Rate-Limiting Process

In this section we discuss the basic mechanisms of pattern formation in growth processes under the influence of a diffusion field. For simphcity we consider the sohdification of a pure material from the undercooled melt, where the latent heat L is emitted from the solidification front. Since heat diffusion is a slow and rate-limiting process, we may assume that the interface kinetics is fast enough to achieve local equihbrium at the phase boundary. Strictly speaking, we assume an infinitely fast kinetic coefficient. 

Caesium—graphite decomposes [672] in a sequence of identifiable steps, each involving a reduction in the Cs C ratio. The rate-limiting process is... 

Sections 2.1—2.3 give accounts of kinetic and mechanistic features of the three rate-limiting processes (i) diffusion at a surface or in a gas (including the nucleation step), (ii) reaction at an interface, and (iii) diffusion across a barrier phase, [(ii) and (iii) are growth processes.] In any particular reaction, the slowest of these processes will, at any particular instant, control the rate of product formation. (A kinetic analysis of rate measurements must also incorporate an allowance for the geometric factors.)... 

Usually, activation energies for dissociation are much higher than k T and situations like this, where the apparent activation energy becomes negative for the rate-limiting process, are rare. Nevertheless, the present case illustrates nicely that entropy changes may play an important role and that it is not always the activation energy that dominates the process. 

Applying the concept of the rate-determining step (see Section 5.4.2) one can derive the following kinetic equations for adsorption of A, surface reaction, and desorption of R or S, respectively, as rate-limiting processes ... 

Quantitative information can be drawn from such plots. For the a-th order kinetics the slope is the reaction order a and the intercept is In k. For the catalytic reaction considered above with the surface reaction as the rate-limiting process, linearization of the rate equation (5.4-112) leads to ... 

Finally, it may be difficult to sample all the relevant conformations of the system with fixed. This problem is more subtle, but potentially more serious, as illustrated by Fig. 4.2. Several distinct pathways may exist between A and B. It is usually relatively easy for the molecule to enter one pathway or the other while the system is close to A or B. However, in the middle of the pathway, it may be very difficult to switch to another pathway. This means that, if we start a simulation with fixed inside one of the pathway, it is very unlikely that the system will ever cross to explore conformations associated with another pathway. Even if it does, this procedure will likely lead to large statistical errors as the rate-limiting process becomes the transition rate between pathways inside the set = constant. 

It is well appreciated that dietary fat retards proximal gastrointestinal transit. For hydrophobic drugs with low aqueous solubility, dissolution becomes the rate-limiting process the appreciation that fat might increase bioavailability by increasing time available for dissolution and that mixed triglycerides often provide a better solvent than aqueous media has not been lost on formulation scientists. 

Figure 22.4 Example 22-4 Dependence of mean fractional conversion (/B) on mean residence time (t )-effect of rate-limiting process data of Example 22-3 (cylindrical particles of all same size in BMF SCM n = 1)...

The process of cooking involves a complicated series of chemical reactions, each of which proceeds with a rate constant of k. When boiling an egg, for example, the rate-limiting process is denaturation of the proteins from which albumen is made. Such denaturation has an activation energy Ea of about 40 kJ mol 1. 

The rate-limiting process in Equation (8.20) involves the two species (peroxide and octane) colliding within the car cylinder and combining chemically. Because two species react in the rate-limiting reaction step, we say that the reaction step represents a bi molecular reaction. In alternative phraseology, we say the molecularity of the reaction is two . 

Figure 8.26 An Arrhenius plot of ln(heart beat) (as y ) against 1 IT (as x ) for the diamond-backed terrapin (Malaclemys macrospilota) is linear over a limited range of temperatures, showing that the rate-limiting process dictating its heart rate is activated.

When the pore bottom is covered by an oxide, the change of applied potential occurs almost completely in the oxide due to the very high resistance of the oxide. The rate of reactions is now limited by the chemical dissolution of the oxide on the oxide covered area. When the entire pore bottom is covered with an oxide the rate of reaction is the same on the entire surface of the pore bottom. As a result, the bottom flattens and the condition for PS formation disappears. The change of oxide coverage on the pore bottom can also occur when diffusion of the electrolyte inside deep pores becomes the rate limiting process. Since the current at which formation of an oxide occurs increases with HF concentration, a decreased HF concentration at pore bottom due to the diffusion effect can result in the formation of an oxide on the pore bottom of a deep pore at a condition that does not occur in shallow pores. 

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