Metal-Catalysed Reactions of Hydrocarbons / 05-Introduction to the Catalysis of Hydrocarbon Reactions

Figure 5.4. Dependence of rate on reactant pressure PA for a unimolecular reaction: 1, b = 0.1; 2, b = 1; 3, b = 10.

between, the order will seem to be a positive fraction, but a single exponent of the pressure can only describe the change in rate over a very limited range.

When there are two reactants A and B, and the product X is not absorbed, the mechanism becomes

The form of the dependence of rate on PA depends on the size of the term bB PB ; when this is about the same as bA PA , curves of the type shown in Figure 5.5 result. The highest rate occurs when θA equals θB , and since PB is known the position of the maximum reveals the value of bA /bB . There are two important limiting cases.

Figure 5.5. Dependence of rate on reactant pressure PA for a bimolecular reaction: PB = 1; 1, bA = 0.1; 2, bA = 0.5; 3, bA = 1; 4, bA = 10.

(i) When both reactants are only weakly chemisorbed, the denominator approaches unity, so that

where k = k2bA bB ; the reaction is first order in each reactant. (ii) If A is much more strongly adsorbed than B and/or PA greatly exceeds PB , then bA PA > 1 > bB PB and so

where k = k2bB bA ; the reaction is first order in B and inverse first in A. Once again, when neither limiting situation applies, the rate dependence upon reactant pressures may be approximate by the Power Rate Law expression

where the orders x and y may be either positive or negative fractions. Note however that either exponent may change from positive to negative as the pressure which it qualifies increases (see Figure 5.5).

The principles underlying this treatment are capable of extension to cover a greater number of reactants, inhibition by products, poisoning by adventitious impurities, dissociation of reactants upon adsorption (Section 3.2.4) and many other situations.27 The relevant rate expressions were collected and comprehensively evaluated many years ago by O. A. Hougen and K. M. Watson,28,29 and monographs on chemical kinetics2,15,22,30 often contain a fuller presentation than is thought necessary here.

Contemplation of Figure 5.5 raises another concern. If the curves 1 to 4 are given experimentally by four reactants whose adsorption coefficients bA have the values shown, and if the rates are measured at various values of PA , then very clearly the sequence of ‘activities’ will depend on the value chosen: in particular the reactivity of reactant 4 will depend critically on the pressure of A adopted. This simple observation deprives the concept of ‘catalytic activity’ of any meaning whatsoever, if the rates are measured only at a single pressure of each reactant.31

5.2.5. Effect of Temperature on Rate and Rate Constant

Precise determination of the temperature dependence of the rate of a catalysed reaction is beset by difficulties. We have already met one possible complication, namely, that, as temperature increases, mass transport limitation may set in: there may be a change in the rate limiting step, and in the fraction of surface not covered by strongly-held by-products, and hence available for reaction. We shall turn to the most important source of uncertainty in a moment.

According to the Arrhenius equation, which was of course devised to describe a homogeneous gas-phase reaction, the rate constant k, which is the rate at unit pressure of the reactants, i.e.

is the product of two terms, one of which is independent (or only weakly dependent) upon temperature, and the other of which is exponentially dependent on temperature. In the formalism of the Transition State (or Absolute Rate) Theory, the first of these, the pre-exponential factor A, becomes the product of an entropy term (exp( S=/R), a frequency factor (kB T/h) and perhaps a transmission coefficient (κ) for safety’s sake. The second term contains only the activation energy, i.e. exp (− H =/RT). Exact treatments of this important theory will be found all physical chemistry texts and elsewhere. Those who find its logic difficult to follow should take the advice of the mathematician d’Alembert: Allez en avant et la foi vous viendra. Its application to chemisorption and catalysed reactions has often been discussed; it requires the identification of the rate-limiting step and the inclusion of the unknown concentration of active centres, but we shall not be able to make a great deal of use of it. There are comparatively few systems to our understanding of which it has made an important contribution. It is however proper to appreciate that ‘activation energy’ can be regarded in two complementary ways: (1) by applying the Arrhenius equation to experimental results, it is given by

reactants to the point of almost no return at the top of the potential energy barrier, that is to say, to the transition state.32

Now for a homogeneous gas-phase reaction, the two are more or less equivalent, but for heterogeneous reactions the rate constant k is somewhat elusive: as we have used it, it rests on the validity of the Langmuir-Hinshelwood model, and its numerical value depends upon making the best (if not correct) choice of mechanism. Even so, to evaluate it at a sufficient number of temperatures involves considerable work, made easier now by the availability of microprocessor-controlled reactors and on-line computers. Nevertheless in much of the prior literature quite understandably the rate is used instead of the rate constant to obtain what must be called an apparent activation energy Ea : but changing temperature affects not only the rate of each reacting unit (i.e. k), but also the concentrations of the adsorbed reactants θA and θB . Consider first a unimolecular reaction (equation (5.13)). Simple chemisorption is always exothermic, and so θA will decrease as temperature rises; since the adsorption coefficient bA is an equilibrium constant, by applying the

where HA —O is the standard heat of adsorption and C A an integration constant. If θA remains close to unity over the whole temperature range, the increase in rate will simply be due to its effect on k: thus

Et being the true activation energy. If however the adsorption is weaker, θA will decrease significantly, and by joining equations (5.23) and (5.24) we find

This is because the value of the product θAθB is decreased by increasing PA and the reaction is inverse first order in A (see Figure 5.5). Equations of this type can be written to describe a variety of situations. The first systematic analysis was undertaken by M. I. Temkin in 1935;33 he arrived at the general expression

where x and y are the respective orders (equation (5.21)); this clearly invokes Power Rate Law concepts, but as we have seen these harmonise with the Langmuir– Hinshelwood formalism in limiting cases.

It will also be apparent that any other factor besides temperature that changes the coverages by adsorbed species will alter the measured activation energy: changing the reactant pressures will do this (Figures 5.4 and 5.5), so that measuring at high pressures will give a value close to Et , while at low pressures it will be governed by equation (5.28). Indeed one has only to look at the curves in Figure 5.5, imagining that it is temperature that is responsible for causing bA to change, to realise immediately that the activation energy will depend upon the value of PA that is selected.33 While this idea is expressed for a bimolecular reaction, it applies equally to a unimolecular reaction (Figure 5.4).

The distinction between true and apparent activation enquiries is important to draw for several reasons. (1) In trying to understand how catalyst structure and composition affect activity, there are two factors to consider: a thermochemical factor determining the concentration of reacting species, and a kinetic factor controlling their reactivity. Ea contains both, and only when Et and the relevant heats of adsorption are separated can their individual contributions be assessed.

(2) Ea is not a fundamental characteristic of a catalytic system, because its value may depend on the reactant pressures used.33,34 As we shall see in Section 5.5, there are very helpful correlations to be drawn between kinetic parameters, reactant pressures and orders, and structure sensitivity in the field of hydrocarbon reactions.

5.2.6. Selectivity12

In our mission to understand how and why the structure and composition of a catalytic surface determine performance, it is the nature and amounts of the products formed that provide more useful information about the adsorbed intermediates and mechanism than does the rate of reactant removal. The latter is not without its significance, but product analysis is a richer source of inspiration.

A catalytic system is characterised by the degree of selectivity with which each product is formed; the term ‘selective’ is however used very loosely in the literature, sometimes being applied when the product in question is only a small fraction of the total. Strictly speaking, only when that product is unique can the process properly be called ‘selective’.

With a single hydrocarbon molecule, there may be formed either two or more products simultaneously or two or more products in sequence, viz.

X

In catalytic hydrogenation and hydrogenolysis, both schemes are often encountered: they are easily distinguished because Y is not an initial product in the second scheme, whereas the concentration of the product X will pass through a maximum. Very frequently X is the desired product and conditions have to be sought to maximise its yield, but the two schemes often occur together, viz.

X

(5.G)

A

Y

and this exacerbates the problem of obtaining X in high yield. Figure 5.6 shows some examples of how amounts of product may vary with conversion.Here we are only concerned with products detectable in the fluid phase; fuller development of these schemes defining the role of adsorbed intermediates will be undertaken later. Selectivity is then expressed simply as the fraction that one product forms of the total, or

Product ratios, e.g. PX / PY , are often cited, but are generally less useful.

There is a further type of selectivity that arises when two reactants are present,

The selectivities of X and Y are then determined by the relative strengths of adsorption of A and B as well as their inherent reactivities.

Figure 5.6. Dependence of product pressures Px and Py, and of selectivities Sx and Sy on conversion for the reaction schemes 5.E, 5.F and 5.G.

Mathematical treatments of product composition as a function of conversion have been presented.12,35

5.2.7. Kinetic modelling26,36 –39

Kinetic modelling is the art of deducing the best possible rate expression by comparing its predictions with the experimental results, and hence inferring a likely mechanism. This careful definition is predicated on Karl Popper’s precept that it is impossible to prove that a theory or model is correct, because there may be a better one around the corner: it is only possible to negate an hypothesis, by showing that it fails to accord with observation. With modern computational facilities it is easy to compare the merits of various theoretical rate equation based on different mechanisms with experimental results, and for a given equation to find the values of the constants giving best fit, by calculating the mean standard deviation or other statistical parameter. Several words of caution are however necessary before blind faith in what the machine says dulls our critical facilities.

First, it is necessary to examine by eye a plot of the calculated curve of rate versus the pressure of a reactant, and superimpose the experimental points. The reason for doing this is because a fairly good overall fit may result from a very good fit over a part of the range, tending to a very poor fit in another (Figure 5.7). In deciding in which range a good fit makes most sense, variable experimental error must be kept in mind, e.g., very fast and very slow rates may not be measured as precisely as moderate ones. A computer will not know about experimental error

Figure 5.7. Hydrogenolysis of n-butane on Rh/TiO2 at 429 K: comparison of direct (O) and semilogarithmic ( ) plots of rate versus hydrogen pressure. In this and the next figure, the curves are calculated by the rate expression ES5B to be derived in Chapter 13.

unless specifically told. Comparison at low rates is made easier if semi-log or log-log (Figure 5.8) plots are used.

Second, the mechanism on which the rate expression is based should be both plausible and comprehensive, that is, it should take note of other relevant observations in the literature and should not omit any reasonable possibilities. There is a potential conflict here, because the greater the number of disposable parameters in the equation, the better automatically becomes the fit with the results. As the great mathematician Cauchy said: With five constants I can draw an elephant; with six I can make it wag its tail. It is desirable to restrict the number to three even it entails some over-simplification.

Third, the values of the disposable constants must be sensible, and their temperature dependence must be rational, giving linear plots of ln k or ln b vs. T −1, and afford heats of adsorption having positive values of – Ha and true activation energies of reasonable magnitude. These criteria are of extreme importance for this reason. One of the surprises for the novice modeller is the great variety of mathematical expressions, and therefore mechanisms, that can generate curves giving good fits to the points (Figure 5.9): but the value of a constant such as the adsorption coefficient of a reactant can vary astronomically with the form of the

Figure 5.8. Double-logarithmic plots of the rates of hydrogenolysis and of isomerisation of n-butane as a function of hydrogen pressure at 533 K with Pt/SiO2 as catalyst.

expression.36,40 Clearly therefore the statistical quality of the fit is by itself valueless in selecting a mechanism, and not all best values of that constant are equally acceptable. The precious gift of common sense based on experience has to be applied if mathematical modelling is to have any significance. Further illustrations of the use of the procedure will appear in Chapter 13.

5.3. THE CONCEPT OF REACTION MECHANISM16,41

The term ‘mechanism’ has been freely used in the foregoing sections, but without definition: this is because it admits of no simple description, there being many formulations considered by their begetters to be adequate and satisfactory. Mechanistic discussion is like peeling an onion: it is possible to go through a never-ending series of ever more profound analyses without ever reaching the end,

Figure 5.9. Dependence of rate of hydrogenolysis of n-butane on hydrogen pressure at 609 K using PtRe/Al2 O3 ; experimental points fitted to three rate expressions. ES5B, ; ES2, - - -; ES3, · · · . The formulation of these rate expressions is explained in Chapter 13.

and tears may be shed in the process. This Section is based on a part of an article written many years ago,42 and its central theme runs as follows.

Any mechanistic analysis must be made in terms of a model, and every model is limited by its frame of reference. The kind of answer we get depends upon the language in which the question is framed; the value of the answer is determined by the care that has gone into defining the nature of the conceptual model and by the symbolism employed to express it.

So for example we may ask, ‘Is ethene associatively adsorbed during its metalcatalysed hydrogenation?’ and we may hope to obtain a straight ‘yes or no’answer; but if the question is ‘How is ethene adsorbed ?’ we have to expect a more discursive reply, as we express our answer in terms of the many structural formula considered in Chapter 4.

In that article42 it was suggested that at an elementary level a mechanism is understood if the following are established beyond reasonable doubt:

The nature of all the participating species.

The qualitative modes of their interaction contributing significantly to the total reaction.

Quantitative aspects of these interactions expressed on a relative but not absolute basis.