Specific heterogeneous rate constant

In electrochemical literature the standard rate constant fe is often designated as fes h or fe9, called the specific heterogeneous rate constant or the intrinsic rate constant. According to eqns. 3.5 and 3.6, we have... 

Note that the term with E° still changes with the analyte concentration, the transfer coefficient a and the specific heterogeneous rate constant ks h (cf., eqn. 3.7). 

Note that the specific heterogeneous rate constant is not part of this equation, because the reaction is forced to proceed at a rate determined by the applied current. 

Using the Marcus theory, the a value (see -> charge-transfer coefficient) can be predicted, and its dependence on the potential applied. For low - over potentials, and when neither Ox nor Red are specifically adsorbed on the electrode surface, a should be approximately equal to 0.5. Further, the theory describes the relation between homogeneous and heterogeneous rate constants characteristic of the same redox system. An interesting prediction from Marcus theory is the existence of a so-called inverted region for the homogeneous electron transfer reactions, of importance to the phenomenon of... 

The involvement of lysine amino acid residues on cytochrome c in the heterogeneous reactions with functionalized electrodes seems to have been established. Importantly, it is now thought that the proposed protein-promoter complex is more likely to be dynamic, as revealed by the results of a recent investigation (28) of site-specific 4-chloro-3,5-dinitrophenyl (CDNP)-substituted cytochrome c. It was found that monosubstitution of either Lys 13 or Lys 72 did not result in any significant change in its electrochemical response, whereas two modifications greatly decreased the heterogeneous rate constant, and complete loss of electrochemical activity was observed upon modification of more lysines. It was proposed that the electrode reaction occurred in numerous rotational conformations. Therefore, for the mono-... 

A heterogeneous rate constant value corresponding to the specific substrate potential was extracted by fitting each current-distance curve to the theory. 

The constants g, r and d can take on various values to express any given boundary condition. Thus, if we set d = 0 and r 0, we are left with the general form of the Dirichlet condition and specifically with r = 1 and g = 0 we have the Cottrell condition, while Eq. (6.94) expresses Robin conditions. The constant r expresses the heterogeneous rate constant (this formula only considers a single species, so an irreversible reaction is implied). 

It is independent of potential and of the applied current density, but inversely proportional to the exchange current density, because the current-potential relationship is linear in this region (cf. Section 5.2.4). In this region the Wagner number is also inversely proportional to the heterogeneous rate constant of the metal deposition reaction. Thus, fast reactions have low value of the Wagner number and tend to lead to primary current distribution. This is a rather unique situation in electrochemistry, where poor catalytic activity (i.e., low specific rate constant) is an advantage. 

It is necessary to note the limitation of the approach to the study of the polymerization mechanism, based on a formal comparison of the catalytic activity with the average oxidation degree of transition metal ions in the catalyst. The change of the activity induced by some factor (the catalyst composition, the method of catalyst treatment, etc.) was often assumed to be determined only by the change of the number of active centers. Meanwhile, the activity (A) of the heterogeneous polymerization catalyst depends not only on the surface concentration of the propagation centers (N), but also on the specific activity of one center (propagation rate constant, Kp) and on the effective catalyst surface (Sen) as well ... 

Tris-allyl-neodymium Nd(//3-C3I Ishdioxane which performs as a single site catalyst in solution polymerization was heterogenized on various silica supports which differed in specific surface area and pore volume. The catalyst was activated by MAO. In the solution polymerization the best of the supported catalysts was 100 times more active (determined by the rate constant) than the respective unsupported catalyst [408]. 

In addition to the studies in which supported catalysts are exclusively used for gas-phase polymerizations one study is available in which the supported catalyst is optimized in a solution process prior to its application in the gas phase. Tris-allyl-neodymium [Nd(/ 3- C3H5)-dioxane] which is a known catalyst in solution BD polymerization is heterogenized on various silica supports differing in specific surface area and pore volume. The catalyst is activated by MAO. In solution polymerization the best of the supported catalysts is 100 times more active (determined by the rate constant) than the respective unsupported catalyst [408]. In addition to the polymerization in solution, the supported allyl Nd catalyst is applied for the gas-phase polymerization of BD [578,579] the performance of which is characterized by macroscopic consumption of gaseous BD and in-situ-analysis of BD insertion [580]. 

In summary, it can be seen for the three-step reaction scheme of this example that the net rate of the overall reaction is controlled by three kinetic parameters, KTSi, that depend only on the properties of the transition states for the elementary steps relative to the reactants (and possibly the products) of the overall reaction. The reaction scheme is represented by six individual rate constants /c, and /c the product of which must give the equilibrium constant for the overall reaction. However, it is not necessary to determine values for five linearly independent rate constants to determine the rate of the overall reaction. We conclude that the maximum number of kinetic parameters needed to determine the net rate of overall reaction is equal to the number of transition states in the reaction scheme (equal to three in the current case) since each kinetic parameter is related to a quasi-equilibrium constant for the formation of each transition state from the reactants and/or products of the overall reaction. To calculate rates of heterogeneous catalytic reactions, an addition kinetic parameter is required for each surface species that is abundant on the catalyst surface. Specifically, the net rate of the overall reaction is determined by the intrinsic kinetic parameters Kf s as well as by the fraction of the surface sites, 0, available for formation of the transition states furthermore, the value of o. is determined by the extent of site blocking by abundant surface species. 

Similar SECM experiments can be performed using a simple (unassisted) IT process [41]. In this case, both the top and the bottom phases contain the same ion at equilibrium. The micropipet tip is used to deplete concentration of this common ion in the top solvent near the ITIES. The depletion results in the IT across the ITIES, which produces positive feedback. Any solid surface (or a liquid phase containing no specific ion) acts as an insulator in this experiment. The mass transfer rate for IT measurements by SECM is similar to that for heterogeneous ET measurements, and the standard rate constants in excess of 1 cms-1 should be measurable. 

As a first approximation a convective term in the film region has been negleted, u is the superficial gas velocity and u f denotes the gas velocity at minimum fluidization conditions. Tne specific mass transfer area a(h) is based on unit volume of the expanded fluidized bed and e OO is the bubble gas hold-up at a height h above the bottom plate. Mathematical expressions for these two latter quantities may be found in detail in (20). The concentrations of the reactants in the bubble phase and in film and bulk of the suspension phase are denoted by c, c and c, respectively. The rate constant for the first order heterogeneous catalytic reaction of the component i to component j is denoted... 

If the specific rate of the chemical step is slow compared to the encounter rate, that is, kf < kE lO M" s , then the rate of the elementary reaction, A + B products, is governed by chemical reaction. [Similar considerations enter in the case of heterogeneous reactions, for example, dissolution or adsorption, where, for spherical particles, the quantities of interest are k (cm s ) and D/a (cm s ), where k is a rate constant and a the radius]. 

In this section the same industrial reactor data of the Polymer Corporation, Sarnia, Ontario, Canada is used. Tables 6.27, 6.28 and 6.34 give the specification of the reactor. The feed conditions to the reactor are given in Table 6.27. Table 6.35 (columns 3 and 4) shows apparent kinetic data discussed in the previous section. These constants are not intrinsic rate constants. They were used as starting values in an iteration scheme to obtain intrinsic kinetics suitable for heterogeneous models. 

Models of parallel pseudo first-order reactions consider the case when two interactions with different rate constants proceed simultaneously. Such situations can be attributed to different kinds of receptor sites or to different states of the analyte [8,11]. In the first case the model can describe heterogeneity of the sensor surface the second may concern a macromolecular analyte that can be present in various conformations, protonation states, etc. Besides two sets of rate constants, the models also require specification of proportion p between the two fractions of the receptor or analyte. For the model considering two kinds of receptors, the following equations are obtained ... 

Recently it was proposed that the apparently slow heterogeneous electron-transfer rates for such proteins as cytochrome c, cytochrome b5, plasto-cyanin, and ferredoxin are an artifact of the experimental approach (25). Instead of assuming that protein molecules react at a planar and essentially homogeneous surface, it is assumed instead that movement of the protein occurs predominantly by radial diffusion to very small, specific sites. These sites are presumed to facilitate very rapid electron transfer at the reversible potential while the rest of the surface remains inactive. Thus, the modified electrode surface behaves like an array of microelectrodes. If this theory is used to treat previous data, much higher electron-transfer rate constants are obtained. Although this theory deserves more detailed scrutiny, it may serve... 

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