Macroscopic diffusion control

MACROSCOPIC DIFFUSION CONTROL MICROSCOPIC DIFFUSION CONTROL Magic acid,... 

MICROSCOPIC DIFFUSION CONTROL MACROSCOPIC DIFFUSION CONTROL MICROSCOPIC REVERSIBILITY CHEMICAL REACTION DETAILED BALANCING, RRINCIRLE OF CHEMICAL KINETICS MICROTUBULE ASSEMBLY KINETICS BIOCHEMICAL SELF-ASSEMBLY ACTIN ASSEMBLY KINETICS HEMOGLOBINS POLYMERIZATION... 

The term macroscopic diffusion control has been used to describe processes in which the rate of reaction is determined essentially by the rate of mixing of the reactant solutions. The nitration of toluene in sulpholane by the addition of a solution of nitronium fluoroborate in sulpholane appears to fall into this class (Ridd, 1971a). Obviously, if a reaction is subject to microscopic diffusion control when the reactants meet in a homogeneous solution, it must also be subject to macroscopic diffusion control when preformed solutions of the same reactants are mixed. However, the converse is not true. The difficulty of obtaining complete mixing of solutions in very short time intervals implies that a reaction may still be subject to macroscopic diffusion control when the rate coefficient is considerably below that for reaction on encounter. The mathematical treatment and macroscopic diffusion control has been discussed by Rys (Ott and Rys, 1975 Rys, 1976), and has been further developed recently (Rys, 1977 Nabholtz et al, 1977 Nabholtz and Rys, 1977 Bourne et al., 1977). It will not be considered further in this chapter. 

While considering the influence of the encounter rate on chemical reactivity a microscopic and macroscopic diffusion control should be mentioned. In microscopic diffusion control, the reactants exist together in a homogeneous solution and the reaction occurs on every encounter. 

The term macroscopic diffusion control describes processes in which the rate of the reaction is determined by the rate of mixing of the reactant solutions. 

Special effects arise for a solution reaction that is extremely rapid, in which case the rate may depend on the rate with which the reactant molecules diffuse through the solvent. Two effects are to be distinguished, macroscopic diffusion control and microscopic diffusion control. If a rapid bimolecular reaction in solution is initiated by mixing solutions of the two reactants, the observed rate may depend on the rate with which the solutions mix, and one then speaks of mixing control or macroscopic diffusion control. 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

The objective is to reduce volatiles to below 50-100-ppm levels. In most devolatilization equipment, the solution is exposed to a vacuum, the level of which sets the thermodynamic upper limit of separation. The vacuum is generally high enough to superheat the solution and foam it. Foaming is essentially a boiling mechanism. In this case, the mechanism involves a series of steps creation of a vapor phase by nucleation, bubble growth, bubble coalescence and breakup, and bubble rupture. At a very low concentration of volatiles, foaming may not take place, and removal of volatiles would proceed via a diffusion-controlled mechanism to a liquid-vapor macroscopic interface enhanced by laminar flow-induced repeated surface renewals, which can also cause entrapment of vapor bubbles. 

Typical feedback approach curves for DMPPD oxidation in solutions of pH 10.20, 10.78, and 11.24 are shown in Figure 8, along with the behavior for diffusion-controlled positive and negative feedback. The tip electrode process was defined by Eq. (36), while the unbiased macroscopic substrate was poised at a potential where the reverse reaction occurred at a diffusion-controlled rate ... 

Depending on the conditions in which a process is conducted and its features, any of the five steps may be the slowest one. Hence, the rate of the catalytic process may be limited by one of them. An interfacial chemical reaction may proceed only with continuous molecular or convective diffusion of the reactants to the surface on which the given reaction is proceeding, and also with continuous reverse diffusion of the products. The rate of a process as a whole will be determined by the rate of its slowest step. If the rate of a reaction on the surface of a catalyst is greater than that of diffusion, the rate of the process as a whole will be determined by the rate of diffusion. The observed macroscopic kinetics of the reaction will obey equations that can be obtained by considering only processes of diffusion and will not reflect the true rate of the chemical reaction at the interface. Such a process is a diffusion-controlled one. It is most frequently described by a first-order reaction equation, since the rate of diffusion is directly proportional to the concentration. 

When macroscopic diffusion models are compared to experimental data, high values of tortuosity (corresponding to low values of Detr) are obtained (Tables I and II). For a random porous medium, tortuosity values due to windiness ofthe diffusional path should be between 1 and 3 (Pismen, 1974 Bhatia, 1986). Because the overall tortuosities predicted by the macroscopic models are much larger, other physical properties ofthe matrix must influence the rate of protein diffusion within the matrix. Microgeomet-ric models and percolation theory have been used to study the factors that might control protein diffusion within these polymer matrices. 

For both dyes in aqueous solution (10 mM phosphate) the forward rate is almost diffusion controlled, so that the stability of the dimer is reflected in the reverse rate. For thionine and proflavin the measured rate constants at 22 C are listed below. For thionine in ethanol and propanol the effect on the forward rate is much too large to be attributed to changes in any of the macroscopic properties of the solutions. To interpret these results specific interactions at the molecular level must be considered. 

This treatment of the rate of diffusion-controlled encounters in solution was developed originally by Smoluchowski [13,a] for the rates of coagulation of colloidal solutions, and was later applied to reactions between molecules. It assumes that the diffusive motions of molecules can be treated like those of macroscopic particles in a continuous viscous fluid. A simplified version is as follows. 

The connection of random-walk calculations based on concentration-gradient theory may be summoned as follows. As we have noted (Section 2.3), the Equation for diffusion-controlled rate constants based on Pick s law agrees with that derived from a random-walk model (Section 2.3). Pick s law is in fact a macroscopic consequence of the random-walk model of molecular-scale processes or, to put it the other way round, the random-walk model is an interpretation of Pick s experimental law [4]. 

A special feature of the kinetics of reactions initiated by electron pulses is that in the early stages the reaction proceeds mainly in relatively small confined regions of the solution, known as spurs , within which the concentrations of electrons, radicals and excited molecules are larger by orders of magnitude than those that obtain in the bulk solution. That this is the state of affairs may be expected from the physics of absorption of electrons by liquids sketched above. Much experimental evidence on product yields shows good agreement with a model based on the assumption of equilibrated macroscopic values of rate constants within the spurs, coupled with diffusion-controlled transfer of product molecules to the bulk solution. A brief account of the theory follows. 

---