bar 250–300°C, 250–500 bar 100–200°C, 200–400 bar

Fe3O4 (Cr, K oxide) Cr2O3/Al2O3

500–600°C, 1.4 bar 500–600°C, 1 bar

Ag/support Ag cryst. V2O5/support

200–250°C, 10–22 bar c.600°C 400–500°C, 1–2 bar

V2O5/TiO2 V2O5-K2S2O7/SiO2 Bi/Mo oxides

400–450°C, 1.2 bar

Bi molybdate (U, Sb oxides) Pt/Rh nets

400–450°C, 10–30 bar

CuCl2/Al2O3

200–240°C, 2–5 bar

H3PO4/SiO2 Al2O3/SiO2 or H3PO4/SiO2

300°C, 40–60 bar 300°C, 40–60 bar

noble metals Aldehydes and ketones to alcohols Esters to alcohols Nitriles to amines Dehydrogenation

Ethylbenzene to styrene Butane to butadiene Oxidation

Ethylene to ethylene oxide Methanol to formaldehyde Benzene or butane to maleic anhydride o-Xylene or naphthalene to phthalic anhydride Propene to acrolein

350–450°C, 1.5 bar

Ammoxidation

Propene to acrylonitrile Methane to HCN

800–1700°C, 1 bar

Oxychlorination

Vinyl chloride from ethylene + HCl/O2 Alkylation

Cumene from benzene and propene Ethylbenzene from benzene and ethylene

From J. Hagen, Industrial Catalysis, Wiley VCH, Weinheim, 2006.

Catalysis

Fig. 2.28 Classification of catalytic materials.

For instance, metal catalysts are not in their bulk form, but they are generally dispersed on a high surface area insulator (support), such as Al2O3 or SiO2. Sometimes, both the metal and the support act as a catalyst, which is referred to as bi-functional catalysts. An example is platinum dispersed on alumina used in 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 the activity, selectivity or stability in order to prolong the catalyst’s life. The promoter is often added in small amounts and by itself has little activity. There are various types of promoters, depending on how they improve the catalyst. Textural promoters are inert substances, 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 relatively high melting point. Structural promoters change the chemical composition, produce lattice defects, alter the electronic structure and the chemisorption strength. An inhibitor is the opposite of a promoter. When added in small amounts, it diminishes activity, selectivity or stability. Inhibitors are useful for reducing the activity of a catalyst for an undesirable side reaction. For instance, silver supported on alumina is an excellent oxidation catalyst, widely used in the production of ethylene oxide from ethylene. However, at the same conditions, complete oxidation to carbon dioxide and water also occurs, decreasing selectivity to ethylene oxide. Addition of halogen compounds (eg, dichloroethane) to the catalyst inhibits the complete oxidation and results in much better selectivity.

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Catalytic Kinetics

Metal oxides usually consist of bulk oxides. As semiconductors, metal oxides catalyze the same kind of reactions as metals but in processes requiring higher temperatures. Often a mixture of various oxides is applied to increase the catalytic activity. For example, transition metals, such as MoO3 and Cr2O3, are good catalysts for polymerization of olefins; a mixture of copper and chromium oxides, named copper chromite, is used for hydrogenation and a mixture of iron and molybdenum oxide (ie, iron molybdate), is used for formaldehyde formation from methanol. Many heterogeneous catalysts require application of supports. This is a means of spreading expensive materials and providing the necessary mechanical strength, heat sink/source, optimization of bulk density and dilution of an overactive phase. There are also geometric (eg, increase of surface area, optimization of porosity, crystal and particle size) and chemical functions (eg, improvement of activity, minimization of sintering, and poisoning, as well as beneficial effect of spillover) provided by the supports.

2.2.2 Kinetics in Heterogeneous Catalysis A physico-chemical understanding of catalytic processes is essential for proper reactor modeling. According to the very definition of catalysis, it is a kinetic process and thus reliable kinetic models that describe the rates of catalytic reactions are of vital importance for solving applied problems in mathematical modeling, 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, ie, 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 a reaction on a molecular level. Reaction kinetics is in this sense 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 reflecting ability of a catalyst to affect the rate of a thermodynamically feasible reaction. For example, at a temperature of 200–400°C, a mixture of CO and O2 can react completely over Pt/SiO2 catalyst, while in the presence of SiO2, there would be no reaction.

Catalysis

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 turnover rate (frequency). The turnover frequency (TOF) is defined as the number of molecules reacted per site per unit time and depends on the kinetics (concentration of reagents, T). TOF ¼

Amount reacted mol ¼ time mass of catalyst ðactive siteÞ gðmol Þs *

(2.7)

The TOF is a useful concept when a comparison of catalysts is made at the same conditions; however, it should be used with caution for several reasons. First, it is difficult to determine the true number of active sites even if the situation is somewhat easier for metals, as a chemisorption technique could be used to measure the exposed surface area. In addition, TOF is not an intrinsic characteristic of a catalyst, since it depends on reaction conditions and can even change during an experiment when the reaction rate is changing as a function of concentration. There is also some confusion between different branches of catalysis in a way TOF (or similar concepts) are defined. In enzymatic catalysis, the turnover rate is referred to as a turnover number (TON), defined as the maximum number of molecules of substrate that an enzyme can convert to product per catalytic site per unit of time. The concept of TON is also applied in homogeneous (organometallic) catalysis with a different meaning, defining the number of moles of substrate that a mole of catalyst can convert before becoming inactivated. It is apparently clear that TON (dimensionless) can be infinite if there is no deactivation. The concept of TOF in organometallic catalysis is used to refer to the turnover per unit time. In order to avoid misunderstanding related to TOF it might be suggested to report instead the values of reaction rates and reaction constants. For some heterogeneous catalytic reactions (eg, so-called structure insensitive reactions on metals), the rate is independent of size, shape, and 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’s lifetime 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 the ability to direct conversion of the reactant(s) along one specific path 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,

59

60

Catalytic Kinetics

conversion and nature of the catalyst. Therefore, selectivity should be referred to under specific conditions. Integral selectivity is defined as the ratio of the desired product per consumed reactant and differential selectivity is the ratio of the rate of desired product formation to the rate of the reactant consumption. Another important issue besides activity and selectivity is the catalyst’s stability. A catalyst with good stability will change very slowly over the course of time under the conditions of use. Indeed, it is only in theory that the catalyst remains unaltered during the reaction. Actual practice is far from this ideal, as the progressive loss of activity could be associated with coke formation, attack of poisons, loss of volatile agents, or changes of crystalline structure, which cause a loss of mechanical strength. Due to the extreme importance of catalyst deactivation, the kinetic aspects of this phenomenon will be treated in a separate chapter.

2.2.3 Elementary Steps Heterogeneous catalysts in industrial applications have a variety of different geometrical shapes and porosity. Therefore, an understanding of mass and heat transfer in heterogeneous catalysis is essential for the description of the processes involved in getting the reactants to the catalyst surface and removal or addition of energy from the catalyst particles. The following steps are usually considered in heterogeneous catalysis: 1. External diffusion: Transfer of the reactants from the bulk fluid phase to the fluid-solid interface and external surface of the catalyst particle. 2. Internal diffusion (if particle is porous): Transfer of the reactants into the catalyst particle. 3. Adsorption: Physisorption and chemisorption of the reactants on surface sites of the catalyst particle. 4. Surface reaction: Chemical reaction of adsorbed species to produce adsorbed products; this is the intrinsic or true chemical reaction step. This step can be preceded by formation of precursors on the surface and surface diffusion of them as well as other adsorbed species to the reaction sites. 5. Desorption: Release of adsorbed products by the catalyst. This is the opposite of adsorption (Fig. 2.29). 6. Internal diffusion: Transfer of products to outer surface of the catalyst particle. 7. External diffusion: Transfer of products from the fluid-solid interface into the bulkfluid stream. The potential energy diagram (Fig. 2.30) demonstrates the importance of adsorption in heterogeneous catalysis, which could be related to some extent to the concept of catalysis by approximation in enzymatic catalysis. In the similar way as with enzymatic catalysis, the solid catalyst does not affect the thermodynamics of the reaction. Therefore, first the reaction conditions (temperature,

Catalysis

O2 CO2

CO Precursor Adsorption

Desorption

Surface reaction Surface diffusion

Fig. 2.29 Adsorption, surface reaction and desorption steps.

Potential energy

Non-catalytic Energy

Reaction

Adsorption

Desorption

Fig. 2.30 Potential energy diagrams in heterogeneous catalytic processes. (From http://spaceflight.esa. int/impress/text/education/Images/Catalysis/Image_001.png).

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 can be initiated to find a suitably active and selective material. The first kinetic approach in catalysis based on a physical chemical understanding is probably due to I. Langmuir, the Nobel Prize in Chemistry winner in 1932, who applied the mass action law to reactions on solid surfaces, ie, supposing that the rate of an elementary reaction is proportional to the surface concentration (coverage) of reactive species adsorbed on the surface.

61

62

Catalytic Kinetics

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 surface 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. Adsorption on heterogeneous surfaces assuming ideal adsorbed layers will be considered first. In case of chemisorption, a chemical bond is formed between molecules and the surface (Table 2.2). 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

Table 2.2 Comparison between physisorption and chemisorption Properties Chemisorption Physisorption

Adsorption temperature Adsorption enthalpy Crystallographic specificity Nature of adsorption Saturation Adsorption kinetics

Virtually unlimited range Wide range(40–800 kJ/mol) Marked difference between crystal planes Often dissociative and irreversible in many cases Limited to a monolayer Activated process

Near or below Tbp of adsorbate (Xe < 100 K, CO2 < 200 K) Heat of liquifaction (5–40 kJ/mol) Independent of surface geometry Nondissociative and reversible Multi-layer occurs often Fast, nonactivated process

Catalysis

energy is 30–70 kJ/mol for molecules and 100–400 kJ/mol 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. For chemisorption, the potential energy curve is dominated by a much deeper chemisorption minimum at shorter values of d, which is the distance between the surface and adsorbed species (Fig. 2.31). The structure of solid surfaces is fairly complicated (Fig. 2.32) and the adsorbed species can form different types of complexes on the surfaces (some examples are given in Fig. 2.33). 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.

E(d)

d (nm) Physisorption Chemisorption

Fig. 2.31 Potential energy curves for chemisorption and physisorption.

Fig. 2.32 Structure of solid surfaces.

63

64

Catalytic Kinetics

Fig. 2.33 Examples of the complexes, which could be formed on the solid surfaces (M stands for a metal site).

2.2.4 Adsorption Isotherms—Ideal Surfaces 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

(2.8)

The desorption rate can be expressed by: r ¼ 

dθA ¼ k θ A dt

(2.9)

where θA is the coverage (surface concentration), while the adsorption rate is given by: r+ ¼

dθA ¼ k + PA θ0 dt

(2.10)

where θ0 is the coverage of vacant (empty) sites. Taking into account the balance equation, which relates occupied and vacant sites: θA + θ0 ¼ 1

(2.11)

dθA ¼ k + PA ð 1  θ A Þ dt

(2.12)

the adsorption rate is expressed as:

In case of equilibrium the rate of the forward reaction is equal to the rate of reverse one: k + PA ð1  θA Þ ¼ k θA

(2.13)

Catalysis

leading finally to an expression for the coverage of A at equilibrium: θA ¼

k + PA KPA ¼ k + k + PA 1 + KPA

(2.14)

where the equilibrium constant is the ratio between adsorption and desorption coefficients: K ¼ k + =k

(2.15)

Plotting the surface coverage as a function of partial pressure (Fig. 2.34) according to Eq. (2.14), it becomes clear that two limiting regions exist. At low values of partial pressure, the surface coverage is expressed θA  KPA being linearly proportional to pressure, while at high pressures, the coverage is approaching unity. The value of K is increased by a decrease in temperature, as the adsorption is predominantly exothermic, and by an increase in the strength of adsorption. Dissociative adsorption requires several sites on the catalyst surface. For a diatomic molecule (H2, O2, N2, etc.) adsorption occurs on two adjacent sites: A2 + 2Z $ 2AZ

(2.16)

Adsorption and desorption rates are expressed as: r + ¼ k + PA θ20 , r ¼ k θ2A

(2.17)

The balance equation is the same as for associative adsorption, thus at equilibrium: k + PA ð1  θA Þ2 ¼ k θ2A

(2.18)

Fig. 2.34 Surface coverage as a function of partial pressure at different values of the equilibrium constant.

65

66

Catalytic Kinetics

giving an equation for the coverage at equilibrium for dissociative adsorption: pffiffiffiffiffiffiffiffiffi KPA pffiffiffiffiffiffiffiffiffi θA ¼ 1 + KPA

(2.19)

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, equilibrium constants for the following reactions: A + Z $ AZ, B + Z $ BZ

(2.20)

could be directly written: KA ¼

θA θB ; KB ¼ PA θ 0 PB θ0

(2.21)

resulting in: θA ¼ KA PA θ0 ; θB ¼ KB PB θ0

(2.22)

The balance equation for the surface sites requires: 1 ¼ θA + θB + θ0 ¼ θ0 ð1 + KA PA + KB PB Þ

(2.23)

leading to the expressions for surface coverage: θA ¼

KA PA KB P B ; θB ¼ 1 + KA PA + KB PB 1 + KA PA + KB PB

(2.24)

In a more general case for multi-component adsorption analogously to the derivation above, it can be directly written: θA ¼

KA P A KD PD X X ,…, θD ¼ ,… 1+ Ki Pi 1+ Ki Pi i

(2.25)

i

When adsorption requires several sites on the catalyst surface: Aj + jZ , jAZ

(2.26)

similarly to dissociative adsorption of diatomic molecules, the surface coverage takes the following form: θA ¼

ðKA PA Þ1=j 1 + ðKA PA Þ1=j

(2.27)

Adsorption on ideal surfaces from a mixture of two molecules A2 and B, one of which dissociates, can be easily written: pffiffiffiffiffiffiffiffiffiffiffiffi KB PB KA PA pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ; θB ¼ (2.28) θA ¼ 1 + K A P A + KB P B 1 + KA PA + KB PB

Catalysis

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 gas-phase noncatalytic reactions. In the enzymatic, homogeneous and three-phase heterogeneous catalytic reactions, concentrations in the liquid-phase are used, while for heterogeneous gas-phase reactions it is more convenient to operate with partial pressures.

2.2.5 Adsorption Isotherms—Real Surfaces Although the Langmuir theory of adsorption is used frequently for technical process development, it is a crude approximation. Adsorbed molecules can 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 the spatial arrangement and the heat of adsorption depends on coverage (Fig. 2.35). 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

60

Differential heat of adsorbed CO (kJ/mol)

55 50 45 40 35 30 25 20 15 10 5 0 0,0000 0,0005 0,0010 0,0015 0,0020 0,0025 0,0030 0,0035 0,0040 Adsorbed amount of CO (mmol/g)

Fig. 2.35 Differential heats of CO adsorption on Pd/CNF catalyst. (From S. Sahin, P. Mäki-Arvela, J.P. Tessonier, A. Villa, S. Reiche, S. Wrabetz, D.S. Su, R. Schlögl, T. Salmi, D.Yu. Murzin, Palladium catalysts supported on N-functionalized hollow vapor-grown carbon nanofibers: the effect of the basic support and catalyst reduction temperature, Appl. Catal. A: Gen. 408 (2011) 137–147. Copyright 2011 Elsevier).

67

68

Catalytic Kinetics

and the second one is defined as induced nonuniformity. On biographically 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 neighboring site, or discrete. However, if in an elementary surface reaction only one adsorbed particle is involved, the difference in these distributions cannot be observed. The physical nature of the biographical (intrinsic or a priori) nonuniformity, 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 Freundlich, Zeldovich, and Roginskii to this field were also essential.

Yakov Zeldovich

Simon Roginskii

Herbert Freundlich

Mikhail Temkin

Catalysis

In the special model of nonuniformity, several assumptions were made. The Gibbs activation energy ΔG6¼ is considered to be a linear function of the standard Gibbs energy of adsorbed species ΔG°a   (2.29) ΔG6¼ ¼ α ΔGa° + const, where the value of transfer coefficient α (Polanyi parameter) is 0  α  1. Often, the value of the transfer coefficients is 0.5. The relationship between kinetics and thermodynamics exemplified by this parameter α is a general feature of chemical reactions and will be discussed in Section 3.4. The model assumes that each site is characterized by a definite adsorption energy of a particular substance and there is a certain distribution of the total number of sites with this energy. The values of adsorption energy on different sites lie between a certain minimum     and a certain maximum value, ΔGa° min  ΔGa°  ΔGa° max . The number of sites dl     with standard Gibbs energy of adsorption within (ΔG°a) and ΔGa° + d ΔGa° is given by an exponential dependence: ΔGa°   dl ¼ C e RΘ d ΔGa°

(2.30)

where С and Θ are constants. The values of Θ could be positive, negative or infinity, reflecting cases of Freundlich, Zeldovich-Roginski, and Temkin isotherms, respectively. In case when Θ ! 1 instead of (2.30) the following distribution is valid   (2.31) dl ¼ Cd ΔGa° Such 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: aρ Ω¼ (2.32) 1 + aρ where ρ is the pressure corresponding to adsorption equilibrium and a is the adsorption coefficient. Defining ξ ¼ ln b ¼  ln a, where b is the desorption coefficient, leads to ξ ¼ ΔGa° =RT , as ΔGa° ¼ RT ln a. Numbering the sites in the sequence of increasing ξ (decreasing adsorption strength) and defining the relative number of site s ¼ 1/L, where L is the total number of sites (0 < s < 1), the distribution of sites function φ(ξ)dξ, which is the number of sites with ΔG°a/RT within ξ + dξ can be defined: ξ < ξo

φðξÞ ¼ 0

ξo  ξ  ξ1 φðξÞ ¼ A exp ðγξÞ ξ > ξo

φðξÞ ¼ 0

(2.33)

69

70

Catalytic Kinetics

where ξo is lowest value of ξ and ξ1 is the highest value of ξ, A is a constant, γ ¼ T =Θ and for even distribution γ ¼ 0, eg φðξÞ ¼ A. The value of A is determined by the normalized condition: 1 ð φðξÞdξ ¼ L (2.34) 1

where L is the total number of sites on the unit surface. Taking into account the limiting values of the logarithm of desorption coefficient instead of Eq. (2.34), the normalization condition is: ξð1

φðξÞdξ ¼ L

(2.35)

ξ

Finally, the expressions for the constant A are: A¼

eγξ1

γL  eγξ0

(2.36)

if γ 6¼ 0 and for evenly nonuniform surfaces: A¼

L ξ1  ξ0

(2.37)

A

x1

x0

For discrete nonuniformity, the total surface coverage is determined by summation of equilibrium adsorption isotherms of Langmuir type: X as p θ¼ (2.38) 1 + as p s while for continuous nonuniformity, when all values in a range of adsorption energies are present, integration gives the surface coverage: ð1 θ¼ 0

as p ds 1 + as p

(2.39)

Catalysis

When the pressure is small, then a0p ≪ 1, where a0 is the highest value of a (eg at s ¼ 0). Such conditions are similar to uniform surfaces at low pressures and the surface coverage is proportional to pressure, obeying Henry’s law. Then: ð1 θ ¼ p ads

(2.40)

0

At high equilibrium pressure, a1p >> 1, where a1 ¼a at s ¼ 1 (lowest a) and the fraction of vacant sites is given by: ð1 ds 1  θ ¼ 1=p (2.41) a 0

being proportional to the reciprocal value of partial pressure similar to ideal surfaces. An important conclusion follows from these considerations that nonuniform surface is not manifested itself in the region of small and large coverages. For evenly nonuniform surfaces it follows from Eq. (2.31) that: l ¼ CΔGa° + C 0

(2.42)

where C0 is an integration constant. Then the Gibbs energy is defined as: ΔGa° ¼

L C0 s C C

(2.43)

leading to an expression for the adsorption coefficient: ln a ¼ 

L C0 s+ CRT CRT

(2.44)

Introducing f as f ¼ L/CRT and a0 as ln a0 ¼ c 0 =CRT, one arrives at: a ¼ a0 efs

(2.45)

where for the smallest value of a at s ¼ 1: a1 ¼ a0 ef

(2.46)

where the value of f is called the nonuniformity exponent: a0 a1

(2.47)

a0 efs p ds 1 + a0 efs p

(2.48)

f ¼ ln Integrating now Eq. (2.39): ð1 θ¼ 0

71

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Catalytic Kinetics

and keeping in mind an expression for the standard integral: ð ax e 1 dx ¼ lnðb + c eax Þ b + c eax ac we arrive at s ¼ 1 and s ¼ 0 correspondingly to the following values:  a0 p  ln 1 + a0 ef p  fpa0 a0 p lnð1 + a0 pÞ  fpa0

(2.49)

(2.50) (2.51)

and finally to the quasi-logarithmic isotherm: θ¼

1 1 + a0 p ln f 1 + a1 p

(2.52)

For strongly nonuniform surfaces a0 ≫ a1, there is a region of medium coverage where a0p ≫ 1, a1p ≪ 1, which means that for the most strongly adsorbing sites, the probability of being occupied is almost 1 and for the weakly adsorbing sites almost 0. Then the quasilogarithmic isotherm is simplified to the so-called logarithmic isotherm or Temkin adsorption isotherm. Such adsorption isotherm corresponds to a linear decrease of heats of adsorption with coverage (Fig. 2.36) θ¼

1 ln ða0 pÞ f

(2.53)

A comparison of the experimental data for ammonia adsorption on titanium silicate and calculations using the quasi-logarithmic adsorption isotherm is presented in Fig. 2.37. For derivation of adsorption isotherms in the case of exponential nonuniformity, the desorption exponent ξ ¼ ln b ¼  ln a should be preferably used as an integration variable:

Fig. 2.36 Differential heats of adsorption as a function of coverage for ideal and nonuniform surfaces.

Catalysis

Ammonia adsorption (mmol/g)

2.5

2.0

1.5

1.0

0.5 0

50

100

150

200

250

Pressure (Torr)

Fig. 2.37 Comparison of the experimental data for ammonia adsorption on titanium silicate and calculated using the quasi-logarithmic adsorption isotherm. (From N.V. Kul’kova, M.Yu. Kvyatkovskaya, D.Yu. Murzin, Liquid-phase ammoximation of cyclohexanone. 3. Ammonia adsorption on ammoximation catalysts, Khim. Prom. (1997) 28).

The probability that a site is occupied is given by Eq. (2.37): ap σ¼ 1 + ap

(2.54)

and adsorption equilibrium is given by a rearranged form of Eq. (2.39): ð1

ð1 θ ¼ σds ¼

1

01+

0 ξ

1 ap

ds

(2.55)

As e ¼ 1=a, then: 1 ð

φðξÞdξ 1 ξ 1 1 + e p For this distribution function, eg, exponential nonuniformity: 1 θ¼ L

1 θ¼ L

(2.56)

ξð1

Aeγξ dξ 1 ξ ξ0 1 + e p

(2.57)

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Catalytic Kinetics

The ratio of the probabilities of a site to be either free or occupied u ¼ 1=ap ¼ ð1=pÞeξ ¼ ð1  σ Þ=σ, can be described as:   1 1 d eξ ¼ eξ dξ (2.58) p p then du ¼ d ðξÞu and dðξÞ ¼ du=u. Further substitutions eγξ ¼ pγ eξγ ¼ pγ uγ and eξ ¼ pu give:

A θ ¼ pγ L

1 að1 p

uγ1 du 1+u

(2.59)

1 a0 p

This equation can be expressed through elementary functions only in some special cases. For the range of medium coverage a0p ≫ 1 and a1p ≪ 1, the limits are 1/a0p ! 0 and 1/a1p ! 1. Therefore, the expression for the surface coverage becomes: 1 ð A γ uγ1 du θ¼ p L 1+u

(2.60)

0

It can be further simplified in the case 0 < y < 1 as the solution for the standard integral is known: 1 ð

0

uγ1 du π ¼ 1+u sin ðγπ Þ

(2.61)

As eξ1 ¼ 1=a1 and eξ0 ¼ 1=a0 , and a0 ≫ a1, eξ0 could be neglected compared to eξ1 in: A¼

eγξ1

γL ¼ Lγaγ1 ; A=L ¼ γaγ1  eγξ0

(2.62)

Then an expression for the surface coverage is: θ ¼ γaγ1 pγ

π γπ ða1 pÞγ ¼ C ða1 pÞγ ; θ ¼ C ða1 pÞγ ¼ sin ðγπ Þ sin ðγπ Þ

(2.63)

which corresponds to the Freundlich adsorption isotherm. According to this isotherm, there is a nonlinear decrease in the heat of adsorption as a function of coverage (Fig. 2.36). The Temkin and Freundlich adsorption isotherms have been successfully applied to treat experimental data, not only for adsorption of gases (hydrogen) on metal surfaces and ions on metal electrodes, but also the adsorption of proteins to functional surfaces. The explicit expression for reaction kinetics in the case of biographic (intrinsic) nonuniform

Catalysis

surfaces can be derived only in a few cases, which, in fact, limits the possibility of wider application of the a priori nonuniform surfaces. 2.2.5.1 Adsorption Isotherms—“Induced” Nonuniformity 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, ie, 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 probability of their interactions and formation of surface polyatomic structures increase. These structures can be rather stable and 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 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, ie, lateral interactions result in ordering of adsorbed

75

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Catalytic Kinetics

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 tunneling 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 interactions between two nearest-neighbor species are often in the range of 4–16 kJ/mol, 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 neighbors. Nevertheless, commonly applied simplified models 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 two-dimensional array of surface sites and a site is assumed to be either vacant or occupied by a single adsorbate. The regular lattice systems with nearestneighbor 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 nearest-neighbor interaction energy are treated as though molecules were distributed randomly among the sites.

Ernst Ising

William L. Bragg

Evan J. Williams

Catalysis

Ralph Howard Fowler

Edwin A. Guggenheim

The simplest representation of adsorbate-adsorbate-adsorbent interactions is the Fowler-Guggenheim isotherm, which is formally similar to the Bragg-Williams approximation. This isotherm differs from the Langmuir one in the exponential term, accounting for the interactions: θ νθ (2.64) e 1θ For multi-component chemisorption, we will consider the monolayer of Ni species of molecules (or atoms) of type i and Nj species of molecules (or atoms) of type j with the total amount of J sorts of molecules (or atoms) (i 6¼ j). For multi-component chemisorption, the canonical partition function Q in the one-dimensional case is given by: aP ¼

Q ¼ Qi ðNi , L, T Þ

J Y



Qj Nj , L, T



i ¼ qN i

j¼1, i6¼j

Ω1 X

N eEi =kT qj j

1

Ωj X

eEj =kT

(2.65)

1

where qi is the single-molecule partition function without interactions, Ei is the total energy of nearest-neighbor interactions of Ni molecules (or atoms), and Ωi is the total number of configurations of Ni molecules (or atoms) on L sites. Eq. (2.65) can be rewritten: X N

Q ¼ qi

i

Nii



J Y 1

gðNi , L, Nii Þe

ωii Nii =kT +

J P

ωij Nij =kT

j¼1, i6¼i

J P ω N =kT ω N =kT + ωjl Njl =kT jj jj ji ji X   N l¼1, l6¼i, l6¼j qj j g Nj , L, Nij e

Nij

(2.66)

77

78

Catalytic Kinetics

where g is a degeneracy factor, allowing the possibility that several physically distinguishable states may have the same energy. In Eq. (2.66), ωij 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: ωii Nii =kT +

i Q ¼ qN i e

J P j¼1, i6¼i

ωij Nij =kT X Nii



J Y

N

ωjj Njj =kTωji Nji =kT +

qj j e

gðNi , L, Nii Þ

J P

l¼1, l6¼i, l6¼j

ωjl Njl =kT X

  g Nj , L, Nij

(2.67)

Nij

1

The degeneracy factor for adsorbed species of i-type is given by: X L! ! gðNi , L, Nii Þ ¼ J J X Y Nii Nj !Ni ! Nj ! L  Ni  j¼1, i6¼j j¼1, i6¼j

(2.68)

Similar equations can be derived for species of j-types. An expression for the average interaction energy follows from the Bragg-Williams approximation:

and

N ii ¼ ðcNi =L ÞðNi =2Þ

(2.69)

  N ij ¼ ðcNi =L Þ Nj =2

(2.70)

where the factor 2 is used for preventing the counting of each pair ii or ij twice. Using ln y! ¼ y ln y  y, the chemical potential can be obtained: μi ¼ kT ð@ ln Q=@Ni ÞNi , LT and therefore: μi ¼ kT ln qi  ln L  Ni   ωij Ni =2LkT +

J X

J X

(2.71) !

Nj

+ ln Ni

j¼1, i6¼j

ωjj Nj =2LkT

(2.72)

j¼1, i6¼i

The chemical potential of a gas molecule (μgi) is expressed, setting the lowest energy state as energy zero:   μgi ¼ kT ln qgi T =Pi d (2.73)

Catalysis

The equilibrium condition is the equality of chemical potentials in the gas phase and in the adsorbed layer; therefore, an adsorption isotherm follows naturally: ai Pi ¼

θi 1  θi 

J X

evii θi θj

J Y

evij θj

(2.74)

j¼1, j6¼i

j¼1, j6¼i

where ai is the adsorption coefficient of i, θ is the degree of covering (θi ¼ Ni/L) and: vii ¼ ωii c=2kT

(2.75)

vij ¼ ωij c=2kT

(2.76)

Eq. (2.74) 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 Sommerfeld 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 twodimensional 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 form of the model of the surface electronic gas provides a 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 theory of metals for the twodimensional case leads to the equation: ε ¼ ε0 

η2 h2 L θ 4πm∗

(2.77)

where η is the effective charge acquired by an adsorbed particle, h is the Plank constant, ε is the heat of adsorption, and m* is the effective electron mass. For the treatment of multi-component chemisorption, let us assume that on the surface Ni adsorbed species Ai exists with the effective charge ηi and Nj species of Ai

79

80

Catalytic Kinetics

with the effective charge ηj, where i 6¼ j. It can also be assumed that η is not a function of coverage. The total quantity of electrons in a subsurface layer Ne can be expressed as follows: Ne ¼ Ne0 + ηi Ni +

J X

ηj Nj

(2.78)

j¼1, j6¼i

where N0e is the value of Ne at θ ¼ 0. The value of maximum surface kinetic energy Es is, therefore: ðEs Þmax ¼

J X h2 Nj h2 Ne0 h2 Ni η + η + i 4πm∗ s 4πm∗ s j¼1, j6¼i j 4πm∗ s

(2.79)

where s is the surface area. When an additional amount of particles Ni is adsorbed an additional amount of electrons, ηi 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: Ei ¼ ε

0

h2 Ni Ni  η2i Ni  4πm∗ s

J X

ηi ηj

j¼1, j6¼i

h2 Nj Ni 4πm∗ s

(2.80)

where ε0 is the value of E at θ ¼ 0. The sign “+” corresponds to the case of repulsive interactions, and sign “” for attractive interaction. A thermodynamic approach results in:     (2.81) ln Pa ¼ H cai =RT  Scai =R + Ha6¼i =RT  Sa6¼i =R where H cai is the configurational enthalpy and Sca is the configurational entropy of the adsorbed layer. Sca coincides with that for the simple adsorption, thus: Sca ¼ R ln

θ0 θist θi θ0st

(2.82)

where θsti is the standard coverage, which can be defined in the following way θsti ¼ θst0 ¼ 0:5. The site balance is given: θ0 ¼ 1  θi 

J X

θj

(2.83)

j¼1, j6¼i

  From Eqs. (2.80) to (2.83) and H cai =R ¼ ε0  ε =kT , where εi ¼ Ei/Ni, the adsorption isotherm for the adsorption of i component of gas mixture can be obtained:

Catalysis

ai Pi ¼

θi 1  θi 

J X

e θj

ϖ ii η2i θi C=T

J Y

eϖij ηi ηj θj C=T

(2.84)

j¼1, j6¼i

j¼1, j6¼i

where h2 (2.85) 4kπm∗ and ϖ ii(ϖ ij) can be either +1 (repulsive interactions) or 1 (attractive). Eq. (2.85) is of the same type as Eq. (2.74); however, it gives more possibilities for predicting (or estimating) the values of the interaction parameter, which remains unclear within the framework of the Bragg-Williams lattice gas. In the case of formation of clusters on the solid surfaces, the surface of adsorbent has (i) Nn molecules and N(i) j clusters of i molecules, where i varies from unity to imax, imax being 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: C¼

J   Y   ð jÞ Q ¼ Qn NnðiÞ , L, T Qj Nj , L, T j¼1, n6¼j J Ωn Ωn   Y   X X ð jÞ exp E n ð jÞ =kT qj Nj exp E j =kT ¼ qn Nn 1 1 j¼1, n6¼j

(2.86)

where qn is the single-cluster partition function without interactions, Eðn jÞ is the total (j) nearest-neighbor energy of interactions of N(j) n clusters, Ωn is the total number of con(j) figurations of Nn clusters on L sites, T is the temperature, and k is the Boltzmann constant. Replacing the energy of interaction by average interaction energy we arrive at: ! J X X Q ¼ qn Nn exp wnn N nn =kT + wnj N nj =kT gðNn , L, Nnn Þ Nnn j¼1, n6¼j ! J J Y X X    qj Nj exp wjj N jj =kT  wjn N jn =kT  wjl N jl =kT g Nj , L, Nnj Nnj j¼1 l¼1, l6¼n, l6¼j (2.87) where g is a degeneracy factor allowing for the possibility that several physically distinguishable states may have the same energy; Nnn is the average interaction energy, and w is the potential energy of an interaction. For the degeneracy factor of adsorbed clusters of n-type, it holds that:

81

82

Catalytic Kinetics

X gðNn , L, Nnn Þ ¼

L! L  Nn 

J X j¼1, n6¼j

!

Nj !Nn !

J Y

(2.88) Nj !

j¼1, n6¼j

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: an Pn ¼ 1  θðn1Þ 

θðn1Þ J X

ðiÞ

Nj T 1 =L

J   Y   ðiÞ exp νnj θðn1Þ T 1 exp νjj Nj T 1 =L j¼1, j6¼n

j¼1, j6¼n

(2.89) where an is the adsorption coefficient of n and θ(1) n is the degree of covering of singlets and: vnj ¼ wnj c=2k

(2.90)

vjj ¼ wjj c=2k

(2.91)

Defining the coverage of singlets through the total coverage of species of n-type: θðn1Þ ¼ θn =α0n where a0n ¼

I X

(2.92)

iNnðiÞ =Nnð1Þ

(2.93)

i¼1

we finally arrive at: an Pn ¼

J     Y θn =a0n 1 0 1 0 exp ν T θ a =a exp ν T θ a =a nj n n n jj j j j J X j¼1, j6¼n 0 0 1  θn an =an  θj aj =aj j¼1, j6¼n (2.94)

where an ¼

I X

NnðiÞ =Nnð1Þ

(2.95)

i¼1

aj ¼

I X

ðiÞ

ð1Þ

Nj =Nj

(2.96)

i¼1

The system of adsorption isotherms (2.94) allows for the discussion of the critical phenomena in the multi-component adsorption layer. When there is an interaction between

Catalysis

the adsorbed particles and the adsorbate, segregation occurs. For the sake of clarity, we will present as an example below the derivation for 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. Each point of the bifurcation set for the two-component case corresponds to two types on the surface θa and θa + Δθa,θb and θb + Δθb, where a and b indicates two different types of adsorbed molecules. Using the logarithmic form of the system (2.94):     θa + dθa 1  ðθa + dθa Þαa =α0a  ðθb + dθb Þαb =α0b  ln ln θa 1  θa αa =α0a  θb αb =α0b     ¼ νaa αa =α0a dθa =T + νab αb =α0b dθb =T (2.97) and



   θb + dθb 1  ðθa + dθa Þαa =α0a  ðθb + dθb Þαb =α0b  ln ln θb 1  θa αa =α0a  θb αb =α0b     ¼ νab αa =α0a dθa =T + νbb αb =α0b dθb =T

(2.98)

Resolving Eqs. (2.97), (2.98) in series and dropping all terms higher than the first power, we obtain:     dθa =θa + αa =α0a dθa + αb =α0b dθb = 1  θa αa =α0a  θb αb =α0b     (2.99) ¼ vaa αa =α0a dθa =T + vab αb =α0b dθb =T     dθb =θb + αa =α0a dθa + αb =α0b dθb = 1  θa αa =α0a  θb αb =α0b     ¼ vab αa =α0a dθa =T + vbb αb =α0b dθb =T (2.100) therefore:     dθa =θa  dθb =θb ¼ ðvaa  vab Þ αa =α0a dθa =T + ðvab  vbb Þ αb =α0b dθb =T Further manipulations give:     2 + αa =α0a θa + αb =α0b θb = 1  θa αa =α0a  θb αb =α0b   ¼ ðvaa + vab Þðαa =αa Þdθa =T + ðvaa + vbb Þ αb =α0b dθb =T

(2.101)

(2.102)

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: dθa =dθb  θa =θb ¼ 0

(2.103)

83

84

Catalytic Kinetics

and: αb =α0b θb ¼ αa =α0a θa ðvab  vaa Þ=ðvab  vbb Þ

(2.104)

Introducing the quasi-total coverage: αb =α0b θb + αa =α0a θa ¼ θ

(2.105)

we arrive at:

 2   vaa vab  vaa vbb =ð2vab  vaa  vbb Þ (2.106) 2=θ + 1=ð1  θÞ ¼ ðTcr Þ1 2vab pffiffiffi pffiffiffi The left side of Eq. (2.106) gives a minimum at θcr ¼ 2=1 + 2, thus leading to the equation for the critical temperature: 2ν2  νaa νab  νaa νbb Tcr ¼  pabffiffiffi2 1 + 2 ð2νab  νaa  νbb Þ

(2.107)

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 temperature programmed desorption (TPD) spectra as well as for reaction of gases on metal surfaces. The heat of adsorption within the framework of the model, developed by D. King is given by: 2 3 1 6 qads ðθÞ ¼ q0 + zω41  n 2

1  2θ 7 h io0:5 5 1  4θð1  θÞ 1  eω=kB T

(2.108)

where ω is the heat of adsorption at zero coverage and z is the maximum possible number of nearest neighbors.

Sir David King

Catalysis

2.2.5.2 Adsorption Isotherms—Multi-Centered Adsorption 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 with full competition nor with nonexisting 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 Fig. 2.38. 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, ie, 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. Concentration of sites rather than the coverage will be used below. A small molecule (B, typically hydrogen) adsorbs either in molecular form or dissociatively: 0

A + 0 ¼ A∗ B + i∗ ¼ iB∗ Organic species (relatively large) Dissociated hydrogen atom (relatively small)

Fig. 2.38 Semi-competitive adsorption.

(2.109)

85

86

Catalytic Kinetics

where ∗0 denotes a cluster consisting of m primary sites (*). For the rapid adsorption steps, the quasi-equilibria are expressed by: c 0 KA ¼ A* (2.110) c*cA and KB ¼

i cB* c*i cB

(2.111)

The total balance of primary sites is given by: c0 ¼ mcA*0 + cB* + c*

(2.112)

where c0 is the total concentration of primary sites on the surface. For a larger molecule (A), the site balance is written as: cA*0 + c*0 ¼ αðc0  cB* Þ=m

(2.113)

where α is the adsorption-competition parameter of the larger molecule and c*0 is the concentration of vacant clusters of sites accessible for the adsorption of A. The adsorption semi-equilibria are inserted in the balance (2.113), and the relation of vacant clusters of sites ðc*0 Þ and vacant primary sites (c*) is obtained:  h i (2.114) c*0 ¼ α=m c0  ðKB cB Þ1=i c* =ð1 + KA cA Þ This expression is inserted into the balance of primary sites. After straightforward algebraic steps, the coverage of free sites becomes: α θfree ¼   (2.115) P 1=i P Korg, j Corg, j Korg, j Corg, j + ð1  αÞðKH CH Þ1=i 1 + ðKH CH Þ + 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 (σ order), the number of arrangements is lowered by a factor of σ, leading to the number of arrangements of a first particle equal to φL, where φ is the multiplicity of the system (φ ¼ g/σ). 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.

Catalysis

Below we will consider multi-centered species and will take into account interactions between adsorbed species. The number of possible positions for N molecules is given by: N Y M ¼ ð1=N !Þ ðηL  ðj  1Þnð jÞÞ

(2.116)

j¼1

Consequently, the partition function for the adsorbed layer is: Q¼



gN L N N!

N Y

eEj =kT

ηLððj1Þnð jÞÞ L

(2.117)

j¼1

where qN (gN!) is the partition function for one adsorbed particle and Ej is the total nearest-neighbor energy of interactions of Nj molecules (atoms). The free energy of the adsorbed layer is given by: AH ¼  kT(ln Q ¼ kT ωN =kT + N ln q  ln N! + N ln L +

N X j¼1

) j1 ln η  nð jÞ L

(2.118) In Eq. (2.118), the total interaction energy is replaced by multiplication of the potential energy of interactions by the number of molecules. A new variable x is then introduced: j1 m; Δx ¼ m=L L which satisfies the following boundary conditions: x¼

(2.119)

j ¼ 1; x ¼ 0 j ¼ N ; x ¼ ðN  1Þm=L ¼ θ

(2.120)

where θ is the surface coverage and m is the number of elementary sites that one molecule occupies. Summation in Eq. (2.118) can be replaced by integration, since N and L are very large numbers: AH ¼ kT ln Q  ð h i L θ x ¼ kT ωN =kT + N ln q  ln N ! + N ln L + dx ln η  nðxÞ m 0 m (2.121)

87

88

Catalytic Kinetics

The differential of a definite integral at its upper limit is equal to the intergrand at this limit. As ln y! ¼ y ln y  y, then: d d ln y!  ln y dx dx Taking θ ¼ Nm/L, the chemical potential can be obtained:

(2.122)

μ ¼ kT ð@ ln Q=@N ÞN ,L, T

(2.123)

The chemical potential of a gas molecule (μgi) is expressed with the lowest energy state assigned as energy zero:   μgi ¼ kT ln qgi =c (2.124) Concentration c in the gas phase is defined by: (Pid/T) ¼ 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: aa Pa ¼

θa eωθA =kT ηm  θa nðθa Þ

(2.125)

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: aP ¼ θ=ðϕm  nðθÞÞ

(2.126)

Variation of the screening parameter with coverage is approximately linear: n ¼ ν  φθ

(2.127)

where ν is the screening parameter at zero coverage. Values for several systems are given in Table 2.3. Let us consider an interacting adsorbed layer, which consists of three distinct species: activated complexes of arbitrary configuration (total number of different kind of complexes ¼ Mt), multi-centered adsorbed species (total number of different types Mm) Table 2.3 Values of parameters for different adsorption with shielding System

η

m

ν

ϕ

1. 2. 3. 4.

1 2 1 1

2 2 4 7

3 7 9 19

1 3 5 12

Two-centered symmetrical species on a chain Two-centered symmetrical species on a square lattice Four-centered symmetrical species on a square lattice Seven-centered molecules (benzene) on a lattice of equilateral triangles

Catalysis

and uni-centered adsorbed species (total number ¼ Mh). The partition function for the adsorbed layer is given by: Qa ¼

Mt  Ma  Mh Y Y Y  Nh  i ftiNti cti faNa c fh chi a j i i¼1

j¼1

(2.128)

h¼1

The number of ways in which multi-centered molecules can be distributed is formulated: LN Y C ¼ ti Nti ! Nti

k¼1

"

 #   k1 k ηti  nt p θp  nt t θp , m i L ii l p¼1 p P X θp

(2.129)

where p refers to species, which are already adsorbed on the surface. For instance, for the species of third type, the middle term in Eq. (2.129) is given by: θ1 θ2 n31 ðθ1 Þ + n32 ðθ1 , θ2 Þ m1 m2

(2.130)

An expression similar to (2.129) can be formulated for many-centered species. The number of ways of distributing a single-centered species follows directly from statistical mechanics: ! Mt Ma X X L mti Nti  maj Naj ! i¼1

Ch ¼ L

Mt X

mti Nti 

i¼1

j¼1 Ma X

maj Naj 

j¼1

Mh X h¼1

! Nh

(2.131)

Mh Y ! Nh h¼1

The chemical potential of activated complexes of one-centered molecules and adsorbed species is given by: 8 0 1mti Ma Mh X X > > > > " # B1  θ aj  θh C > Ma B C m N ti > j¼1 aj > @ A > 1  θ a > j : j¼1

wit CNt =2LkT 

Ma X j¼1

wajt CNj =2LkT 

Mh X h¼1

wht CNh =2LkT

9 > > > > > = > > > > > ;

(2.132)

89

90

Catalytic Kinetics

8 0 1maj Ma Mh X X > > > > θ aj  θh C " # B1  > Ma B C θ m a a > i j j¼1 > @ A > 1 θ aj > : j¼1

waj aj CNj =2LkT 

Ma X

wal aj CNl =2LkT  waj t CNt =2LkT 

j¼1, j6¼l

Mh X

waj h CNh =2LkT

h¼1

Ma Mh X X X fh wh CNh =2LkT 1 θ aj  θh  μh ¼ kT ln θh j¼1 h¼1 !) Ma Mt X X  waj h CNj =2LkT  whti CNt =2LkT j¼1

> > > > > ;

(2.133)

!

(

9 > > > > > =

(2.134)

i¼1

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 complex are also neglected. Equilibrium condition for adsorption again implies equality of the chemical potential in the gas-phase and the adsorbed phase: !maj Ma X θ aj 1  θ aj j¼1

aj Pj ¼ maj ηaj 

X θ aj

  exp ωjj θaj

!maj ! Ma Mh X X   1 nai naj θa …θaj θ aj  θh

m aj Ma  Y

r¼1, r6¼j

 exp ωjr θar

Mr Y

j¼1

h¼1

  exp ωjh θh

(2.135)

h¼1

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-centered molecule, which adsorbs onto a single site: θA exp ðω1 θA + ω2 θB Þ  aA PA ¼  ϕm  k00 θA + k000 θ2A ð1  θB Þm

(2.136)

Catalysis

aB PB ¼

θB exp ðω2 θA + ω3 θB Þ 1  θA  θB

(2.137)

Numerical calculations for multi-centered adsorption over nonuniform surfaces revealed that the multi-centered nature of adsorbed species masks the influence of nonuniformity; 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-centered adsorption on nonuniform surfaces, with inclusion of lateral interactions with an average energy ε: pffiffiffiffiffiffiffiffiffiffi aP ¼ θ exp θ=2ð1  θÞ exp ðεθ=kT Þ=ηm 1  θ (2.138) This isotherms is reduced for ideal surfaces to:

pffiffiffiffiffiffiffiffiffiffi aP ¼ θ exp θ=2ð1  θÞ=ηm 1  θ

(2.139)

Semi-competitive adsorption model with a partial shielding was developed by Cabrera and Grau. The schematic top view of adsorption is given in Fig. 2.39. In this concept, a large molecule is considered to be interacting with xi -sites. In addition, si -sites (closely adjacent to those xi -sites) are expected to be additionally covered by an organic molecule, since its molecular size is much larger than the distance between neighboring -sites. The si -sites are supposed to be inaccessible for the

Fig. 2.39 Schematic top view representation of occupied-sites (shaded regions) and covered-sites by adsorbed molecules of hydrogen and large organic compounds (dashed regions), with uncovered sites between them. (From M.I. Cabrera, R.J. Grau, J. Mol. Catal. A: Chem. 287 (2008) 24–32. Copyright 2008 Elsevier).

91

92

Catalytic Kinetics

adsorption of another organic molecule being available for adsorption of small (hydrogen) molecules by virtue of their small size. In essence, an organic molecule effectively covers (xi + si) -sites, si being only accessible for hydrogen adsorption. Dissociative adsorption of coverage is assumed for illustration purposes. Two forms of site balance expressions could be then applied: xL θ L + θ H + θ ¼ 1

(2.140)

U ðxL + sL ÞθL + θU H + θ ¼ 1

(2.141)

U where θL corresponds to coverage of large organic molecules, θU H , θ are vacant site coverage for adsorption of hydrogen and the organic molecule. The quasi-equilibrium approximation gives:  xL (2.142) θH ¼ ðKH CH2 Þ0:5 θ ; θL ¼ KL CL θU

The ratio between the -occupied sites and -covered sites is defined: f¼

xL θL ðxL + sL ÞθL

(2.143)

when there is just one type of organic molecule. In a more general case, f is given by: X xi θi (2.144) f ¼X ðxi + si Þθi The site balance can be rewritten:  xL   + f 1 + ðKH CH2 Þ0:5 θU x L K L CL θ U ¼1

(2.145)

Finally, the connection between vacant site coverage and the uncovered sites is:     1  f 1  1 + ðKH CH2 Þ0:5 θU θ ¼ (2.146) 1 + ðKH CH2 Þ0:5 Explicit equations for the adsorption isotherms can be derived for the special cases (eg, f equal to unity). 2.2.5.3 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 are 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

Catalysis

mode of adsorption can be dependent on the concentration, as demonstrated in selective hydrogenation of α,β unsaturated aldehyde-cinnamaldehyde by Moulijn and Kapteijn. It was 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 [1]. This reaction over nonchiral catalysts produces racemic mixtures of optical isomers when the substrate contains a prochiral center. Cinchonidine (Fig. 2.40), being a 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 π-bonding of aromatic quinoline rings, the ring system being almost parallel to the metal surface (Fig. 2.40). The adsorbed parallel form of the modifier would be involved in 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 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.

Parallell (actor) M + q∗

Mq∗

p∗ ≠ q∗

Tilted (spectator) M + p∗

Mp∗

N N

OH H

(A) Fig. 2.40 Different adsorption modes (A) of cinchonidine (B).

(B)

93

94

Catalytic Kinetics

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. It is important to remember when dealing with bulky organic molecules adsorbed on supported metal catalysts that in the classical kinetic treatment, surfaces are 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. This requires special approaches, which will be discussed in Chapter 7.

2.3. ORGANOCATALYSIS This type of catalysis corresponds to application of low molecular weight organic molecules, which in substoichiometric amounts catalyze chemical reactions. The organocatalysts could be achiral or chiral (Fig. 2.41). Organocatalysts in principle can be Lewis and Brønsted bases, as well as Lewis and Brønsted acids; however, most of the reported organocatalysts are involved in Lewis base

Fig. 2.41 Examples of organocatalysts. (From I.R. Shaikh, J. Catal. (2014) 402860. Copyright © 2014 Isak Rajjak Shaikh. Distributed under the Creative Commons Attribution License).

Catalysis

Fig. 2.42 The Hajos-Parrish reaction.

catalyzed reactions. In many organocatalytic reactions, the organocatalyst possesses not only a Lewis base (nitrogen or phosphorus atom), but also a Brønsted acidic site. Current interest in organocatalysis is focused on asymmetric catalysis with chiral catalysts, or enantioselective organocatalysis (Fig. 2.42). In the asymmetric aldol reaction (Fig. 2.42), naturally occurring chiral proline is the chiral catalyst giving 93% enantiomeric excess when just 3% proline is added to an achiral triketone. Kinetic analysis of this reaction was performed in order to elucidate the reaction mechanism. First order dependence on the catalyst concentration, absence of any nonlinear effects (Fig. 2.43) and independence of enantioselectivity on substrate concentration (Fig. 2.44) suggested that only one proline molecule is involved in formation of the transition state. The proposed mechanism of the Hajor-Parrish reaction involves the nucleophilic addition of the neutral enamine to carbonyl group (Fig. 2.45).

Fig. 2.43 Dependence of ee of compound 3 on enantiomeric excess of proline. (From H. Linh, S. Bahmanyar, K.N. Houk, B. List, J. Am. Chem. Soc. 125 (2003) 16–17. Copyright 2003 American Chemical Society).

95

Fig. 2.44 Dependence of ee of compound 3 on triketone concentration. (From H. Linh, S. Bahmanyar, K. N. Houk, B. List, J. Am. Chem. Soc. 125 (2003) 16–17. Copyright 2003 American Chemical Society).

Fig. 2.45 Suggested mechanisms for the Hajor-Parrish reaction. (From C. Allemann, R. Gordillo, F.R. Clemente, P.H.-Y. Cheong, K.N. Houk, Acc. Chem. Res. 37 (2004) 558. Copyright 2004 American Chemical Society).

Catalysis

Kinetic analysis of this and some other organocatalytic reaction mechanisms will be presented in Chapter 5. Organocatalysts were also used for Diels-Alder, Mannich, Michael reactions, as well as for α-amination, α-aminoxylatoin or α-alkylation of aldehydes, to name a few. Heterogenization of organocatalysts through impregnation, occlusion in porous materials (ship-in-a-bottle), as well as grafting or tethering through covalent bonds is the current trend in research with organocatalysts.

2.4. EXAMPLES AND EXERCISES 2.4.1 Answer the following questions: (a) What is the role of desorption in the process of catalysis? (b) Is chemical adsorption an activated process? (c) Is physical adsorption an activated process? (d) Why does physisorption decrease with temperature increase? (e) Can a catalyst change the composition of a gas mixture in equilibrium? (f) What are the differences and similarities between homogeneous and heterogeneous catalysis? To which category does biocatalysis belong? 2.4.2 Indicate the true and the false statements among the following: (1) The activity of the catalyst refers to the rate at which it causes the reaction to proceeds to equilibrium (2) The chemical adsorption occurs at high temperature and can be reversible (3) The catalyst increases the activation energy (4) Porous catalysts have large external surface areas compared to their internal surface areas (5) The reaction rate constant is a function of reactant concentration (6) The deactivation of the catalyst refers to the decrease in the catalyst activity

T

F

T T T

F F F

T T

F F

2.4.3 Hydrogenation of an organic compound using a heterogeneous catalyst at 100°C in a batch mode gave the following data: In the first run, TOF was 190 min1. When the catalyst was filtered and reused TOF 15 min1 was obtained. After filtering the catalyst again and calcination it in air at 450°C for 4 h, the measured TOF was 189 min1 in the third run. Explain this behavior. 2.4.4 Derive equations for the surface coverage of adsorbed species (adsorption isotherms) for a gas mixture containing A, R, S and B2 (adsorbs with dissociation). 2.4.5 The decomposition of methyl amine (CH3NH2) to HCN has been studied on platinum. (a) Provide a feasible pathway for the reaction. (b) What other products would you expect? (c) How could cyanogen (NC-CN) be formed?

97

98

Catalytic Kinetics

2.4.6 The rate of gas phase benzene hydrogenation passes through a maximum with temperature increase. Propose an explanation for this behavior, which happens when the reaction is far from equilibrium. 2.4.7 A certain catalyst based on Pt catalyzes the hydrogenation of double bonds in olefins. Can this metal be used for dehydrogenation of alkanes? 2.4.8 Discuss the various catalytic steps in homogeneous catalytic reactions. 2.4.9 State reasons that enzymes are often used as general acid or base catalysis. 2.4.10 Enantioselective hydrogenation of ethyl benzoylformate on platimum can be done in the presence of a modifier: cinchonidine. Derive an adsorption isotherm for the reactant in the presence of cinchonidine (can adsorb with different adsorption modes). Solution: A side and top view of ethyl benzoylformate on Pt(111) surface is shown below, indicating that the number of Pt atoms covered by this molecule is in the range of 9–11 atoms.

Figure: The adsorption geometries for ethyl benzoylformate on the Pt(111) surface. Carbon atoms are colored grey, hydrogen atoms white, oxygen atoms red and platinum atoms blue.

Adsorption of the EBF (A) follows a classical multi-site adsorption behavior: 1. A + m* , Am* where Am* denotes the adsorbed substrate and m the number of sites required for adsorption. Adsorption of chinchonidine (M) is respectively: 2. M + p* , Mp* 3. M + q* , Mq* where Mp* denotes the parallel adsorption mode of ()-cinchonidine, while Mq* denotes the tilted adsorption mode of ()-cinchonidine, which appears on the catalyst surface as a spectator.

Catalysis

In the case of asymmetric catalysis, adsorption of the reactant (A) together with the modifier leads to reactant-modifier interactions, which are assumed to be essential for transferring the chirality to a prochiral reactant. 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: 4. Am* + Mp* + f * , AM ðmpf Þ* Here f represents extra sites, which might be required for the substratemodifier complex to be formed compared to the site requirement for them separately. The use of the quasi-equilibrium hypothesis for the adsorption steps implies that the concentrations of Am*, Mp*, and Mq* are expressed by: CAm* ¼ K1 CA C m* ; CMp* ¼ K2 CM C p* ; CMq* ¼ K2 CM C q* ; CAM ðmpf *Þ ¼ K4 CAm* CMp* C f * ¼ K1 K2 K4 CA CM C ðf + p + mÞ* All these expressions are inserted into the total balance of metal sites, where c0 and c* denote the total concentration of accessible sites and the concentration of vacant sites, respectively. After inserting the quasi-equilibria into the total balance of sites, the coverages of all species can be expressed via the fraction of vacant sites, θ ¼ C*=C0 :   p1 q1 mK1 CA θm C0m1 + CM pK2 θp C0 + qK3 θq C0 ðm + p + f 1Þ

+ ðm + p + f ÞK1 K2 K4 CA CM θðf + p + mÞ C0

+θ¼1

, It is possible to substitute the following parameters: K10 ¼ mK1c(m1) 0 (p1) (q1) (m+p+f1) 0 0 0 K2 ¼ pK2c0 , K3 ¼ qK3c0 , K4 ¼ (m + p + f)K1K2K4(c0 ). Consequently, the site balance obtains a very convenient form:   K10 CA θm + CM K20 θp + K30 θq + K40 CA CM θðf + p + mÞ + θ ¼ 1 2.4.11 List some advantages and disadvantages of using immobilized enzymes. Hints Advantages: Stability at higher temperature and larger pH range, easier recovery and reuse, absence of contamination, possibility to apply in flow reactors. Disadvantages: Impact of mass transfer, lower concentration of enzymes when immobilized, potential changes of the shape of the active sites.

99

Catalytic Kinetics

2.4.12 Activity of an immobilized enzyme as a function of the loading passes through a maximum. Experimental data of enzyme loading on a solid support are presented in the figure. Explain such behavior. 25

Enzyme loading (mg/g)

100

20

15

10

5

0 0

5

10

15

20

25

30

Starting [E] (mg/mL)

REFERENCE [1] D.Yu. Murzin, P. Ma¨ki-Arvela, E. Toukoniitty, T. Salmi, Asymmetric heterogeneous catalysis: science and engineering, Catal. Rev. 47 (2005) 75–256.