Intraparticle diffusion reaction rate

When intraparticle diffusion is rate limiting, the kinetic behaviour of a chemically reacting system is generally different from that which would prevail if chemical reaction were rate limiting. It is therefore extremely important to develop criteria to assess whether intraparticle diffusion effects may be neglected and thus define the conditions of experiment which would reveal true chemical kinetics rather than overall kinetics disguised by intraparticle diffusion effects. 

Intraparticle diffusion limits rates in triphase catalysis whenever the reaction is fast enough to prevent attaiment of an equilibrium distribution of reactant throughout the gel catalyst. Numerous experimental parameters affect intraparticle diffusion. If mass transfer is not rate-limiting, particle size effects on observed rates can be attributed entirely to intraparticle diffusion. Polymer % cross-linking (% CL), % ring substitution (% RS), swelling solvent, and the size of reactant molecule all can affect both intrinsic reactivity and intraparticle diffusion. Typical particle size effects on the... 

The contribution of intraparticle diffusion to rate limitation can be seen from dependence of kobsi on the particle size of catalysts. Fig. 10 shows the effect of particle size on Kbsd for iodide displacement reactions (Eq. (4)) with catalysts 34, 35, and 41149). 

The activity of polymer-supported crown ethers is a function of % RS as shown in Fig. 11 149). Rates for Br-I exchange reactions with catalysts 34, 35, and 41 decreased as % RS increased from 14-17% to 26-34%. Increased % RS increases the hydro-philitity of the catalysts, and the more hydrated active sites are less reactive. Less contribution of intraparticle diffusion to rate limitation was indicated by less particle size dependence of kohMi with the higher % RS catalysts149). 

If controlled by mass transfer in the liquid, the exchange rate is proportional to the specific surface area and thus is inversely proportional to the particle radius or diameter. If controlled by intraparticle diffusion, the rate is, in addition, inversely proportional to the distance diffusion has to cover from particle surface to center, and so is inversely proportional to the square of the particle radius or diameter. If exchange were controlled by a reaction at the exchange site, the rate would be independent of particle size. Again, the comparison is independent of specific models or equations. 

The term in brackets is a dimensionless group that plays a key role in determining the limitations that intraparticle diffusion places on observed reaction rates and the effectiveness with which the catalyst surface area is utilized. We define the Thiele modulus hT as... 

If there were no limitations placed on the reaction rate by intraparticle diffusion (i.e., if the reactant concentration were C0 throughout the pore), the reaction rate would be given by... 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

The most difficult problem to solve in the design of a Fischer-Tropsch reactor is its very high exothermicity combined with a high sensitivity of product selectivity to temperature. On an industrial scale, multitubular and bubble column reactors have been widely accepted for this highly exothermic reaction.6 In case of a fixed bed reactor, it is desirable that the catalyst particles are in the millimeter size range to avoid excessive pressure drops. During Fischer-Tropsch synthesis the catalyst pores are filled with liquid FT products (mainly waxes) that may result in a fundamental decrease of the reaction rate caused by pore diffusion processes. Post et al. showed that for catalyst particle diameters in excess of only about 1 mm, the catalyst activity is seriously limited by intraparticle diffusion in both iron and cobalt catalysts.1... 

Chen et al. [54] have reported a model for the assessment of the combined effects of the intrinsic reaction kinetics and dye diffusion into phosphorylated polyvinyl alcohol (PVA) gel beads. The analysis of the experimental data in terms of biofilm effectiveness factor highlighted the relevance of intraparticle diffusion to the effective azo-dye conversion rate. On the basis of these results, they have identified the optimal conditions for the gel bead diameter and PVA composition to limit diffusion resistance. 

The porous structure of either a catalyst or a solid reactant may have a considerable influence on the measured reaction rate, especially if a large proportion of the available surface area is only accessible through narrow pores. The problem of chemical reaction within porous solids was first considered quantitatively by Thiele [1] who developed mathematical models describing chemical reaction and intraparticle diffusion. Wheeler [2] later extended Thiele s work and identified model parameters which could be measured experimentally and used to predict reaction rates in... 

If intraparticle diffusion controls the overall reaction rate, the Thiele modulus will be large (0 > 2) and then the effectiveness factor 77 is approximately 0. From eqn. (10) defining the Thiele modulus, it follows that, for a given reaction, the effectiveness factor will be... 

Table 1 summarizes how intraparticle diffusion affects parameters involved either explicitly or implicitly in the expression for the overall rate of reaction. 

The usual experimental criterion for diffusion control involves an evaluation of the rate of reaction as a function of particle size. At a sufficiently small particle size, the measured rate of reaction will become independent of particle size. The reaction rate can then be safely assumed to be independent of intraparticle mass transfer effects. At the other extreme, if the observed rate is inversely proportional to particle size, the reaction is strongly influenced by intraparticle diffusion. For a reaction whose rate is inhibited by the presence of products, there is an attendant danger of misinterpreting experimental results obtained for different particle sizes when a differential reactor is used, because, under these conditions, the effectiveness factor is sensitive to changes in the partial pressure of product. 

If, however, both reactions were influenced by intraparticle diffusion effects, the rate of reaction of a particular component would be given by the product of the intrinsic reaction rate, fecg, and the effectiveness factor, Tj. Substituting eqn. (6) for the effectiveness factor gives (for a first-order isothermal reaction) the overall rate as 0tanh< >. As is often the case, the molecular weights of the diffusing reactants are similar and can be... 

Research with pilot scale units has shown that the major resistances to mass transfer of reactant to catalyst are within the liquid film surrounding the wetted catalyst particles and also intraparticle diffusion. A description of these resistances is afforded by Fig. 14. Equating the rate of mass transfer across the liquid film to the reaction rate, first order in hydrogen concentration... 

Fig. 2. Reactant concentration as a function of distance from the center of a catalyst particle for fast mixing (A) and slow mixing (B). In both figures, (I) represents a reaction rate limited by intrinsic reactivity at the active site, (2) represents a reaction rate limited by mass transfer, and (3) represents a reaction rate limited by a combination of intraparticle diffusion and intrinsic reactivity. (Reprinted with permission from Ref.73). Copyright 1981 American Chemical Society)...

At one extreme diffusivity may be so low that chemical reaction takes place only at suface active sites. In that case p is equal to the fraction of active sites on the surface of the catalyst. Such a polymer-supported phase transfer catalyst would have extremely low activity. At the other extreme when diffusion is much faster than chemical reaction p = 1. In that case the observed reaction rate equals the intrinsic reaction rate. Between the extremes a combination of intraparticle diffusion rates and intrinsic rates controls the observed reaction rates as shown in Fig. 2, which profiles the reactant concentration as a function of distance from the center of a spherical catalyst particle located at the right axis, When both diffusion and intrinsic reactivity control overall reaction rates, there is a gradient of reactant concentration from CAS at the surface, to a lower concentration at the center of the particle. The reactant is consumed as it diffuses into the particle. With diffusional limitations the active sites nearest the surface have the highest turnover numbers. The overall process of simultaneous diffusion and chemical reaction in a spherical particle has been described mathematically for the cases of ion exchange catalysis,63 65) and catalysis by enzymes immobilized in gels 66-67). Many experimental parameters influence the balance between intraparticle diffusional and intrinsic reactivity control of reaction rates with polymer-supported phase transfer catalysts, as shown in Fig. 1. 

The affinity of the polymer-bound catalyst for water and for organic solvent also depends upon the structure of the polymer backbone. Polystyrene is nonpolar and attracts good organic solvents, but without ionic, polyether, or other polar sites, it is completely inactive for catalysis of nucleophilic reactions. The polar sites are necessary to attract reactive anions. If the polymer is hydrophilic, as a dextran, its surface must be made less polar by functionalization with lipophilic groups to permit catalytic activity for most nucleophilic displacement reactions. The % RS and the chemical nature of the polymer backbone affect the hydrophilic/lipophilic balance. The polymer must be able to attract both the reactive anion and the organic substrate into its matrix to catalyze reactions between the two mutually insoluble species. Most polymer-supported phase transfer catalysts are used under conditions where both intrinsic reactivity and intraparticle diffusion affect the observed rates of reaction. The structural variables in the catalyst which control the hydrophilic/lipophilic balance affect both activity and diffusion, and it is often not possible to distinguish clearly between these rate limiting phenomena by variation of active site structure, polymer backbone structure, or % RS. 

The most important point about the alkyl halide reactivities in triphase catalysis is that the reactions which have the highest intrinsic rates are the most likely to be limited by intraparticle diffusion. The cyanide ion reactions which showed the greatest particle size and cross-linking dependence with 1-bromooctane had half-lives of 0.5 to 2 h and with benzyl bromide had half-lives of 0.13 to 0.75 h. The reactions of 1-bromooctane and of benzyl chloride which were insensitive to particle size and cross-linking had half-lives of 14 h and 3 h respectively. Practical triphase liquid/ liquid/solid catalysis with polystyrene-bound onium ions has intraparticle diffusional limitations. 

Spacer chains affect intrinsic reactivity as well as intraparticle diffusion. Rates for Br-I exchange reactions with spacer-modified catalyst 41 were larger than those with catalyst 35 containing no spacer (Fig. 11). An aliphatic spacer makes the catalyst more lipophilic and the intrinsic reactivity of the active site larger, though the intraparticle diffusity of an inorganic reagent is reduced. It is not known at this time how intrinsic reactivity contributes to the rate increase. 

The activity of polymer-supported crown ethers depends on solvent. As shown in Fig. 11, rates for Br-I exchange reactions with catalysts 34 and 41 increased with a change in solvent from toluene to chlorobenzene. Since the reaction with catalyst 34 is limited substantially by intrinsic reactivity (Fig. 10), the rate increase must be due to an increase in intrinsic reactivity. The reaction with catalyst 41 is limited by both intrinsic reactivity and intraparticle diffusion (Fig. 10), and the rate increase from toluene to chlorobenzene corresponds with increases in both parameters. Solvent effects on rates with polymer-supported phase transfer catalysts differ from those with soluble phase transfer catalysts60. With the soluble catalysts rates increase (for a limited number of reactions) with decreased polarity of solvent60), while with the polymeric catalysts rates increase with increased polarity of solvent74). Solvents swell polymer-supported catalysts and influence the microenvironment of active sites as well as intraparticle diffusion. The microenvironment, especially hydration... 

The kinetics of the reaction of solid sodium iodide with 1-bromooctane were studied with a 95 % RS graft of polyethylene oxide) 6-mer methyl ether on 3 % CL polystyrene as catalyst (51)176). The rates were approximately first order in 1-bromooctane and independent of the amount of excess sodium iodide. The rates varied with the amount of the solid catalyst used, but there was not enough data to establish the exact functional dependence. All experiments employed powdered sodium iodide, magnetic stirring, and 75-150 pm catalyst beads. Thus the variables stirring speed and particle size, which normally are affected by mass transfer and intraparticle diffusion, were not studied. Yanagida 177) favors a mechanism of transfer of the sodium iodide by dissolution in the solvent (benzene) and diffusion to the catalyst particle... 

---