Von Smoluchowski rate law

The approximate rate constant in Eq. 6.16d is independent of floccule size, which results in a considerable simplification of the von Smoluchowski rate law ... 

Equation 6.18 is graphed in Fig. 6.6 for the cases q = 1, 2, 3. The number density of primary particles, pj(t), decreases monotonically with time as these particles are consumed in the formation of floccules. The number densities of the floccules, on the other hand, rise from zero to a maximum at t = (q - l)/2KDp0, and then decline. This mathematical behavior reflects creation of a floccule of given size from smaller floccules, followed by a period of dominance, and finally consumption to form yet larger particle units as time passes. Both experimental data and computer simulations, like that whose visualization appears in Fig. 6.1, are in excellent qualitative agreement with Eq. 6.18 when they are used to calculate the pq(t).13,14 Thus the von Smoluchowski rate law with a uniform rate coefficient appears to capture the essential features of diffusion-controlled flocculation processes. 

If the von Smoluchowski rate law (Eq. 6.10) is to be consistent with the formation of cluster fractals, then it must in some way also exhibit scaling properties. These properties, in turn, have to be exhibited by its second-order rate coefficient kmn since this parameter represents the flocculation mechanism, aside from the binary-encounter feature implicit in the sequential reaction in Eq. 6.8. The model expression for kmn in Eq. 6.16b, for example, should have a scaling property. Indeed, if the assumption is made that DJRm (m = 1, 2,. . . ) is constant, Eq. 6.16c applies, and if cluster fractals are formed, Eq. 6.1 can be used (with R replacing L) to put Eq. 6.16c into the form... 

The special case of Eq. 6.27 that obtains where m = n, that is, k°n= 2K , is trivially scale invariant, so this property ought to be implicit in the corresponding solution of the von Smoluchowski rate law, given in Eq. 6.18. That this is the case can be seen by noting the large-time limit of p,(t),... 

Equation 6.28 represents the simplest effect of scaling on the second-order rate constant in the von Smoluchowski rate law no effect whatsoever. More generally, if kmn satisfies the homogeneity condition... 

Given that Eq. 6.1 (with D 2) applies to reaction-controlled flocculation kinetics, Eq. 6.54 implies that MM(t) [or MN(t)] must also exhibit an exponential growth with time. Therefore, by contrast with transport-controlled flocculation kinetics, a uniform value of the rate constant kmn cannot be introduced into the von Smoluchowski rate law, as in Eq. 6.17, to derive a mathematical model of the number density p,(t). Equations 6.22 and 6.24 indicate clearly that a uniform kinil leads to a linear time dependence in the... 

Fuchsian kinetics lead to the model form of the von Smoluchowski rate law that is obtained by introducing Eq. 6.52 into Eq. 6.10 ... 

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 partition ratio kBl(k d + kB) defines the efficiency of product formation from the encounter complex (see also Section 3.7.4). For the limiting case k dobserved rate constant of reaction approaches the rate constant of diffusion, kx kd. In 1917, von Smoluchowski derived Equation 2.27 from Fick s first law of diffusion for the ideal case of large spherical solutes. 

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