Reaction Versus Diffusion

Essentially all of the quirks and imperfections that make solid-state systems interesting— point defects, dislocations, grain boundaries, inclusions, voids, surfaces—fall within the scope of materials kinetics. This focus on solid-state processes and heterogeneity—what many would call microstructural development—is what makes materials kinetics unique. In order to tackle this topic, we will need to borrow a lot of concepts from chemical reaction kinetics, which we will cover in Chapter 3 of this textbook, but we will also learn many other concepts that are not usually covered in traditional chemical-based treatments of kinetics. In particular, we will spend a lot of time on solid-state diffusion and transport (Chapter 4). Compared to the gas and liquid phases, transport of matter in the solid phase tends to be slower and more difficult thus, atomic transport processes such as diffusion become much more important in determining kinetic behavior in solid-state systems. 

TABLE 1.1 Typical Atomic Diffusivities for Solid, Liquid, and Gas States 

In addition to diffusivity D, another kinetic parameter that we will frequently encounter is the reaction rate constant k. The reaction rate constant is used to quantify the relative ease of a chemical reaction or, in some cases, a localized reconfiguration or charge transfer process. As with diffusion processes, there are a number of mathematical expressions (rate laws) available to describe the speed of various reaction 

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Murphy W. M., Oelkers E. H., and Lichtner P. C. (1989) Surface reaction versus diffusion control of mineral dissolution and growth rates in geochemical processes. Chem. Geol. 78, 357-380. 

REACTION-CONTROLLED MINERAL-FLUID ISOTOPE EXCHANGE Chemical reaction versus diffusion... 

Figure 12.2 is a plot of the effectiveness factor r] versus the Thiele modulus hT. For low values of hT (slow reaction, rapid diffusion), the effectiveness factor approaches unity. For values of the Thiele modulus above 2.0, tanh hT 1 and the effectiveness factor may be approximated by... 

Carrier-mediated passage of a molecular entity across a membrane (or other barrier). Facilitated transport follows saturation kinetics ie, the rate of transport at elevated concentrations of the transportable substrate reaches a maximum that reflects the concentration of carriers/transporters. In this respect, the kinetics resemble the Michaelis-Menten behavior of enzyme-catalyzed reactions. Facilitated diffusion systems are often stereo-specific, and they are subject to competitive inhibition. Facilitated transport systems are also distinguished from active transport systems which work against a concentration barrier and require a source of free energy. Simple diffusion often occurs in parallel to facilitated diffusion, and one must correct facilitated transport for the basal rate. This is usually evident when a plot of transport rate versus substrate concentration reaches a limiting nonzero rate at saturating substrate While the term passive transport has been used synonymously with facilitated transport, others have suggested that this term may be confused with or mistaken for simple diffusion. See Membrane Transport Kinetics... 

If a graph of /p versus Vv gives a straight line, then the reaction is diffusion controlled. Prepare such a graph and use it to find the diffusion coefficient of the reactant for this one-electron oxidation. The area of the working electrode is 0.020 I cm2 and the concentration of reactant is 1.00 mM. 

The two contrasting approaches, the macroscopic viewpoint which describes the bulk concentration behavior (last chapter) versus the microscopic viewpoint dealing with molecular statistics (this chapter), are not unique to chromatography. Both approaches offer their own special insights in the study of reaction rates, diffusion (Brownian motion), adsorption, entropy, and other physicochemical phenomena [2]. 

Figure 18. Effectiveness factor rj of a first-order reversible reaction versus the Weisz modulus ip (related to the forward rate constant k+). Influence of intraparticle diffusion on the effective reaction rate (isothermal reaction in a sphere, equal diffusivitics i,e = Die, equilibrium constant as a parameter).

The relative values of Rex versus R xn were estimated by employing TR ESR and computer simulation to estimate the concentrations of products of spin exchange and chemical reaction between the same reagents. " It was assumed that the exchange is strong and chemical reactions are diffusion controlled. The unique feature (Scheme 12.14) of the isotopic asymmetric biradical ( " N-0- N) allows the... 

One of the most striking features of CCT is the exceptionally fast rate at which it takes place. The molecular weight of a polymer can be reduced from tens of thousands to several hundred utilizing concentrations of cobalt catalyst as low as 100—300 ppm or 10 3 mol/L. The efficiency of catalysis can be measured as the ratio between the chain-transfer coefficients of the catalyzed reaction versus the noncatalyzed reaction. The chain-transfer constant to monomer, Cm, in MMA polymerization is believed to be approximately 2 x 10 5.29 The chain-transfer constant to catalyst, Cc, is as high as 103 for porphyrins and 104 for cobaloximes. Hence, improved efficiency of the catalyzed relative to the uncatalyzed reaction, CJCu, is 104/10 5 or 109. This value for the catalyst efficiency is comparable to many enzymatically catalyzed reactions whose efficiencies are in the range of 109—1011.18 The rate of hydrogen atom transfer for cobaloximes, the most active class of CCT catalysts to date, is so high that it is considered to be controlled by diffusion.5-30 32 Indeed, kc in this case is comparable to the termination rate constant.33... 

A.V. Straube, M. Abel, and A. Pikovsky. Temporal chaos versus spatial mixing in reaction-advection-diffusion systems. Phys. Rev. Lett., 93 174501, 2004. 

Spiral wave dynamics Reaction and diffusion versus... 

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