For the conversion A e-> B, the overall activation energy is a complex function of the reaction enthalpies and activation energies of the individual elementary
3 - - CHEMICAL KINETICS OF CATALYZED REACTIONS 97
E obs
..I r v
Fig. 3.6. E n e r g y d i a g r a m f o r a s u r f a c e r e a c t i o n .
reaction steps. We will illustrate this by assuming that the surface reaction is the rate-determining step (3.17) and that the backward reaction can be neglected.
The rate expression for this single-site reaction can now be written as:
k2NT KA
PAr = (3.45)
1 +K A PA + K B PB where KA = K1 and KB = 1/K3.
The dependence of the reaction rate (3.48) on pressure and temperature is determined by contributions of both the numerator and denominator. Often, rate expressions are presented in a power law form:
nA nB
(3.46)
r = k p a P B
Comparison with Eqn. (3.45) suggests that the powers nA and nB will not be constants. They can be extracted as follows:
31nr
n i = ~ (3.47)
3 1 n p i
Applying Eqns. (3.45) to (3.47), and using the adsorption equilibrium relation- ships (Eqns. (3.14) and (3.15)), together with Eqn. (3.16), yields:
nA = 1 - 0A and n B = -- 0B (3.48)
Hence, the apparent reaction orders are related to the fractional surface cover- ages. From Eqn. (3.48), it follows that nA varies from 0 to 1, and nB f r o m - 1 to 0, depending on the conditions.
98 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 H O N S
Like the pressure dependence, the temperature dependence is also often expressed in an empirical form, in which an apparent overall rate constant is used (like in Eqn. (3.44)). The observed (or apparent) activation energy can be expressed as:
o b s
=RT2( ~lnr )
Ea 3 T p (3.49)
The observed activation energy can now be derived from Eqn. (3.45) in a similar way as the derivation of the reaction powers as a function of surface coverage:
EObS = E + ( 1 - 0 A ) A H a a2 A - 0 B A H B (3.50) The observed activation energy contains contributions from the rate-determin- ing step (Ea2) and from the adsorption enthalpies of A and B, the latter depending on the fractional occupancies. Obviously, E~ will depend on the experimental conditions. Therefore, it is not surprising that a wide range of values have been reported for the same reaction system.
3.7.3 F o r w a r d R e a c t i o n ~ L i m i t i n g C a s e s
Based on the previous analysis of the pressure and temperature dependence of the reaction order and the observed activation energy, four different cases can be distinguished:
(i) Strong Adsorption of A
Strong adsorption of A results i n K A PA >> 1 and K B PB, and hence Eqn. (3.45) reduces to:
r = k 2 N T (3.51)
Physically this means that the whole catalyst surface is covered with A (0A --4 1, 0B ~ 0). Therefore, varying the partial pressure of A does not influence the reaction rate. The reaction is said to be zero order in A (and B). The overall activation energy is E ~~ = Ea2 (see Fig. 3.6), provided the concentration of active sites NT is temperature independent.
(ii) Weak Adsorption of A and B
When A and B are only weakly adsorbed, K A p A and K B PB << 1 and 0A, 0B --4 0, SO the rate expression becomes:
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 99
G ~
=G2+ aliaT
A#(g)
/ \
/ \
/ \
/
/ A #* ,,
/ ",,,
A(g)+* , , " ' : ~ ",,
B*
Fig. 3.7. Energy diagram for the catalytic conversion of A to B. * Denotes surface vacancy and # transition state. AHR is the reaction enthalpy. Included is the energy diagram for the gas phase
conversion (dashed curve).
r = k 2 N T K A PA (3.52)
The reaction is now first order in A and zero order in B. The observed overall activation energy will be lower than in the previous case.
Eobs = a E~2 + AHA (3.53)
since the enthalpy of adsorption of A, AHA, is negative; adsorption is an exo- thermic process. This result can be understood from the energy diagram in Fig.
3.7. Due to the low occupancy of A, for reaction to occur, adsorption of gaseous A is required, and hence A gains adsorption enthalpy. Subsequent surface reaction requires overcoming the activation energy.
(iii) Strong Adsorption of B
In this c a s e K B PB >> 1 and K A PA, and 0A --~ 0, 0 B -- ) 1~ and the rate expression becomes:
k2NT KA PA
r = (3.54)
KB PB
The reaction is first order in A and -1 in B (B decreases the reaction rate strongly by competitive adsorption). The surface is nearly completely covered with B.
Therefore, the initial state for the reaction is gaseous A and adsorbed B (see Fig.
3.8). For A being able to react, firstly a molecule of B must desorb with accomp-
100 3 -- CHEMICAL KINETICS OF CATALYZED REACTIONS
G ~ =G2+ AHA-AHB A~*
A(g)+ * +S(g)
A(g)+B*
Fig. 3.8. Energy diagram for the catalyzed conversion of A to B in the case of strong adsorption of B
anying desorption enthalpy. Subsequently, A adsorbs, gaining adsorption enthalpy, and reacts through the surface reaction, where the activation energy barrier has to be overcome. Thus, the observed activation energy is higher than in the previous cases:
obs = Ea2 + AHA - AHB
Ea (3.55)
Note that -AHB > -AHA as a consequence of the stronger adsorption of B.
(iv) Intermediate Adsorption of A and B
For intermediate values of K A P A and K B pBexpressions (3.48) and (3.50) apply for the reaction order and the observed activation energy, respectively. The reaction order will range between 0 and I for A, and between-1 and 0 for B. The observed overall activation energy will have intermediate values between the two extremes of cases (ii) and (iii):
Ea2 + AHA < E ~ E~2 + ( 1 - 0A) AHA- OB ~ B < Ea2 + ~ A - ~ B (3.56) Transitions between the different situations discussed above will occur as a function of temperature, because the occupancies will vary. With increasing temperature the adsorption equilibrium constants decrease resulting in de- creased occupancies. Thus, if starting with strongly adsorbing B at low temper- ature as example, a gradual transition can be envisaged upon increasing the temperature, during which:
- the order of A remains nearly 1
- the order of B changes f r o m - 1 to 0, and
- the observed overall activation energy will change from (3.55) to (3.53).
3 -- CHEMICAL KINETICS OF CATALYZED REACTIONS 101 In addition, large changes in partial pressure can result in changes in the apparent reaction order of a component. At low PA the reaction is first order in A, while at high PA the rate approaches a limit, as can be expected for Langmuir adsorption, and the reaction becomes zero order in A (see Eqn. (3.48)).
Summarizing, the overall observed activation energy and the apparent re- action orders of the components depend on the degree of coverage of the active sites, which in turn depends on the temperature and partial pressures.
Analogously, limiting cases can be distinguished for the dual-site model, in which the order of A can even become negative (see for example Eqn. (3.27)).
This is common for dissociation reactions.
An excellent illustration of the LHHW theory is catalytic cracking of n-alkanes over ZSM-5 [8]. For this reaction, the observed activation energy decreases from 140 to -50 (!) kJ/mol when the carbon number increases from 3 to 20. The decrease appeared to linearly depend on the carbon number as shown in Fig.
3.11. This dependence can be interpreted from a kinetic analysis that showed that the hydrocarbons (A) are adsorbed weakly under the experimental conditions.
The initial rate expression for a rate-determining surface reaction applies (3.30), which in the limiting case of weak adsorption of A reduces to Eqn. (3.52). The activation energy is then represented by equation (3.53).
Measurement of the adsorption enthalpy AHA revealed a linear decrease with carbon number. By applying Eqn. (3.53) the activation energy for the surface reaction, E~2, was estimated. The data in Fig. 3.9 clearly shows that Ea2 has reasonable (and positive) values, while it remains fairly constant for n > 8, supporting the kinetic interpretation.
Until now it has been assumed that the rate-determining step remains the same with changing reaction conditions. However, it can also change, especially under the influence of temperature [4], but also as a result of pressure changes [21.
Suppose the rate-determining step at low temperature is the desorption step.
Generally, this implies that the activation energy barrier for desorption is the
kJ/mol
200 _ ~ , : , , . 9 : ~,
100 ~ Ea2
-100 alia
. . . ""-:-.-.; . .
-200
0 5 10 15 20
Fig. 3.9. Observed activation energies, reaction activation energies, and adsorption enthalpies in the cracking of n-alkanes over ZSM-5 (adapted from [8]).
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.