102 3 - - C H E M I C A L K I N E T I C S O F C A T A L Y Z E D R E A C T I O N S
In lob s
adsorption r.d.s.
"%%%~/
desorption r.d.s.
J
1/T
Fig. 3.10. Change of observed activation energy due to changing rate-determining step as a function of temperature.
highest. A temperature increase will enhance the rate of this step more than the other steps with lower activation energies. Then another step, for instance a dissociation step, can now become rate limiting. The temperature dependence of the overall rate will behave as depicted in Fig. 3.10. This illustrates that the observed overall activation energy decreases with increasing temperature upon a change in the rate-determining step.
In these cases one cannot comply with the assumption of only one rate determining step. To obtain an adequate rate expression, valid over the whole temperature range under consideration, two or even more steps should be assumed not to be in
quasi-equilibrium,
and are, hence, rate determining.Energy diagram 3.7 is of course very similar in heterogeneous, homogeneous and biocatalysis, since kinetics is similar. A difference to be taken into account is that in the liquid phase adsorption is to be considered with respect to the liquid phase.
An essential difference between catalysis and gas phase kinetics is the absence of adsorption complexes in the latter case. A schematic comparison between a gas phase reaction energy diagram and one for a catalytic reaction is included in Fig. 3.7.
3 m CHEMICAL KINETICS OF CATALYZED REACTIONS
Rate
103
Heat of adsorption
Fig. 3.11. V o l c a n o c u r v e for t h e o v e r a l l r e a c t i o n r a t e as a f u n c t i o n of t h e h e a t o f a d s o r p t i o n .
Volcano plots, as in this figure, have been measured for very different re- actions, e.g., the formic acid decomposition, the ammonia synthesis reaction, hydrodesulfurization, or hydrodenitrogenation reactions.
As explained above, the increase of the reaction rate is due to the increased site coverage with the reactant (Eqn. (3.30)). A limit of the increase is reached once the optimum surface coverage has been reached. The rate is then controlled by the rate of product formation. The reason for the decrease in reaction rate with a further increase in adsorption strength is the increased activation energy for desorption; not only will the reactants be adsorbed more strongly, but the products will also be held more strongly (Eqn. (3.48)). Clearly, an optimum for the interaction strength between the catalytically active surface and the adsorbates exists, resulting in a maximum in the rate of reaction (the Sabatier principle). To the left of the maximum, the reaction has a positive order in the reactants, whereas to the right the order has become zero or even negative (see Eqn. (3.48). This kinetic dependence on the interaction with a catalyst can be used to test whether a Volcano plot is due to Sabatier's effect or not.
In practice, the variation of the rate of reaction with adsorption strength and the occurrence of a maximum in this rate have the following consequence for kinetic modelling of heterogeneous catalysts. Usually, the assumption of a homogeneous surface is not strictly valid. It would probably be more realistic to assume the existence of a certain distribution in the activity of the sites. However, certain sites will contribute most to the reaction, since these sites activate the reactants most. This might result in apparently uniform reaction behaviour, and can explain why Langmuir adsorption often provides a good basis for the reaction rate description. This also implies that adsorption equilibrium constants determined from separate adsorption experiments can only be used in kinetic expressions when coverage dependence is explicitly included. Otherwise, these constants have to be extracted from the rate data.
Several authors have derived rate expressions for non-uniform catalyst surfaces. Boudart and Dj6ga-Mariadassou [2] show that relations are obtained with a mathematical similarity to those obtained for a uniform surface. In the rate expression for ammonia production, the Temkin isotherm has been used for a long time. This isotherm accounted for a, supposedly, heterogeneous adsorption
104 3 m C H E M I C A L K I N E T I C S O F C A T A L Y Z E D R E A C T I O N S
behaviour [1]. Recently, however, it has been shown that the LHHW approach can account for the data over a pressure range of -~ 300 bar [9] without the assumption of heterogeneity. This and other examples demonstrate the useful- ness of the LHHW approach for reactions of practical importance.
3 . 9 C O N C L U D I N G R E M A R K S
The preceding sections indicate how a useful approximate reaction rate expression can be derived for catalyzed reactions, starting from an assumed kinetic model of elementary reaction steps. The derivation is based on the following assumptions:
- the reaction system is in a steady state
- the surface for adsorption and reaction is uniform;
- the number of active sites is constant, independent of reaction conditions;
- adsorbed species do not interact, apart from their reaction paths.
The form of the resulting expression differs from the gas-phase reaction rate expressions due to the presence of a denominator representing the reduction in rate due to adsorption phenomena. The individual terms of this denominator respresent the distribution of the active sites among the possible surface comp- lexes and vacancies. Expressions of this type are termed the Langmuir-Hinshel- wood-Hougen-Watson (LHHW) rate expressions in heterogeneous catalysis and Michaelis-Menten expressions in biocatalysis.
The steady-state approach generally yields complex rate expressions. A sim- plification is obtained by the introduction of one or several rate-determining step(s) and
quasi-equilibrium
steps, and further by the initial reaction rate ap- proach. For complex reaction schemes, identifying the most abundant reaction intermediates ("mari") and making use of the site balance can simplify the kinetic models and rate expressions.In practice, useful relations result even for the non-ideal heterogeneous surfaces of solid catalysts. Some reasons can be:
- similarity of mathematical relations for uniform and non-uniform adsorption models;
- Sabatier's principle of the optimum site activity. Optimum sites contribute most to the reaction, resulting in an apparently uniform behaviour.
An interesting further development in describing the kinetics of heterogeneous- ly catalyzed reactions is the so-called microkinetics approach, whereby inde- pendent information about adsorbed species from temperature programmed desorption and spectroscopic studies are used to predetermine rate and equili- brium constants of elementary processes, thus enabling the prediction of the overall rate. Especially for metal catalyzed reactions this gives good results [9].
More information about reaction kinetics related to catalysis can be found in Refs. [10-131.
3 - - CHEMICAL KINETICS OF CATALYZED REACTIONS 105 In this chapter the aspects of kinetic m o d e l s e l e c t i o n / d i s c r i m i n a t i o n and p a r a m e t e r estimation and the experimental acquisition of kinetic data are not dealt with, since they fall outside its scope. Moreover, in i n t e r p r e t i n g the observed t e m p e r a t u r e d e p e n d e n c e of the rate coefficients in this chapter w e are a s s u m e d to be dealing w i t h intrinsic kinetic data. As will be s h o w n in C h a p t e r 8, parasitic p h e n o m e n a of mass and heat transfer m a y interfere, disguising the intrinsic kinetics. Criteria will be presented there to avoid this experimental problem.
N O T A T I O N Ao
C Ci
Ea Eo h kB
kbarrier
ki
Ki Km Km K'm
n Hi
NT
Pi 1"
t" i
R
S
t T
V
Vm
preexponential factor
constant in Freundlich isotherm concentration
activation e n e r g y
total e n z y m e concentration Planck's constant
Boltzmann constant
n u m b e r of molecules reacting per unit time reaction rate constant for reaction i
equilibrium constant of reaction i Michaelis constant
Modified Michaelis constant (inhibition) Michaelis constant inhibitor
constant in Freundlich isotherm reaction order in i
total concentration of active sites partial pressure of c o m p o n e n t i reaction rate (overall)
reaction rate of reaction i ideal gas constant
n u m b e r of nearest neighbours of active site time
t e m p e r a t u r e
reaction rate (biocatalysis)
m a x i m u m reaction rate (biocatalysis)
8-1
a t m n mol m -3 J mo1-1 mol d m -3
Js
j K -1
S-1
s -1 (for first order reaction)
a t m -1 o r m 3 mol -~
mol d m -3
mol d m -3
mol d m -3
mol (g cat) -1 or mol ( m 2 cat) -1 atm, kPa
a t m s -1 or mol rn -3 s -1 a t m s -1 or mol m -3 s -1
J mo1-1 K-I
S
K
mol s -1 d m -3 mol s -1 d m -3 Greek
(2
AG
constant in T e m k i n isotherm
Gibbs free energy change J mo1-1
106 3 - - C H E M I C A L K I N E T I C S O F C A T A L Y Z E D R E A C T I O N S
AH AS 0i vi
E n t h a l p y change E n t r o p y change
fraction of total n u m b e r of sites occupied b y i stoichiometric n u m b e r of e l e m e n t a r y step i
J mo1-1 J mo1-1 K-1
Subscripts
0 initial or at zero coverage + f o r w a r d reaction
- b a c k w a r d reaction eq equilibrium
g gas phase
obs observed / a p p a r e n t Superscripts
0 s t a n d a r d conditions
# transition state
* surface species REFERENCES
1 M.I. Temkin, "The Kinetics of Some Industrial Heterogeneous Catalytic Reactions' in Advances in Catalysis, Vol. 28, Academic Press, New York, 1979, p. 173.
2 M. Boudart and G. Dj6ga-Mariadassou, Kinetics of Heterogeneous Catalytic Reactions, Prince- ton University Press, Princeton, NY, 1984.
3 C.N. Satterfield, Heterogeneous Catalysis in Industrial Practice, 2nd ed., McGraw-Hill, New York, 1991, p.61.
4 G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design 2nd ed., Wiley, New York, 1991.
5 0 . A . Hougen and K.M. Watson, Chemical Process Principles, Vol. III, Wiley, New York, 1947.
6 R.W. Maatman, Site Density and Entropy Criteria in Identifying Rate-Determining Steps in Solid-Catalyzed Reactions, in Advances in Catalysis, Vol. 29, Academic Press, New York, 1980, p. 97.
7 R.A. van Santen and H. Niemantsverdriet, Chemical Kinetics and Catalysis, Plenum, 1995.
8 J. Wei, Ind. Chem.Eng. Res., 33 (1994) 2467-2472.
9 P. Stoltze and J.K. Norskov, An interpretation of the high-pressure kinetics of ammonia synthesis based on a microscopic model. ]. Catal. 110, 1 (1988).
10 J.M. Thomas and W.J. Thomas, Introduction to the Principles of Heterogeneous Catalysis, Aca- demic Press, London, 1967.
11 R. Mezaki and H. Inoue, Rate Equations of Solid-Catalysed Reactions, University of Tokyo Press, Tokyo, 1991.
12 F. Kapteijn and J.A. Moulijn, Kinetics and transport processes, in: Handbook of Heteroge- neous Catalysis Vol. 3, Ch. 6.1, G. Ertl, H. Kn6zinger and J. Weitkamp (Eds.), VCH, Weinheim, p. 1189 (1997).
13 L. Stryer, Biochemistry, W.H. Freeman, New York, Ch. 8, 1995.
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