Introduction to heterogeneous catalysisPer Stoltze

Department of Chemistry and Applied Engineering Science Aalborg University

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ContentsIntroduction......................................................................................................................6

Definition of catalysis..................................................................................................6Catalysis and process design........................................................................................6Catalysis and kinetics...................................................................................................6The basis for catalysis..................................................................................................7Status of the study of catalysis.....................................................................................7

Chalenges................................................................................................................8Caveats........................................................................................................................8

Reaction mechanism.......................................................................................................10Complications ...........................................................................................................10

Kinetic equations are non−linear...........................................................................10Inerts.....................................................................................................................11Non−consecutive steps..........................................................................................11Elusive intermediates............................................................................................11Undetectable steps.................................................................................................12Dead ends..............................................................................................................12Linear dependence between reaction steps.............................................................12

Consistency................................................................................................................13Guidelines..................................................................................................................13

Kinetics..........................................................................................................................16Reversibility..............................................................................................................16The reaction rate........................................................................................................17Rate laws...................................................................................................................17Forward and backward rate........................................................................................18Stoichiometric matrix................................................................................................18

Example ...........................................................................................................18Properties..........................................................................................................19Rate..................................................................................................................19Equilibrium equation........................................................................................20

Rate constant..............................................................................................................20Rate limiting steps.....................................................................................................21Most abundant reaction intermediate..........................................................................22Reaction order............................................................................................................22The activation energy.................................................................................................23

Activation energy vs reaction energy.....................................................................23Graphical determination........................................................................................24Analytical determination.......................................................................................24Activation energy for composite reactions.............................................................24Compensation effect..............................................................................................25Two reactions in series .........................................................................................26

Rate and conversion...................................................................................................27Pseudo−first order kinetics.........................................................................................27

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Analogy between reactors..........................................................................................28Solving the reaction scheme...........................................................................................30

The full solution........................................................................................................30Steady state approximation........................................................................................30The quasi−equilibrium approximation.......................................................................31Irreversible step approximation..................................................................................31MARI approximation.................................................................................................32

Adsorption .....................................................................................................................34Introduction...............................................................................................................34The Lennard−Jones picture........................................................................................34The potential energy surface......................................................................................35Motion of adsorbed molecules...................................................................................37

Example A2*....................................................................................................38Catalysis.........................................................................................................................39

Langmuir−Hinshelwood mechanism..........................................................................40Example................................................................................................................40Example................................................................................................................40Example................................................................................................................41

Eley−Rideal mechanism............................................................................................41Example................................................................................................................41Example................................................................................................................42

Eley−Rideal or Langmuir−Hinshelwood....................................................................42The principle of Sabatier............................................................................................42Structure sensitivity...................................................................................................44Physisorption.............................................................................................................45The BET isoterm.......................................................................................................45Chemisorption...........................................................................................................46

The langmuir isoterm............................................................................................46Non−dissociative adsorption.............................................................................46Dissociative adsorption.....................................................................................47Competitive chemisorption...............................................................................48

Sticking.....................................................................................................................49σ as a rate..............................................................................................................49The activation energy for $\sigma$.......................................................................49Flux and exposure.................................................................................................50The exposure.........................................................................................................50

Temperature programmed desorption........................................................................51Kinetics.................................................................................................................52

First order kinetics............................................................................................52Redheads equation............................................................................................53Second order kinetics........................................................................................55Zero order desorption.......................................................................................56

Catalyst structure and texture.........................................................................................59Catalyst structure.......................................................................................................59Pore structure.............................................................................................................59Catalyst models..........................................................................................................60

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The terrace−ledge−kink model...................................................................................60Defects excluded in the TLK−model.....................................................................61

The stereographic map...............................................................................................61Body−centered cubic lattice.......................................................................................63Face−centered cubic lattice........................................................................................64Hexagonal close−packed lattice.................................................................................66Adsorbates and impurities..........................................................................................66The Wulff construction..............................................................................................68Homogeneous catalysts..............................................................................................69

Preparation of catalysts...................................................................................................70Shape.........................................................................................................................70Precipitation...............................................................................................................73Pelletizing..................................................................................................................73Fusion........................................................................................................................74Catalyst supports........................................................................................................75Impregnation..............................................................................................................76Metal sponges and colloids .......................................................................................77

Poisoning and deactivation.............................................................................................78Deactivation...............................................................................................................78

Sintering................................................................................................................78Fouling..................................................................................................................79Dynamic poisoning...............................................................................................80

Microkinetic modelling..................................................................................................81Limitations.................................................................................................................82Level of the model.....................................................................................................83

The full reaction dynamics................................................................................83Mean field model..............................................................................................83

Types of models.........................................................................................................84Phases in a simulation................................................................................................85Input parameters........................................................................................................86Test of the model.......................................................................................................87Significant parameters...............................................................................................87Applications of microkinetic modelling.....................................................................87Ammonia synthesis....................................................................................................88

Solution for θ* ......................................................................................................88Calculation of the equlibrium constants.................................................................89Calculation of reaction enthalpies..........................................................................90Calculation of the rate...........................................................................................90Practical calculation..............................................................................................91Parameters.............................................................................................................91Stability of intermediates.......................................................................................92The NH3 concentration.........................................................................................92

Coverage by * ...................................................................................................93Coverage by N*................................................................................................94Coverage by N2*..............................................................................................94Variations with temperature..............................................................................95

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The reaction rate....................................................................................................96The turnover frequency.........................................................................................96Lifetime of intermediates......................................................................................96Activation enthalpy...............................................................................................97Reaction orders.....................................................................................................98

Why does it work ?....................................................................................................99Experimental methods..................................................................................................102

Titration of active sites.............................................................................................102Rate measurements..................................................................................................103

Ideal reactors................................................................................................................105Rate.........................................................................................................................105The tank reactor.......................................................................................................105The semibatch reactor..............................................................................................106

The plugflow reactor...........................................................................................108The isothermal plugflow reactor.....................................................................108

Fit of kinetic data.........................................................................................................110The simple approach................................................................................................110A more general method............................................................................................112

Kinetic measurements.........................................................................................112Thermodynamics.................................................................................................113Choice of kinetic expression................................................................................114Fitting the parameters .........................................................................................114Checking the fit...................................................................................................115

Bibliography............................................................................................................116

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1 Introduction

1.1 Definition of catalysis

A catalyst was defined by J. J. Berzelius in 1836 as a compound, which increases the rateof a chemical reaction, but which is not consumed by the reaction. This definition allowsfor the possibility that small amounts of the catalyst are lost in the reaction or that thecatalytic activity is slowly lost.

However, the catalyst affects only the rate of the reaction, it changes neither thethermodynamics of the reaction nor the equilibrium composition.

Catalysis is of crucial importance for the chemical industry, the number of catalystsapplied in industry is very large and catalysts come in many different forms, fromheterogeneous catalysts in the form of porous solids over homogeneous catalystsdissolved in the liquid reaction mixture to biological catalysts in the form of enzymes.

1.2 Catalysis and process design

The thermodynamics frequently limits the concentration of a desired product. As the catalyst does not affect the thermodynamics of the reaction, it is futile to searchfor a catalyst to improve the situation. Instead the reaction conditions (temperature,pressure and reactant composition) must be optimized to maximize the equilibriumconcentration of the desired product. Once suitable reaction conditions have beenidentified, the reaction rate is found to be too low, frequently by orders of magnitude.And the search for a suitable catalyst begins.

1.3 Catalysis and kinetics

The study of the kinetics of heterogeneous catalyzed reactions consists of at least threerather different aspects.

� Kinetics studies for design purposes. In this field, results of experimental studies aresummarized in the form of an empirical kinetic expression. Empirical kineticexpressions are useful for design of chemical reactors, quality control in catalystproduction, comparison of different brands of catalysts, studies of deactivation and of

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poisoning of catalysts.� Kinetics studies of mechanistic details. If a reasonable and not too detailed reactionmechanism is available, an experimental kinetic study may be used to determinedetails in the mechanism. Mechanistic considerations may be very valuable as aguidance for kinetic studies. � Kinetics as a consequence of a reaction mechanism. The deduction of the kineticsfrom a proposed reaction mechanism generally consists in a reasonablystraightforward transformation, where all the mechanistic details are eliminated untilonly the net gas−phase reaction and its rate remains. This approach may be used toinvestigate if a proposed mechanism consistent, what the reaction rate is and if it isconsistent with available experimental data.

For the three aspects of the study of kinetics, the optimal experimental and theoreticalapproach is quite different.

1.4 The basis for catalysis

The modern basis for the understanding of catalysis is� spectroscopy of catalysts and catalyst models.� kinetic data for catalytic reactions� quantum−chemical calculations for reactants, intermediates and products.� calculation of the thermodynamics of reactants, intermediates and products frommeasured spectra and quantum−chemical calculations.� micro−kinetic modelling.

Modern approaches to the study of reaction mechanisms consists of two approaches,experiments on well defined systems and detailed calculations for individual moleculesand intermediates.

The studies of well defined systems consists of spectroscopic studies of individualmolecules and measurements of the rate of catalytic reactions on single crystal surfaces. as well as structure and reactivity of well−defined catalyst models.

The computations consist of electron structure calculations including calculations fortransition state as well as large Monte Carlo simulations.

1.5 Status of the study of catalysis

One of the most fascinating aspects of heterogeneous catalysis is that it is largely anempirical science. The application of catalysis has been a necessity for the chemical

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industry for at least 150 years, while the experimental techniques for investigation ofcatalysis at the atomic level did not become routine until less than 25 years ago and thecomputational techniques are even younger and have hardly become routine yet. For thisreason vast amounts of emipirical knowledge exists and awaits systematic investigation.

1.5.1 Chalenges

The challenges in the deduction of reaction mechanisms from spectroscopic studies are� The pressure. Spectroscopic studies of molecules adsorbed on single crystal surfacesare made in ultra−high vacuum and computations are made in the limit of zeropressure. The pressure must be extrapolated by at least 12 orders of magnitude.� The temperature. Computations are made at zero temperature and a properthermodynamics must be constructed.� The structure. The catalyst consists of small particles stabilized by a structuralpromoter. This challenge may be overcome by studies of suitable catalyst models � The conversion. The changes in gas phase concentrations which may be reached in asingle crystal reactor is generally very low. This dictates that measurements areperformed in the limit of zero product concentration. In this limit the kinetics may beentirely different from the kinetics at higher conversions.

1.6 Caveats

Before we proceed, it may be useful to list some of the key problems in the study of thekinetics of catalytic reactions.

We cannot deduce the kinetics from the net reaction. For the reaction

aA + bB ⇔ cC + dDthe kinetics is in general not

r � kpA

po

a pB

po

b � k

K

pC

po

c pD

po

d

although this kinetics predict the correct equilibrium

K �pC

po

c pD

po

d

pA

po

a pB

po

b

where po is the reference pressure.

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In the absence of solid evidence it is dangerous argue by analogy. As an example, the reaction

H2 + I2 → 2HIhas a simple mechanism and a reaction rate of the form

r � kpH

2

po

pI2

po� k

K

pHI

po

2

while the reaction H2 + Br2 → 2HBr

proceeds by a chain mechanism and has a complicated kinetics.

A reaction with a simple kinetics does not necessarily have a simple mechanism. As an example, the reaction

2N2O5 ↔ 4NO2 + O2

has a simple kinetics of the form r = kpN2O5

but has a rather complex mechanism

1 N2O5 ⇔ NO3 + NO2

2 NO3 + NO2 → NO2 + O2 + NO3 NO + NO3 → 2NO2

A simple mechanism such as

1 A+* ↔ A*2 B2+2*↔ 2B*3 *+B*→ AB+2*

may have a very complex kinetics.

Very different reaction mechanisms may predict the same overall reaction rate. Even if we have reliable data for the overall reaction rate over a large range ofreaction conditions we may be unable to distinguish between two differentreaction mechanisms.

Many mechanistic details cannot be deduced from an experimental determinationof the form of the kinetic expression.

As an example, the Temkin−Pyzhev rate expression for ammonia synthesisreproduces the experimentally observed kinetics quite well. However, this rateexpression was originally derived from a proposed mechanism which had boththe wrong key intermediates and the wrong rate−limiting step.

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2 Reaction mechanism

A net reaction such as A2 + 2B ↔ 2AB

often consists of a number of steps. Short−lived intermediates may be formed by somesteps and consumed in other steps, e.g

A2 + B ↔ A2B A2B + B ↔ 2AB

Evidently, we can always subdivide the steps further and introduce hypotheticalintermediates, e.g

A2 + B ↔ A2B A2B + B ↔ A2B2 A2B2 ↔ 2AB

This leads to the introduction of the concept of an elementary step. A step in a reactionmechanism is elementary if it is the the most detailed, sensible description of the step.A step, which consists of a sequence of two or more elementary steps is a composite step.

The question if a step in a reaction is an elementary step obviously depends on howdetailed the available information is. The reaction mechanism deduced from a few, crudemeasurements of the reaction rate may consist of a small number of elementary steps. If we then decide to investigate the reaction through quantum chemical calculations, wewill most likely find that many of these steps are in fact composite. The key features of amechanistic kinetic model is that it is reasonable, consistent with known data andamenable to analysis.

The description of a net reaction as a sequence of elementary steps is the mechanism forthe reaction.

2.1 Complications

There are a number of features a reaction mechanism may have, which greatlycomplicates the situation.

2.1.1 Kinetic equations are non−linear

For mechanisms where all steps consist of unimolecular reaction steps, the kinetics of thereaction is available analytically for arbitrarily large mechanisms. However, kinetic

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expressions for elementary steps are not necessarily first order in the concentration ofreactants. In a mechanism consisting of several steps, steps may even have differentsame order.

2.1.2 Inerts

An inert adsorbate does not have a well defined chemical potential and if inert surfacespecies are present, the model is not soluble without additional assumptions on thebehavior of the inert.

Adsorbed inerts with constant coverage are better described by an adjustment of thenumber of adsorption sites. Adsorbed inerts with variable coverage are better describedas reactants.

2.1.3 Non−consecutive steps

The reaction mechanism does not necessarily consist of a sequence consecutive steps. Apart from the trivial case where consecutive steps are written in random order, somemore interesting possibilities are

One or more steps have been written "backwards"E.g step 2 in the mechanism:1 A_2 + * ↔ A_2* 2 2A* ↔ A_2* 3 B + * ↔ B* 4 A* + B* ↔ AB + *

The steps may not appear to be consequtiveThe mechanism has been written such that all steps except one has the form

ni2A1 + ni2A2 + ... + * = mi1B1 + mi2B2 + ... C*where A1, A2, .. B1, B2 .. are all gases and C* is an adsorbed molecule.

Parallel steps Parallel steps convert the same reactants into the same products through differentroutes.

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2.1.4 Elusive intermediates

In a reaction mechanism, short−lived intermediates may be formed by some steps andconsumed by other steps. The mechanism may contain intermediates, which have notbeen observed experimentally.

The introduction of an hypothetical intermediate in the mechanism is in many cases anecessity to link observed the intermediates formed from the reactants with the observedintermediates formed form the products. If the calculated concentration of thehypothetical intermediate is too small and the lifetime too short to allow the experimentalobservation, the introduction of the hypothetical intermediate is of no consequence for the agreement between the model and experimental results. The introduction ofhypothetical intermediates in excess of the absolutely necessasary is not sensible.

2.1.5 Undetectable steps

A mechanism may contain steps that are irrelevant as they are of no consequence for theconsistency of the mechanism and of no consequence for the reaction rate. Someexamples of undetectable steps are � A slow reaction step that is short circuited by a sequence of equilibrium steps.

The net rate of the slow step is then zero.� A fast step in series with a slow step� A slow step in parallel to a fast step.

2.1.6 Dead ends

One or more steps may form a dead end in the form of an intermediate formed throughan elementary reaction and consumed exclusively by the reverse of this step. Althoughthe dead−end will not contribute to the overall reaction rate, the step may affect thekinetics if the intermediate is strongly adsorbed on the surface. The poisonous effect ofH2O in ammonia synthesis is an example.

2.1.7 Linear dependence between reaction steps

A reaction mechanism may have linearly dependent reaction steps. This may happen fortwo reasons.

First, the same reaction step occurs more than once with different kinetic parameters, e.g

A* + B* ↔ AB* + * with A=109 and E#=6 kJ/mol.

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A* + B* ↔ AB* + * with A=1013 and E#=52 kJ/mol

where the first equation describes a low barrier, low temperature channel and the seconddescribes a high barrier, high temperature channel.

Second, steps may be combined, such as the following steps that occur in the water−gasshift reaction

1 H2O + * ↔ H2O* 2 H2O* + * ↔ OH* + H* 3 OH* + * ↔ H* + O* 4 2OH* ↔ H2O* + O*

Although linear dependence among the reaction steps complicates the analysis of amechanism, the mechanism is physically meaningful provided the mechanism isstochiometrically and thermodynamically consistent.

2.2 Consistency

For a proposed reaction mechanism, there must be a sequence of steps that leads fromreactants to products. This requirement is implicit in the definition of a reactionmechanism.

All intermediates occur as reactant for at least one step and as product for at least onestep. This requirement is essentially the definition of an intermediate.

Further all reaction step must have a thermodynamics and all slow steps must have a rate.

If an reactant, intermediate or product participate in two or more steps, the stoichiometryof molecule must be independent of the way the intermediate is formed. This is theprinciple of stoichiometric consistency.

If two or more different sequences of steps lead from reactants to products, thesesequences must describe the same gas phase thermodynamics. This is the principle ofthermodynamic consistency.

2.3 Guidelines

There are some rules of thumb which can guide the formulation of reaction mechanisms.

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The reaction enthalpy of each step is moderate.If the reaction enthalpy is large and positive, the activation energy in the forwarddirection must be large and the reaction rate will be negligible. Either the step inquestion is in reality an irrelevant byway in the mechanism or the overall reactionwill have negligible rate. If the reaction enthalpy is large and negative, the activation energy in thebackward direction must be large and the reaction rate will be negligible. Eitherthe step in question is in reality an irrelevant byway in the mechanism or theoverall reaction may have problems establishing equilibrium.

For any step, the number of reactants and product moleculesIf the number of molecules is large, the activation entropy will be large and wehave the same complications as for large reaction enthalpies. Actually theproblem is a little worse, because it will go away at high temperatures.

For any step the number of broken or formed bonds are small.

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3 Kinetics

The rate of chemical reactions can be described at two levels: dynamics and kinetics.Dynamics is the description of the rate of transformation for individual molecules. Themolecule has a well−defined energy, it may even start in a well−defined quantum state.There is no temperature. Temperature is a property of a large number of molecules, notindividual molecules.

The detailed microscopic description of a chemical reaction in terms of the motion of theindividual atoms taking part in the event is known as the reaction dynamics. The study ofreaction dynamics at surfaces is progressing rapidly these years, to a large extent becausemore and more results from detailed molecular beam scattering experiments arebecoming available.

Kinetics is the description of the rate of reaction for a large number of molecules. Themolecules have a temperature, although the temperature may change in the course of thereaction. The energy is well−defined, but the energy is a statistical average.

3.1 Reversibility

For a reaction, e.g. A2+ 2B → 2AB

the reverse reaction 2AB → A2 + 2B

will proceed through the same mechanism, although the sequence and the direction ofeach of the elementary reactions is reversed This is known as the principle ofmicroscopic reversibility

The cause of this principle is that in the kinetic description, we explicitly assume that theintermediates equilibrate at the reaction temperature. This implies that the intermediateshave no memory how they are formed. A formed by dissociation of AB is identical to Aformed by dissociation of A2.

As the reaction proceeds through the same steps in the forward and in the backwardreaction, while the rate of the individual steps may differ by many orders of magnitude itis convenient to consider two classes of steps. Fast steps have a high rate in both forward and backward direction, while slow stepshave a low rate in the forward direction, in the backward direction or both.

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3.2 The reaction rate

For net a reaction, say, A2 + 2B ↔ 2AB

with mechanismA2 ↔ 2A A + B ↔ AB

each step in mechanism proceeds with some rate r+ in the forward direction and some rate r− in the backward direction. The net rate of the step is obviously r= r+ − r−

The rate is a function of temperature, pressure and concentrations r(T,p,xi). The rategenerally decreases with time as the composition approaches quilibrium.

While the forward reaction rate for the net reaction may depend on the concentration ofboth reactants and products, the forward rate of each elementary step can depend only onthe concentration of reactants for this step.

This leads to the expression of the rate of as the number of times the reaction proceedsper second, the turnover frequency. The rate is thus a rate for the reaction, not the ratefor the reactants or for the products.

If ni is the number of moles produced of product number i and νi is the stochiometriccoefficient for product number i, the turnover frequency is

r � ni�i

This turnover frequency is obviously the same for all products. The reactants havenegative stochiometric coefficients and are "produced" with negative rate, so the turnoverfrequency is actually the same for all reactants and products.

3.3 Rate laws

In the simplest case the reaction rates are proportional to the coverages e.g for themechanism

A2 + 2* ↔ 2A* B + * ↔ B*

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A* + B* ↔ AB + 2* the rates are

r 1� k+1

pA2

po

�*02 � k−1

�A*2

r 2� k+2

pB

po

�*

� k−2

�B*

r 3� k+3

�A*

�B*

� k+3

pAB

po

�*2

The assumption that rates are proportional to coverages eliminates some, but not all,hysteresis phenomena.

3.4 Forward and backward rate

For each step the rate, r = r+ − r−, is obviously the difference between a forward rate, r+,and a backward rate, r−.

For each of the gases we have an formation rate. For many applications, it is sufficient todetermine the rate of formation for each of the gases. However, if we want to determinethe kinetic parameters for the net reaction, the form of the rate expression must bedetermined.

3.5 Stoichiometric matrix

For a systematic treatment of mechanisms, we need a suitable mathematical device. Onpossibility is to use a stoichiometric matrix to represent the mechanisms in symbolicform.

We write the reaction mechanism using a the stoichiometric matrix, α. For a mechanismconsisting of G gases, S adsorbates including free sites, and R reactions, α is a R by G+Smatrix.

We use the convention than αrc < 0 if c is a reactant of step r, αrc > 0 is a product of stepr and αrc = 0 if c does not participate in step r.

We will frequently need products or sums running over subsets of the molecules. We willuse the convention that the molecules are enumerated with gases number 1,...,G, freesites is number G+1 and adsorbates are number G+2,...,G+S.

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3.5.0.1 Example

As an example we will consider the mechanism for ammonia synthesis. We include Ar inthe gas phase to illustrate the effect of inerts. This mechanism is rich enough to illustratemost of the features discussed below.

For this mechanism R=7 (steps 1 to 7), G=4 (N2, H2, NH3, and Ar), S=7 (∗, N2* , N*,NH*, NH2, NH3, and H* ).

The stoichiometric matrix is

c=1 c= 2 c=3 c=4 c= 5 c=6 c= 7 c=8 c=9 c=10 c=11

r=1 −1 0 0 0 −1 1 0 0 0 0 0

r=2 0 0 0 0 −1 −1 2 0 0 0 0

r=3 0 0 0 0 1 0 −1 1 0 0 −1

r=4 0 0 0 0 1 0 0 −1 1 0 −1

r=5 0 0 0 0 1 0 0 0 −1 1 −1

r=6 0 0 1 0 1 0 0 0 0 −1 0

r=7 0 −1 0 0 −2 0 0 0 0 0 2

3.5.0.2 Properties

αrc has a number of interesting properties:

Surface sites are conserved �r � 1

Rr r � rc

� 0 for c=G+1,...,G+S

All elements in the stoichiometric matrix are integers. The use of non−integer stoichiometric coefficients is unnecessary and greatlycomplicates the treatment.

Gas inerts have αrc =0 for r=1,...,R.

3.5.0.3 Rate

The rate is calculated as a turnover frequency i.e as a number of molecules produced1

per site per second.

1 The distinction between reactants and products depends on which gases are present in the initial mixture. From now on, we use the word product for all gases wheninformation on the initial mixture is unknown or irrelevant

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The reaction rate for step r is r r

� r +r� r −r

r +r� kr � c � 1,

rc 0

G pc

po

� rc � c � G � 1,

rc 0

G � S �rc

� rc

r −r� kr

K r

� c � 1, rc 0

G pc

po

rc � c � G � 1,

rc 0

G � S �c

� rc

For each of the gases we have an formation rate

r c� �

r � 1

R � rcr r

this rate is evidently negative for the reactants.

3.5.0.4 Equilibrium equation

As a consequence of this choice of sign for αrc theequilibrium constants are

K r� � c � 1

G pc

po

rc � c � G � 1

G � S �c

rc for r=1, ..., R

3.6 Rate constant

At equilibrium the net rate is by definition zero. If we compare the rate equation for anelementary step, e.g the rate equation

r 1� k+1

pA2

po

�*2 � k1

�A*2

and the equilibrium equation

K 1

pA2

po

�*2 � �

A*2

for the step A2 + 2* ↔ 2A* we find that the forward and backward rate constants arerelated

k � � k+

K

Usually both k+ and k− have Arrhenius form

k � Aexp� H#

RTat least for small or moderate variations of T. In this equation Ais the preexponentialfactor while H# is the activation energy

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H# � RT2 d ln k

dT

If k+ and k− have Arrhenius form, the equilibrium constant, K,

K � k+

k−

will have Arrhenius form as will any product or quotient of rate and equilibriumconstants. The activation energies and the energy of reaction will be related by

H−# � � H � H+

#

3.7 Rate limiting steps

It is often the situation that most reaction steps in a mechanism are fast, while a singlestep is much slower than the other. In this situation, the slow step is called the ratelimiting step (RLS) or the rate controlling step as it determines the rate of the overallreaction.

Let us return to the question why the slowest step controls the overall rate of the reaction.As an example consider the net reaction

A2 + 2B ↔ 2ABwith mechanism

A2 + B ↔ A2B A2B + B ↔ 2AB

and assume that the first step is much slower than the second. In this case we will havethat

r1+, r1− << r2+, r2−

However, A2B is a common intermediate in these two steps. If A2B is an intermediate,rather than a product, the concentration of A2B must be essentially constant, i.e the netrate of formation of A2B must be essentially zero

r1+ − r1− − r2+ + r2− = 0

A rearrangement yieldsr1+ − r1− = r2+ − r2−

i.e the net rate of the fast step is the same as the net rate of the slow step.

To summarize, the first step converts A2 + B. to A2B at some rate and dissociates a littleA2B back into A2 + B, while the second step rapidly converts A2 + B into 2AB and 2ABback into A2 + B. Still the massbalance and the fact that A2B is an intermediate results inthe net reaction for both steps. The following figure provides an illustation of thisprinciple:

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3.8 Most abundant reaction intermediate

For many reaction mechanisms one of the reaction steps is much slower than all othersteps and this slow step determines the net rate of the reaction. There is an analogoussituation with respect to the concentration of reaction intermediates.

In many reaction mechanisms there are several intermediates, but frequently theconcentration of one of the intermediates is much larger than the concentration of allother intermediates. This intermediate is then called the most abundant reactionintermediate (MARI).

It may be tempting to speculate that the most abundant reaction intermediate should beeither the product of the first reaction step or one of the reactants for the rate limitingstep. This is not the case.

3.9 Reaction order

The forward rate typically has the form

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r1+

r1−

r2+

r2−

r

r +� k

pA

po

A pB

po

B

...

where αA is the reaction order for A.

The reaction orders for the forward and backward reactions are related through theequilibrium equation. If we write the backward rate as

r −� k

pA

po

�A pB

po

�B

...

and the equilibrium equation as

K � pA

po

�A pB

po

�B

...

we haveαA = βΑ + γΑ αΒ = βΒ + γΒ

and

r � kpA

po

A pB

po

B

... � k

K

pA

po

A

� �A pB

po

B

� �B

...

� kpA

po

A pB

po

B

1 � 1

K

pA

po

�B pB

po

�B

...

The thermodynamics of the overall reaction thus provides a connection between thereaction orders for the forward and backward reaction orders.

There is no simple connection between the reaction orders for the overall reaction and thestochiometry of the rate limiting step. For the same reason, the experimental reactionorders for the overall reaction provides essentially no information on the reactionmechanism.

3.10 The activation energy

The rate−limiting step can usually be described as an energy barrier the system mustcross.

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The rate constant then has the temperature dependence:

k � Aexp� E#

kBT

where A is the pre−exponential factor and E is the activationenergy.

3.10.1 Activation energy vs reaction energy

The rate constants, k+ and k− and the equilibrium constant, K, are thus related by

K � k+

k−

and the activation energies, E+# and E−

#, and the energy of reaction, ∆E, are related by∆ E = E+

# − E−#

3.10.2 Graphical determination

24

The activation energy can be determined graphicallyfrom experimental determination of the rate constantthrough an Arrhenius plot.

3.10.3 Analytical determination

The activation energy can be determined analytically from an expression for the rate:

E+# � kBT2 d ln r +

dT

3.10.4 Activation energy for composite reactions

While the Arrhenius form of the rate constant for an elementary step is easy to justify, the situation is more complicated for composite steps or even complete reactionmechanisms.

The complexity occurs because even if all the rate constants for the elementary stepshave Arrhenius form, the overall rate constant will in general not have Arrhenius form. We can force the net rate to have Arrhenius form only if we allow the apparent activationenergy to depend on the reaction conditions.

Only in the case that the activation energy is more or less independent of the reactionconditions can we determine the activation energy for the mechanism and only in thiscase can the activation energy be interpreted as an energy barrier. Fortunately, for mostreactions of practical interest, the dependence of the activation energy on the reactionconditions is weak except at extreme reaction conditions.

The question is then how to define the activation energy for a composite reaction. Theobvious answer is that as the activation energy is determined experimentally through anArrhenius plot, we should use the Arrhenius plot to define the activation energy for

25

composite reactions.

3.10.5 Compensation effect

The activation energy for a reaction is sometimes measured under different reactionconditions. An example might be a measurement using a very active catalyst at amoderate temperature and a measurement using a less active catalysts at highertemperatures. The increase in temperature partially compensates for the lower activity.

We might expect to find the sameactivation energy, however, one usuallyfinds different activation energies anddifferent preexponential factors:

A large value for the activation energy iscorrelated with a large prefactor and alllines in the Arrhenius plot intersect in asingle point, the isokinetic point.

The correlation between activation energyand preexponential factor is known as the compensation effect.

There are a number of possible explanations for the compensation effect:� The most interesting case is when the compensation effect is caused by differences inthe transition state. In this case the compensation effect is a real physical effect.� A trivial explanation is when each of the measurements has been made over a narrowrange of temperatures. The slopes of the lines are then very uncertain and the observedcompensation effect may be statistical insignificant. � If the kinetic expression used to convert the measured rate into a rate constant is notquite right, the conversion will be more or less in error. If error depends ontemperature, the Arrehenius plot will display a fictitious compensation effect.

3.10.6 Two reactions in series

One of the simplest cases of two consequtive reation steps is the Michaelis−Mentenkinetics:

A + * ↔ A*

26

A* → B Assuming that the second step is irreversible and using the steady−state approximation,we can determine the coverage

k1

p

po1 � � � k1

K 1

� � k2

� � 0

and the reaction rate

r � k2

� � k1 k2

k1

p

po� k1

K1

� k2

From the rate we can calculate the activation enthalpy for the reaction

kBT2 d lnr +

dT� H1

# � H2# � k1

p

poH1

# � k1

K1

H1# � � H � k2H2

#

k1

ppo � k1

K1

� k2

The activation enthalpy has two limits. � When the first step is rate−limiting, k1 <<k2 and H# = H1#� When the second step is rate−limiting, k1 >> k2, and H# = H2

#+∆ H(1−θ)

Going back through the calculation, we see that the first term comes from k2 and iscaused by the rate limiting step speeding up when the temperature is increased. Thesecond term comes from the differentiation of θ. When the temperature is increased, therate will change partially due to changes in θ. The second term describes this effect.

For two consequtive reactions the reaction enthalpy for the overall reaction is the sum ofthe reaction enthalpies for each of the steps

� H � � H1 � � H2

For the activation enthalpy, there is no similar sum−rule. The activation enthalpy for theoverall reaction is not the sum of the activation enthalpies for each step, actually theactivation enthalpy for the overall reaction is only slightly larger than the activationenthalpy for the slowest step.

27

3.11 Rate and conversion

When we measure a reaction rate, we are usually not measuring the rate itself, instead wemeasure a conversion, i.e a change in concentration from the inlet to the outlet of areactor or the change in concentration over some time interval.

Generally the reaction rate depends on the concentration in a non−linear fashion. Thechange in concentration divided by the contact time or some other rubbish is not ameasure of the reaction rate.

We need to know or to assume the dependence of the rate on the concentration in orderto calculate the reaction rate from the observed change in concentration.

In general this is difficult for realistic models:

In the next section we will look at the pseudo−1−order kinetics, which is just about theonly realistic kinetic model that can be treated by hand. We will return to the treatmentof realistic models later.

3.12 Pseudo−first order kinetics

The pseudo−1−order reaction has the kinetics:d x � xe

dt� � k x � xe

where xe is the equilibrium concentration.

If the concentration is x1 at t=0 and x2 at t=t we can solve the differential equation�x1

x2 dx

x � xe

� � kt

The solution is

28

lnx2

� xe

x1� xe

� � kt

and after rearrangement we getx2

� xe � x1� xe exp � kt

For a catalyst bed with mass m and flow F, the contact time is proportional to

t � m

F

The pseudo−1−order reaction cannot easily be generalized to a pseudo−nth−orderreaction.

d x � xe

dt� � k x � xe

n

If n is even the rate is negative for all concentrations and the composition after infinitetime is zero, independent of the value of xe.

3.13 Analogy between reactors

There is an important equivalence between position in a reactor andthe contact time. The upper instrument is placed at the exit of the catalyst bed. Themiddle instruments moves along the catalyst bed with the same linearvelocity as the gas flowing in the catalyst bed. When the yellowinstrument reaches the end of the reactor, the green and the yellowinstruments show the same concentrations illustrating the equivalenceof position and contact time.

The lower instruments monitors a closed container sealed off at timezero. The blue and the yellow instruments show the sameconcentrations from time zero until the yellow instruments reachesthe end of the catalyst bed, illustrating the equivalence between flow

and batch reactors.

When we calculate the conversion for a given reaction, the important parameters are thetime and the reaction rate. It does not matter if we let the reaction run in a closedcontainer or if the gas flows while it reacts.

29

30

4 Solving the reaction scheme

4.1 The full solution

In this case we don’ t make any approximations in the reaction scheme. We consider aLangmuir Hinshelwood mechanism with stoichiometric matrix αrc.

The kinetic model for this mechanism consists of the forward and backward rates.

r � r� kr � c � 1,

rc 0

G pc

po

� rc

� c � G � 1, rc 0

G � S �rc

� rc

r � r� kr

K r

� c � 1, rc 0

G pc

po

rc

� c � G � 1, rc 0

G � S �rc

rc

and the balance over sites

�c � G � 1

G � S �c

� 1

The solution to this system of equations evidently gives a correct description ofboth equilibrium, steady state and transient behavior.

If the reaction mechanism contains more than one or at most two steps, the full solutionbecomes very complicated and we will have to solve for the rates and coverages bynumerical methods. Although the full solution contains the steady state behavior as aspecial case, it is not generally suitable for studies of the steady state as the transientsmay make the simulation of the steady state a numerical nightmare.

4.2 Steady state approximation

In the steady state approximation the net rate of formation for all intermediates is set tozero.

In the steady state approximation the net rate of formation for all intermediates isexplicitly set to zero,

31

�r � 1

Rr r � rc

� 0 for c = G+1,...,G+S

Assuming that the net rate of formation for intermediates is zero does not imply that thecoverage by the intermediates is small.

For a mechanism with S−1 intermediates in addition to free sites the steady stateequations is a system of S equations, at most (S−1) are linearily independent.

The steady state approximation eliminates transient behavior in the kinetics. However, itis only the transient behavior of the rates and coverages that has been eliminated. Theexpression for the rate obtained through the steady state approximation is perfectlysuitable for the simulation of e.g the conversion through a catalyst bed or most aspects ofthe transient behavior of a reactor.

4.3 The quasi−equilibrium approximation

If all steps except one are fast, we can use the quasi−equilibrium approximation: For thefast steps we use the corresponding equilibrium equations instead of the kineticequations.

K 1� � c � 1

G pc

po

1c � c � G � 1

G � S �c

1c

... � ...

r r� kr � c � 1,

rc 0

G pc

po

� rc

� c � G � 1, rc 0

G � S �rc

� rc � kr

K r

� c � 1, rc 0

G pc

po

rc

� c � G � 1, rc 0

G � S �rc

rc

... � ...

K R� � c � 1

G pc

po

Rc � c � G � 1

G � S �c

Rc

This approximation will in most cases provide a very significant simplification inparticular for large reaction mechanisms. In the quasi−equilibrium approximation thetransient behavior is eliminated. Further, the description of changes in rate−limiting stephas been lost.

32

4.4 Irreversible step approximation

In the irreversible step approximation, we neglect the forward or backward rate for oneof the steps. For small mechanisms the irreversible step approximation may be usedalone, for larger mechanisms it is usually combined with the quasi equilibriumapproximation

K 1� � c � 1

G pc

po

1c � c � G � 1

G � S �c

1c

... � ...

r r� kr � c � 1,

rc 0

G pc

po

� rc

� c � G � 1, rc 0

G � S �rc

� rc

... � ...

K R� � c � 1

G pc

po

Rc � c � G � 1

G � S �c

Rc

This approximation is very crude as we have lost the description of the approach toequilibrium. The model will now continue to convert all reactants right across theequilibrium, often ending in a spectacular numerical instability when the concentration ofa reactant becomes negative.

Although this approximation is useless for the quantitative modeling of reactions, it hastwo important uses in the analysis of reaction mechanisms:� If we want to determine the limiting behavior of a kinetic model very far from

equilibrium, the irreversible step approximation is the appropriate limit. � If we have difficulties making sense of a complicated reaction mechanism, theirreversible step approximation may provide a simplification, which allows us tounderstand the mechanism well enough to choose a better approximation.

4.5 MARI approximation

The most−abundant reaction intermediate (MARI) approximation is a furtherdevelopment of the quasi−equilibrium approximation. Often one of the intermediates ismuch more abundant than all other intermediates, the coverages by the less−abundantintermediates may then be neglected in the balance over sites.

If we assume that A* is the most stable intermediate and the balance over sites becomes θ* + θA* = 1

33

In the MARI approximation, we have made one of the intermediates MARI at allreaction conditions. This has two consequences 1. We have lost the description of a change in MARI. 2. We have explicitly assumed that all intermediates except the MARI are much less

abundant than free sites. In particular the latter consequence is troublesome as an estimate of the validity of thisaspect of the MARI approximation amounts to solving the problem without using theMARI approximation. In other words, if the validity of the MARI approximation can beverified, only if it is superflous.

The MARI approximation can be used for quantitative modeling, if we have verified thatit is valid at the reaction conditions we are considering.

The MARI approximation is very much used for the analysis of reaction mechanisms,both when we have difficulties in formulating a kinetic model for a complicated reactionmechanism and when we want to derive a limiting form of a kinetic model.

34

5 Adsorption

5.1 Introduction

When a gas reacts with a solid, the most common situation is that a new compound isformed and that the crystal structure of the solid is destroyed, Fe reacts with H2S to formFeS, Al reacts with O2 to form Al2O3 etc. However, there are reactions where the crystal lattice of the solid is only slightly odified:absorption, where the gas enters the bulk of the solid, and adsorption, where the gasrmains at the surface. There are two main classes of adsorption: physisorption and chemisorption.

The generic case is of course the case, where the crystal structure of the solid isdestroyed.

5.2 The Lennard−Jones picture

In adsorption the gas molecules form some kind of bond with the surface, the strength ofthis bond depends on both the surface and on the gas molecule. The nature of the bondvaries from very weak van der Waals interactions to very strong chemical bonds.

35

Gas solid reactions

Generic Absorption Adsorption

Chemisorption Physisorption

While the binding energy is very variable, thequalitative form of the potential energy surfacewas deduced by Lennard−Jones in 1932.

The abscissa is the reaction coordinate and mayloosely be interpreted as the height over thesurface. As the molecule approaches the surface,it first feels a weak, long−rangedattraction. Near the surface there is an energyminimum and closer to the surface there isrepulsion in the form of a steep increase inenergy.

In reality, the potential energy surface is multidimensional and depends on the height, theposition and orientation of the molecule. The Lennard−Jones picture is obviously an oversimplification, but is still an extremely useful way of analyzing a reaction, and it hasbeen shown in a number of cases to work remarkably well.

5.3 The potential energy surface

If we consider the energy as a function ofthe height, z, above the surface and theinteratomic distance, r, in the molecule, wecan represent the energy as a contour plot.

The existence of adsorption sites causesproblems for the Lennard−Jones picture, theinteraction energy depends on manycoordinates, not just the height above thesurface.

The transition state is at the arrow.

36

A vibrating molecule might follow the blackcurve. Only the start of this curve is shown.The continuation may be either simple orextremely complicated.

The outer part of the potential energy surfacedescribes the motion of the molecule far from thesurface. The motion along z is a smoothtranslation, while the motionalong r is just the vibration of the free molecule.

The inner part of the potential energy surfacedescribes the motion of the chemisorbedmolecule.

The motion along z is now a stiff vibration, whilethe motion along r has become softer due to theweakening of the intermolecular bond.

5.4 Motion of adsorbed molecules

For most adsorbates the motion along the surface is at reasonably low temperatures betterapproximated by 2 vibrations around an equilibrium position.

37

For a three−dimensional gas 3 translational degrees of freedom exist. Upon adsorptionthese 3 dimensional degrees of freedom are transformed into one degree of freedomorthogonal to the surface and 2 degrees of freedom parallel to the surface. The degrees offreedom in the adsorbed state, which originates from the translation of the gas−moleculewill be referred to as parallel and orthogonal frustrated translation in the following.

The degree of freedom orthogonal to thesurface is of vibrational nature. If theelectronic structure of the surface does notaffect the movement of the molecule, the2 degrees of freedom parallel to thesurface will be translational and theadsorbed molecule will behave as a two−dimensional gas. However, as a

considerable variation in binding energy exists over the unit−cell. Assuming that thepotential varies smoothly over the unit cell we can approximate it by

V x � 1

2E0 1 � sin kx

The free translation has a very high entropy, while the entropy of a vibration is moderate.For this reason the adsorption entropy is usually negative. At a given pressure theequilibrium between gas and adsorbate will shift towards desorption when thetemperature is increased.

For surface species one or more of the rotational degrees of freedom may be restricted.Such frustrated rotations are better described as vibrations.

5.4.0.1 Example A2*

In the gas−phase the A2 molecule has three translational, one vibrational and a rotationaldegree of freedom.

In the adsorbed state these degrees of freedom aretransformed into one rotational and 5 vibrational degreesof freedom.

38

A rotation of the molecule around the vertical axis.This is essentially one of the rotations in the gas phase.

A vibration, which is the remains of the other rotation inthe gas phase.

A vibration parallel to the surface. This is the remains of the free translation along x

A vibration parallel to the surface. This is the remains of the free translation along y.

A vibration orthogonal to the surface. This is the remains of the free translation along z.

An intramolecular vibration.This is essentially the intramolecular vibration in the freemolecule.

6 Catalysis

As a catalyst takes part in the chemical reaction, but is not consumed by the reaction, thecatalyst must be a reactant in one of the first steps in the mechanism and a product in oneof the last steps. The reaction proceeds in a cyclic fashion, where the catalyst, or morecorrectly the catalytic sites, are regenerated and used again and again.

39

The activity of a catalyst can be written as a product of two factors, the number of activesites and the turnover frequency. The turnover frequency is the subject of this section,we will return to the number of active sites later.

6.1 Langmuir−Hinshelwood mechanism

When we consider a catalytic reaction, we may imagine that the reaction mechanismconsist of many different steps. Each of these steps may be of different types, we mayimagine that adsorbed species react with each other, that surface species may migrate intothe bulk, that reactive radicals desorp and then react in the gas−phase etc. Fortunately,real reaction mechanisms appear to be rather simple.

The Langmuir−Hinshelwood mechanisms form an important class of reactions. Thesemechanisms consist of the following types of steps:

� Adsorption from the gas−phase� Desorption to the gas−phase� Dissociation of molecules at the surface� Reactions between adsorbed molecules

The questions � if the reaction has a Langmuir−Hinshelwood mechanism� and what is the precise nature of the reaction stepscannot be solved without either experimental or computational studies.

6.1.1 Example

The reaction A2 + 2B ⇔ 2ABmay have the following mechanism

A2 + * ⇔ A2* A2* + * ⇔ 2A*

B + * ⇔ B*

A* + B* ⇔ AB* + *

AB* ⇔ AB + *

40

6.1.2 Example

If AB* is formed through the following steps A2* + B* ⇔ A2B* + *

A2B* + B* ⇔ 2AB* the mechanism remains a Langmuir−Hinshelwood mechanism.

6.1.3 Example

If we have evidence that the reaction appears to proceed asA2 + 2* ⇔ 2A*

B + * ⇔ B*

A* + B* ⇔ AB* + *

AB* ⇔ AB + *then we cannot immediately conclude that the mechanism is not a Langmuir−Hinshelwood mechanism. Actually, to rule out that this mechanism is not a Langmuir−Hinshelwood mechanism, we need to rule out that the first step may be a compositereaction

A2 + * ⇔ A2*

A2* + * ⇔ 2A* with a vanishing small equilibrium concentration of A2.

6.2 Eley−Rideal mechanism

The Eley−Rideal mechanisms form an important class of reactions. These mechanismsconsist of the following types of steps:

� Adsorption from the gas−phase� Desorption to the gas−phase� Dissociation of molecules at the surface� Reactions between adsorbed molecules� Reactions between gas and adsorbed molecules.

The last type of steps cannot occur in a Langmuir−Hinshelwood mechanism.

41

6.2.1 Example

The reaction A2 + 2B ⇔ 2AB

may have the following Eley−Rideal mechanismA2 + * ⇔ A2*

A2* + * ⇔ 2A*

A* + B ⇔ AB + *where the last step is the direct reaction between the adsorbed molecule A* and the gas−molecule B.

6.2.2 Example

However, without further evidence we cannot conclude that the above mechanism is anEley−Rideal mechanism. The last step may be composite and consist of the followingsteps

B + * ⇔ B*

A* + B* ⇔ AB* + *

AB* ⇔ AB + *with a vanishing small equilibrium concentration of B* . The mechanism is thenLangmuir−Hinshelwood and not Eley−Rideal.

6.3 Eley−Rideal or Langmuir−Hinshelwood

If we can � vary the coverage of A* and monitor this variationor if we can � vary the ratio between A2 and B we can discover if the mechanism is Eley−Rideal or Langmuir−Hinshelwood.

The trick is that the step B + * ⇔ B* requires a free site. If we measure the reaction rateas a function of the coverage by A* , the rate will initially increase for both mechanisms.

42

For the Eley−Rideal mechanism, the rate will increasewith increasing coverage until the surface is completelycovered by A* .

However, for the Langmuir−Hinshelwood mechanism therate will go through a maximum and end up at zero, whenthe surface is completely covered by A* . This happensbecause the step B + * ⇔ B* cannot proceed when A*blocks all sites.

6.4 The principle of Sabatier

When different metals are used to catalyse the same reaction, it is generelly observed thatthe reaction rate can be correlated with the position of the metal in the periodic table:

This plot is, for obvious reasons, called a volcano curve and the principle that the pointswill fall on a smooth curve is called the principleof Sabatier

It is generally found that the activation energyfor dissociation of simple diatomic moleculesdecrease when going left from the noble metalsin the periodic table. This can be described mostsimply in terms of an increased interactionbetween the anti−bonding adsorbate states andthe metal d−states.

During the adsorption of simple diatomicmolecules the anti−bonding molecular orbitals are gradually filled. For CO, for instance,the anti−bonding 2π∗ states are partly filled for the chemisorbed molecule and fills evenmore during the dissociation process, and a similar picture holds for the other simple gasmolecules.

The transfer of electrons to the anti−bonding molecular orbitals is in general most facilewhen the metal work function is small. Low work functions are found for the most opensurfaces. The work function can also be lowered locally by adsorbed electropositivespecies like alkali atoms.

During the electron transfer the anti−bonding molecular states must be close to the

43

Fermi−level of the metal, and when the metal has d−states around the Fermi level, as inthe transition metals there will be a strong covalent interaction between the anti−bondingstates and the metal d−states.

Trends in dissociative energies and activationenergies for dissociation as a function of thenumber of d−electrons. The results are calculatedin the Newns−Anderson model including thecoupling between an adsorbate level$\epsilon_a$ and the metal d−band.

The interaction tends to stabilize the adsorbingmolecule more for metals towards the left in thetransition metal series which have fewer delectrons than ten, the largest effect being aroundthe middle of the series where the number of delectrons is around five. We can use the model to begin understandingwhich metals are active and which are not. Weknow roughly how the stability of theintermediates and the barrier for N2 dissociationvaries through the periodic system. If we includethese variations in the kinetic model, we get thefollowing variations in the ammonia activity:

The calculated ammonia production for a fixedset of reaction conditions as a function of thenumber of d−electrons.

For the elements left of Fe, the kinetics is similarto the kinetics over Fe, but the rate is low due tothe strong bonding of N* . For the elements rightof Fe, the rate is low due to the low stickingcoefficient for N2, the coverage by N* is low andthe predicted kinetics is somewhat different fromthe kinetics over Fe.

Fe (and the other elements with approximately 7 d−electrons) are the optimum choice notbecause the sticking probability is high or because there is much free surface, but becauseit is the best compromise between the two effects.

These model calculations have treated all the transition metals as equivalent except forthe d−band occupancy. This is of cause a gross oversimplification and much moredetailed calculations are needed for a detailed picture. The simple description does,

44

however, give a physical picture of the main trend.

Going to the finer details the interaction energy does, for instance, depend on the d−bandwidth, even in the simple Newns−Anderson model. The main effect is that the narrowerthe band the stronger the interaction. This is an additional reason why, in the calculationsdescribed in the previous section, the open surfaces have lower activation energies thanthe more close packed ones. The surface atoms in an open surface have a lower metalcoordination number and since the band width is roughly proportional to the square rootof the coordination number, the band width is smaller.

6.5 Structure sensitivity

The reactivity of different facets of a metal may be different. It depends on both themetal and on the reaction, if the reaction rate will be different on different facets and ifthe difference is small or large. Reactions where the rate varies significantly from onefacet to another are called structure sensitive.

It is rather surprising that most reactions appear to be more or less structure insensitive.Some possible explanations are� Sites consist of only a few atoms and the local geometry of the site is more or

less the same independent of the structure of the facet. � The surface reconstructs and the the pure surface has more or less the samestructure independent of the orientation of the bulk orientation.� The adsorbate layer reconstructs and has a structure which is more or lessindependent of the structure of the pure surface. � The coverage under reaction conditions is high and differences in adsorbate−adsorbate interactions balance the differences in adsorption energy.

6.6 Physisorption

In physisorption the bond is a van der Waals interaction and the adsorption energy istypically 5−10 kJ/mol. This is much weaker than a typical chemical bond and thechemical bonds in the adsorbing molecules remain intact. However, the van der Waalsinteractions between adsorbed molecules is not much different from the van der Waalsinteraction between the molecules and the surface. For this reason many layers ofadsorbed molecules may be formed.

45

6.7 The BET isoterm

The Brunauer−Emmett−Teller (BET) isoterm follows from the following assumptions1. The adsorption takes place on a lattice.2. The first adsorbate layer is adsorbed on the solid surface, the second adsorbate layer is

adsorbed on the first etc. Except, of course, for the first layer, a molecule can only beadsorbed on a given site in layer number n, if the same site is occupied in layer n−1.

3. At the saturation pressure p0 the number of adsorbed layers is infinite.4. The adsorption enthalpy is H1 for molecules in the first layer and HL for molecules in

the following layers.

The BET isoterm isp

N p0� p

� 1

nmC

� C � 1 p

nMC p0

where n is the amount of gas adsorbed at pressure p nm is the amount of gascorreponding to one monolayer. p0 is the saturation pressure, n is infinite at p=pm.

C is a constant,

C � exp� H1

� H L

kBT

where H1 is the adsorption enthalpy for the first layer and HL is the adsorption enthalpyfor the following layers.

6.8 Chemisorption

For chemisorption the adsorption energy is comparable to the energy of a chemical bond.The molecule may chemisorp intact or it may dissociate. The chemisorption energy is30−70 kJ/mol for molecules and 100−400 kJ/mol for atoms.

The number of sites is constant and the competition for the adsorption sites has importantconsequences for the macroscopic kinetics. This is the reason for treating the surfacesites as if they were a reactant in the reaction equations.

The competition for adsorption sites is very important for the kinetics of a heterogeneouscatalytic reaction. For this reason sites, * , are included as a reactant in the kineticmodel. As a site must be either free or occupied by one of the surface intermediates,there is a conservation law for the coverages

46

� �X

� 1

where θX is the coverage by the intermediate X.

In writing this equation we have implicitely defined θX=1 to be saturation. With thisconvention, coverages may be interpreted as probabilities.

An isoterm is the coverage, θ, considered as a function of temperature and pressurte,θ(T,p).

In the next two sections we give a simple and intuitive derivation of the Langmuirisoterm before we present a much more general derivation.

6.8.1 The langmuir isoterm

An adsorption isoterm describes the coverage, θ function of temperature and pressure,θ(T,p).

The simplest, useful adsorption isoterm is the Langmuir isoterm. This isoterm was firstderived by Irving Langmuir in 1916 and is one of the oldest concepts in surface scienceand catalysis.

The Langmuir isoterm occurs when� The adsorbate forms a monolayer.� There is zero or one adsorbed molecule at each site.� All sites have the same adsorption energy.� There is no interaction between sites.

6.8.1.1 Non−dissociative adsorption

If we consider the reaction

A + * ↔ A*and for a moment assume that the mechanism for the reaction is equal to the net reaction. The adsorption rate is proportional to the pressure, p, and to the coverage by free sites,1−θ, while the he desorption rate is proportional to the coverage by A* . Theproportionality constants are actually the rate constants.

At equilibrium the adsorption rate balances the desorption rate

47

k f

p

po1 � � � kb

�Solving this equation with respect to θ leads to the Langmuir isoterm

� � Kp

po

1 � Kp

po

where

K � k f

kb

6.8.1.2 Dissociative adsorption

If we consider the reactionA_2 + 2* ↔2A*

and for a moment assume that the mechanism is identical to the net reaction, theadsorption rate will be proportional to the pressure, p, and to the probability, (1−θ)2 offinding a vacant pair of sites. The desorption rate is proportional to the probability, θ2, of finding an occupied pair ofsites.

At equilibrium the rate of adsorption equals the rate of desorption:

k f

p

po1 � � 2 � kb

� 2

Solving this equation with respect to the coverage, θ, gives the Langmuir isoterm

� � Kp

po

1 � Kp

po

6.8.1.3 Competitive chemisorption

If instead of n molecules of A* adsorbing on S sites, we have n1 molecules of A* , n2

molecules of B* etc we obtain the following equations

48

K1

�*

pA

po� �

A*

K 2

�*

pB

po� �

B*

... � ...

6.9 Sticking

Sticking is a process where a gas molecule collides with a surface and ends in anadsorbed state. There are two interesting aspects of this process, the probability that themolecule is adsorbed and the state of the adsorbed molecule. The probability that anatom or molecule hitting the surface will adsorp is known as the sticking coefficient. Ifthe sticking coefficient is $\sigma$, the adsorption rate is σ .

6.9.1 σ as a rate

In the following we will usually calculate turn−over frequencies, i.e the number ofreactions per site per second. The adsorption rate, σ , is the rate per m2 per second.

One may obtain an expression for the sticking coefficient, σ , by equating the rate ofadsorption in the kinetic model to the expression used to define σ .

If the rate constant is k, the turnover frequency in the forward direction is

r � kp

po1 � �

If we consider a surface with area A (m2) and density of sites d (mol m−2), we can equatethe rate of adsorption (mol/s) in the two models

r � kp

po1 � �

dA � pA �2 � mkBT

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which after simplification gives us the desired transformation between rate constant,sticking coefficient, rate and turn−over frequency.

From this equation it is obvious that while the rate constant, k, generally does not dependon the coverage, the sticking coefficient must depend on the coverage. The value usuallyreported for the sticking coefficient is the initial sticking coefficient σ0 corresponding toθ0.

6.9.2 The activation energy for $\sigma$

There is a further complication in the interpretation of the sticking coefficient. If k hasArrhenius form, the activation energy for the sticking coefficient is larger than theactivation energy for k by ½ kBT. This difference is significant in most situations.

6.9.3 Flux and exposure

A partial pressure, p, of a gas with molecular weight M, corresponds to a flux, f,

p

2 � mkBT

where T is the temperature.

In words, the flux is the number of collisions per second pr m2. Even at atmosphericpressure the flux is huge. Each atom in a surface exposed to the air is hit by close to 109

gas molecules per second.

For all catalytic reactions the flux is − in very round numbers − a million fold largerthan the reaction rate. Except for laboratory experiments under vacuum, the reaction rateis saturated with respect to the flux and an increase in the flux will have a marginal effecton the rate for reactions at 1 atm or above.

However, many reactions are performed at high pressures. The reason is purelythermodynamic. High pressures are applied to shift the equilibrium in favor of thedesired product.

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6.9.4 The exposure

Integrating the flux over time gives the exposure, i.e the total number of collisions perm2. Informally exposures are often reported in Langmuir or as a number of monolayers.One Langmuir is 1 torr second. Within a factor of 2 or so, 1 L corresponds to 1 collisionper surface atom. The exact number of collisions can of course be worked out when if weknow the details. This is precisely what has been done if an exposure is reported as anumber of monolayers. 1 L is one collision per surface atom.

6.10 Temperature programmed desorption

The principle in TPD is that the surface is cleaned and then exposed to a gas at arelatively low temperature. When the desired coverage has been reached, the pressure isreduced, the temperature is increased linearily and the flux of desorbing molecules ismeasured.

The principle in TPD is that the surface with adsorbed molecules or atoms is placed infront of a mass spectrometer.

The temperature of the surface is then increased linearly with time. The final temperaure

51

is high enough to desorp all the adsorbed atoms and molecules but should not be so highthat the sample is damaged.

The desorption is performed in vacuum. Once a molecule has desorbed it is rapidlyeliminated by the pumps.

The events in TPD are easy to visualize.� At low temperatures the coverage is high, but the rate constant is low and thedesorption rate is low. As the temperature increases, the rate constant and thus thedesorption rate increases. � At intermediate temperatures the desorption rate is high as both the coverage and therate constants are high. � At high tempererature the molecules have desorbed, the coverage is virtually zero andalthough the rate constant is enourmous, the desorption rate is virtually zero.

6.10.1 Kinetics

When the surface is heated in a vacuum the rate equation for the schemeA2 + * ↔ A2* A2* + * ↔ 2A*

becomes d

�dt

� � K1 k2

� 2

If K1 and k2 have Arrhenius form, the kinetic equation becomes

d�

dt� � A1 A2exp

� E1 � E2

kB T0 � � t

� 2

where β is the heating rate.

This equation is usually integrated using e.g a Runge−Kutta algorithm. The parametersare T0, A1A2 and E1+E2 addition to the initial value θA.

Note that only the product of the preexponential factors and the sum of the energies enterinto the equation. It is a very general problem in the interpretation of TPD spectra thatthe A’s and E’s appear so that the A or E for a single step cannot be determined.

As the A’s and E’s often are lumped, the generic rate expression for TPD has the form

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d�

dt� � k

� n

where the order, n, is 1 or 2, or, in rare cases, 0.

6.10.1.1 First order kinetics

The kinetic equation is

d�

dt� � k

�This is the kinetic equation one would expect for e.g for the mechanism A + * → A*

The generic shape of the desorption peak for TPD, first order kinetics:

The peak is not symmetric around the peak temperature(Parameters: A=$10^9$~$s { −1} $, E = 100 kJ/mol, initial coverage 0.95):

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The peak becomes larger, but the peak temperature remains constant when the initialcoverage is increased. This is characteristic for first order kinetics and is important fordiagnostic purposes (Parameters: A = 109 s−1, E = 100 kJ/mol, initial coverage 0.1 (lowerpeak), 0.3, 0.5, 0.7, 0.9 (upper peak).)

6.10.1.2 Redheads equation

The rate equation for TPD is

d�

dt� � k

� n

where k has Arrhenius form

k � Aexp � E#

kBT

TPD is one of the simpler experimental techniques and the rate equation makes itobvious that it should be possible to determine the activation energy for desorption froma TPD spectrum. However, a general problem in TPD is that we cannot measure thecoverage during the experiment. Also determination of the absolute reaction rate is ratherdifficult.

One possibility is to determine the activation energy from a detailed simulation of theexperiment. While this is doable, we will here rewrite the rate equation for TPD toeliminate the rate and the coverage.

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The derivative of the rate constant with respect to temperature is

dk

dT� E#

kBT2k

and the derivative of the temperature, T, with respect to time, t, is the heating rate

dT

dt� �

If we combine these equations, we can determine the derivative of k with respect to t.

From the rate equation we can determine the second derivative of the coverage:

d2 �dt 2

� � dk

dT

dT

dt

� n � nk� n � 1 d

�dt

This second derivative of the coverage is the first derivative of the rate. The peak in theTPD spectrum is in reality the maximum in rate and thus a zero for the second derivative

d2 �dt 2

� 0

By straightforward algebra we find

E#

kBT2k � � n � nk2 � 2n � 1

where T is the temperature of the peak maximum.

If and only if n=2n−1, i.e n=1, we obtain Redheads equation

E#

kBT� AT� exp

E#

kBT

where T is the peak temperature.

If we know that the desorption follows first order kinetics (n=1) and we have an educatedguess on the value of A (e.g from transition state theory), Redheads equation makes iteasy to determine E from a measurement of the peak temperature, T.

In a TPD experiment, the difference between the smallest and the largest applicable

55

heating rate is often less than a factor of 10. If we don’ t know A, we could measure theTPD spectrum using a range of heating rates and then use Redheads equation todeterming both E and A. However, a variation by a factor of 10 is usually too little for areliable determination.

Finally, a caution about the Redhead equation. The requirement n=1 is not explicit in theequation. For this reason nothing prevents that the equation is applied in situationswhere the order of the desorption is unknown or, even worse, when the order is knownto be different from 1. To describe E obtained in this fashion as an approximation is anunderstatement, completely wrong is a better description.

6.10.1.3 Second order kinetics

The kinetic equation is d

�dt

� � k� 2

This is the kinetic equation one would expect for e.g for the mechanismA2 + * ↔ A2* A2* + * ↔ 2A*

The generic shape of the desorption peak for TPD, second order kinetics. The peak is notsymmetric around the peak temperature. (Parameters: A=109 s−1, E = 100 kJ/mol, initial

56

coverage 0.95.

When the preexponential factor is increased, the peak moves towards lower temperatures.When the binding energy increases, the peak becomes broader and moves towards highertemperatures.

6.10.1.4 Zero order desorption

The kinetic equation is

d

�dt

� � k

This is the kinetic equation one would expect e.g when the adsorbate rapidly movesaround on the surface, but desorption takes place from a small number of defects. Thismakes the desorption rate independent of the coverage.

The generic shape of the desorption peak for TPD, zeroth order kinetics. (Parameters:A=109 s−1, E = 100 kJ/mol, initial coverage 0.95.)

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58

7 Catalyst structure and texture

7.1 Catalyst structure

Many, but not all heterogeneous catalysts used in industry, consist of small metalparticles on a support. As the catalytic reaction proceeds at the surface of the metal particles, the catalysts havebeen prepared to expose a large metal area, typically 10 − 100 m2 per gram of catalyst.

The function of the support is primarily to increase and stabilize the area of the metalparticles.

The size of the metal particles may be reported in three closely related ways� The area of the metal surface� The diameter of the metal particles� The dispersion, defined as the fraction of all metal atoms that are present at thesurface.

7.2 Pore structure

A catalyst is porous and contains pores. The pores are of course of a very irregular shape.

When the pore volume is measured, one usually find pores of two distinct sizes: � Macro pores have a diameter of 100 nm or more. Macropores are formed as cracksbetween between crystallites. � Micropores have a diameter of 1 nm or less. Micropores are formed by the roughnessof the surface.

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7.3 Catalyst models

In the study of reaction mechanisms, industrial catalysts are complicated to work with.For this reason, mechanistic studies are often performed on model systems, such as singlecrystals. For this purpose, single crystals are available of most metals in sizes of e.g 10by 10 mm.

Experience has shown that it is often much easier to study a number of single crystalsurfaces, deduce the reaction mechanism and then verify this for the real catalyst, than toattempt to deduce the reaction mechanism directly from studies of the catalyst.

7.4 The terrace−ledge−kink model

The surface of real crystals are not perfect. At high temperatures thermal disorder willlead to a non−zero concentration of defects. At low temperatures, defects created at hightemperatures may be frosen at the surface, defects in the bulk may extend to the surface.Also the orientation of the surface with respect to the crystal lattice may be oriented sothat an atomically flat arrangement of the atoms is impossible.

The simplest model of a surface with defects is the terrace−ledge−kink model. In thismodel only 4 types of defects are assumed present in the surface:

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A kink is a defect formed at the end of theledges.

If we tilt the surface to make the ledge horizontal, it should be obvious that the onlydifference between ledges and terraces is the width.

Adatoms are a limiting case of a terrace made up of a single atom.

7.4.1 Defects excluded in the TLK−model

Some of the defects excluded from the TLK model are cavities of missing atoms, surfacedislocations and bulk defects extending to the surface. The success of the TLK−modeldemonstrates that although such defects do exist and are interesting, they are in manycontexts not terribly important.

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7.5 The stereographic map

If we look at a sphere, all possible surface orientations will be present:

Only a few of the more important orientations have been shown.However, depending on the symmetry of the lattice, we have already shown some orientations where the corresponding surfaceplanes are equivalent.

As a example, for the face centered cubic lattice the (100), (010) and (001) planes lookidentical and the (110), (101) and (011) planes look identical. But the sphere also has abackside with a (−100) plane, a (0−10) plane etc what about them ? The answer comes inthe form of the stereograpic map. The stereographic map is an area on the unit sphere,which contains one representative for each equivalent surface orientation.

For the simple cubic, the body centered cubic and for the face centered cubic lattices thecorners of the stereographic map are (100) (110) and (111).

For BCC and FCC (100) (010) and (001) are equivalent. For the hexagonal closepackedlattice, (100) and (010) are equivalent, but (100) and (001) are not eqivalent and thestereographic map is considerably larger for HCP than for SC, BCC and FCC.

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7.6 Body−centered cubic lattice

Besides the conventionel, cubic cell, the BCC lattice can be build from a primitive cell.The primitive cell is akward for many purposes. First it is a parallelipiped and not cubic.Secondly, the crystallograhic directions are defined with respect to the conventional cell.

The figure shows the conventionel unit cell for BCC, this cell is cubic and contains 2atoms. Each atom has 8 neighbors.

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The BCC(100) surface.

Each surface atom has 4 neighbors in the surface plane and 1neighbor in the plane below.

The BCC(110) surface. Each surface atom has 4 neighbors in the surface plane and 2neighbors in the plane below. The (110) plane is the mostopen of the three basal planes for BCC.

The BCC(111) surface.This surface is very open, both the atoms in the first and inthe second layer have lost neighbors. The atoms in the firstlayer have 3 atoms in the second layer and 1 neighbor in thethird layer. The atoms in the second layer have 3 neighbors inthe first layer, 3 in the second layer and 1 in the third layer.

7.7 Face−centered cubic lattice

Besides the conventionel, cubic cell, the FCC lattice can be build from a primitive cell.The primitive cell is awkward for many purposes. First it is a parallelipiped and notcubic. Secondly, the crystallograhic directions are defined with respect to theconventional cell.

The figure shows the conventional cell for FCC. This cell is cubic and contains 4 atoms.

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The FCC(100) surface. Atoms in the first layer have 8neighbors.

The FCC(110) surface. This surface is the most open of the three basal planes forFCC. Atoms in the first layer have 7 neighbors and atoms inthe second layer have 11 neighbors.

The FCC(111) surface. Atoms in the first layer have 9 neighbors. (111) is the most closepacked of the three basal planes forFCC.

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7.8 Hexagonal close−packed lattice

Both FCC and HCP are closepacked structures, FCC has an ABC stacking, HCP hasABA stacking.

The HCP(100) surface.

The HCP(001) surface.

The HCP(110) surface.

7.9 Adsorbates and impurities

Adsorbates and other foreign atoms may form patterns on the surface of a crystal.Fortunately, the most common patterns are rather simple and a simple notation is used todescribe these lattices.

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p(2x2) on FCC(111). The "P" (primitive) indicates oneforeign atom and (2x2) indicates the size of the unit cellrelative to the original cell.

p(2x3) on FCC(111). Again "P" indicates one foreign atomand (2x3) indicates the size of the new unit cell. Thedifference between p(2x2) (above) and p(2x3) (left) is in thesize of the unit cell. Note that p(2x2) has a { \em higher}concentration of foreign atoms than p(2x3).

c(2x2) on FCC(100). The "C" (centered) indicates two foreignatoms in the cell and (2x2) indicates the size of the unit cellrelative to the original cell.

Note that c(2x2) corresponds to a concentration of foreignatoms, which is exactly twice that of p(2x2).

p(1x1) on FCC(100). One foreign atom ("P") for each (1x1)atom in the surface.

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7.10 The Wulff construction

The surface energy is different for different facets. For a crystallite consisting of a givennumber of atoms, the equilibrium shape is the shape which minimizes the surface (Gibbs)energy.

If we know the surface energies, the equilibrium shape can easily be determined from theWulff construction:

First, in 2 dimesions: In a polar coordinate system, draw avector parallel to the normal of the surface and withlength proportional to the energy of the surface. At theendpoint of the vector, draw a tangent line. Repeat for allsurfaces. The equilibrium shape is the area limited by thetangent lines.

Don’ t forget to use the symmetry of the crystal. For acubic crystal the surface energy for (100) will give you 6vectors and the surface energy for (111) will give you 8.

It can easily happen that a particular surface has such ahigh energy, that is will not be present at all in the finalconstruction.

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The construction in 3 dimensions is in principlethe same. You use a sperical (not polar)coordinate system, you draw the vectors and thenthe tangent plane (not line).

The figure below shows the Wulff constructionfor a typical FCC metal. The object is a cubo−octahedron. The larger hexagonal facets have a(111) orientation and the smaller, square facetshave a (100) orientation.

7.11 Homogeneous catalysts

Heterogeneous catalysts have a number of advantages and disadvantages compared tohomogeneous catalysts.

� In heterogeneous catalysis, the catalyst and the product can easily be separated. � The solubility of some homogeneous catalysts is rather small, � Some heterogeneous catalysist have problems with diffusion in the pore system.� Homogeneous catalysis can usually be performed at mild reaction conditions. Thehigh temperature in heterogeneous catalysis reduces the selectivity. � Heterogeneous catalysis can be performed over much larger ranges of reactionconditions, this is important when the equilibrium constrains the reaction to extremeconditions. � Homogeneous catalysis is possible in liquid phase and can be performed for reactantsthat are only stable in solution. Heterogeneous catalysis in solution is difficult.

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8 Preparation of catalysts

The details of the manufacturing process for a given catalyst is a buisness secret of thecatalyst manufacturer and only the broad principles are available in the open literature.

The secrecy of the manufacturing processes is maintained partially by discouragingoutside investigators through patent protection. However, the most important aspect ofthe protection is probably that examination of the finished catalyst offers littleinformation on how it was prepared and no information on why a particular preparationprocedure was used.

For a catalyst the desired properties are� high and stable activity � high and stable selectivity� controlled surface area and porosity� good resistance to poisons� good resistance to high temperatures and temperature fluctuations.� high mechanical strength� no uncontrollable hazards

Once a catalyst system has been identified, the parameters in the manufacture of thecatalyst are� If the catalyst should be supported or unsupported.� The shape of the catalyst pellets. The shape (cylinders, rings, spheres,

monoliths) influence the void fraction, the flow and diffusion phenomenaand the mechanical strength.� The size of the catalyst pellets. For a given shape the size influences only theflow and diffusion phenomena, but small pellets are often much easier toprepare.

Catalyst based on oxides are usually activated by reduction in H2 in the reactor.Some catalysts are prereduced at the factory and transported to the plant covered by athin layer of oxide.

8.1 Shape

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Pellet. Catalyst pellets for fixed bed reactorsare typically 1.5 − 10 mm in diameter.

Ring. Typical sizes are 6−20 mm. The centralchanel significantly improves the transportof the gas both in the pellet and in thecatalyst bed.

Extrudate. Typical sizes are 1−5 mm in diameter and10−30 mm in length. Extrudates are only used for impregnatedcatalysts.

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Pellet with several chanels. Typical sizes are20−40 mm in diameter and 10−20 mm inheight. Compared to the ring, the use of several,smaller chanels changes the trade−offbetween mechanical strength of the pelletand mass− and heat−transport phenomena.

Fragment. Typical sizes are 1−12 mm. \parFragments are usually prepared by crushingof catalysts manufactured by fusion.

Sphere. Typical sizes are 3−12 mm. Mainlyused for impregnates catalysts.

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Monolith.

Typical sizes are from 50x100x200 mm upto 250x250x250 mm or more.

The gas flows along the large chanels,which constitutes almost all of the volume.Monoliths are used when the flow rate of thegas is high, or when the gas contains dust orsoot.

8.2 Precipitation

Catalysts are prepared by precipitation typically by mixing one solution with anothersolution or a suspension. The precipitate is washed and filtered (or filtered and washed), dried, calcined and crushed to a fine powder. A binder, often graphite or stearic acid isadded and the powder is tabletted.

Alternatively, the catalyst may be shaped before calcination by extruding the filter−cake.

The filtering and washing are particularily troublesome. A clogged filter will disrupt theproduction line and may be difficult to fix in a safe way.

The wash liquid may contain small concentrations of transition metals or organic aminesin addition to larger concentrations of less troublesome compounds. Disposal of thisliquid may add significantly to the total manufacturing cost for the catalyst.

8.3 Pelletizing

Many catalyst are manufactured as powders which are pelletized as one of the last stepsin the manufacturing process.

The pellets are prepared by compressing the powder. The powder flows through a feedermechanism, and is filled into a rotating die, where it is compressed by pistons.

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At the start of the cycle, the lower piston is in the upper position. The pellet prepared inthe previous cycle is pushed out as the die passes under the feeder mechanism. Catalyst powder fills the boring as the die moves under the feeder mechanism while thelower piston moves down. The feeder mechanism scrapes the upper end of the boringclean. At this moment, the position of the lower piston defines the amount of catalystpowder in the finished pellet.

The upper piston moves down by a fixed length. At this moment, the position of theupper piston defines the dimension of the finished pellet. The pressure applied to thepellet and the density of the finished pellet are implicitely defined by the mechanicalproperties of the powder and by the movement of the pistons.

The cycle is completed when the upper piston moves up and the lower piston pushes thefinished pellet out of the die.

Before the catalyst powder is pelletized, "binders" such as graphite or stearic acid areadded. While stearic acid might act as a binder at room temperature, graphite is betterknown as a lubricant than as a glue. One might suspect that the "binders" work not asglue but by improvement of the flow properties of the powder.

8.4 Fusion

Catalysts based on metal oxides may be prepared by fusion, crushing and screening.Large pieces are returned to the crusher, small pieces are returned to the furnace. The finished catalyst consists of irregular pieces of a narrow range of sizes.

Fusion is only possible for catalysts that are conductors at high temperatures. Mostoxides are insulators at room temperature and become conductors at higher temperatures.

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The catalyst mass is placed as a powder in a electro−furnace. The mass is heated by passing a large currentthrough 3 graphite electrodes. Each electrode is equippedwith a mechanism for raising and lowering the electrode. At the start of the cycle, crushed catalyst mass is filled intothe furnace and the electrodes are lowered. The voltage isfixed and the current is regulated through the height of theelectrodes. The molten mass is stirred by the electromagnetic fieldsgenerated by the huge current flowing between theelectrodes. At the end of the process, the electrodes are raised and themolten catalyst is poured out of the furnace.

The furnace is operated at a rigid schedule. The thermal insulation is provided by a thicklayer of catalyst mass. If the furnace spends too long time at high temperatures, toomuch of the catalyst mass will melt.

The catalyst has no porosity before reduction. The pore−system is created when the metaloxide is reduced during the activation of the catalyst. The lack of porosity beforereduction makes the reduction tricky and the catalyst may be supplied in pre−reducedform. The pre−reduced catalyst is first completely reduced then passivated by gentleoxidation to allow it to be transported and loaded safely.

8.5 Catalyst supports

Al2O3

The most widely used supports are gamma−Al2O3 and eta−Al2O3 both have a higharea and a good stability.

The structure of both gamma−Al2O3 and eta−Al2O3 is a more or less skew FCCpacking of oxygen ions with Al−ions distributed somewhat disordered in theholes between the O−ions. The disorder is the reason why Al2O3 can disolve otherions.

Al2O3 is made by precipitation of Al−hydroxide followed by calcination.

SiO2 Kieselguhr has area 20−40 m2/g. Silica gel has areas up to 700 m2/g.

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Silica gel is prepared by adding acid to a solution of Na4SiO4 and Na2SiO3.

Activated carbonActivated carbon has areas up to 1200 m2/g.

Activated carbon is prepared by oxidation of porous carbon and containssignificant amounts of oxygen.

TiO2

TiO2 is more expenside and less used than Al2O3 and SiO2

ZrO2

ZrO2 is rather expensive and is only used as support for catalysts operating atextremely high temperatures.

MgOThe mechanical strength is poor. Not used in industry.

ZnOZnO has some tendency to reduction. Not used in industry.

Cr2O3

Cr2O3 has some tendency to reaction with H2O. High surface area facilitates theoxidation of Cr(III) to Cr(VII). Proper destruction of the used catalyst is ratherexpensive.

8.6 Impregnation

Impregnated catalysts are prepared by impregnating a metal salt on a porous support. Themetal loading in the finished catalyst is typically 1−5 %.

When liquid is slowly added to a porous solid powder, the liquid is first absorbed in thepores and the powder will flow as if it is dry. When the pores have been filled the outsideof the grains rather suddenly become wet, the grains will tend to stick together and thepowder will form lumps instead of flowing freely. The situation when the pores havebeen filled but the outside of the grains is dry is callled incipient wetness and can easilybe detected by shaking or stirring the powder.

If you want a hands−on illustration of this effect, put a handful of dry sand into a cup andadd water one spoonfull at a time while shaking the cup. At first the surface of the sand isdull. At some point the surface of the sand suddenly becomes glistening, and the sand isquite sticky. This is the incipient wetness. Add one more spoonful, and the sand becomeswet and starts to flow like mud.

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Catalysts are prepared by impregnation by spraying a solution of a metal salt onto pelletsof a porous support until incipient wetness. The pellets are then dried and calcined totransform the metal into insoluble form.

The metal salt can be deposited homogeneously through the pellet or most of the metalmay be deposited near the outside of the pellet. The distribution of the metal iscontrolled through the pH or through addition of chelating agents to the impregnationliquid.

Compared to precipitation, impregnation offers a number of advantages� The pellets are shaped before the metal is added.� The filtering and the wash of the catalyst are eliminated.� Small metal loadings are easily prepared.� Impregnation offers some control over the distribution of the metal in pellets.�and disadvantages� High metal loadings are not possible.� A good impregnation solution may be impossible to find.

8.7 Metal sponges and colloids

Highly porous metal sponges can be made in aqueous solutions by dissolution ofaluminium alloys by a strong base. The metal sponges are typically used forhydrogenation in laboratory syntheses.

Raney nickel is made by dissolving an (Al,Ni)−alloy containing 50 % Ni in 20 % NaOHin water. The nickel sponge is black, the area is 80−100 m2/g. The nickel contains someAl and Al2O3. The most used Raney metals are Ni, Co and Fe.

Metal colloids can be prepared by reduction of metal salts in solution using CO,hydroxylamine, oxalic acid, formaldehyde, citric acid or sodium citrate.

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9 Poisoning and deactivation

A model for the changes in reactivity for a reaction on a catalytic surface in the presenceof adsorbed inactive atoms. The model is based on a mean field description of theformation of partly disordered structures for the adsorbed atoms.

If other gases than the reactants and products of the catalytic reaction are present and ifthese foreign atoms are chemisorbed much more weakly than the intermediates of thecatalytic reaction, the presence of the foreign gas will be of no consequence for thekinetic of the catalytic reaction. In this case the foreign gas behaves as an inert.

If the foreign gas is chemisorbed much more strongly than the intermediates, the surfacewill to a large extent become covered by adsorbed foreign atoms. In this case the foreigngas behaves as a poison and the consequences for the kinetics will mainly be a largedecrease in the reaction rate.

The most complex situation occur when the binding energy for the foreign atoms is aboutequal to the binding energy for the reaction intermediates. The coverage of foreignatoms on the surface will then depend on the reaction conditions. The kinetics of the reaction under dynamic poisoning may be quite complicated.

In the present chapter we formulate a simple model for the changes in reactivity of the catalytic reaction under the conditions of a partial poisoning by adsorption of foreignatoms. For the understanding of the kinetic phenomena an approximate but transparentmodel is preferable.

9.1 Deactivation

A number of different processes contribute to the loss of catalytic activity.

9.1.1 Sintering

The catalytic reaction takaes place at the surface of the catalyst. For this reason thecatalyst must have a large area and the active component is present in the catalyst in theform of small particles, which are much less stable than large particles.

A catalyst will slowly loose catalytic activity due to growth of the particle and loss ofsurface area.

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There are two distinct mechanisms for sintering:The crystallites may continously emit and collect atoms.Since the atoms will be more stable at the surface of alarge particle than at the surface of a small particle, theatoms will on average migrate from the small particles tothe large.

The particles may move and when two particles come into contact, they may merge into a single, larger particle.

The loss of catalytic activity due to particle growth is irreversible.

9.1.2 Fouling

Catalytic activity may be lost due to the formation of carbon or due to the deposition ofimpurities or of dust in the catalyst.

The formation of carbon is in some cases reversible. The catalyst can be taken of streamand the carbon removed by oxidation.

The reactant gas may contain impurities, a few examples are metalorganic compounds inhydrodesulphurization or $H_2S$ in steam reforming. These impurities react with thecatalyst and reduces the catalysts activity. Fortunately, the reaction between catalystand impurity is often very strong and the impurity is completely absorbed in the first fewpercent of the catalyst bed. The catalyst bed is then designed to be a little larger thaninitially necessary, and the catalyst temperature is initially slightly lower than thenominal operating temperature. Over the lifetime of the catalyst, the temperature of thecatalyst is slowly increased to compensate for the loss of catalytic activity due to fouling.

Fouling by dust carried into the reactor can occur, a few examples are removal of$NO_x$ from flue gases or removal of CO from exhaust from car exhaust.In this case the catalyst must be manufactured in a shape, which will allow the dust topass through the reactor.

For most catalysts a much more serious problem with dust may occur, if the catalyst bedis improperly loaded or if there is a problem with excessive vibration in the reactor. In

79

this case dust may be formed by attrition. In this case the resulting loss of catalystactivity is overshadowed by two much more serious problems. � The dust may accumulate somewhere in the reactor or immediately downstream of the

reactor and rather suddenly block the gas flow. � The catalyst dust is highly reactive and may cause a violent explosion when pipes orheat exchangers are opened for maintenance.

9.1.3 Dynamic poisoning

The more interesting situation occurs when the catalyst is partially and reversiblypoisoned by impurities in the reactant gas. The degree of loss of catalyst activity thendepends on the operating conditions.

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10 Microkinetic modelling

The link between the microscopic description of the reaction dynamics and themacroscopic kinetics that can be measured in a catalytic reactor is a micro−kineticmodel. Such a model will start from binding energies and reaction rate constantsdeduced from surface science experiments on well defined single crystal surfaces andrelate this to the macroscopic kinetics of the reaction.

If one can understand what the basic parameters of the reactants and the surface are thatdetermine the reaction dynamics (activation barriers etc.) then given a micro−kineticmodel one has a knowledge of the factors determining the catalytic activity of thecatalyst.

The kinetics of a catalytic reaction is usually measured in a reactor under conditionsrelevant to the industrial process. The measured overall rates can then be fitted to amathematical model, the macroscopic kinetics. This is extremely convenient for process design purposes.

If the aim is to explore the mechanism of the reaction and understand which are theimportant parameters of the catalyst determining the activity, then a micro−kinetic model is needed. A micro−kinetic model is based on a detailed mechanism andindependent information about the rates of the elementary steps involved and the stabilityof the intermediates. The micro−kinetic model is the synthesis of all the basic knowledgeabout a reaction over a given catalyst.

A kinetic model consists of a description of the elementary steps at the atomic level ispresented. Input data for elementary steps are taken from available single crystal studies.The model is successfully tested against kinetic data for a working catalyst.

The purpose of our studies is not to present a kinetic model, which will reproduce one ora few of the aspects of water−gas shift reaction very accurately. The point we make isthat a physically reasonable treatment of the proposed reaction mechanism with kineticand thermodynamic data measured for Cu single crystals leads to a reasonably accuratedescription of most aspects of the observed kinetics.

The number of parameters in the model is large. One could suspect that only a fewparameters are critical, but we cannot know { \em a priori} which. Rather than trying todetermine this at an early state we concentrate on the determination of reasonable valuesfor all parameters. When this has been done, we can test the model, against independentexperimental data and we can perform a proper sensitivity analysis for the inputparameters. We then backtrack and concentrate on the determination of accurate values

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for the critical parameters. The advantage of this iterative scheme is that a much morecomplete sensitivity analysis can be made on the full model compared to an analysis for amodel where some parameters have been eliminated at an early state.

The starting point for microkinetic modeling is the detailed reaction mechanism. Thus,while a conventional kinetic model is formulated as the rate for an apparent gas phasereaction, the surface species are explicitly included in a microkinetic model.

In a microkinetic model, the simplest feasible model at themolecular level is formulated based on available data. The model is evaluated throughsimulation of kinetic data for the catalytic reaction at high pressures and the results of thesimulation is compared to existing kinetic datasets.

Successful models are useful for visualization of the reaction as well as for detailedinvestigation of kinetic and mechanistic features.

10.1 Limitations

1. We will limit ourself to catalytic reactions. Surface sites consumed in the adsorption

82

Kinetic data Electron structure Single crystals

Important aspects

Model

Evaluation

InterpretationsVisualization

of reactants must be regenerated in the desorption of products. The net production ofany surface intermediate must be zero.

2. We describe the reactions at the molecular level. The mass balance for the atoms isonly implicitly described through the mass balances for the molecules. We don’ tdescribe the structure of the molecules.

3. The gas phase is assumed ideal.4. Diffusion limitations and temperature gradients are neglected.5. The reaction mechanism is a Langmuir−Hinshelwood mechanisms comprising G

gases, S surface species and R elementary steps, where G, S and R are arbitrary.

10.2 Level of the model

The modelling of a surface chemical reaction can take place at many different levels ofcomplexity. At the most fundamental level the time development of the reaction systemis followed in detail and at the most approximate level only gross averages areconsidered.

10.2.0.1 The full reaction dynamics

At the most fundamental level one follows the time development of the system in detail. The reactants are started in a specific initial (quantum) state and the equation of motionare propagated to give the final state. The equation of motion of the system is the time dependent Schröinger equation, or, if the atoms involved are heavy enough (not H or Li)Newtons equation. The starting point is the adiabatic potential energy surface on whichthe process takes place. For some reactions electronic excitations during the reaction areimportant and must be included in addition to the electronically adiabatic dynamics.

The adiabatic potential energy surface is the ground state electronic energy of the systemas a function of all the degrees of freedom of the system.

Such a detailed description could in principle be made of every elementary step in agiven reaction, while a dynamical simulation of a whole chemical process includingseveral elementary steps is usually impossible. Typically intermediates in a reactioncan have lifetimes which are many orders of magnitude larger than typical times for a dynamical simulation (a few picoseconds). Another point is that if the aim is to get anelementary reaction rate for a system at a given temperature, the full dynamicalapproach may be too detailed.

10.2.0.2 Mean field model

In this description only average properties are considered. The rate of a givenelementary step involving adsorbates A and B are assumed given by

83

r = k θAθB

where θA and θB re the coverages of A and B, respectively, and the rate constant kusually has an Arrhenius form. The activation energy and prefactor may still depend onthe surroundings, but only in an average way through the coverages.

In the case where there is one slow step in the reaction mechanism, the solution for therate of the catalytic reaction is straightforward.

When the reaction mechanism has more than one slow step, the solution for the rate ismore complicated, but the final expression is remarkably similar.

The kinetic model may be formulated using kinetic equations for all steps or usingequilibrium equations for all but the slowest steps. The latter approach reduces thecomputational effort and leads to a kinetic expression, which is far easier to analyze.However, if a step, which is slow in reality, is modeled by an equilibrium equation, themicro−kinetic model becomes unrealistic and it may in some cases be the simplest totreat a problematic step by a kinetic equation.

10.3 Types of models

Even with the limitations listed in the previous section, there are a very large number ofmodels. We will our attention to the following 6 models:

Molecules have a name, a stoichiometry and a stability. One of the reaction steps is assumed rate limiting, this step has a forward rateconstant. This model maps onto the quasi equilibrium approximation,Section \ref{ sec:solve/quasieq} .

In this model the equilibrium constants for all steps are calculated fromstoichiometry and from the stability of the molecules. The rate of the fast steps isnot available.

Molecules have a name and a stoichiometry. Each step has a thermodynamics. One of the reaction steps is assumed ratelimiting, this step has a forward rate constant.

This model is a trivial variation the previous model. The equilibrium constants forthe steps are among the input parameters. The stability of the intermediates andthe rate of the fast steps are not available.

Molecules have a name, a stoichiometry and a stability. Each reaction step has a forward rate constant. This model maps onto the steadystate approximation, Section \ref{ sec:solve/steady} .

84

In this model the equilibrium constant for the steps are calculated from thestoichiometry and from the stability of the molecules.

Molecules have a name and a stoichiometry. Each step has a forward and backward rate constant. This model is a trivialvariation of the previous model. The equilibrium constants for the steps arecalculated from the forward and backward rate constants. The stabilities of themolecules are not available.

Molecules have a name, a stoichiometry and a stability. For each configuration of molecules at the surface, there are a number of possibleevents. The events occur randomly with a characteristic rate for each type ofconfiguration. This model is the kinetic Monte Carlo simulation.

In this model the equilibrium constants for the steps are calculated from thestability of the molecules. The rates are calculated by simulation.

Molecules have a name and a stoichiometry. For each configuration of molecules at the surface, there are a number of possibleevents. Each event has a thermodynamics and occur randomly with acharacteristic rate for each type of configuration.

This is a trivial variation of the previous model. The equilibrium constants areamong the input parameters. The rate is calculated by simulation. The stability ofthe molecules is not available.

Hybrids between these 6 models are possible, e.g by specifying the thermodynamics bythe thermodynamics for some steps and the stability for some of the molecules. However,these hybrids are much more difficult to work with than the 6 models defined above, inparticular for large mechanisms.

10.4 Phases in a simulation

Simulations consists of 3 phases: initialization, equilibration and production. At theinitialization, a configuration is generated or read from a file. The system is thenequilibrated by propagation in time. When the system has equilibrated, the productionphase starts. Data are stored on disk when the production phase starts and thenafter each major time step.

85

The major time step is an abstraction. The program internally builds up the major timesteps through a series of minor time steps.

It is a complication that in KMC the minor time steps are of varying length. As we wantthe output from the simulation program stored at constant time intervals, special care isrequired.

For mechanisms, there are two important aspects, stoichiometric and thermodynamicconsistency. However, we must make the description of the kinetic model moreconcrete before we can implement the approximation schemes and arrive at a solublemodel.

86

Minor timestep

Major timestep

Initialization

Production

Data Data Data Data Data Data

Equilibration

10.5 Input parameters

The input into a micro−kinetic model may be measured or calculated.

Measured data for the catalystThe input parameters for a microkinetic model may be taken from measuredadsorption and reaction rates for the catalyst, measured heats of adsorptiontogether with thermodynamic data for the gas (or liquid−) phase above thecatalyst.

This gives information of direct interest for the catalyst system considered, butoften the interpretation of the experiments is difficult due to the fact that usuallythe state of the active surface is not known and may vary with the conditions ofthe experiment.

Measured data for catatalyst modelsThe input parameters can be taken from measurements on model systems. If thestructure of the catalyst is known and one has a suspicion which is the activecrystal surface and do experiments on this model with all the chemical phase,then one can isolate this phase, usually in the form of a single and structuralcharacterization tools available in surface science.

Calculated dataElectronic structure theory has developed to a point where realistic bond energiesand activation barriers can be calculated. Typically the model catalysts used insuch calculations are even more idealized than in the surface science experiments(perfect surfaces, ordered overlayers etc.), but the insight into the details of thepotential energy surface of the reaction is much greater.

10.6 Test of the model

Fragments of the kinetic model may be tested by detailed simulation of the sameexperiments as have been used to determine the parameters in the model. Failure toreproduce these experiments is entirely possible if a wrong reaction mechanism has beenassumed.

A sensitive test can be made by simulation of a reactor through numerical integration ofthe rate expression. This will test the model at higher pressures, frequently at highercoverages, in the presence of all reactants, intermediates and products.

The model can be further tested for internal consistency. Steps treated through kineticequations should be slow at least under some conditions or the model may obviously be

87

simplified. The significant parameters should obviously be determined rather directly from experiments. The coverages of the intermediates calculated under reactionconditions should not be greatly different from the coverages in the experiments used todetermine the parameters.

10.7 Significant parameters

While the number of parameters may be large, only a few of the parameters are usuallysignificant. The significant parameters are easily determined by calculation of thesensitivity of the calculated rate at typical conditions to a small variation in the value ofeach parameter. For the NH3 synthesis, for instance, the rate of dissociation of N2 andthe binding energy for N* are the most significant parameters and the kinetics ofdesorption of N2 in a TPD experiment is rather closely related to the kinetics of NH3−synthesis.

10.8 Applications of microkinetic modelling

Once the kinetic model passes the tests, the model may be used to understand details ofthe catalytic reaction, such as the origin of the macroscopic activation energy, thereaction orders or the structure of empirical kinetics.

In spite of the shortcomings of the modelling, the real strength is that they can be used tounderstand { \em variations} in the catalytic activity from one system to another. Thestability of the intermediates and the activation barriers are among the input parameters for the micro−kinetic model, and it is straight forward to calculate the effects of changes in stability for some or all the intermediates.

10.9 Ammonia synthesis

1) N2(g) ↔ N2*2) N2 + * ↔ 2N*3) N* + H* ↔ NH* + *4) NH* + H* ↔ NH2* + *5) NH2*+ H* ↔ NH2*+ *6) NH3* ↔ NH3(g) + *7) H2(g) + 2* ↔ 2H*

88

In this mechanism, the second step is known experimentally to be slow.

The competition for adsorption sites is very important for the kinetics of a heterogeneouscatalytic reaction. For this reason sites, * , are included as a reactant in the kinetic model.

As a site must be either free or occupied by one of the surface intermediates, there is aconservation law for the coverages � �

X� 1

where θX is the coverage by the intermediate X.

In writing this equation we have implicitely defined θX =1 to be saturation. With thisconvention, coverages may be interpreted as probabilities.

For each of the fast steps, we get an equilibrium equation, e.g

K 1

pN2

po

� � � �N2

For a slow step we get an kinetic equation, e.g

r � k2

�N2

�*

� k2

K 2

�N2

which expresses the net rate, r, as the difference between the forward rate, r+, and thebackward rate, r−.

10.9.1 Solution for θ*

When the mechanism has only one slow step, the system of equilibrium equations andthe rate equation may be solved with respect to θ∗ �

*� 1

1 � K1 PN 2

p0

� pNH 3po

12

K 3K 4 K 5K 6K 7

32 pH 2

32

� pNH 3

K 4 K 5K 6 K 7 pH 2

� pNH 3

K 5K 6K 7 po12 pH 2

12

� pNH 3

K 6po� K 7

12

pH 2

12

po12

The coverages by intermediates may be expressed by θ* and the partial pressures of thereactants and products.

89

�N 2 *

� K1

PN2

po

�*

�N *

� pNH 3po

K 3K 4 K 5K 6K 7

32 pH 2

32

�*

�NH *

� pNH 3

K 4 K 5K 6 K 7 pH 2

�*

�NH 2 *

� pNH3

K 5 K 6 K 7 po12 pH 2

12

�*

�NH3* � pNH 3

K 6po

�*

�H *

� K 7

1

2pH 2

12

po1

2

�*

r � 2 k2K1

pN 2

po� pNH 3

2 po

K g pH 2

�*2

10.9.2 Calculation of the equlibrium constants

The equilibrium constants are expressed in terms of the molecular partition functions

qtrans� 2 � mkBT

h2

3

2

V

qrot� kBT

2B

qvib� 1

1 � exp� � �kBT

90

q � qtransqrot qvibexp� E g

kBT

10.9.3 Calculation of reaction enthalpies

The enthalpy of formation for reactants and intermediates may be calculated from theexpression for the partition function

H# � kBT2 d ln q

dT

The enthalpy obviously depends on temperature. However, even at T=0 the enthalpy ofan intermediate may differ substantially from the electronic binding energy due to the zero−point motion in vibrational degrees of freedom.

10.9.4 Calculation of the rate

The solution for the rate of the catalytic reaction is straight forward

r � 2K 2K 1

pN 2

po� 1

K g

NH3

po

2 H2

po

3 �{2 � }

The parenthesis has a simple interpretation in terms ofthe thermodynamics of the gas phase while the interesting part ofthe kinetics comes from the factor θ*

2.

10.9.5 Practical calculation

The calculation above closely follows the sequence of calculations when a reactionmechanism is turned into a rate expression. In principle, such a derivation is done oncefor a proposed reaction mechanism and the rate expression is then used repeatedly tocalculate the reaction rate at different reaction conditions.

The derivation of the rate expression takes, say, one hour of computation and the analysisof the rate expression takes, say, one week of programming followed by one month of

91

computation. For this reason the derivation of the rate expression is usually repeated afew times to check for errors.

Once we have derived a rate expression and carefully checked the derivation, thesequence of steps in the calculation of the rate is quite different from the the sequence ofsteps in the derivation of the rate expression.

The main steps in the calculation of the rate from the rate expression are:Properties of intermediates are deduced from spectroscopic data. This is done once foreach project.� At the actual reaction temperature, the molecular partition function for all

intermediates are calculated from the properties of each intermediate. � The equilibrium constant for each step is alculated from the molecular partitionfunctions. � The coverage by free sites is calculated from the equilibrium constants and the partialpressures of the gases. � The coverage by intermediates are calculated from the the coverage by free sites, theequilibrium constants and the partial pressures of the gases.� The reaction rate is calculated from the rate constant, the equilibrium constants andthe coverage by intermediates.

10.9.6 Parameters

The input parameters for the model are the thermodynamics of the gas phase,chemisorption energy and spectroscopic properties for the intermediates, the kineticparameters for the rate limiting step and the number of active sites on the catalyst. Noreference to experimental data for catalytic reaction rates are made in the determinationof the input parameters.

Only few of the input parameters are critical for the prediction of high pressure reactionrates. The critical kinetic parameters are the parameters for N2 adsorption anddissociation on Fe(111) and on K/Fe.

Fe(111) K/Fe

EN2 −38 kJ/mol −51 kJ/mol

EN −91 kJ/mol −91 kJ/mol

H2# 47.4 kJ/mol 47.4 kJ/mol\

A2 1.37 × 10 11 s−1 5.89 × 10 10 s−1

92

EN2 and EN are the ground state energies of N2* and N*, respectively. H2# and A2 are the

activation energy and the prefactor, respectively, for reaction step 2.

10.9.7 Stability of intermediates

10.9.8 The NH3 concentration

NH3 concentration in acatalyst bed operating at100 atm, 673 K,inlet 25 % N2 and 75 %H2, outlet 19.2 % NH3

corresponding to 75 %approach to equilibrium.

Significant variation in coverages occur from inlet to outlet. The most drastic variationstake place in the first approx 3 % of the bed. In this reaction zone the reaction rate ishigh.

It is a general feature at all temperatures and pressures that the coverages and the reaction

93

rate change dramatically when the NH3 concentration increases from zero to a fewpercent of the equilibrium concentration.

10.9.8.1 Coverage by *

Coverage by * through acatalyst bed operating at100 atm, 673 K,inlet 25 % N2 and 75 %H2, outlet 19.2 % NH3

corresponding to 75 %approach to equilibrium.

Throughout the bed the coverage by free sites is small. In the present case the coverageby free sites defreases from 5×10−2 at inlet to 2×10−4 at outlet. This is remarkable as thesynthesis of NH3 proceeds through the trapping of N2 on the free sites.The most important parameter in the calculation of θ* is the groundstate energy for N*

which is determined from experiments.

The NH3 synthesis rate is proportional to θ*2. Since the calculated synthesis rate is in

reasonable agreement with independent measurements, it is unlikely that the low valueof θ* should be significantly in error.

94

10.9.8.2 Coverage by N*

Coverage by N*through a catalyst bedoperating at 100 atm,673 K, inlet 25 % N2

and 75 % H2, outlet19.2 % NH3

corresponding to 75 %approach toequilibrium.

The surface species N*, NH*, NH2* and NH3* are all mainly formed from NH3. Forthis reason the coverage by these four species is close to zero at inlet.

The sequence of coverages under synthesis is θN* > θNH* > θNH2* > θNH3*. This sequence isdetermined by entropy rather than enthalpy and is independent of temperature andpressure.

10.9.8.3 Coverage by N2*

Coverage by N2* througha catalyst bed operatingat 100 atm, 673 K,inlet 25 % N2 and 75 %H2, outlet 19.2 % NH3

corresponding to 75 %approach to equilibrium.

N2 is a weakly adsorbed species. It is mainly formed from N2. The coverage by N2 issmall under synthesis conditions.

95

We will later find that N2* is also too small to influence the macroscopic kineticparameters. The kinetics of NH3−synthesis might thus have been adequately describedwithout explicitely including N2* in the mechanism.

However, the existence of N2* is important for the understanding of two aspects of theadsorption of N2 on Fe:� the origin of the uniquely low sticking coefficient for N2 � the description of chemisorption of N2 at low temperature.

Neglecting N2* is thus conceptually less simple than the explicit consideration of N2* inthe reaction scheme.

10.9.8.4 Variations with temperature

θ* increases rapidly with temperature far from equilibrium. Closer to equilibrium theincrease is more moderate. The increase is most pronounced at low temperatures andlow pressure.

θΝ* decreases with temperature. The variation is most pronounced at smal consersions. Atzero conversion, θΝ* is essentially zero at all temperatures.

θΗ* decreases with temperature at higher conversions, while it increases withtemperature at higher conversions. This complicated behaviour is caused by thedecreasing tendency of H2 to adsorb with increasing temperature combined with thecompetition between H* and the much more strongly adsorbed N*.

96

10.9.9 The reaction rate

Reaction ratescalculated for acatalyst operating at100 atm, 673 K andinlet gas withcomposition 25 % N2

and 75 % H2. Theexit NH3

concentration is 19.2% corresponding to75 % approach toequilibrium.

Significant variation in coverages occur from inlet to outlet. The most drastic variationstake place in the first approx 3 % of the bed. The reaction rate is high at inlet due to therelatively high coverage by free sites followed by a rapid decrease in the reaction rate asNH3 is formed. It should again be stressed that N* is mainly formed from NH3 and notN2.

10.9.10 The turnover frequency

For a catalyst operating at 673 K, 100 atm and 28 % approach to equilibrium theturnover frequency is 0.029 s−1. This can be further interpreted by analyzing the forwardand backward rate. Each active site turns over 0.031 times each second, 0.030 times inthe forward rate and 0.001 times in the backward direction.

Each turnover results in the synthesis of two NH3 molecules. Taking the surface coverageinto account the turnover frequency at these conditions corresponds to the synthesis of 71NH3 molecules per free site per second.

At inlet conditions the turnover frequency corresponds to each site synthesizing 250 anddecomposing 0 NH3 molecules per second.

10.9.11 Lifetime of intermediates

We do not know the forward and backward rate of the fast steps. For this reason we canonly estimate the lifetimes of the intermediates.

97

Assuming that the sticking coefficient for N2* is 10−2 and refering to a catalyst operatingat 673 K, 100 atm and 10 % to equilibrium, the coverage by N2* is typically 4×10,from which the average lifetime for N2* can be estimated to be 1×10−13 s.

For an N−atom on the surface, the upper limit is essentially zero at inlet conditions.At 673 K, 10.1 MPa, and 10 % approach to equilibrium, the upper limit to the lifetime is1.10 s for N* , 0.18 s for NH*, 24 ms for NH2* and 0.4 ms for NH3* .

10.9.12 Activation enthalpy

The activation enthalpy for the catalytic reaction may be calculated from the micro−kinetic model as

H# � kBT2 d lnr +

dT

where r+ is the forward rate of NH3 synthesis

r +� K 2K1

pN 2

po

�*2

The result is H# = H2

# + H1

− 2H1θN2*

− 2 (H3+H4+H5+H6+3/2 H7)θN*

− 2 (H4+H5+H6+H7)θNH*

− 2 (H5+H6+1/2 H7)θNH2*

− 2 H6 θNH3*

− H7 θH*

The terms in the activation enthalpy can be interpreted as follows:

98

Enthalpy Reaction

−H1 N2* ↔ N2 + *

−(H3 + H4 + H5 + H6 + 3/2H7) N* + 3/2 H2 ↔ NH3 + *

−(H4 + H5 + H6 + H7) NH* + H2 ↔ NH3 + *

−(H5 + H6 + 1/2H7) NH2* + 1/2 H2 ↔ NH3 + *

−H6 NH3* ↔ NH3 + *

−1/2H7 H* ↔ 1/2 H2 + *

This table allows us to interpret the activation enthalpy for the synthesis as the activationenthalpy for N2 + 2* → 2N* plus the averaged enthalpy to be supplied to create two freesites. The average is formed by weighing by the coverage for each intermediate.

The activation enthalpy for synthesis is not constant but depends on the temperature andthe surface coverages.

10.9.13 Reaction orders

The reaction orders for N2, H2 and NH3 can be defined as

99

� i� d ln r +

d lnpi

po

No assumptions on αN2 αH2 or αNH3 have been made.

From the forward rate of NH3

r +� k2K1

pN 2

po

�*2

the reaction orders can be calculated

α N2 = 1−2θN2*

α H2 = 3θN* + 2θNH* + θNH2* −θH*

αNH3 = −2θN*−2θNH* −2θNH2*−2θNH3*

The reaction orders are not constant but depend on the surface composition.

As θN2* is very small under all experimentally feasible synthesis conditions the reactionis always first order in N2.

At very small conversions θH* is large and αH2 is close to −1. This indicates that the

100

reaction is inhibited by H2 at low conversion.

At high conversion θN* is close to 1, αH2 is close to 3 and αNH3 is close to −2. At theseconditions H2 increases the reaction rate by decreasing θN*.

10.10 Why does it work ?

One of the important conclusions of the microkinetic modeling is that even large changesin some parameters do not affect the overall agreement between model and experimentmuch.

The reason is that a larger sticking probability through the principle of detailed balanceimplies that the TPD rate increases. To get the correct TPD peak, the N bonding energyon the surface must then be increased, this leads to a smaller coverage of free sites whichcompensates for the larger sticking probability.

To see that this is a more general conclusion consider the kinetics in more detail. Usingthe rate equations given above we can write the rate of NH3 production as

r � k � 2 K1 K 2

pN2

po� �

N *2

where we have also used the equilibrium of reaction step (1) to express the forward ratedirectly through the gas phase N2 pressure. Likewise, using that all steps after step (2)are also in equilibrium we can express the coverage of free sites as

�*

� K g

K 1K 2

pH2

po

3

2

pNH 3

po

12

�N *

giving the following expression for the synthesis rate:

r � k � 2

�N *2 K g

pN2

po

pH2

po

3

pNH3

po

� 2 � 1

101

r � r TPD K g

pN2

po

pH2

po

3

pNH3

po

� 2 � 1

In the last equation we have introduced the N2 TPD rate,

r TPD� k � 2

�N *2

and it is seen that the synthesis rate can always be expressed as the product of the TPDrate and the approach to equilibrium.

If, therefore, one uses a consistent description of the synthesis rate and the TPD rate,including the principle of detailed balance and makes a reasonable fit to the TPDspectrum in the relevant coverage and temperature range, one cannot avoid getting agood description of the synthesis rate. Since the N2 TPD peak is in the same temperaturerange as typical synthesis temperatures it is, in hindsight, not so surprising as firstthought, that something good has come out of relatively simple modeling. The key to thesuccess is consistency.

102

11 Experimental methods

Although experimental methods are outside the scope of this manuscript, we haveactually touched upon the computational aspects of a number of experimental methods.

Pure surfaces to be used as catalyst models may be prepared in a number of ways:� A filament may be purified by heating to high temperature in UHV. For mostapplications it is a complication that the area of the filament is small.� Metal films may be prepared by evaporation. For some applications it is acomplication that the crystallography andsometimes even the area of the film is notwelldefined.

A catalyst may be examined in its working state. However, most catalysts are extremelyreactive in its working state and it may be necessary to perform measurements before thecatalyst is activated or after it has been passivated.� In in−situ measurements the catalyst is examined in its reactive state. The advantage is

that the catalyst is known to be in the relevant state and this can easily be verified bydirect measurement of the catalytic activity. The disadvantage is that the catalyst mustbe inside some kind of reactor and there is not much choice with respect to pressure,temperaure or gas composition.� In ex−situ measurements the catalyst is outside a proper reactor and it is in a statewhich is more or less different from the active state. The advantage is that we have awide choice of temperature, pressure and gas composition. The disadvantage is thatwe don’ t really know how far the state is from the active state and we have no easyway of finding out.

11.1 Titration of active sites

The density of active sites at the catalyst surface is of course of great interest.

The density of sites may be determined by two methods� In volumetric chemisorption the catalyst is activated, gas is removed by evacuation ata relatively high temperature and the catalyst is then exposed to a known amount ofgas. From the final pressure and the volume of the cell, the amount of adsorbed gasmay be calculated. � If the desorption rate is measured quantitatively in a TPD experiment, the number ofactive sites in the surface can be determined by integration of the desorption rateacross the peak.

103

Temperature programmed reaction

Temperature programmed desorption is one limit of a more general technique,temperature programmed reaction (TPR).

In TPR the catalyst is heated in a reactive gas and this opens the possibility of adsorptionof the reactive gas and the reaction between the adsorbate and the gas in addition to thedesorption of the adsorbate.

In an intermediate form, TPD is performed in a flow of an inert gas instead of in avacuum.

11.2 Rate measurements

Reaction rates are measured in either a plugflow or in a backmix reactor.

104

105

12 Ideal reactors

12.1 Rate

The reaction rate for a catalytic reaction obviously depends on the amount of catalyst,pressure, flow, temperature and composition of the gas.

Reaction rates are often reported on different basis � The rate per m3 of catalyst bed.� The rate per m2 of catalyst area. � The rate per kg of catalyst.

The space velocity is the flow reported as the volume flow (m3/s) divided by the volumeof the catalyst. The rate of reaction may be reported as the space velocity correspondingto a particular concentration of product in the exit gas. This is very convenient for aquick estimate of the necessary size ot the size of the catalyst bed, but very awkward forany other purpose.

The reaction rate for the catalyst is the product of the density of sites and the turn overfrequecy.

12.2 The tank reactor

The flow in Fi with temperature Ti and mole fractions x1i, x2i, ..., xni, the flow out is Fe

with temperature Te and mole fractions x1e, x2e, ..., xne. The mixture in the tank has thesame temperature and composition as the product.

As the temperature of the flow in and out of the reactor may be different, the enthalpy ofcompound j hji} at temperature Ti may be different from the enthalpy hje at temperatureTe. The heat transfered to the reactor is �Q .

The mass and energy balances, are

106

In + Produced = Out

1 Fix1i + rν1V = Fex1e

... ... + =

n Fixni + rνnV = Fexne

sum Fi + r�

i

�i V = Fe

energy �Q � F i

�jx ji h ji

+ 0 =�

jFex jeh je

The equations for the tank reactor are "larger" than the equations for the plugflowreactor. However, there is no integration for the tank reactor.

For the adiabatic tank reactor, �Q � 0 . Unless the reaction happens to bethermoneutral, Ti and Te will be different.

For the isothermal tank reactor Ti=Te and hji=hje for j=1... n.

12.3 The semibatch reactor

The equations controlling the operation of ideal reactors are the energy balance and amass balance for each reactant and product.

We assume that n reactants, products and inerts participate in a reaction withstoichiometric coefficients ν1, ν2, ... νn$.

The reaction rate ([mol/s/m^3]) is

r � 1

V �i

d ni

dt

where V is reaction volume (m3) and ni ([mol]) is the amount of compound i.

We use the convention that heat, work and flow is positive when they go into the system.Heat released from the system is a negative amount of heat transfered to the system etc.

The mole fractions add up to unity �

ixi

� 1

Initially, the reactor is partly filled with some of the reactants in the liquid phase. The reaction starts when other reactants are added, either as liquid or as gas.

107

Some of the reaction mixture may be withdrawn from the reactor during the reaction. As a result of the flow into and out of the reactor, the volume of the reaction mixturemay change.

The reaction rate is r = r(T,c1... c_n)

For compound j, the molecular mass is M j, the heat capacity is Cj and the amount in thereactor is nj.

As the temperature of the flow in and out of the reactor may be different, the enthalpy hji

of compound j at temperature Ti may be different from the enthalpy hje at temperature Te.

The density, ρ, of the reaction mixture depends on the reaction temperature and thecomposition ρ = ρ(Te, c1, ..., cn)

The mass and volume of the reaction mixture ism � �

n j M j

V � m�If j is added a gas the rate of addition of compound i is

�n ji� F i x ji

where xji is the mole fraction of j in the gas and Fi ([mol/s]) is the gas flow into thereactor. However, if it is added as a liquid�n ji

� �V i c ji

where cji ([mol/m^3]) is the concentration of j in the feed and �V i [m3/s] is the flow rateinto the reactor.

The rate of withdrawal for compound j is �n je� c je

�Ve

where cje is the concentration of j in the product and is �Ve the flow rate out of thereactor.

The mass energy balance (”Initial + in + produced = out + final”) for component j is

n j � �n ji � r �j V

� �n je � n j � dn j

and the energy balance is

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�n j h je � �Qdt � � �n j h ji dt � 0 � � �n jedt � �

n j � dn j h je � C p dT

The solution for the mass and volume of the reaction mixture aredm

dt� � �n ji M j � Vr

� �j M j

� � �n jeM j

dV

dt� 1� �

M j

dn j

dt� V � dT

dt

The temperature develops as

dT

dt� � �n ji h ji

� h je � �Q�n j C j

12.3.1 The plugflow reactor

We want to calculate the mole fractions, x1,..., xG , coverages θG+1,...θG+S, and flow F[mol/s] through a catalyst bed operating at temperature T [K] and pressure p [Pa].

The parameters for the catalyst is the porosity ε, the skeletal density ρ [kg/m3],the density of sites ρs [mol/m2] and the specific area a [m2/kg]. The parameters for thecatalyst bed is the cross sectional area A [m2] and the length L [m].

The mass of the bed ism= LA(1−ε)ρ

A slice of length dl has massdm = A(1−ε)ρ dl

contains a gas volumedVg = A ε dl

and the amount of gas

dn � p

RTdV g

� � p

RT 1 � � �dm

The contact time dt is

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dt � dV g

F� �

F

�1 � � dm

The rate of reaction [mol s−1} for gas g in the slice is�sa

�r � 1

Rr r � rg dm

The mole fractions add up to unity�ixi

� 1

12.3.1.1 The isothermal plugflow reactor

We consider a small slice of volume dV during the time dt. The mass and energybalances for the slice are

In + Produced Out

1 Fx1dt + rν1dtdV = (F+dF)(x1+dx1)dt

... ... + ... = ...

n Fxndt + rνndtdV = (F+dt)(xn+dxn)dt

sum Fdt + r�

i

�i dtdV = (F+dF)dt

energy �Qdt � F�

ixi hi dt + 0 = F � dF

�i

xi � dxi hi dt

From the sum we find

dF

dV� r

�i

�i

Substituting this equation into the mass balances for each of thecompounds we find

dx1

dV� r �

1� x1

�i

�i

F... � ...

dxn

dV� r �

n� xn

�inyi

F

Substituting into the energy balance we find

110

�Q � �0

Vr

�i

�i hi dV

Even for simple reactions analytical solution is impossible. Instead we use an ODEsolver to integrate the equations. The boundary values are the inlet flow and inletconcentrations.

111

13 Fit of kinetic data

The core problem in experimental kinetic studies is to determine kinetic parameters frommeasurements.

A prerequisite for kinetic studies is a reliable knowledge of the thermodynamics of thesystem. The reaction always takes the system closer to equilibrium, the rate is zero atequilibrium and has opposite signs on each side of the equilibrium composition. The ratethus depends on the thermodynamics. However, kinetic measurements are not suitablefor determination of thermodynamic parameters. Before we can begin studies of thekinetics, we must know the thermodynamics either from calculation or frommeasurements.

Kinetic studies may be performed on several levels.� We may assume something simple, such as a pseudo−first order reaction determinethe rate constant.� Suppose we know the reaction conditions and the desired exit composition. We canthen use a small sample at these conditions and adjust the flow until the desired exitcomposition is reached. We then know a reliable values for F/m.� We may fit the rate constant, k.� We may fit the activation energy, E#, and preexponential factor, A.� We may fit the activation energy, E#, the preexponential factor, A, and one or morereaction orders, αi.

13.1 The simple approach

Let us assume we have a first order reaction A → B with kinetics dc

dt� � kc

where c is the concentration of A. The solution is c � c0 exp � kt

r � � kc0exp � kt

We then measure the rates, r, at different temperatures, T, and times t:

112

T t r

... ... ...

... ...

... ... ...

... ...

... ... ...

... ...

We can easily rewrite Equation \ref{ eq2} asln � r � ln kc0

� ktFor each temperature we prepare a graph

t, ln � rand determine the rate constant, k, by linear regression. Using the value of k at each ofthe temperatures, we then prepare an Arrhenius plot

ln k ,1

Tand determine A from the intercept, ln(A), and the activation energy, E#, from the slope,−E#/R, by linear regression.

You have without doubt seen this approach sketched in introductory textbooks. Mostlikely you have also tried applying it to your measurements and discovered that it failedmiserably. Why does the method work for the "experimental" data in the textbook whenit fails for your data ?

The major problems are 1. If the values of r are uncertain, the values of k will be uncertain and most likely

slightly wrong. When we then use these values to determine A and E#, the values will

be uncertain and wrong. We have no idea how wrong and how uncertain A and E# are.2. Calculation of the uncertainty for A and E# is quite complex. Most likely we have

roughly the same errorbar for each of the concentrations, and roughly the sameerrorbar for each of the temperatures, but this implies different, asymmetrical"errorbars" for ln{ c} and 1/T.

3. For kinetic expressions which are just a little more complicated than first oderkinetics, the rate cannot be integrated by hand. Very complex kinetic expressions areeasy to treat by numerical methods, but analytical treatment is only possible forextremely simple kinetic expressions.

and the minor problems are1. In the linear regression using the integrated rate law, we actually determined −k from

the slope and ignored that kc0 was available from the intercept. But we know c0. We

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thus fitted k twice in one fit and did this as if the two values were independent. Inother words, we did not even do the fit properly.

2. In almost all situations we measure the inlet and outlet concentrations, together withthe values of the flow, temperature and pressure, this is enough to calculate the rate.The rate itself can only be measured in exceptional cases.

13.2 A more general method

Let us consider a more complicated reaction|ν1| A1 + |ν2| A2 ...→ ... |νn| An

13.2.1 Kinetic measurements

We then perform the kinetic measurements. We select pressures, p, flows, F,temperatures, T, and inlet concentrations x1i,...,xni We select a key component $r$ andmeasure the exit concentration, xre, of this component. The exit concentration of all othercomponents may then be determined from a mass balance, the inlet concentrations andxre.

It is important that we select a reference point and return to this point frequently. Thisshould be the first and last point we measure, the first point we measure on each mondaymorning, the last point we measure on friday afternoon and, say, every seventh point wemeasure during the week. During the measurements the reference point is used to checkfor problems with the catalyst or with the equipment. The result should always be almostthe same or we have a major experimental problem. During the analysis, the referencepoint is used to determine the uncertainty in the measurements.

If the conversion is virtually zero for a given set of reaction conditions, it will bevirtually zero at most other reaction conditions and you can fit this with A∼0 and anyvalue of E# and α1,..αn. Similarly, if the exit composition is near the equilibriumcomposition, you can fit the data with a large value for A. For this reason, kineticmeasurements made at very small conversions and kinetic measurements made nearequilibrium do not contain much information. The amount of catalyst and the reactionconditions must be selected to give exit compositions which are significantly differentfrom both the input composition and the equilibrium compositions. It is frequentlynecessary to make several short series of kinetic measurements to determine a properexperimental procedure.

When you have determined a suitable range of reaction conditions, you can perform thekinetic measurements without any prior knowledge on the values of the kineticparameters. Just proceed with the measurements and leave the columns xrc and (xre−xrc)2

114

blank in the table below.

However, if you have an idea on the valued for A, E# and α1,..αn use these values withthe program and the reaction conditions p, F, T, x1i,...,xni to calculate an expected exit concentration xrc of the key component. Ifthe agreement between xre and xrc is too poor you may want to check your reactor andreconsider your experimental procedures before you waste too much time.

The table of experimental results may look like this:

p T F x1i ... xni xre xrc (xre−xrc)2

p0 T0 F0 x10 ... xno ... ... ...

p1 T1 F1

F2

T2 F1

F2

p2 T1 F1

F2

T2 F1

F2

x11 ... xn1 ... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

p0 T0 F0 x10 ... xno ... ... ...

p1 T1 F1

F2

T2 F1

F2

p2 T1 F1

F2

T2 F1

F2

x12 ... xn2 ... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

... ... ...

p0 T0 F0 x10 ... xno ... ... ...

In an experiment, the pressure is usually controlled manually, while the temperature andthe flow are controlled electronically. For this reason, changes in pressure are made lessfrequently than changes in flow or temperature.

115

After and adjustment of the flow, the transient usually takes minutes while changes intemperature and the resulting transient frequently takes several hours. For this reasonchanges in temperature are made less frequently than changes in flow.

13.2.2 Thermodynamics

The equilibrium equation is

K � � i � 1

n pi

po

�i

We verify that we know K, either from tabulated data in the literature or frommeasurement.

13.2.3 Choice of kinetic expression

We then assume some form of the kinetic expression, e.g

r � k � i � 1

n pi

po

i � k

K� i � 1

n pi

po

�i

� i

and verify that we do not have any serious reason to believe that this form for the kineticsis wrong.

It is important that we don’ t make the expression more complex than necessary. If weknow the reaction must be first order in one of the components or zero’ th order inanother component, this information should be fixed a priori in the kinetic expression wewill use in the following.

It may be tempting to leave parameters you already know as free parameters. However, aleast−squares fit often uses extra flexibility to generate a slightly better fit and a drasticreduction of the reliability of the fitted values. What will frequently happen is thefollowing: Suppose the true values are A=1 and B=2. You know that A is 1, but youdon’ t know the value of B. You leave A free "just to see if it comes out right" and obtainthe fit A=1.2 ± 0.1, B=2.4 ± 0.2$. However, if you fix A=1, you find a slightly poorer fitbut a considerably better value for B, e.g B=2.1 ± 0.1.

We verify that we have a program which can treat the kinetic expression we haveselected for the reactor we are going to use.

116

13.2.4 Fitting the parameters

If you have an idea on the valued for A, E# and α1,...,αn use these values with theprogram and the reaction conditions p, F, T, x1i,...,xni to calculate an exit concentration xrc

of the key component. (If you don’ t know the values for A, E# and α1,...,αn, just make awild guess to get started.) Calculate �

xr e� xrc

2

Most likely you will find that the calculated exit concentration, xrc, do not reproduce themeasured exit concentration, xre.

We then start the fitting procedure 1. Make a small random variation of A, E# and α1,...,αn. 2. If the new parameters are unacceptable, keep the old paramets and go to step 1. 3. Use the program to update xrc and Σ ( xre−xrc)2 4. If this value is better than before, keep the new parameters else keep the old

parameters. 5. Go to step 1.

It may be necessary to adjust the meaning of small in step 1 as the fit improves.

13.2.5 Checking the fit

When the fit is not improved through a long series of guesses, there are two possibilities

The fit is good. This is fine, but you should at least consider the possibility that the kinetic modelis too general or has too many parameters.

The fit is bad. Use the table to prepare plots (p,(xre−xrc)2), (T,(xre−xrc)2), (x1i,(xre−xrc)2) etc. Theseplots will give valuable hints on exactly what it is in the experimental data thatyou kinetic model cannot fit.

Don’ t forget to check that all your measurements of your reference points givesalmost the same value. You might have a problem with the catalyst or with thereactor.

Don’ t forget to plot (t,(xre−xrc)2), where t is the number of hours from the start ofthe experiment until each measurement is made. This plot may send a loudmessage about experimental problems.

117

118

13.3 Bibliography

Catalysis

Charles L. Thomas: Catalytic processses and proven catalysts. Academic Press(1970).

Bruce C. Gates: Catalytic chemistry. John Wiley and Sons (1992)

B. C. Gates, J. R. Katzer, and G. C. A. Schuit. Chemistry of catalytic processes.McGraw−Hill Book Company, 1979.

Robert J. Farrauto and Calvin H. Bartholomew: Fundamentals of industrialcatalytic processes. Blackie Academic and Professional, London (1998)

J. A. Moulijn, P. W. N. M. van Leeuwen, and R. A. van Santen: Catalysis. Anintegrated approach to homogeneous, heterogeneous and industrial catalysis.Studies in Surface Science and Catalysis. volume 79. Elsevier, 1993.

R. A. van Santen and J. W. Niemantsverdriet: Chemical kinetics and catalysis.Plenum (1995).

Charles N. Satterfield: Heterogeneous catalysis in industrial practice.McGraw−Hill. 2nd edition (1991).

G. A. Somorjai: Introduction to surface chemistry and catalysis. John Wileyand Sons, 1994.

J. M. Thomas and W. J. Thomas: Principles and practice of heterogeneouscatalysis. VCH (1997).

R. J. Wijngaarden, A. Kronberg, K. R. Westerterp: Industrial Catalysis. Wiley−VCH

Encyclopedias

J. R. Anderson and M. Boudart (editors): Catalysis: Science and technology.Springer. (1981−1997). 11 volumes.

Volume 1 (1981)A brief history of industrial catalysis.An introduction to the theory of catalytic reactors.Catalytic activation of dinotrogen.

119

The Fischer−Tropsch synthesis.Catalytic reforming of hydrocarbons.

Volume 2 (1981)History of concepts in catalysis.Crystallography of catalysts types.Catalytic kinetics: Modelling.Texture of catalysts.Solid acid and base catalysts.

Volume 3 (1982)History of coal liquefaction. Catalytic activation of dioxygen. Catalytic activation of carbon monoxide on metal surfaces. Chemisorption on nonmetallic surfaces. Chemisorption of dihydrogen.

Volume 4 (1983)Catalytic processed in organic conversions. Nature and estimation of functional groups on solid surfaces. Kinetics of chemical processes on well−defined surfaces.

Volume 5 (1984)Catalytic steam reforming. Automobile catalytic converters. Infrared spectroscopy in catalytic research. X−ray techniques in catalysis.

Volume 6 (1984) Catalyst deactivation an regeneration. Catalytic olefin polymerization. Metal catalysed skeletal reactions of hydrocarbons on metal catalysts. Dispersed metal catalysts.

Volume 7 (1985) The history of the catalytic synthesis of ammonia. The electron microscope of catalysts. Surface structural chemistry.

Volume 8. (1987) The historical development of catalytic oxidation processes. Catalytic methathesis of alkenes. Physico−chemical aspects of mass and heat transfer in heterogeneouscatalysis. Small scale laboratory reactors. EPR methods in heterogeneous catalysis.

Volume 9 (1991) Determination of mechanism to heterogeneous catalysis. Dynamics relaxation methods in heterogeneous catalysis. Dynamics of heterogeneously catalyzed reactions.

Volume 10 (1996) Application of NMR methods to catalysis. Glossary of terminology used to catalysis.

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Volume 11 (1996)Hydrotreating catalysis. Science and technology.

G. Ertl, H. Knözinger and J. Weitkamp (editors). Handbook of heterogeneouscatalysis. Wiley−VCH. (1998). 5 volumes.

Experimental

J. W. Niemantsverdriet: Spectroscopy in catalysis. VCH (1993).

D. P. Woodruff and T. A. Delchar. Modern techniques of surface science.Cambridge University Press (1986).

Historical

Olaf A. Hougen and Kenneth M Watson. Chemical process principles. Part 1.Material and energy balances. John Wiley and Sons (1943)

Olaf A Hougen, Kenneth M. Watson and Roland A. Ragatz. Chemical processprinciples Part 2: Thermodynamics. John Wiley and Sons. 1st edition (1947)2nd edition. (1959).

Olaf A Hougen and Kenneth M. Watson: Chemical process principles Part 3:Kinetics and catalysis. John Wiley and Sons (1947).

H. Heinemann: A brief history of industrial catalysis. in John R. Anderson andMichel Boudart (editors): Catalysis. Science and technology. Volume 1. Springer(1981).

M. Boudart. Kinetics of chemical reactions. Prentice Hall, 1968.

Introductions

Gary Attard and Colin Barnes: Surfaces. Oxford Chemistry Primers, number 59(1998).

Ian S. Metcalfe: Chemical Reaction engineering. A first course. OxfordChemistry Primers, number 49. Oxford University Press (1997).

Michael Bowker: The basis and applications of heterogeneous catalysis.Oxford Chemistry Primers. Volume 53. Oxford University Press (1998).

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Kinetics

Sidney W. V. Benson: Thermochemical kinetics. John Wiley and Sons (1968).

Michel Boudart and G. Djega−Mariadassou: Kinetics of heterogeneous catalyticreactions. Princeton University Press (1984).

James A. Dumesic, Dale F. Rudd, Luis M. Aparicio, James E. Rekoske, AndresA. Trivino: The microkinetics of heterogeneous catalysis. American ChemicalSociety (1993)

Reaction engineering.

J. J. Carberry. Chemical and catalytic reaction engineering. McGraw−Hill,1976.

G. F. Froment and K. B. Bischoff. Chemical reactor analysis and design. JohnWiley and Sons, 2nd edition, 1990.

Octave Levenspiel: Chemical reaction engineering. John Wiley and Sons 2nd

edition (1972) .

L. D. Schmidt. The engineering of chemical reactions. Oxford UniversityPress, 1998.

H. Scott Fogler. Elements of chemical reaction engineering. Prentice−Hall, 2nd

edition, 1992.

Theory

Stephen Elliot: The physics and chemistry of solids. John Wiley and sons.(1998).

R. I. Masel: Principle of adsorption and reaction on solid surfaces. JohnWiley and Sons, 1996.

R. A. van Santen and J. W. Niemantsverdriet: Chemical kinetics and catalysis.Plenum (1995).

R. A. van Santen: Theoretical heterogeneous catalysis. World Scientific (1991).

Andrew Zangwill: Physics at surfaces. Cambridge University Press (1988)

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V. P. Zhdanov. Elementary physicochemical processes on solid surfaces.Plenum, 1991

123