Linearization of rate equation

Theory of Chemical Relaxation 64 Linearization of Rate Equations 64 Relaxation Time 67... 

THEORY OF CHEMICAL RELAXATION Linearization of Rate Equations... 

As mentioned earlier, one of the salient features of relaxation techniques for measuring fast reactions is the fact that due to small perturbations, all rate equations are reduced to first-order reactions. This linearization of rate equations is derived below and is taken entirely from Bernasconi (1976). 

The constants of rate equations of single reactions often can be found by one of the linearization schemes of Fig. 7-1. Nonhnear regression methods can treat any land of rate equation, even models made up of differential and algebraic equations together, for instance... 

To get the equilibrium sticking coefficient we assume that at an ambient pressure Pq the adsorbate is in equilibrium at a temperature T with partial coverages Hq, m, and Iq. We then increase the pressure slightly to p = Pq- - AP and linearize the rate equations in the increase in the precursor coverages Am = (m) —m and Al = (/) — Iq. If adsorption into and desorption from the precursors is much faster than transitions from the precursors into the adsorbed state, we can ignore terms proportional to An = n) -6 on the right-hand side of Eqs. (70-72) and also assume that the precursors will be in a steady state. It has been shown that the sticking... 

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... 

Equivalent or stoichiometric concentrations. The reaction 2A + B = P + Q was studied with [A]o = 2[B]o. A plot of [A] 1 versus time was linear. What rate equations does this result suggest What experiments could be designed to distinguish among them ... 

A distinction is to be drawn between situations in which (a) the flow pattern is known in detail (b) only the residence time distribution is known or can be calculated from tracer response data. Different networks of reactor elements can have similar RTDs, but fixing the network also fixes the RTD. Accordingly reaction conversions in a known network will be unique for any type of rate equation, whereas conversions figured when only the RTD is known proceed uniquely only for linear kinetics, although they can be bracketed in the general case. 

The speed of burning of quickmatch is related to the chemical composition and the mass of the composition on the string. The linear burning rate equation enables a crude estimate of the speed of burning to be made. For example, a typical relationship already seen for gunpowder is given by equation (10.1),... 

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations n (t), n t) due to which the reaction rate K(t) turns out to be concentration-dependent. (In particular, it depends also on initial reactant concentration.) As it is demonstrated below, in the fluctuation-controlled kinetics (treated in the framework of all joint densities) such fundamental steady-state characteristics of the linear theory as a recombination profile and a reaction rate as well as an effective reaction radius are no longer useful. The purpose of this fluctuation-controlled approach is to study the general trends and kinetics peculiarities rather than to calculate more precisely just mentioned actual parameters. 

Attempts at solution reported in the literature are based on linearization of the equation. An expression for the reaction rate is chosen such that the... 

It was previously normal practice to use linear forms of rate equations to simplify determination of rate constants by graphical methods. For example, the logarithmic version of the first-order rate law (Table 3.1), Equation 3.17a, allows k to be determined easily from the gradient of a graph of In Ct against time, by fitting the data to the mathematical model, y = a + bx ... 

Assuming that the system A + B Y is functional, one can linearize the rate equation of any one-step equilibrium. Equations (4.17) and (4.19) are first-order equations that result from linearizing the rate equation of a one-step equilibrium like Eq. (4.1). The symbol Tis the relaxation time of the system. 

The DO data from Runs D2-16 (150°C) and D2-29 (100°C) were fit to a number of rate equations, with linear correlation coefficients from 0.95-0.99 for first order kinetics. However, the uncertainties and lack of adequate data do not permit a rigorous statistical evaluation. Therefore, as a first approximation, a first order rate equation with respect to DO concentration was assumed. The data can then be expressed by the equation ... 

For (x - A)=0, as in the absence of non-linear terms (x=A=0), the pumped modes k acquire a Planck-type distribution, p=0, modified only by the term Sk, and consistent with the set of rate equations, Eq. (13),... 

A linear algebraic system of rate equations for the fast species results, which can be solved a priori. Hence a strongly reduced (in the number of species to be treated) system is obtained. This concept originates from astrophysical applications and from Laser physics. It is in some instances also referred to as collisional-radiative approximation , for the fast species, lumped species concept , bundle-n method or intrinsic low dimensional manifold (ILDM) method in the literature. We refer to [9,12,13] for further references on this. 

Figure 12.2. Arrhenius plots for reactions with shift in rate control but no change in form of rate equation (schematic). Left shift to different step within same pathway right shift to alternative, parallel pathway. Dashed lines are linear extrapolations into respective other temperature regions (from Helfferich and Savage [14]).

The non-linearity of the equations (5.1.2) to (5.1.4) prevents us from the use of analytical methods for calculating the reaction rate. These equations reveal back-coupling of the correlation and concentration dynamics - Fig. 5.1. Unlike equation (4.1.23), the non-linear terms of equations (5.1.2) to (5.1.4) contain the current particle concentrations riB t) due to which the... 

At a maximum generated pressure of 250 psi, the burning time of the propellant can be estimated from the linear burning rate, equation (4.3) ... 

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