The Laplace Transform in Kinetic Calculations

All the enumerated stages of the matrix solution of an ODE system can be, of course, also realized by the Maple suits. However, provided there is the function dso Ive which copes with linear systems without difficulty, using of the classical matrix method is unnecessary. 

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The Laplace Transform in Kinetic Calculations 2.3.1 Brief Notes from Operational Calculus... 

The initial conditions are CD = CD(0) at t = 0 and CR = 0 at t = 0. Efforts to obtain analytical solutions are tedious and unnecessary. By applying the change in concentrations (or mass) in the donor and receiver solutions with time to the Laplace transforms of Eqs. (140) and (141), the inverse of the simultaneous transformed equations can be numerically calculated with appropriate software for best estimates of a, (3, and y. It is implicit here that P Pap, Pbh and Ke are functions of protein binding. Upon application of the transmonolayer flux model to the PNU-78,517 data in Figure 32, the effective permeability coefficients from the disappearance and appearance kinetics points of view are in good quantitative agreement with the permeability coefficients determined from independent studies involving uptake kinetics by MDCK cell monolayers cultured on a flat dish... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

Figure 4.6 presents the concentrations of A, B and C calculated by eqs. (4.70), (4.73) and (4.74), respectively, for the case in which k- = k = 3k2 = 6k 2-For rate equations that are linear with respect to the reactants, the Laplace transform method allows us to solve the kinetic equations. In the case of non-linear rate laws, these can be made linear by using excess concentrations of certain reactants, leading to pseudo-first-order equations, which can be solved. 

A number of kinetic models of various degree of complexity have been used in chromatography. In linear chromatography, all these models have an analytical solution in the Laplace domain. The Laplace-domain solution makes rather simple the calculation of the moments of chromatographic peaks thus, the retention time, the peak width, its number of theoretical plates, the peak asymmetry, and other chromatographic parameters of interest can be calculated using algebraic expressions. The direct, analytical inverse Laplace transform of the solution of these models usually can only be calculated after substantial simplifications. Numerically, however, the peak profile can simply be calculated from the analytical solution in the Laplace domain. 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

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