Direct-computation rate methods

Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time f, (rate)f, is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first-order, or pseudo-first-order in analyte, the rate at time f is given as... 

Curve-Fitting Methods In the direct-computation methods discussed earlier, the analyte s concentration is determined by solving the appropriate rate equation at one or two discrete times. The relationship between the analyte s concentration and the measured response is a function of the rate constant, which must be measured in a separate experiment. This may be accomplished using a single external standard (as in Example 13.2) or with a calibration curve (as in Example 13.4). 

Miscellaneous Methods At the beginning of this section we noted that kinetic methods are susceptible to significant errors when experimental variables affecting the reaction s rate are difficult to control. Many variables, such as temperature, can be controlled with proper instrumentation. Other variables, such as interferents in the sample matrix, are more difficult to control and may lead to significant errors. Although not discussed in this text, direct-computation and curve-fitting methods have been developed that compensate for these sources of error. ... 

In 1984 Krauss and Stevens described tests and applications of the effective potential method used to gain knowledge of the electronic structure of the molecules in order to analyze the accuracy of the experimentally deduced dissociation energies of refractory metal salts [3]. They used the development of ab initio theoretical methods for the calculation of potential energy surfaces, which further allowed the direct computation of certain rate constants. Transition state theory was also utilized for this computation of some rate constants. However, as discussed by Krauss and Stevens, as of the mid 1980 s computational techniques were not yet readily applied to atmospheric science. Computing power and theoretical methods since these seminal reports have been greatly advanced. 

Direct-computation methods, on which most kinetic analyzers rely, use one or more signal readouts to calculate generally the sought reaction rate. 

Once these factors have been determined, the remaining problem is to compute L and r since these quantities establish the neutron-leakage rates for the finite system. A first approximation to L has already been given, and a few remarks have been made about corrections for directional effects. Some methods have been developed for including anisotropic effects in the computation of the migration area, and these are reported in a subsequent section. 

In order to obtain crack-tip quantities such as the strain energy release rate g, the complex stress intensity factor K, and the mode-mixity xp, the following procedure may be adopted first, the strain energy release rate Q is directly computed via a contour integral evaluation - the J-integral method, or the VCCT second, the modulus of K, can be computed from Equation (10) and third, the crack surface displacements may be substituted in Equation (8) and with the knowledge of e, the parameter a is computed. Finally, the stress intensity factors may be expressed as ... 

We have discussed simulations that were intended to elucidate aspects of crystal growth under diverse conditions. In most cases a direct simulation of growth using realistic conditions is impractical. The growth rate may be many orders of magnitude slower than that required to produce observable crystalline material in the available computer time. We have described several methods to obtain information about the crystallization process in this situation. 

Following the early studies on the pure interface, chemical and electrochemical processes at the interface between two immiscible liquids have been studied using the molecular dynamics method. The most important processes for electrochemical research involve charge transfer reactions. Molecular dynamics computer simulations have been used to study the rate and the mechanism of ion transfer across the water/1,2-dichloroethane interface and of ion transfer across a simple model of a liquid/liquid interface, where a direct comparison of the rate with the prediction of simple diffusion models has been made. ° ° Charge transfer of several types has also been studied, including the calculations of free energy curves for electron transfer reactions at a model liquid/liquid... 

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