Activation energy diffusion-limited

The iGLE also presents a novel approach for studying the reaction dynamics of polymers in which the chemistry is driven by a macroscopic force that is representative of the macroscopic polymerization process itself The model relies on a redefined potential of mean force depending on a coordinate R which corresponds locally to the reaction-path coordinate between an n-mer and an (n -t 1 )-mer for R = nl. The reaction is quenched not by a kinetic termination step, but through an (R(t))-dependent friction kernel which effects a turnover from energy-diffusion-limited to spatial-diffusion-Iimited dynamics. The iGLE model for polymerization has been shown to exhibit the anticipated qualitative dynamical behavior It is an activated process, it is autocatalytic, and it quenches... 

J mol ). This is additional evidence in favor of rate limitation by inner diffusion. However, the same reaction in the presence of Dowex-50, which has a more open three-dimensional network, gave an activation energy of 44800 J mol , and closely similar values were obtained for the hydrolysis of ethyl acetate [29] and dimethyl seb-acate [30]. The activation energy for the hydrolysis of ethyl acetate on a macroreticular sulphonated cationic exchanger [93] is 3566 J mol . For the hydrolysis of ethyl formate in a binary system, the isocomposition activation energy (Ec) [28,92] tends to decrease as the solvent content increases, while for solutions of the same dielectric constant, the iso-dielectric activation energy (Ed) increases as the dielectric constant of the solvent increases (Table 6). 

Film-free conditions It has been observed for many metals that the magnitude of / i, (see Section 1.4) increases with temperature and that the activation energy for dissolution is low, suggestive of a diffusion-limited anode process when the migration of corrosion products away from the surface is rate controlling. Some examples of the value of the activation energy for this process are given in Table 2.4. 

Minimize the effects of transport phenomena If we are interested in the intrinsic kinetic performance of the catalyst it is important to eliminate transport limitations, as these will lead to erroneous data. We will discuss later in this chapter how diffusion limitations in the pores of the catalyst influence the overall activation energy. Determining the turnover frequency for different gas flow velocities and several catalyst particle sizes is a way to establish whether transport limitations are present. A good starting point for testing catalysts is therefore ... 

The presence of diffusion limitations has a strong effect on the apparent activation energy one measures. We can express both the rate constant, k, and the diffusion constant, Defr, in the Arrhenius form ... 

In the absence of diffusion limitation, the overall activation energy is... 

Hence, the apparent activation energy is half the value we would obtain if there were no transport limitations. Obviously it is important to be aware of these pitfalls when testing a catalyst. Indeed, apparent activation energies generally depend on the conditions employed (as discussed in Chapter 2), and diffusion limitation may further cause them to change with temperature. 

When solving for the apparent activation energy in the diffusion limited region we obtain... 

Normally the activation energy for diffusion in the gas phase is much smaller than the activation energy for a catalyzed reaction, and hence, according to Eqs. (38) and (46), the overall or apparent activation energy for the diffusion-limited process is half of what it would be without transportation limitation. If we plot the rate as a function of reciprocal temperature one observes a change in slope when transport limitations starts to set in. 

Figure 5.37. Arrhenius plot illustrating the effect on the apparent activation energy of pore diffusion and transport limitations through the stagnation layer surrounding a catalyst...

At low temperatures diffusion will be rapid compared to chemical reaction and diffusional limitations on the reaction rate will not be observed. In this temperature regime, one will observe the intrinsic activation energy of the reaction. However, since chemical reaction rates increase much more rapidly with increasing temperature than do diffusional processes, at... 

This situation is termed pore-mouth poisoning. As poisoning proceeds the inactive shell thickens and, under extreme conditions, the rate of the catalytic reaction may become limited by the rate of diffusion past the poisoned pore mouths. The apparent activation energy of the reaction under these extreme conditions will be typical of the temperature dependence of diffusion coefficients. If the catalyst and reaction conditions in question are characterized by a low effectiveness factor, one may find that poisoning only a small fraction of the surface gives rise to a disproportionate drop in activity. In a sense one observes a form of selective poisoning. 

It appears that the rate constant for most of the boards has a smaller temperature dependence than the initial maximum rate, the corresponding "activation energy" E3 being around or less than 5 kcal/mol. An important conclusion is that the rate is diffusion limited. This has to be compared to a mean activation energy around 20 kcal/mole for the initial maximum rate of heat release for the commercial boards. As a consequence thereof the total heat release extrapolated over infinite time does increase to a significant extent with temperature from 150 to 230°C. 

On the other hand, the low temperature dependance of the rate constants with activation energies around 5 kcal/mole indicates a diffusion limited reaction rate which could refer to diffusion of oxygene into the fibers of the board, i.e. into the fiberwalls. The corresponding negative activation energy for the groundwood based hardboard and the effect of fire retardants there upon are difficult to understand. 

Diffusion of particles in the polymer matrix occurs much more slowly than in liquids. Since the rate constant of a diffusionally controlled bimolecular reaction depends on the viscosity, the rate constants of such reactions depend on the molecular mobility of a polymer matrix (see monographs [1-4]). These rapid reactions occur in the polymer matrix much more slowly than in the liquid. For example, recombination and disproportionation reactions of free radicals occur rapidly, and their rate is limited by the rate of the reactant encounter. The reaction with sufficient activation energy is not limited by diffusion. Hence, one can expect that the rate constant of such a reaction will be the same in the liquid and solid polymer matrix. Indeed, the process of a bimolecular reaction in the liquid or solid phase occurs in accordance with the following general scheme [4,5] ... 

Hydrogen transfer from the gas phase to the liquid phase becomes rate limiting with very fast hydrogenations (or with insufficient agitation). The observed reaction rate is then equal to the rate of gas-liquid mass transfer of hydrogen and becomes first order in hydrogen and independent of substrate concentration. The activation energy decreases to that of a diffusion process. 

Only large-pore zeolites exhibit sufficient activity and selectivity for the alkylation reaction. Chu and Chester (119) found ZSM-5, a typical medium-pore zeolite, to be inactive under typical alkylation conditions. This observation was explained by diffusion limitations in the pores. Corma et al. (126) tested HZSM-5 and HMCM-22 samples at 323 K, finding that the ZSM-5 exhibited a very low activity with a rapid and complete deactivation and produced mainly dimethyl-hexanes and dimethylhexenes. The authors claimed that alkylation takes place mainly at the external surface of the zeolite, whereas dimerization, which is less sterically demanding, proceeds within the pore system. Weitkamp and Jacobs (170) found ZSM-5 and ZSM-11 to be active at temperatures above 423 K. The product distribution was very different from that of a typical alkylate it contained much more cracked products trimethylpentanes were absent and considerable amounts of monomethyl isomers, n-alkanes, and cyclic hydrocarbons were present. This behavior was explained by steric restrictions that prevented the formation of highly branched carbenium ions. Reactions with the less branched or non-branched carbenium ions require higher activation energies, so that higher temperatures are necessary. 

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