Differential rate equations

With these reaction rate constants, differential reaction rate equations can be constructed for the individual reaction steps of the scheme shown in Figure 10.3-12. Integration of these differential rate equations by the Gear algorithm [15] allows the calculation of the concentration of the various species contained in Figure 10.3-12 over time. This is. shown in Figure 10.3-14. 

A second common approximation is the steady-state condition. That arises in the example if /fy is fast compared with kj in which case [i] remains very small at all times. If [i] is small then d[I] /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). Eor equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equihbrium condition, namely, equations 24a and b. 

Equation (2-4) is the stoichiometric equation for an elementary first-order reaction, and Eq. (2-5) is the corresponding differential rate equation. 

In Chapter 1 we distinguished between elementary (one-step) and complex (multistep reactions). The set of elementary reactions constituting a proposed mechanism is called a kinetic scheme. Chapter 2 treated differential rate equations of the form V = IccaCb -., which we called simple rate equations. Chapter 3 deals with many examples of complicated rate equations, namely, those that are not simple. Note that this distinction is being made on the basis of the form of the differential rate equation. 

Note the rate constant symbolism denoting the forward (fc,) and backward (/c i) steps.] The differential rate equation is written, according to the law of mass action, as... 

Evidently simple first-order behavior is predicted, the reactant concentration decaying exponentially with time toward its equilibrium value. In this case a complicated differential rate equation leads to a simple integrated form. The experi-... 

This consists of two consecutive irreversible first-order (or pseudo-first-order) reactions. The differential rate equations are... 

The differential rate equation is - dCf /dt = kCffiR, and the mass balanee equation is Ca i A = r r- Eliminating Ca between these equations and integrating gives the usual second-order integrated equation, whieh can be written in this form ... 

The differential rate equations of a complex reaetion, expressing rates as functions of concentrations, are usually simpler in form than are the corresponding integrated equations, whieh express concentrations as funetions of time moreover, it is always possible to write down the differential rate equations for a postulated kinetie seheme, whereas it may be difficult or impossible to integrate them. Of course, we usually measure concentration as a funetion of time. If, however, we can measure rates, we may use the differential equations directly. 

In 1950 French " and Wideqvist independently described a data treatment that makes use of the area under the concentration-time curve, and later authors have discussed the method.We introduce the technique by considering the second-order reaction of A and B, for which the differential rate equation is... 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

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. 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-(ej-48. pp. 76-81 some numerical calculations. Farrow and Edelson and Porter... 

Sections 3.1 and 3.2 considered this problem Given a complex kinetic scheme, write the differential rate equations find the integrated rate equations or the concentration-time dependence of reactants, intermediates, and products and obtain estimates of the rate constants from experimental data. Little was said, however, about how the kinetic scheme is to be selected. This subject might be dismissed by stating that one makes use of experimental observations combined with chemical intuition to postulate a reasonable kinetic scheme but this is not veiy helpful, so some amplification is provided here. 

According to the law of mass action the differential rate equation is... 

This chapter takes up three aspects of kinetics relating to reaction schemes with intermediates. In the first, several schemes for reactions that proceed by two or more steps are presented, with the initial emphasis being on those whose differential rate equations can be solved exactly. This solution gives mathematically rigorous expressions for the concentration-time dependences. The second situation consists of the group referred to before, in which an approximate solution—the steady-state or some other—is valid within acceptable limits. The third and most general situation is the one in which the family of simultaneous differential rate equations for a complex, multistep reaction... 

The route from kinetic data to reaction mechanism entails several steps. The first step is to convert the concentration-time measurements to a differential rate equation that gives the rate as a function of one or more concentrations. Chapters 2 through 4 have dealt with this aspect of the problem. Once the concentration dependences are defined, one interprets the rate law to reveal the family of reactions that constitute the reaction scheme. This is the subject of this chapter. Finally, one seeks a chemical interpretation of the steps in the scheme, to understand each contributing step in as much detail as possible. The effects of the solvent and other constituents (Chapter 9) the effects of substituents, isotopic substitution, and others (Chapter 10) and the effects of pressure and temperature (Chapter 7) all aid in the resolution. 

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]. 

Since the measurements of conductance change are not directly related to the composition of the solution, as an alternative method numerical integration of the differential rate equations implied by the proposed mechanism was employed. The second order rate coefficients obtained by this method are... 

The size distributions in polymers may be derived in a more elegantly im pressive manner through the use of differential rate equations for each species. After introduction of the inevitable simplifying assumptions, the final results are the same as those found above. 

Again one way of dealing with this is to replace those differential rate equations, having low time constants (i.e., high K values) and fast rates of response, by quasi-steady-state algebraic equations, obtained by setting... 

Rates of reaction Rate constants and their temperature dependence leading to a differential rate equation... 

The odd-oxygen concept can be understood from a look at the differential rate equations for the mechanism ... 

Differential rate equation The equation relating the rate of change of concentration with time of a reactant or product. 

However, the average rates calculated by concentration versus time plots are not accurate. Even the values obtained as instantaneous rates by drawing tangents are subject to much error. Therefore, this method is not suitable for the determination of order of a reaction as well as the value of the rate constant. It is best to find a method where concentration and time can be substituted directly to determine the reaction orders. This could be achieved by integrating the differential rate equation. 

Problem 1.1 Write the differential rate equations of the following reactions ... 

A third order reaction can be the result of the reaction of a single reactant, two reactants or three reactants. If the two or three reactants are involved in the reaction they may have same or different initial concentrations. Depending upon the conditions the differential rate equation may be formulated and integrated to give the rate equation. In some cases, the rate expressions have been given as follows. 

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