Reactant diffusion process coefficient

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

The following, well-acceptable assumptions are applied in the presented models of automobile exhaust gas converters Ideal gas behavior and constant pressure are considered (system open to ambient atmosphere, very low pressure drop). Relatively low concentration of key reactants enables to approximate diffusion processes by the Fick s law and to assume negligible change in the number of moles caused by the reactions. Axial dispersion and heat conduction effects in the flowing gas can be neglected due to short residence times ( 0.1 s). The description of heat and mass transfer between bulk of flowing gas and catalytic washcoat is approximated by distributed transfer coefficients, calculated from suitable correlations (cf. Section III.C). All physical properties of gas (cp, p, p, X, Z>k) and solid phase heat capacity are evaluated in dependence on temperature. Effective heat conductivity, density and heat capacity are used for the entire solid phase, which consists of catalytic washcoat layer and monolith substrate (wall). 

Since the study of diffusion-limited reactions in solution seeks to discover more about the nature of the reaction path, the nature of the encounter pair, the energetics of the reaction and possibly the rate of reaction of the encounter pair, ftact, it is to be recommended that experimentalists actively seek to measure the diffusion coefficients of the reactants (or similar species), as well as any other parameters which may have an important bearing on the rate coefficient. By so doing, some of the uncertainty in estimating encounter distance may be removed and inconsistencies between diffusion coefficients measured independently and those obtained from an analyses of rate coefficient time dependence may provide valuable insight into the nature of the diffusion process at short distances. 

Strictly the mathematical expression to be used for the diffusion process should take account of these constraints however, this kind of counter-diffusion involving two reactants and two products in proportions determined by the stoichiometry of the process is of a complexity which has not yet been considered theoretically. In the absence of such a theoretical treatment, Equation (3) was applied using diffusion coefficients reported in the literature for each of the components for diffusion at room temperature. A small correction for the effect of the temperature gradient in the boundary layer on the diffusion coefficient was made in a manner discussed later. 

The temporal evolution of the species grating component is determined by the chemical reaction and protein diffusion processes. When there is no chemical reaction in the detection time window, and the molecular diffusion coefficient (D) is time-independent, the temporal profile of the species grating signal can be calculated by the molecular diffusion equation. The Fourier component at a wavenumber of q of the concentration profile decays with a rate constant Dq2 for both the reactant and the product. Hence, the time development of the TG signal can be expressed by [15-19,23]... 

Transport Coefficients The reactivity of zeolites is determined by the ability of molecules (reactants and products) to diffuse to and from an active site. Therefore, knowledge about the diffusion process is essential for understanding zeolite chemical activity. The diffusion process is characterized by a diffusion coefficient which can be calculated two ways. The first makes the use of the Einstein relationship that relates mean-square displacement of a molecular CoM position... 

To summarize, the proton transfer reaction can be broken into three distinct parts Diffusion of the reactants to within the radius of the ionic atmosphere accelerated diffusion to within the encounter distance and subsequent conversion of the encoimter complex to products. For reactions in which the equilibrium is rapidly established within the encounter complex, the rate equations are dominated by the diffusion process. This results in the loss of information about the dynamics of the encounter complex. For such a reaction some information can be obtained about the ionic radius by varying the ionic strength and using an electrostatic theory (such as is done for Deby-Hiickel activity coefficients) to calculate the effect of shielding by the ions. ... 

Diffusion process at a constant electrode potential. Assuming that Reaction (2-II) is a totally reversible reaction, and the reductant is insoluble (CR(0,f) = 1). According to the Nernst Eqn (2.24), the oxidant s surface concentration should be constant if the electrode potential is held as a constant. In this case, Co(0, t) = Cq = constant (Cq is the reactant concentration at electrode surface). Using the other three conditions as (1) the diffusion coefficient (Do) is constant, independent on the reactant concentration (2) at the beginning of reaction (t=0), the reactant concentration is uniform across the entire electrolyte solution, that is,Co(x,0)= C and (3) at any time, the reactant concentration at unlimited distance is not changed with reaction process, that is, Co(°o,t) = C, Eqn (2.40) can be resolved to give the expression of Co(x,t) ... 

The rate of a solid-state reaction is controlled either by the chemical combination at the reaction interface or by the transport of reactants to the reaction zone. In diffusion-controlled reactions, in systems including only plane surfaces, unidirectional diffusion processes, and constant diffusion coefficients the thickness of the product layer y is related to reaction time t by the well known parabolic rate law... 

Table 1.6 Characteristic quantities to be considered for micro-reactor dimensioning and layout. Steps 1, 2, and 3 correspond to the dimensioning of the channel diameter, channel length and channel walls, respectively. Symbols appearing in these expressions not previously defined are the effective axial diffusion coefficient D, the density thermal conductivity specific heat Cp and total cross-sectional area S, of the wall material, the total process gas mass flow m, and the reactant concentration Cg [114].

The relation between E and t is S-shaped (curve 2 in Fig. 12.10). In the initial part we see the nonfaradaic charging current. The faradaic process starts when certain values of potential are attained, and a typical potential arrest arises in the curve. When zero reactant concentration is approached, the potential again moves strongly in the negative direction (toward potentials where a new electrode reaction will start, e.g., cathodic hydrogen evolution). It thus becomes possible to determine the transition time fiinj precisely. Knowing this time, we can use Eq. (11.9) to find the reactant s bulk concentration or, when the concentration is known, its diffusion coefficient. 

Surface reactions are in principle multistep processes, and previously we saw how diffusion to the surface of a porous pellet or diffusion within the pores of a pellet can lead to rate expressions with rate coefficients much different than k". However, so far all were first order in the reactant partial pressure. 

We emphasize that we are interested in the fluid-phase diffusion coefficient of the reactant A, which we call D, and also the solid-phase diffusion of this species D,4s (bold and subscript s). The diffusion of reactant A in either situation can limit the reaction process. 

When the activation process is comparable with or slower than the rate of approach of reactants to form encounter pairs, it is no longer satisfactory to say that the reactants can not co-exist within a distance R of one another. Because the rate of reaction, /eact, of the activation process is finite, so too is the lifetime (and hence concentration) of encounter pairs non-zero. The inner boundary condition, which describes reaction of A and B together in the diffusion analysis, is unsatisfactory. Collins and Kimball [4] suggested an alternative boundary condition and the remainder of this section analyses their work following Noyes [5]. Firstly, the boundary condition is developed and then included in the diffusion equation analysis to obtain the density distribution. Finally, the rate coefficient is obtained. 

In Chaps. 3 and 4, estimates of encounter distances and mutual diffusion coefficients from similar experiments to those of Buxton et al. [18] are discussed. The complications to the analysis of diffusion-controlled rate processes in solution when the reactants interact strongly with one another or the reaction can occur over distances much larger than typical encounter distances do not lead to markedly different time-dependent rate coefficient expressions to the Smoluchowski form. Indeed, replacing R in eqn. (29) by an effective encounter distance, Reff, allows the compactness of the Smoluchowski rate coefficient to be extended to other situations. Means of estimating Reff are discussed in Chaps. 3, 4, 5 (Sect. 4.3), 8 (Sect. 2.6) and 9 (Sects. 4 and 6). 

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