Reaction infinitely rapid

The electron transfer reaction at the electrode may be (a) rapid or (b) slow. For case (a) it is of interest to distinguish between two possibilities, viz., an infinitely rapid (aj) and a reasonably rapid electron transfer (a2). 

By considering the limits of 1/y as zero and infinity corresponding to infinitely rapid desorption or slow reaction and very slow desorption or rapid reaction, respectively, show that... 

To find the second-order rate coefficient for the reaction of A and B subject to the encounter pair reacting with a rate coefficient feact, the method developed in Sect. 3.7 can be used. Using eqn. (19), the rate coefficient, k(t), can be defined in terms of the diffusive current of B towards the central A reactant. But the partially reflecting boundary condition (22) equates this to the rate of reaction of encounter pairs. The observed rate coefficient is equal to the rate at which the species A and B could react were diffusion infinitely rapid, feact, times the probability that A and B are close enough together to react, p(R). 

At the limit in an infinitely rapid reaction a and 6 in the reaction zone tend to zero so that nowhere (except in an infinitely narrow zone) can a and b be simultaneously nonzero. Thus, in the case of combustion, by finding the distribution of p in space from the solution of the Unear equation (15) with conditions (16), we easily find the fields for a and b as well ... 

This equation, with the appropriate modification of X, is exactly the same as that for a straight pore, Eq. (6.4.1). Moreover, the boundary conditions are the same, for at the surface x — Z, a = a, and at the central plane daldx = 0 by symmetry. It follows that the effectiveness factor for the slab, defined in the same way as the ratio of the total rate of reaction under diffusion limitation to the rate when diffusion is infinitely rapid, is again... 

When the system passes the intersection at a high velocity, that is, the above condition is not met even approximately, it will usually jump from the lower R surface (before S along the reaction coordinate) to the upper R surface (after S). That is, the system behaves in a nonadiabatic (or diabatic) fashion, and the probability per passage of electron transfer occurring is small (i.e., k< C1). The nuclear coordinates of the system change so rapidly that it cannot remain at equilibrium. At the nonadiabatic limit, the time interval for passage between the two states at point S approaches zero, that is, (tf — t,) —> 0 (infinitely rapid), and the probability density distribution functions that describe the initial and final states remain unchanged ... 

The boundary conditions on a surface where a chemical reaction occurs depend on the specific physical statement of the problem. In the special case of an infinitely rapid heterogeneous chemical reaction, the corresponding boundary condition has the form... 

We divide both sides of (3.1.13) by ks and let the parameter ks tend to infinity. As a result, we arrive at the limit boundary condition (3.1.10), corresponding to the diffusion regime of reaction. This passage to the limit adequately illustrates the notion of an infinitely rapid reaction, which was used previously. 

For second-order reactions, the asymptotic solution for infinitely rapid reaction [P.V. Danckwerts, Trans. Faraday Soc., 46, 701 (1950)] is given by the awkward parametric form... 

We ean summarize the major results of this section in terms of the three enhancement faetor equations—(8-139), (7-82) and (7-83) for the pseudo-first-order reaetion, the infinitely rapid second-order reaction, and the true second-order reaction, respectively. Via A all mass-transfer coefficients under reaction conditions can be expressed in terms of their pure mass-transfer relatives, so correlations developed for the mass-transfer coefficient ki can be used for estimation of k. These three eases probably eonstitute the large majority of gas-liquid reaetions one is likely to eneounter. Some additional cases are discussed by Fromont and Bischoff [G.F. Froment and K.B. Bischoff, Chemical Reactor Analysis and Design, 2 ed., John Wiley and Sons, New York, NY (1990)]. 

If the interaction were instantaneous and without mass-transfer rate limitations, then we would have a simple transmission of whatever input of adsorbate that was admitted to the bed, with the input function at the exit of the bed at a time equal to the residence time in the bed. A step-function input is shown in Curve A. More commonly, we have breakthrough curves such as B in Figure 9.1, there the mass transport and/or adsorption processes are not infinitely rapid, or C in the figure, where rates are slow and even some reversible reaction may occur. 

Collins and Kimball [4] suggested that the chemically activated process which leads to the formation of products from the encounter pair occurs at a rate proportional to the probability that the encounter pair exists. Defining an encounter pair as a pair of reactants which lie within a distance of R to (R + 8R) of one another, and since the probability that B is within this range of distances about A is p R), then the rate of reaction of encounter pairs is feactP( )- act is the second-order rate coefficient for the reaction of A and B when they are almost in contact and close enough to react with each other. It is the rate coefficient for reaction between A and B if the rate of diffusion were infinitely rapid. It has unit of dm mol" s" . From eqn. (7), the rate at which B diffuses towards A to form encounter pairs is 47r(f + 5/ ) D(3p/3r) R+5jj. For sufficiently small 5R (e.g. <0.01 nm), the term in Si is unimportant and this becomes the diffusive flux to the encounter separation 4irR D dp/dr) ji from Fick s first law. Providing the probability of A and B existing as an encounter pair rapidly reaches a steady-state value, the rate of formation and rate of reaction of the encounter pairs may be equated, i.e. 

Whereas each fuel molecule burns at the ideal (adiabatic) flame temperature, the reaction heat is transferred to surrounding gases, liquids, and solid objects as combustion proceeds. Only by infinitely rapid combustion, or by combustion in a perfectly insulated chamber, can the adiabatic flame temperature be reached. 

Let us consider an irreversible and infinitely rapid gas-liquid reaction... 

No catalyst has an infinite lifetime. The accepted view of a catalytic cycle is that it proceeds via a series of reactive species, be they transient transition state type structures or relatively more stable intermediates. Reaction of such intermediates with either excess ligand or substrate can give rise to very stable complexes that are kinetically incompetent of sustaining catalysis. The textbook example of this is triphenylphosphine modified rhodium hydroformylation, where a plot of activity versus ligand metal ratio shows the classical volcano plot whereby activity reaches a peak at a certain ratio but then falls off rapidly in the presence of excess phosphine, see Figure... 

The broadening of the distribution with increasing a can also be noted by the XwfXn value. Equations 2-167 and 2-169 show that the difference between the number- and weight-average degrees of polymerization increases very rapidly with increasing extent of reaction. At the gel point the breadth of the distribution Xw/Xn is enormous, since Xw is infinite, while X has a finite value of 4 (Fig. 2-19). Past the gel point the value of Xw/Xn for the sol fraction decreases. Finally, at a = p — 1, the whole system has been converted to gel (i.e., one giant molecule) and Xw/Xn equals 1. 

Experimental Systems. Rate-of-reaction studies were carried out utilizing both finite and infinite bath techniques. Test solutions were prepared at the desired ionic strength, temperature, and initial pH. These solutions were stirred rapidly with a motor-driven polyethylene-coated stirring blade. For each test, a carefully measured quantity of carbon was added in the dry form. The finite bath technique consisted of recording pH values as a function of time after carbon addition. All... 

This coalescing and redispersion may be either rapid (possibly so rapid that, compared with the reaction rate, the interaction may be considered practically infinite) or it may be slow. The point, however, is What is rapid and what is slow ... 

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