Rate constant spheres

The simplest approach to computing the pre-exponential factor is to assume that molecules are hard spheres. It is also necessary to assume that a reaction will occur when two such spheres collide in order to obtain a rate constant k for the reactants B and C as follows ... 

To a first approximation, the activation energy can be obtained by subtracting the energies of the reactants and transition structure. The hard-sphere theory gives an intuitive description of reaction mechanisms however, the predicted rate constants are quite poor for many reactions. 

NflaNsc) < 10,000 Nrs < 1-0. Constant sphere diameter. Low mass-transfer rates. [146] p. 214... 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

In the first step the hydrated ion and ligand form a solvent-separated complex this step is believed to be relatively fast. The second, slow, step involves the readjustment of the hydration sphere about the complex. The measured rate constants can be approximately related to the constants in Scheme IX by applying the fast preequilibrium assumption the result is k = Koko and k = k Q. However, the situation can be more complicated than this. - °... 

According to the Marcus theory [64] for outer-sphere reactions, there is good correlation between the heterogeneous (electrode) and homogeneous (solution) rate constants. This is the theoretical basis for the proposed use of hydrated-electron rate constants (ke) as a criterion for the reactivity of an electrolyte component towards lithium or any electrode at lithium potential. Table 1 shows rate-constant values for selected materials that are relevant to SE1 formation and to lithium batteries. Although many important materials are missing (such as PC, EC, diethyl carbonate (DEC), LiPF6, etc.), much can be learned from a careful study of this table (and its sources). 

The Marcus treatment applies to both inorganic and organic reactions, and has been particularly useful for ET reactions between metal complexes that adopt the outer-sphere mechanism. Because the coordination spheres of both participants remain intact in the transition state and products, the assumptions of the model are most often satisfied. To illustrate the treatment we shall consider a family of reactions involving partners with known EE rate constants. 

In the case of other systems in which one or both of the reactants is labile, no such generalization can be made. The rates of these reactions are uninformative, and rate constants for outer-sphere reactions range from 10 to 10 sec b No information about mechanism is directly obtained from the rate constant or the rate equation. If the reaction involves two inert centers, and there is no evidence for the transfer of ligands in the redox reaction, it is probably an outer-sphere process. 

However, some quantitative interpretation of the rates of outer-sphere reactions may be made. It is possible to determine the rate constant, kn for the reaction of two complex ions and [M Lg] (Eq. 9.26). 

Let us consider the electron transfer between two rigid metal ions located some distance x from each other in the bulk of the solution. It is assumed that the inner-sphere reorganization of the donor D and acceptor A does not take place. The experiments show that the rate constants of these reactions differ by many orders of magnitude and the processes have an activated character even for identical ions D and A. The questions to be answered are Why does the electron exchange between identical ions in the solution require activation What is the reaction coordinate ... 

FIGURE 34.4 Dependence of electrochemical rate constant on the electrode potential for outer-sphere electron transfer. An exponential increase in the normal region changes for the plateau in the activationless region. 

The first estimations of for photoinduced processes were reported by Dvorak et al. for the photoreaction in Eq. (40) [157,158]. In this work, the authors proposed that the impedance under illumination could be estimated from the ratio between the AC photopotential under chopped illumination and the AC photocurrent responses. Subsequently, the faradaic impedance was calculated following a treatment similar to that described in Eqs. (22) to (26), i.e., subtracting the impedance under illumination and in the dark. From this analysis, a pseudo-first-order photoinduced ET rate constant of the order of 10 to 10 ms was estimated, corresponding to a rather unrealistic ket > 10 M cms . Considering the nonactivated limit for adiabatic outer sphere heterogeneous ET at liquid-liquid interfaces given by Eq. (17) [5], the maximum bimolecular rate constant is approximately 1000 smaller than the values reported by these authors. 

Fig. 1. Mean lifetimes of a single water molecule in the first coordination sphere of a given metal ion, th2o> and the corresponding water exchange rate constants, h2o- The tall bars indicate directly determined values, and the short bars indicate values deduced from ligand substitution studies. References to the plotted values appear in the text.

The definition of solvent exchange rates has sometimes led to misunderstandings in the literature. In this review kjs 1 (or fc2lsolvent]), sometimes also referred to as keJ s 1, is the rate constant for the exchange of a particular coordinated solvent molecule in the first coordination sphere (for example, solvent molecule number 2, if the solvent molecules are numbered from 1 to n, where n is the coordination number for the solvated metal ion, [MS ]m+). Thus, the equation for solvent exchange may be written ... 

This formulation assumes that the continuum diffusion equation is valid up to a distance a > a, which accounts for the presence of a boundary layer in the vicinity of the catalytic particle where the continuum description no longer applies. The rate constant ky characterizes the reactive process in the boundary layer. If it approximated by binary reactive collisions of A with the catalytic sphere, it is given by kqf = pRGc(8nkBT/m)1 2, where pR is the probability of reaction on collision. 

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... 

The first density correction to the rate constant depends on the square root of the volume fraction and arises from the fact that the diffusion Green s function acts like a screened Coulomb potential coupling the diffusion fields around the catalytic spheres. 

MPC dynamics follows the motions of all of the reacting species and their interactions with the catalytic spheres therefore collective effects are naturally incorporated in the dynamics. The results of MPC dynamics simulations of the volume fraction dependence of the rate constant are shown in Fig. 19 [17]. The MPC simulation results confirm the existence of a 4> 2 dependence on the volume fraction for small volume fractions. For larger volume fractions the results deviate from the predictions of Eq. (92) and the rate constant depends strongly on the volume fraction. An expression for rate constant that includes higher-order corrections has been derived [95], The dashed line in Fig. 19 is the value of /. / ( < )j given by this higher-order approximation and this formula describes the departure from the cf)1/2 behavior that is seen in Fig. 19. The deviation from the <[)11/2 form occurs at smaller values than indicated by the simulation results and is not quantitatively accurate. The MPC results are difficult to obtain by other means. 

We focus on the effects of crowding on small molecule reactive dynamics and consider again the irreversible catalytic reaction A + C B + C asin the previous subsection, except now a volume fraction < )0 of the total volume is occupied by obstacles (see Fig. 20). The A and B particles diffuse in this crowded environment before encountering the catalytic sphere where reaction takes place. Crowding influences both the diffusion and reaction dynamics, leading to nontrivial volume fraction dependence of the rate coefficient fy (4>) for a single catalytic sphere. This dependence is shown in Fig. 21a. The rate constant has the form discussed earlier,... 

The aquation of [IrCl6]2- to [Er( E120)C1S] and Ir(H20)2Cl4 has been found to activate the complex toward the oxidation of insulin in acidic solutions, with measured rate constant of 25,900 and 8,400 Lmol-1 s 1, respectively.50 The oxidation reaction proceeds via an outer-sphere mechanism. 

Clusters larger than 25 all appear to react without further dramatic variations. The magnitude of the large cluster rate constants are a few percent of the hard sphere collision cross... 

Fig. 2.7. Characteristic rate constants (s 1) for substitution of inner-sphere H20 of various aqua ions. Note The substitution rates of water in complexes ML(H20)m will also depend on the symmetry of the complex (adapted from Frey, C.M. and Stuehr, J. (1974). Kinetics of metal ion interactions with nucleotides and base free phosphates in H. Sigel (ed.), Metal ions in biological systems (Vol. 1). Marcel Dekker, New York, p. 69).

---