Transport-controlled reactions

In this section we consider experiments in which the current is controlled by the rate of electron transfer (i.e., reactions with sufficiently fast mass transport). The current-potential relationship for such reactions is different from those discussed (above) for mass transport-controlled reactions. 

FIG. 19 Normalized concentration profiles (solid lines) of the reactants and products in the DCE (a) or aqueous (b) receptor phase for the reaction between Fc (DCE) and IrClg (aqueous) with 0.1 M CIO4 in both DCE and the aqueous phase. In each case, the reactant concentration in the receptor phase was 1 mM, with 10 mM reactant inside the droplet. Drop times and final sizes were (a) 5.54 s and 0.96 mm, and (b) 6.32 s and 1.00 mm. The theoretical profiles (dashed lines) are for a transport-controlled reaction, with no transfer of the product ions. (Reprinted from Ref. 80. Copyright 1999, Royal Society of Chemistry.)... 

Reactions for common minerals fall in both categories, but many important cases tend, except under acidic conditions, to be surface controlled (e.g., Aagaard and Helgeson, 1982 Stumm and Wollast, 1990). For this reason and because of their relative simplicity, we will consider in this chapter rate laws for surface-controlled reactions. The problem of integrating rate laws for transport-controlled reactions into reaction path calculations, nonetheless, is complex and interesting (Steefel and Lasaga, 1994), and warrants further attention. 

Among the theories proposed, essentially two main mechanisms can be distinguished these are that the rate-determining step is a transport step (e.g., a transport of a reactant or a weathering product through a layer of the surface of the mineral) or that the dissolution reaction is controlled by a surface reaction. The rate equation corresponding to a transport-controlled reaction is known as the parabolic rate law when... 

The morphology of weathered feldspar surfaces, and the nature of the clay products, contradicts the protective-surface-layer hypothesis. The presence of etch pits implies a surface-controlled reaction, rather than a diffusion (transport) controlled reaction. Furthermore, the clay coating could not be "protective" in the sense of limiting diffusion. Finally, Holdren and Berner (11) demonstrated that so-called "parabolic kinetics" of feldspar dissolution were largely due to enhanced dissolution of fine particles. None of these findings, however, addressed the question of the apparent non-stoichiometric release of alkalis, alkaline earths, silica, and aluminum. This question has been approached both directly (e.g., XPS) and indirectly (e.g., material balance from solution data). 

The determination of mechanistic rate laws for soil chemical processes is very difficult since microscopic heterogeneity is pronounced in soils and even for most soil constituents such as clay minerals, humic substances, and oxides. Heterogeneity can be enhanced due to different particle sizes, types of surface sites, etc. As will be discussed more completely in Chapter 3, the determination of mechanistic rate laws is also complicated by the type of kinetic methodology one uses. With some methods used by soil and environmental scientists, transport-controlled reactions are occurring and thus mechanistic rate laws cannot be determined. 

A rather special possibility to attain information about molecular diffusion is provided by catalytic reactions if they proceed in the range of medium Thiele moduli (i.e. in the transition range between intrinsically and transport controlled reactions) [100]. By analyzing the dependence of the effective reaction rate on the catalyst particle size [101. 102] and/or the intrinsic re-... 

In this chapter you will learn that proper assessment of mass transport controlled corrosion reactions requires knowledge of the concentration distribution of the reacting species in solution, certain properties of the electrolyte, and the geometry of the system. A rigorous calculation of mass transport controlled reaction rates requires detailed information concerning these parameters. Fortunately, many of the governing equations have been solved for several well-defined geometries. 

The mass transfer coefficient, K, is defined as the ratio of the mass transport controlled reaction rate to the concentration driving force. The concentration driving force will depend on both turbulent and bulk convection. Bulk convection depends on molecular diffusivity, while the turbulent component depends on eddy diffusivity (4). The mass transfer coefficient considers the combination of the two transport mechanisms, empirically. 

Usually the experimental apparatus can be calibrated in terms of transport-controlled reactions and the rate constant... 

In transport-controlled reactions in the G/L systems, mass transfer is usually... 

In the case of a mass transport-controlled reaction, we have to make the complete transformation of (13.1) into... 

The results of the simulations indicate that the most effective channel propagation occurs when the reactions are transport-controlled (reaction rate constants faster than transport) since in these cases, the channel maximizes its own rate of growth by restricting permeability change to its walls and tip. 

Two fundamentally different regimes can exist 1) those characterized by transport-controlled reaction and 2) those characterized by kinetic rate-controlled reaction (2.)- In the case of transport-controlled reaction, the reaction rate constant is much faster than any of the transport processes involved so that the length scale over which a moving fluid comes to equilibrium is small. In this regime, therefore, the walls of a dissolution channel are essentially discontinuities in permeability while in the kinetic rate-controlled case, where equilibrium between the fluid and the reacting mineral occurs over some distance, the boundaries of a channel are blurred by a more gradual permeability change. 

Figure 2. Unstable growth of a dissolution channel in the transport-controlled reaction regime. Flow is from bottom to top, with the cement-bearing portion of the aquifer shown as shaded. Note the convergence of flow into the dissolution channel. Times are based on the dissolution of calcite using an undersaturation of 1 x 10 M calcite. The Darcian velocity upstream of the front is 1 x 10 ms ( O.Smi/r" ), the dispersion coefficient is 1 x 10 m s U...

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