Diffusion-limited regimes

Consider first the diffusion-limited regime. The simplest experiment to perform is a chronoamperometric measurement, i.e. to monitor the current after a potential step to a value where an electroactive species will undergo electron transfer. This effectively allows us to monitor the rate of reaction, v, as a function of time, through the relationship ... 

From eqn. (11) and the functional dependence of the Thiele modulus on concentration [eqn. (10)], it is also obvious that the overall rate of reaction in the diffusion-limited regime is proportional to the product of Cg and j2. Hence the apparent order of reaction which would be... 

Rayleigh quotient method has been used only in the spatial diffusion limited regime but not in the energy diffusion limited regime (see the next Seetion). 

The quantum thermodynamic factor S is the quantum correction to the Kramers-Grote-Hynes classical result in the spatial diffusion limited regime, derived by Wolynes " ... 

For kinetics studies, powdered samples are less likely to give rise to diffusion-limited regimes in the reaction rates. Even loose powdered catalysts—a further step toward real conditions—can now be studied using diffuse-reflection infrared techniques (62), and it is to be hoped that more studies of this type will be made. 

The rate of diffusive separation, k, was determined from separate experimental measurements of iodine radical diffusion rates in the high pressure diffusion limited regime (19). The rate of excited state deactivation, k i, was calculated from the measured quantum yields at high densities where G> = kd/k i (18). It was assumed that k i is proportional to the inverse diffusion coefficient, D 1 (19,23) as both properties are related to the collision frequency. 

Figure 5 Density relaxations in Monte Carlo simulations of the geometry shown in Fig. 4 with conditions same as in Fig. 3 /3fi = -5.5) (a) Grand canonical simulations. (6) Simulation with mass conservation. The solid line, dotted line, and the open circles are the Kawasaki dynamics, ideal diffusion, and the grand canonical result shown in (a) rescaled by td with ro = 2 gmcs. The inset shows the initiail diffusion-limited regime in the logarithmic scale.

A summary picture is presented in Fig. 5.43. The catalytic process can be carried out in the kinetic and in the diffusion-limited regime. In the figure the former case is represented by the smooth Pt data. The catalyst potential stabilizes at a rather high value ( 1 V) — at lower values the alcohol oxidation would not be able to keep up with the 02-reduction rate. At this potential the Pt surface is covered to an appreciable extent with Oad species (that are not involved in the... 

Electrodeposition in the diffusion-limited regime is very sensitive to asperities on the surface of the... 

Time response. In most situations enzyme kinetics have very little effect on the response time of enzyme-based biosensors. From the analysis given above, it is clear that one should operate these devices under conditions where the analyte concentration within the sensor is much less than Km- For sensors which are in the membrane diffusion limiting regime (section 7.3.1.1 above), the response characteristics of the membrane material will be governing. These depend on the thickness of the membrane and the diffusivity of the analyte in the membrane material. An approximate estimate of the membrane lag time is... 

One can examine the current passed at the modified electrode during a potential step to the diffusion-limited regime for a solution species and then compare the result to the Cottrell behavior at a bare electrode (equation 5.2.11). Usually simple models, like those in Figure 14.5.1, are used. 

Kramers approach to rate theory in the underdamped and spatial-diffusion-limited regimes spurred extensions which were applicable to the much more complex STGLE. Grote and Hynes (23) used a parabolic barrier approximation to derive the rate expression for the GLE in the spatial diffusion limit. Carmeli and Nitzan derived expressions for the rate of the GLE (24) and the STGLE (25) in the underdamped limit. The overdamped limit for the rate in the presence of delta correlated friction was solved using the mean first passage time expression (26,27). A turnover theory, valid for space- and time-dependent friction, has only been recently presented by Haynes, Voth, and Poliak... 

Here T1U is the so called one-dimensional TST estimate for the rate and is mainly determined by the one-dimensional potential of mean force w(q). The depopulation factor Y becomes much smaller than unity in the underdamped limit and is important when the rate is limited by the energy diffusion process. In the spatial-diffusion-limited regime, the depopulation factor Y is unity but the spatial diffusion factor becomes much smaller than unity. The major theme of this review is theoretical methods for estimating the depopulation and spatial diffusion factors. 

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