Coupled differential rate equations

The usual chemical kinetics approach to solving this problem is to set up the time-dependent changes in the reacting species in terms of a set of coupled differential rate equations [5,6]. 

In the second general approach to this problem, an attempt is made to examine in some manner the overall behavior of the entire ensemble of interacting units. By far, the most common approach here, and the one normally taught from textbooks, is to represent the kinetic behavior of a particular system in terms of an applicable set of coupled differential rate equations. These equations, with their associated rate constants, summarize the bulk behaviors of the ingredients involved in an averaged way. For example, the simple two-step transformation A —> B —> C, can be characterized by the set of rate equations ... 

In the preceding examples the isomers A and B were in equilibrium, that is, k2 and kj were assumed to be much greater than and k. However, it is also possible that the various rates are more nearly equal. In that case the coupled differential rate equations must be solved (Baer et al., 1975, 1983). The formation rate of P and P2 is... 

Thus, in circumstances of moderate pumping and not too powerful generation of laser radiation, the system of four coupled differential rate equations reduces to that for level 3 ... 

The system of coupled differential equations that result from a compound reaction mechanism consists of several different (reversible) elementary steps. The kinetics are described by a system of coupled differential equations rather than a single rate law. This system can sometimes be decoupled by assuming that the concentrations of the intennediate species are small and quasi-stationary. The Lindemann mechanism of thermal unimolecular reactions [18,19] affords an instructive example for the application of such approximations. This mechanism is based on the idea that a molecule A has to pick up sufficient energy... 

This set of coupled differential equations can—as also expressed for the activated sludge model concept—be formulated in terms of a matrix. This matrix includes the relationships between the relevant components, processes, expressions, process rates and coefficients (Table 5.3). The mass balances shown in Equations (5.6) to (5.9) can be identified as columns in the matrix. 

From the resulting reactions a set of coupled differential equations can be derived describing the deactivation of P, L and PI and the reaction rate constants can be derived from storage stability data by the use of parameter estimation methods. The storage stability data give the concentration of P+PI (it is assumed that the inhibitor fully releases the protease during analysis due to fast dynamics and the extensive dilution in the assay) and L as a function of time. 

The film at the start of the melting process has an initial thickness estimated from Eq. 6.13. As the solids continue to melt, the film thickness at the barrel solid bed interface increases slightly and decreases the dissipation because the local shear rate decreased. Since the dissipation decreases, the melting rate also decreases. This theory requires the solution of only two coupled differential equations ... 

Using a simple kinetic model, Solomon demonstrated that the spin-lattice relaxation of the I and S spins was described by a system of coupled differential equations, with bi-exponential functions as general solutions. A single exponential relaxation for the I spin, corresponding to a well-defined Tu, could only be obtained in certain limiting situations, e.g., if the other spin, S, was different from I and had an independent and highly efficient relaxation pathway. This limit is normally fulfilled if S represents an electron spin. The spin-lattice relaxation rate, for the nuclear spin, I, is in such a situation given by ... 

In Section 4.7, we discussed the relaxation process of SE s in a closed system where the number of lattice sites is conserved (see Eqn. (4.137)). A set of coupled differential equations was established, the kinetic parameters (v(x,iq,x )) of which describe the rate at which particles (iq) change from sublattice x to x. We will discuss rate parameters in closed systems in Section 5.3.3 where we deal with diffusion controlled homogeneous point defect reactions, a type of reaction which is well known in chemical kinetics. 

Because of the spectral relaxation due to the appearance of a high dipole moment in the charge-transfer state, the dynamics of the TICT state formation has been studied by following the fluorescence rise in the whole A band. In Fig. 5.6 are plotted, in the 10 ns time range, the experimental curve iA(t) at -110°C in propanol (tj = 1.5 x 103 cp) and the decay of the B emission at 350 nm. The solid curve representing the evolution of the TICT state expected in a constant reaction rate scheme shows a slower risetime with respect to that of the recorded A emission. To interpret the experimental iA(t) curves, the time dependence of the reaction rate kliA(t) should be taken into account. From the coupled differential equations for the populations nB(t) and nA(t) of the B and A states (remembering that the reverse reaction B <—A is negligible at low temperatures) ... 

For a system with n components (including nonad-sorbable inert species) there are n — 1 differential mass balance equations of type (17) and n — 1 rate equations [Eq. (18)]. The solution to this set of equations is a set of n — 1 concentration fronts or mass transfer zones separated by plateau regions and with each mass transfer zone propagating through the column at its characteristic velocity as determined by the equilibrium relationship. In addition, if the system is nonisothermal, there will be the differential column heat balance and the particle heat balance equations, which are coupled to the adsorption rate equation through the temperature dependence of the rate and equilibrium constants. The solution for a nonisothermal system will therefore contain an additional mass transfer zone traveling with the characteristic velocity of the temperature front, which is determined by the heat capacities of adsorbent and fluid and the heat of adsorption. A nonisothermal or adiabatic system with n components will therefore have n transitions or mass transfer zones and as such can be considered formally similar to an (n + 1)-component isothermal system. 

Equation (7) was combined with appropriate chemical reaction rate expressions to yield a set of coupled differential equations expressing rates of change in the dissolved O3 and S(IV) concentrations. The equations were then solved numerically with the usual constraints of electroneutrality and the appropriate ionic equilibria given in Table I. 

The heat equation (Eq. 29) is coupled to the equations governing the mechanical response through the temperature dependence of the bulk viscoplastic strain rate (Eq. 3), the craze thickening rate (Eq. 22), and the thermal expansion in Eq. 1. The system of differential equations resulting from the finite element discretization of the energy balance in [9] is modified [57] to... 

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