Rate equations, chemical

Analytical solutions also are possible when T is constant and m = 0, V2, or 2. More complex chemical rate equations will require numerical solutions. Such rate equations are apphed to the sizing of plug flow, CSTR, and dispersion reactor models by Ramachandran and Chaud-hari (Three-Pha.se Chemical Reactors, Gordon and Breach, 1983). 

Suppose we have a system in which a spin can exist in either of two different sites, A or B, and that these are distinguished by different resonant frequencies, coa and coB, and/or by different relaxation times, T2a and T2b If there is no exchange between sites, site A spins and site B spins can be described separately and independently by sets of Bloch equations. When exchange takes place, however, additional rate terms - completely analogous to terms in chemical rate equations - must be added to the Bloch equations. 

Solutions with other chemical rate equations are in P8.03.03, and some numerical cases in P8.03.04-P8.03.06. Such rate equations can be applied to the sizing of plug flow, CSTR and dispersion reactor models. 

The reaction set was numerically modeled using the computer program CHEMK (9) written by G. Z. Whitten and J. P. Meyer and modified by A. Baldwin of SRI to run on a MINC laboratory computer. CHEMK numerically Integrates a defined set of chemical rate equations to reproduce chemical concentration as a function of time. Equilibria can be modeled by Including forward and reverse reaction steps. Forward and reverse reaction rate... 

The concentration A of molecules A is again taken constant and supposed so large that the reverse reactions may be neglected. Alternatively one may imagine that A is continually supplied and B is drained. The reaction scheme (1.1) therefore describes an open system, see VII.4. The chemical rate equation for the concentration of X is... 

Normal usage, however, gives the edge to (3.1) rather than (V.8.6). In the case of a chemical reaction for instance, Q is the volume clearly the chemical rate equations... 

Whatever the nature of the reaction and whether the vessel chosen for the operation be a packed tubular reactor or a fluidised bed, the essence of the design problem is to estimate the size of reactor required. This is achieved by solving the transport and chemical rate equations appropriate to the system. Prior to this however, the operating conditions, such as initial temperature, pressure and reactant concentrations, must be chosen and a decision made concerning the type of... 

If, on the other hand, surface reaction determined the overall chemical rate, equation 3.68 (or 3.69 if an Eley-Rideal mechanism operates) would represent the rate. If it is assumed that a pseudo-equilibrium state is reached for each of the adsorption-desorption processes then, by a similar method to that already discussed for reactions where adsorption is rate determining, it can be shown that the rate of chemical reaction is (for a Langmuir-Hinshelwood mechanism) ... 

When the HSS solution of the chemical rate equations (la)—(lc) first becomes unstable as the distance from equilibrium is increased (by decreasing P, for example), the simplest oscillatory instability which can occur corresponds mathematically to a Hopf bifurcation. In Fig. 1 the line DCE is defined by such points of bifurcation, which separate regions of stability (I,IV) of the HSS from regions of instability (II,III). Along section a--a, for example, the HSS becomes unstable at point a. Beyond this bifurcation point, nearly sinusoidal bulk oscillations (QHO, Fig. 3a) increase continuously from zero amplitude, eventually becoming nonlin-... 

In the last section of our chapter we deal briefly with the problem of fluctuations. As an example, we will discuss the influence of fluctuations on chemical rate equations. 

Often there are cases where the submodels are poorly known or misunderstood, such as for chemical rate equations, thermochemical data, or transport coefficients. A typical example is shown in Figure 1 which was provided by David Garvin at the U. S. National Bureau of Standards. The figure shows the rate constant at 300°K for the reaction HO + O3 - HO2 + Oj as a function of the year of the measurement. We note with amusement and chagrin that if we were modelling a kinetics scheme which incorporated this reaction before 1970, the rate would be uncertain by five orders of magnitude As shown most clearly by the pair of rate constant values which have an equal upper bound and lower bound, a sensitivity analysis using such poorly defined rate constants would be useless. Yet this case is not atypical of the uncertainty in rate constants for many major reactions in combustion processes. 

The Integration of Chemical Rate Equations on a Vector Computer... 

The flow method that has been briefly discussed sometimes offers special advantages in kinetic studies. The basic equations for flow systems with no mixing may be derived as follows let us consider a tubular reactor space of constant cross-sectional area A as shown in Fig. 7.4 with a steady flow of u of a reaction mixture expressed as volume per unit time. Now we will select a small cylindrical volume unit dV such that the concentration of component i entering the unit is C(- and the concentration leaving the unit is C,- + dC-,. Within the volume unit, the component is changing in concentration due to chemical reaction with a rate equal to r(. This rate is of the form of the familiar chemical rate equation and is a function of the rate constants of all reactions involving the component i... 

Also, as in the solution of the chemical rate equations, a linearization error of order 0(AC) appears in the approximate solution. Thus, the same precautions that were taken to preserve accuracy in solving the ordinary chemical rate equations are required in the solution of the atmospheric model equation. 

D Golden, A Baldwin. OLCHEM Chemical-Rate-Equation Integrator. Molecular Physics Laboratory, Stanford Research Institute, 1998. 

Brown, R.L. "A Computer Program for Solving Systems of Chemical Rate Equations , NBSIR 76-1055, Natl. Bur. of Standards, Washington, D.C., 1978. 

An analogous approach has been applied to the epoxy-amine system. The three sets of parameters were derived one set for the chemical rate equation [Eq. (14)], one set for the diffusion rate constant according to Eq. (26), and one set for the T — x relation [Eq. (27)]. As seen in Figure 2.22, the experimental and the calculated DF profiles agree very well for the quasi-isothermal cure at reaction temperatures ranging from 25 to 100°C. 

Oppenheim, L, Shuler, K. E. Weiss, G. M. (1969). Stochastic and deterministic formulation of chemical rate equations. J. Chem. Phys., 50, 460-6. 

The traditional analysis of proton transfer reactions [3, 5-7] is by a kinetic scheme e.g., eq 1, and the ensuing chemical rate equations describing the time dependence of bulk concentrations. This is insufficient for explaining the long time tail. The minimal level of complexity has to involve the spatial diffusion of the proton in the electrostatic field of the anion, as depicted by the time-dependent Smoluchowski equation [15], but with a boundary condition (the back-reaction boundary condition) which describes reversible reactions [10]. 

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