Reaction-controlled diffusion

Reactions that are bimolecular can be affected by the viscosity of the medium [9]. The translational motions of flexible polymeric chains are accompanied by concomitant segmental rearrangements. Whether this applies to a particular reaction, however, is hard to tell. For instance, two dynamic processes affect reactions, like termination rates, in chain-growth polymerizations. If the termination processes are controlled by translational motion, the rates of the reactions might be expected to vary with the translational diffusion coefficients of the polymers. Termination reactions, however, are not controlled by diffusions of entire molecules, but only by segmental diffusions within the coiled chains [10]. The reactive ends assume positions where they are exposed to mutual interaction and are not affected by the viscosity of the medium. 

Termination is a fast chemical reaction controlled by the rates at which the two radical ends encounter each other. The apparent rate coefficient is affected not only by pressure and temperature, but also by system viscosity (a function of solvent choice, polymer concentration and MW) and the lengths of the two terminating radicals. This complex behavior. 

As polymer concentration in the system increases with conversion, the viscosity of the system rapidly increases such that the rate at which two polymer chains encounter each other is slower than the rate of segmental reorientation, and the rate of reaction becomes controlled by how quickly the two chains find each other among the tangle of 

The overall behavior of kt with conversion is often modeled as a composite of the various diffusional processes  

The subscripts SD, TD and RD refer to segmental diffusion, translational diffusion and reaction diffusion, with fct sD set to the low conversion values summarized in Table 3.2. The reaction diffusion term kt,RD is proportional to propagation, with proportionality coefficient Crd estimated from experimental data [10]  

Under the glass effect, kp may itself become diffusion controlled at very high conversions [10, 47], 

Many reactions in glasses are controlled by the diffusion of atoms, molecules, or ions from the surrounding environment into the glass. Since an extensive discussion of these reactions is beyond the scope of this book, only one typical example of a diffusion-controlled process will be presented here. 

All of these processes can be described by a diffusion-controlled model originally derived to explain the tarnishing of metals and hence commonly called the tarnishing model. The derivation of this model is based on the assumptions that (a) the reaction site is immobile, (b) the concentration of reaction sites is independent of time and temperature in the absence of the tarnishing reaction, and (c) the reaction rate is very 

If the experimental conditions prevent direct measurement of the layer thickness, we can convert Eq. 8.24 into a form appropriate for measurements of the average concentration of the reacted species in the sample. Under these conditions, for samples exposed to gas from both surfaces, one can show that the concentration of the reacted species, C, is given by the expression 

Diffusion of atoms, molecules, and ions control many processes in glasses, including ionic diffusion, ion exchange, electrical conduction, chemical durability, gas permeation, and permeation-controlled reactions. Since the mechanisms underlying all of these processes are based on similar principles, a fundamental understanding of diffusion phenomena serves as the basis for understanding all diffusion-controlled properties of glasses. 

Select the glass with the higher conductivity in each pair below. Explain your choice. 

If the rate of a reaction is governed by the encounter frequency, it is said to be diffusion-controlled. This frequency imposes an upper limit on the rate of reaction that can be evaluated by the use of Fick s laws of diffusion. The mathematical expression of this phenomenon was first presented by von Smoluchowski.2 We shall adopt a simple approach,3,4 although more rigorous derivations have been given.5 

The quantity of solute B crossing a plane of area A in unit time defines the flux. It is symbolized by J, and is a vector with units of molecules per second. Fick s first law of diffusion states that the flux is directly proportional to the distance gradient of the concentration. The flux is negative because the flow occurs in a direction so as to offset the gradient  

The proportionality constant in this equation, D, is called the diffusion constant. A typical solute has D 1 X 10 9 m2 s . Values of D do not vary greatly with the solute but are inversely proportional to the viscosity of the solvent. 

Consider the bimolecular reaction of A and B. The concentration of B is depleted near the still-unreacted A by virtue of the very rapid reaction. This creates a concentration gradient. We shall assume that the reaction occurs at a critical distance tab- At distances r tab. [B] = 0. Beyond this distance, at r rAB, [B] = [B]°, the bulk concentration of B at r = °°. We shall examine a simplified, two-dimensional derivation the solution in three dimensions must incorporate the mutual diffusion of A and B, requiring vector calculus, and is not presented here. 

Provided that the bulk concentrations remain constant, the flux J of Eq. (9-3) is also constant. Consider a spherical distribution of potential reactants B around a particular molecule of A. The surface area of a sphere at a distance r from A is 4irr2. Thus, the expression for the flow of B toward A is 

The black sphere approximation permits us to obtain the most simple and physically transparent results for the kinetics of diffusion-controlled reactions. We should remind that this approximation involves a strong negative correlation of dissimilar particles at r ro, where Y (r, t) = 0, described by the Smoluchowski boundary condition 

Defining K t) through equation (4.1.58), we can obtain with the help of equation (4.1.59) that 

Transient kinetics with the time-dependent reaction rate K t), equation (4.1.61), have been observed experimentally more than once (e.g., by Tanimura and Itoh [19]). In the steady-state case, f - oo, we get finally 

The formation of the steady-state recombination profile (4.1.62) occurs for the space dimension d — 2 only. For instance, if J = 2, taking into account the change in the diffusion operator A and the expression for the two-dimensional reaction rate 

In other words, the peculiarity of the rwo-dimensional motion which has led to the zero survival probability of correlated pairs, equation (3.2.26), for randomly distributed particles consists of the complete zerofication of the reaction rate at a great time, K (oo) = 0. The logarithmic dependence of the reaction rate on time does not considerably affect the asymptotic behaviour of macroscopic concentrations. Introducing the critical exponent a 

Consider a dilute solution of two reactant molecules, A and B. Inevitably an A molecule and a B molecule will undergo an encounter, the frequency of such encounters depending upon the concentrations of A and B. If, upon each encounter of A and B, they undergo bimolecular reaction, then the rate of this reaction is determined solely by the rate of encounter of A and B that is, the rate is not controlled by the chemical requirement that an energy barrier be overcome. One way to find this rate is to treat the problem as one of classical diffusion, and so this maximum possible rate of reaction is often called the diffusion-controlled rate. This problem was solved by Smoluchowski. In the following development no provision is made for attractive forces between the molecules.  

Consider spherical molecules A and B having radii and Tb and diffusion coefficients Da and Db- First, suppose that B is fixed and that the rate of reaction is limited by the rate at which A molecules diffuse to the B molecule. We calculate the flux 7(A- B) of A molecules to one B molecule. Let a and b be the concentrations (in molecules/cm ) of A and B in the bulk, and let r be the radius of a sphere centered at the B molecule. The surface area of this sphere is Aitr, so by Pick s first law we obtain 

Assume that chemical reaction occurs at contact, namely, when r = r tb thus, we integrate between limits as shown  

Thus far the single B molecule has been held stationary, and Eq. (4-3) gives the flux of A molecules toward this fixed B. At the same time, however, the B molecule 

The conventional second-order rate equation is = A dCa b. where is in moles per liter per second and Ca, Cb are in moles per liter. Because c = 1000 /A/a and Vc = 1000v Wa, where A a is Avogadro s number, we obtain 

R to P. Also, in the infinite time limit C(t - oo) - 0 because at long time x(0) and /i[ (f)] are not coiTelated. However, this relaxation to zero is very slow, of the order of the reaction rate k, because losing this correlation implies that the trajectory has to go several times between the wells. 

Our focus so far was on unimolecular reactions and on solvent effects on the dynamics of barrier crossing. Another important manifestation of the interaction between the reaction system and the surrounding condensed phase comes into play in bimolecular reactions where the process by which the reactants approach each other needs to be considered. We can focus on this aspect of the process by considering bimolecular reactions characterized by the absence of an activation barrier, or by a barrier small relative to ke T. In this case the stage in which reactants approach each other becomes the rate determining step of the overall process. 

In condensed phases the spatial motion of reactants takes place by diffusion, which is described by the Smoluchowski equation. To be specific we consider a 

It is convenient to define A = -4[A] and B = v4[2 ], where A is the Avogadro number and A and B are molecular number densities of the two species. In terms of these quantities Eq. (14.100) takes the form 

In order to obtain an expression of the bimolecular rate coefficient associated 

Consider a bimolecular reaction in solution as occurring in two steps. In the first step, an encounter complex is formed  

The complex may then either revert to separated reactants or react to form products  

Applying the steady-state assumption to the concentration of the encounter complex  

If the encounter complex reacts to form products much faster than it reverts to reactants, that is, if k k, then k=k kjk =k, that is, the rate is controlled by the rate of formation of the encounter complex. Such a reaction is desalbed as diffusion controlled or encounter controlled. 

Electron-transfer reactions are a class in which diffusion control may be observed. If an electron-donor species D (reductant) is to react with an electron acceptor A (oxidant), they first form an encounter complex, within which the transfer occurs by tunneling at a rate chiefly determined by the height of the barrier between the donor s HOMO and the acceptor s LUMO, and its width, which is determined by the distance of closest approach. If the transfer rate constant is large, the rate-limiting step will be the formation of the encounter complex by diffusion. This picture is an oversimplification. The theory developed by R. A. Marcus, and independently by others, is described in most physical chemistry texts (e.g., Atkins and de Paula, 2010, pp. 856-861). 

The kinetic schemes in this chapter have been written assuming that kt is independent of the sizes of the radicals involved in the termination reaction. This is not 

Strictly true, since the termination rate is limited by the rates at which the radical ends can encounter each other. For a diffusion-controlled reaction, the apparent rate coefficient is affected not only by pressure and temperature, but also by system viscosity (a function of solvent, polymer concentration, and MW) and the lengths of the two terminating radicals. This complex behavior, as well as experimental difficulties in measuring fej, has led to a large scatter in reported values, even at low conversion [7]. Through the application of pulsed-laser experimental techniques [15] and a critical examination of available data [18], the situation is starting to improve. 

Lower values of 10 L mol for dodecyl (meth)acrylate is attributed to shielding of the radicals by the long hain dodecyl ester groups for these monomers fej remains relatively constant up to 60% conversion [64]. Even lower kt values are measured for termination during polymerization of stericaDy hindered monomers such as the itaconates [74] and acrylate trimer species [44]. The variation of low-conversion kt with polymer composition in copolymer systems can be represented by Eq. (55). 

In order to obtain an expression of the bimolecular rate coefficient associated with this process we follow a similar route as in the barrier crossing problem, by considering the flux associated with a steady state that is approached at long time by a system subjected to the following boundary conditions (1) the bulk concentration of B remains constant and (2) B disappears when it reacts with at a distance R  

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Caldin E F, de Forest L and Queen A 1990 Steric and repeated collision effects in diffusion-controlled reactions in solution J. Chem. See. Faraday Trans. 86 1549-54... 

In these circumstances a decision must be made which of two (or more) kinet-ically equivalent rate terms should be included in the rate equation and the kinetic scheme (It will seldom be justified to include both terms, certainly not on kinetic grounds.) A useful procedure is to evaluate the rate constant using both of the kinetically equivalent forms. Now if one of these constants (for a second-order reaction) is greater than about 10 ° M s-, the corresponding rate term can be rejected. This criterion is based on the theoretical estimate of a diffusion-controlled reaction rate (this is described in Chapter 4). It is not physically reasonable that a chemical rate constant can be larger than the diffusion rate limit. 

The temperature dependence of a diffusion controlled reaction is typically described by the Arrhenius relationship ... 

Finally, one must know the effect of catalyst particle size on Kw. For a pore diffusion-controlled reaction, activity should be inversely proportional to catalyst particle diameter, that is directly proportional to external catalyst surface area. 

The electron transfer from a methanol molecule to the activated diazonium ion is obviously a diffusion-controlled reaction. The rate constant is of the same order... 

The properties of barrier layers, oxides in particular, and the kinetic characteristics of diffusion-controlled reactions have been extensively investigated, notably in the field of metal oxidation [31,38]. The concepts developed in these studies are undoubtedly capable of modification and application to kinetic studies of reactions between solids where the rate is determined by reactant diffusion across a barrier layer. 

Fig. 3. Reduced time plots, tr = (t/t0.9), for the contracting area and contracting volume equations [eqn. (7), n = 2 and 3], diffusion-controlled reactions proceedings in one [eqn. (10)], two [eqn. (13)] and three [eqn. (14)] dimensions, the Ginstling— Brounshtein equation [eqn. (11)] and first-, second- and third-order reactions [eqns. (15)—(17)]. Diffusion control is shown as a full line, interface advance as a broken line and reaction orders are dotted. Rate processes become more strongly deceleratory as the number of dimensions in which interface advance occurs is increased. The numbers on the curves indicate the equation numbers.

Kodama and Brydon [631] identify the dehydroxylation of microcrystalline mica as a diffusion-controlled reaction. It is suggested that the large difference between the value of E (222 kJ mole-1) and the enthalpy of reaction (43 kJ mole-1) could arise from the production of an amorphous transition layer during reaction (though none was detected) or an energy barrier to the interaction of hydroxyl groups. Water vapour reduced the rate of water release from montmorillonite and from illite and... 

While it is possible that surface defects may be preferentially involved in initial product formation, this has not been experimentally verified for most systems of interest. Such zones of preferred reactivity would, however, be of limited significance as they would soon be covered with the coherent product layer developed by reaction proceeding at all reactant surfaces. The higher temperatures usually employed in kinetic studies of diffusion-controlled reactions do not usually permit the measurements of rates of the initial adsorption and nucleation steps. 

The kinetic expressions applicable to diffusion-controlled reactions have been discussed in Chap. 3, Sect. 3.3. Mass transport in ionic solids has been reviewed by Steele and Dudley [1182],... 

Kinetic data on the carbonylation of vinyl cations have not been obtained so far, but it is likely to be a diffusion-controlled reaction as in the case of primary alkyl cations (Section IV, A). 

For a monograph on diffusion-controlled reactions, see Rice, S.A. Comprehensive Chemical Kinetics, Vol. 25 (edited by Bamford Tipper Compton) Elsevier NY, 1985. 

Other particular theories are confined to diffusion-controlled reactions (109), to the so called cooperative processes (113), in which the reactivity depends on the previous state, or to resistance of semiconductors (102), while those operating with hydrogen bridges (131), steric factors (132), or electrostatic effects (133, 175) are capable of being generalized less or more. 

One of the calculation results for the bulk copolyroerization of methyl methacrylate and ethylene glycol dimethacrylate at 70 C is shown in Figure 4. Parameters used for these calculations are shown in Table 1. An empirical correlation of kinetic parameters which accounts for diffusion controlled reactions was estimated from the time-conversion curve which is shown in Figure 5. This kind of correlation is necessary even when one uses statistical methods after Flory and others in order to evaluate the primary chain length drift. 

There are three (4) types of diffusion-controlled reactions possible for heterogeneous solid state reactions, viz-... 

For simple diffusion-controlled reactions, we can show the following holds ... 

The series of diffusion-controlled reactions are for the case of the solid state reaction between BaO Si02 as given in 4.9.16. above. These solid... 

The rate of MV formation was also dependent on pH. The bimolecular rate constant, as calculated from the first order rate constant of the MV build-up and the concentration of colloidal particles, was substantially smaller than expected for a diffusion controlled reaction Eq. (10). The electrochemical rate constant k Eq. (9) which largely determines the rate of reaction was calculated using a diffusion coefficient of of 10 cm s A plot of log k vs. pH is shown in Fig. 24. 

In the pulse radiolysis studies on the reaction of MV with TiOj, the sol contained propanol-2 or formate and methyl viologen, MV Ionizing radiation produces reducing organic radicals, i.e. (CH3)2COH or C02 , respectively, and these radicals react rapidly with MV to form MV. The reaction of MV with the colloidal particles was then followed by recording the 600 nm absorption of MV . The rate of reaction was found to be slower than predicted for a diffusion controlled reaction. 

Pulse radiolysis studies showed that the rate of the reaction of MV with a-Fc203, in which an electron is transferred to the colloidal particles, is slower than predicted for a diffusion controlled reaction. For pH > 8, the reaction is incomplete as the reverse reaction Fe203 - - - Fe203 - - MV takes place more efficiently... 

Strictly speaking, the validity of the shrinking unreacted core model is limited to those fluid-solid reactions where the reactant solid is nonporous and the reaction occurs at a well-defined, sharp reaction interface. Because of the simplicity of the model it is tempting to attempt to apply it to reactions involving porous solids also, but this can lead to incorrect analyses of experimental data. In a porous solid the chemical reaction occurs over a diffuse zone rather than at a sharp interface, and the model can be made use of only in the case of diffusion-controlled reactions. 

Here ko = 471 gD is the rate constant for a diffusion-controlled reaction (Smoluchowski rate constant) for a perfectly absorbing sphere. For long times, kf t) approaches its asymptotic constant value kf = kofkD/(kqf + kD) as... 

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