Nonlinear rate laws

To incorporate nonlinear rate laws into the solution procedure for tracing kinetic reaction paths (Section 16.3), we need to find the derivative of the reaction rate with respect to the molalities m,- of the basis species A,. The derivatives are given by,... 

This approach is analytically correct for isothermal reactors and first-order rate laws, since concentration does not appear in the expression for the Thiele modulus. For other (nonlinear) rate laws, concentration changes along the reactor affect the Thiele modulus, and hence produce changes in the local effectiveness factor, even if the reaction is isothermal. Problem 21-15 uses an average effectiveness factor as an approximation. 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

An important property of the stochastic version of compartmental models with linear rate laws is that the mean of the stochastic version follows the same time course as the solution of the corresponding deterministic model. That is not true for stochastic models with nonlinear rate laws, e.g., when the probability of transfer of a particle depends on the state of the system. However, under fairly general conditions the mean of the stochastic version approaches the solution of the deterministic model as the number of particles increases. It is important to emphasize for the nonlinear case that whereas the deterministic formulation leads to a finite set of nonlinear differential equations, the master equation... 

Derive an expression for the pseudo-first-order volumetric rate constant based on molar densities in the rate law, ki with units of inverse time, in terms of a generic nonlinear rate law for irreversible reactions. 

Due to the variable gas velocity and the nonlinear rate law the model equations represent a set of coupled nonlinear algebraic and differential equations of boundary value type which must be solved numerically. For this purpose the nonlinear equations are entirely linearized using the cjuasilinearization technique (12) and the linearized differential equations are solved using the orthogonal collocation method based on shifted Legendre polynomials (13). 

The system we have just analyzed had linear kinetics, a situation that made for easy analysis, but one that prevented the occurrence of any interesting dynamics. We will now increase the complexity by considering a quadratic, and therefore nonlinear, rate law. To make life interesting, we will look at an autocatalytic system in a CSTR ... 

When we think of diffusion acting on a system in which there are concentration inhomogeneities, our intuition suggests that diffusion should act to lessen, and eventually eliminate, the inhomogeneities, leaving a stable pattern with concentrations that are equal everywhere in space. As in the case of temporal oscillation, for a closed system the laws of thermodynamics require that this intuition be valid and that the eventual concentration distribution of the system be constant, both in time and in space. In an open system, however, just as appropriate nonlinear rate laws can lead to temporal structure, like oscillations and chaos, the interaction of nonlinear chemical kinetics and diffusion can produce nonuniform spatial structure, as suggested schematically in Figure 14.1. 

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