Reactant diffusion process

Figure 2.1 Schematic of the electrode electron-transfer and reactant diffusion process in an electrochemical system. Cb(0,t) is the surface concentration of oxidant species, ChlO.t) is the surface concentration of reductant species, Co(x,f) is the bulk solution concentration of oxidant, and Cr(x, ) is the bulk solution concentration of reductant species. In these four expressions of concentration, tis the reaction time. (For color version of this figure, the reader is referred to the online version of this book.)...

Do we expect this model to be accurate for a dynamics dictated by Tsallis statistics A jump diffusion process that randomly samples the equilibrium canonical Tsallis distribution has been shown to lead to anomalous diffusion and Levy flights in the 5/3 < q < 3 regime. [3] Due to the delocalized nature of the equilibrium distributions, we might find that the microstates of our master equation are not well defined. Even at low temperatures, it may be difficult to identify distinct microstates of the system. The same delocalization can lead to large transition probabilities for states that are not adjacent ill configuration space. This would be a violation of the assumptions of the transition state theory - that once the system crosses the transition state from the reactant microstate it will be deactivated and equilibrated in the product state. Concerted transitions between spatially far-separated states may be common. This would lead to a highly connected master equation where each state is connected to a significant fraction of all other microstates of the system. [9, 10]... 

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. 

It is clear that the achievement of equilibrium is assisted by the maximum contact between the reactant and the Uansporting gas, but the diffusion problem is complex, especially in a temperature gradient, when a process known as thermal diffusion occurs. The ordinaty concenuation-dependent diffusion process occurs across dre direction of gas flow, but the thermal diffusion occurs along the direction of gas flow, and thus along the temperature gradient. 

In a discussion of these results, Bertrand et al. [596,1258] point out that S—T behaviour is not a specific feature of any restricted group of hydrates and is not determined by the nature of the residual phase, since it occurs in dehydrations which yield products that are amorphous or crystalline and anhydrous or lower hydrates. Reactions may be controlled by interface or diffusion processes. The magnitudes of S—T effects observed in different systems are not markedly different, which indicates that the controlling factor is relatively insensitive to the chemical properties of the reactant. From these observations, it is concluded that S—T behaviour is determined by heat and gas diffusion at the microdomain level, the highly localized departures from equilibrium are not, however, readily investigated experimentally. 

Thermal-Gradient Infiltration. The principle of thermal-gradient infiltration is illustrated in Fig. 5.15b. The porous structure is heated on one side only. The gaseous reactants diffuse from the cold side and deposition occurs only in the hot zone. Infiltration then proceeds from the hot surface toward the cold surface. There is no need to machine any skin and densification can be almost complete. Although the process is slow since diffusion is the controlling factor, it has been used extensively for the fabrication of carbon-carbon composites, including large reentry nose cones. 

For a catalytic reaction of a reactant from a single fluid phase (either gas or liquid) to take place on a solid catalyst, diffusion processes also play a role, so in the complete process the following steps can be distinguished ... 

An interfacial reaction is accompanied by a diffusion process which provides reactants from the bulk phase to the interface. When the reaction rate is faster than the diffusion rate, the overall reaction has to be governed by the diffusion rate of the reactant. Diffusion, in general, takes place in the stagnant layers (the region on both sides of the interface) and is not disturbed even under the stirring of a bulk region. 

Kinetics of chemical reactions at liquid interfaces has often proven difficult to study because they include processes that occur on a variety of time scales [1]. The reactions depend on diffusion of reactants to the interface prior to reaction and diffusion of products away from the interface after the reaction. As a result, relatively little information about the interface dependent kinetic step can be gleaned because this step is usually faster than diffusion. This often leads to diffusion controlled interfacial rates. While often not the rate-determining step in interfacial chemical reactions, the dynamics at the interface still play an important and interesting role in interfacial chemical processes. Chemists interested in interfacial kinetics have devised a variety of complex reaction vessels to eliminate diffusion effects systematically and access the interfacial kinetics. However, deconvolution of two slow bulk diffusion processes to access the desired the fast interfacial kinetics, especially ultrafast processes, is generally not an effective way to measure the fast interfacial dynamics. Thus, methodology to probe the interface specifically has been developed. 

Electrogenerated chemiluminescence (ECL) is the process whereby a chemiluminescence emission is produced directly, or indirectly, as a result of electrochemical reactions. It is also commonly known as electrochemiluminescence and electroluminescence. In general, electrically generated reactants diffuse from one or more electrodes, and undergo high-energy electron transfer reactions either with one another or with chemicals in the bulk solution. This process yields excited-state molecules, which produce a chemiluminescent emission in the vicinity of the electrode surface. 

These results take into account three possible processes in series mass transfer of fluid reactant A from bulk fluid to particle surface, diffusion of A through a reacted product layer to the unreacted (outer) core surface, and reaction with B at the core surface any one or two of these three processes may be rate-controlling. The SPM applies to particles of diminishing size, and is summarized similarly in equation 9.1-40 for a spherical particle. Because of the disappearance of the product into the fluid phase, the diffusion process present in the SCM does not occur in the SPM. 

The mechanisms considered above are all composed of steps in which chemical transformation occurs. In many important industrial reactions, chemical rate processes and physical rate processes occur simultaneously. The most important physical rate processes are concerned with heat and mass transfer. The effects of these processes are discussed in detail elsewhere within this book. However, the occurrence of a diffusion process in a reaction mechanism will be mentioned briefly because it can lead to kinetic complexities, particularly when a two-phase system is involved. Consider a reaction scheme in which a reactant A migrates through a non-reacting fluid to reach the interface between two phases. At the interface, where the concentration of A is Caj, species A is consumed in a first-order chemical rate process. In effect, consecutive rate processes are occurring. If a steady state is achieved, then... 

Many experimental studies of the rates of oxidation of S(IV) in solution have used either bulk solutions or droplets that are very large compared to those found in the atmosphere. In addition, reactant concentrations in excess of atmospheric levels have often been used for analytical convenience. The use of large droplets increases the diffusion times, whereas higher reactant concentrations speed up the aqueous-phase chemical reaction rates. The combination of these two factors can lead to a situation where the rates of the diffusion processes, either of the gas to the droplet surface or more likely within the aqueous phase itself, become comparable to, or slower than, the chemical reaction rate. If this is not recognized, the observed rates may be attributed in error to the intrinsic chemical reaction rate. 

With a monolith reactor, diffusion from the channels to the catalyst coated on the channel walls is the sole means by which reactants are able to reach the catalyst (Section III). It seems reasonable that a similar diffusion process occurs in a coated filter. 

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). 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

In this section we mov6 on to examine the behaviour of an open system in which the transport of reactants and products relies on molecular diffusion processes. The situation we envisage is that of a reaction zone in which various species diffuse and react. Outside this zone there exists an external reservoir in which the reactants have fixed concentrations. The reservoir provides a source of reactants which can diffuse across the boundary into the reaction zone, and either a source or a sink for the intermediates and products. The reaction-diffusion cell is sketched in Fig. 9.2. 

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. 

The above description is of a thermally propagating steady-state wave. It must be emphasized, however, that the basic feature of a thermal mechanism is not altered by the superposition of molecular diffusion onto the diffusional transport of heat. This applies not only to interdiffusion of reactants and products but also to the diffusion of chain carriers participating in the chemical reaction, provided that the chains are unbranched. The reason for this is that in a wave driven by a diffusion process, the source strength of an entering mass element must continue to grow despite the drain by the adjacent sink region. This growth can occur only if the reaction rate is increased by a product of the reaction, which may be temperature as well as a material product. 

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