Limited Kinetics—Debye Theory

To compute the maximum rate of a bimolecular chemical reaction in solution assume that, after the molecules have diffused together, reaction takes place with each encounter.In order to simplify the mathematics we center our attention, and the coordinate system, on a representative molecule of type B. Molecules of type A diffuse through the solution and, occasionally, encounter the B molecule. Details of molecular structure are suppressed. Both A and B are spherical the distance of closest approach 

Ionic strength can affect reactions of neutral species. An example is the reaction of CO(aq) with hemoglobin [L. H. Parkhurst and Q. H. Gibson, J. Biol. Chem. 242, 5762 (1967)]. Here changing ionic strength greatly alters the activity of the dissolved CO thus changing the rate constant. 

The treatment in this section is modeled after that given by I. Amdur and G. G. Hammes, Chemical Kinetics (New York McGraw-Hill, 1966), pp. 59-64. 

Since reaction occurs whenever r = d, the rate of reaction of A with a representative B molecule is the total flux of A at the surface of a sphere of radius d, as indicated in Fig. 9.13. 

To continue, generalize the ionic flux equation (3.15) to consider effects due to any potential 

---

Salt effects in kinetics are usually classified as primary or secondary, but there is much more to the subject than these special effects. The theoretical treatment of the primary salt effect leans heavily upon the transition state theory and the Debye-Hii ckel limiting law for activity coefficients. For a thermodynamic equilibrium constant one should strictly use activities a instead of concentrations (indicated by brackets). 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

While this is natural in view of the complexity of more detailed approaches to diffusion-limited reaction rates (e.g. kinetic theory), it is nevertheless a mute point as to whether the Debye—Smoluchowski equation represents an adequate description of diffusion and drift of interacting species in solution. 

The A, /c empirical relation found by Kohlrausch can be explained on the basis of the analogy of the gas and solution properties. According to the simple kinetic theory of gasesthe root-mean-square velocity /v is related (46) to /P by s/v = /3P7r where is the density of the gas. In dilute solutions, therefore, the conductivity (or the mobility) of the ions is proportional to /n or 7c. On the other hand, the Debye-Huckel-Onsager (D-H-0) limiting law,... 

---