Outer adiabatic process

According to the Marcus model, the standard rate constant, ks, for reaction (4.9) can be expressed as follows if the reaction is an adiabatic outer-sphere process 6 ... 

The above results for outer-sphere redox reactions were obtained by CHRISTOV /37d/ using an essentially different approach based on the general formulation of reaction rate theory. Equation (63.IV), however, differs from a similar result of DOGONADZE /147,151/ for adiabatic processes, which incorrectly gives the same activation energy as for non-adiabatic processes i.e., it neglects the large resonance interaction. 

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. 

As such, the thermal process in equation (60) proceeds via the same reactive intermediates (arising from an adiabatic electron transfer) as that observed in the photochemical processes in equations (57) and (58). The proposed electron-transfer activation for the thermal retropinacol reaction is further confirmed by the efficient cleavage of benzpinacol with tris-phenanthroline iron(III), which is a prototypical outer-sphere one-electron oxidant195 (equation 61). 

The Marcus classical free energy of activation is AG , the adiabatic preexponential factor A may be taken from Eyring s Transition State Theory as (kg T /h), and Kel is a dimensionless transmission coefficient (0 < k l < 1) which includes the entire efiFect of electronic interactions between the donor and acceptor, and which becomes crucial at long range. With Kel set to unity the rate expression has only nuclear factors and in particular the inner sphere and outer sphere reorganization energies mentioned in the introduction are dominant parameters controlling AG and hence the rate. It is assumed here that the rate constant may be taken as a unimolecular rate constant, and if needed the associated bimolecular rate constant may be constructed by incorporation of diffusional processes as ... 

This transient heat conduction problem can be used as a model for the following real process. A fire resistant wall is rapidly heated on its outer side (x = <5) as a result of a fire. We are interested in the temperature rise over time of the other side of the wall at x = 0. The assumption of an adiabatic surface at x = 0 results in a faster temperature rise than would be expected in reality. This assumption therefore leaves us on the safe side. 

For this virtual test of the furnace, the flow of process fluid inside the pipes was not considered so the outer surface temperature had to be specified based on thermography data and an assumed emissivity of 0.8. To represent the heating of the process fluid traveling through the pipes and its resulting affect on heat transfer, the inlet side was made colder than the outlet side according to process fluid temperature data provided by furnace operators. The four furnace outlets (see Figure 11.17) were assumed to have the same pressure. The refractory wall was assumed to be adiabatic with an emissivity of 0.6. 

Properties of the medium, initial condition and a boundary condition for thermal process, hydrological process, mass transport and geochemistry are shown in the Table 2. Temperature is fixed at 80°C in the inner boundary of buffer material and the outer boundary of hard rock is assumed adiabatic condition. And the buffer material is unsaturated in initial condition on the other hand hard rock is saturated in initial condition. 

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