Laplace transforms, kinetic equations

This is the procedure From the postulated kinetic scheme we write the differential rate equations. Take the Laplace transforms of the differential equations. Solve the resulting set of algebraie equations for the transforms of the concentrations. Then take the inverse transforms to obtain the coneentrations as funetions of time. 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

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. 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

Equation (9.17) is solved by a Laplace transformation. In chemical kinetics and diffusion, the problems may often be formulated in terms of partial differential equations... 

They may be obtained by means of the matrix IET but only together with the kernel E(f) = F(t) specified by its Laplace transformation (3.244), which is concentration-independent. However, from the more general point of view, Eqs. (3.707) are an implementation of the memory function formalism in chemical kinetics. The form of these equations shows the essentially non-Markovian character of the reversible reactions in solution the kernel holds the memory effect, and the convolution integrals entail the prehistoric evolution of the process. In the framework of ordinary chemical kinetics S(/j = d(t), so that the system (3.707) acquires the purely differential form. In fact, this is possible only in the limit when the reaction is entirely under kinetic control. 

The occurrence of partial differential equations in electrochemistry is due to the variation of concentration with distance and with time, which are two independent variables, and are expressed in Fick s second law or in the convective-diffusion equation, possibly with the addition of kinetic terms. As in the resolution of any differential equation, it is necessary to specify the conditions for its solution, otherwise there are many possible solutions. Examples of these boundary conditions and the utilization of the Laplace transform in resolving mass transport problems may be found in Chapter 5. 

USING LAPLACE TRANSFORMS TO SOLVE KINETICS EQUATIONS... 

Most simple kinetics problems involve first-order differential equations, which can be integrated using Laplace15 transforms. As discussed in Section 2.16, for a function/(x), the Laplace transform F(k) is given by... 

When the transport of reactants is controlled by linear diffusion, the kinetic analysis can be performed using convolution potential sweep voltammetry [182]. Here it is more convenient to choose one of the reactant concentrations to be equal to zero, i.e., the initial conditions are recovered at sufficiently negative or positive potentials as in linear potential sweep voltammetry. By using the Laplace transform and the convolution theorem in solving the second Fick equation for each reactant, the convolution current m. 

D Me-S surface alloy and/or 3D Me-S bulk alloy formation and dissolution (eq. (3.83)) is considered as either a heterogeneous chemical reaction (site exchange) or a mass transport process (solid state mutual diffusion of Me and S). In site exchange models, the usual rate equations for the kinetics of heterogeneous reactions of first order (with respect to the species Me in Meads and Me t-S>>) are applied. In solid state diffusion models, Pick s second law and defined boundary conditions must be solved using Laplace transformation. 

For a general rate model including axial dispersion mass transfer (Equation 6.80), (apparent) pore diffusion (Equations 6.82, 6.84 and 6.85), and linear adsorption kinetics (Equations 6.32 and 6.33), Kucera (1965) derived the moments by Laplace transformation, assuming the injection of an ideal Dirac pulse. If axial dispersion is not too strong Pe S>4), the equations for the first and second moments can be simplified to (Ma, Whitley, and Wang, 1996)... 

Often PK equations for common inputs such as first-order absorption zero-order input, or constant-rate infusions are derived from the differential equations describing the kinetics. LSA offers a very attractive alternative to such derivations that is more direct and does not require the use of differential equations or Laplace transforms. The LSA derivations can be done simply by elementary convolution operations (see Tables 16.1 and 16.2) in conjunction with the input-response convolution relationship between concentration, c(t), and the rate of input,/(t) ... 

A very valuable technique, useful in the solution of ordinary and partial differential equations as well as differential delay equations, is the use of Laplace transforms. Laplace transforms (Churchill, 1972), though less familiar and somewhat more difficult to invert than their cousins, Fourier transforms, are broadly applicable and often enable us to convert differential equations to algebraic equations. For rate equations based on mass action kinetics, taking the Laplace transform affords sets of polynomial algebraic equations. For DDEs, we obtain transcendental equations. 

It follows from the last example that when controlling the potential, its value is to be related to i and surface concentration of EAC. The latter term appears in the kinetic equation. For this purpose, it is necessary to know mechanism and kinetic parameters of the charge transfer process. However, this complicated way may be avoided, if the experimentally obtained i(t)-function is utilized. Then, to find concentration profiles, the Laplace transform of this function is to be obtained ... 

In this section the model for the continuous reactor, consisting of the three balances Eqns. (17.12), (17.13) and (17.14) and additional kinetic equations, will be Laplace transformed and linearized to get some insight in the responses to feed changes. 

As complex reactions follow a reaction mechanism involving various elementary steps, the determination of the corresponding kinetic law involves the solution of a system of differential equations, and the complete analytical solution of these systems is only possible for the simplest cases. In slightly more complicated cases it may stiU be possible to resolve the system of corresponding differential equations using methods such as Laplace transforms or matrix methods. However, there are systems which cannot be resolved analytically, or whose... 

METHODS FOR SOLVING KINETIC EQUATIONS 4.3.1 Laplace transforms... 

Using these three properties and tables containing both the Laplace transforms and inverse transforms, it is possible to solve many differential kinetic equations analytically. 

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. 

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