Integrated Forms of the Rate Expression

In the steady-state approach to determining the rate law, solutions containing reactants are pumped separately at a constant flow rate into a vessel ( reactor ), the contents of which are vigorously stirred. After a while, produets and some reactants will flow from the reactor at the same total rate of inflow and a steady state will be attained, in which the reaction will take place in the reactor with a constant concentration of reactants, and therefore a constant rate. This is the basis of the stirred-flow reactor, or capacity-flow method. Although the method has been little used, it has the advantage of a simplified kinetic treatment even for complex systems. 

The manner in which [A] varies with time determines the order of the reaction with respect to A. Since it is usually much easier to measure a concentration than a rate, the form (1.15) is integrated. The three situations a = 0, 1, and 2 account for the overwhelming number of kinetic systems we shall encounter, with a = 1 by far the most common behavior, a = 0, zero-order in A  

The differential (rate) forms are (1.16), (1.18) and (1.20), and the corresponding integrated forms are (1.17), (1.19) (or (1.19a)) and (1.21). The designations [A]q and [A], represent the concentrations of A at zero time and time /. Linear plots of [A], In [A], or [A], vs time therefore indicate zero-, first, or second order dependence on the concentration of A. The important characteristics of these order reactions are shown in Fig. 1.1. Notwithstanding the appearance of the plots in 1.1 (b) and 1.1 (c), it is not always easy to differentiate between first-and second-order kinetics.Sometimes a second-order plot of kinetic data might be mistaken for successive first-order reactions (Sec. 1.6.2) with similar rate constants. 

In the unlikely event that a in (1.15) is a non-integer, the appropriate function must be plotted. In general, for (1.15), a 1 (see Prob. 8) 

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Figure 2.3 compares the rate of conversion of reactant into product for different orders of reaction. Table 2.5 summarizes the differential and integral forms of the rate expressions for reactions with various orders. 

Fortunately, the kinetics of the wall loss, measured from the decay of the reactive species in the absence of added reactant, are generally observed to be first order, so that corrections for these processes can be readily incorporated into the kinetic analyses. When these wall losses are significant, the integrated form of the rate expression (T) for reaction (17) of A + B becomes... 

We should also note that the expressions for f(a) simplify considerably when c is very large. In this case / r and so the concentration of B remains almost unchanged as A reacts the integrated form of the rate expression, eqn. (32), reduces to... 

Quaternlzatlon of amines with alkyl halides exhibit second-order kinetics and for such reactions, the integrated form of the rate expression is ... 

Methods based on simplification of the reaction rate expression. In these approaches one uses a vast excess of one or more of the reactants or stoichiometric ratios of the reactants in order to permit a partial evaluation of the form of the rate expression. They may be used in conjunction with either a differential or integral analysis of the experimental data. 

The stable species referred to in the termination reactions are presumed to arise from reactions of the atoms and free radicals with similar species already adsorbed at the walls. From results of the studies of the low-temperature reaction (400 to 500°C) on silica surfaces which give rates approximately first order in H2 and zero order in 62, it seems quite reasonable to suppose that active sites on the walls are covered with O2 and that H2 can react with them to form a hydride (W -H in initiation step) and liberate an H atom into the gas phase. For the explosion limits the initiation reaction is unimportant however, for the stationary reaction it is an integral part of the rate expression. 

Consider first the tubular reactor. From the material balance (Table 3.5.1), it is clear that in order to solve the mass balance the functional form of the rate expression must be provided because the reactor outlet is the integral result of reaction over the volume of the reactor. However, if only initial reaction rate data were required, then a tubular reactor could be used by noticing that if the differentials are replaced by deltas, then ... 

For an elementary process the ratio of the forward- and reverse-rate constants is equal to the equilibrium constant, Eq. (2-13). Hence the ne< rate of reaction can be expressed in terms of one k and the equilibrium constant. Then the integrated form of this rate expression can be used with... 

Once the form of the rate expression is known, we can determine the reaction rate constant, k(J), by using the integral form of the design equation. Consider file integral form of the design equation (Eq. 6.2.5) ... 

It should be noted that of all these methods and their variations, the direct determination of the desorption rate as a function of concentration and temperature is the most powerful and straightforward. This method is not limited to a simple heating curve or to a simple form of the rate expression. It can, if necessary, encompass concentration dependence in both the heat and the preexponential. The other techniques, such as the use of integrated rate expression, or of the maximum evolution rate are, however, very useful for rapid evaluation of the desorption energy in simple systems. 

We see that the integrated form of the kinetic expression containing the rate constant k is related to the concentration of reactants in the normal way, but those expressions are related indirectly to the measured conductivity T via the expression... 

Rearrangement and integration of Equation 4 gives the final form of the rate expression used to test the kinetic data. 

Also if the kinetic form of the rate expression is simple, then analytical integration is possible. This is so for zero-order, first-order and second-order reactions. This exercise relates to the latter. 

In the foregoing, k denotes the velocity constant for cubic autocatalysis, and a the initial concentration of reactant. Dimensionless time T = k a t = cpt. One integrated form of this rate-expression is ... 

The Integrated form of the rate equation of propagation is expressed as... 

Numerical approaches for estimating reactivity ratios by solution of the integrated rate equation have been described.124 126 Potential difficulties associated with the application of these methods based on the integrated form of the Mayo-kewis equation have been discussed.124 127 One is that the expressions become undefined under certain conditions, for example, when rAo or rQA is close to unity or when the composition is close to the azeotropic composition. A further complication is that reactivity ratios may vary with conversion due to changes in the reaction medium. 

Rate equations of the form f(a) = kt are derived through integrations of specific forms of the generalized expression [ 28] representing the summation of the growth of all nuclei, so that the volume of product at time f, V(t), is given by... 

The appearance of this heterogeneous form for the rate expression reflects the presence of a mass transfer step in series with the reaction step. If the parameter values are known, this ODE for bi i) can be integrated subject to the initial condition that bi=(bi)o at t = 0. The result can then be used to find a (f). 

In equation (13.11), the first term corresponds to the catalysed part of the reaction and the remaining terms, which make a relatively small contribution, apply to the uncatalysed part. Kinetic data at constant acidity were in good agreement with the integrated form of the calculated rate expression. The rate coefficients k2,k, k, and the ratio k. jk were evaluated. Almost linear plots of log 2 versus log [ ] were obtained at four temperatures with slopes close to —1.8. This result suggests that the dominant activated complex is that formed by loss of two ions, viz. 

It is obvious that to quantify the rate expression, the magnitude of the rate constant k needs to be determined. Proper assignment of the reaction order and accurate determination of the rate constant is important when reaction mechanisms are to be deduced from the kinetic data. The integrated form of the reaction equation is easier to use in handling kinetic data. The integrated kinetic relationships commonly used for zero-, first-, and second-order reactions are summarized in Table 4. [The reader is advised that basic kinetic... 

Assume that the reaction follows first-order kinetics. The integral form of the reaction rate expression is given by equation 3.1.8. 

A reaction rate constant can be calculated from the integrated form of a kinetic expression if one has data on the state of the system at two or more different times. This statement assumes that sufficient measurements have been made to establish the functional form of the reaction rate expression. Once the equation for the reaction rate constant has been determined, standard techniques for error analysis may be used to evaluate the expected error in the reaction rate constant. 

The values of a A, and EA must be determined from experimental data to establish the form of the rate law for a particular reaction. As far as possible, it is conventional to assign small, integral values to a2, etc., giving rise to expressions like first-order, second-order, etc. reactions. However, it may be necessary to assign zero, fractional and even negative values. For a zero-order reaction with respect to a particular substance, the rate is independent of the concentration of that substance. A negative order for a particular substance signifies that the rate decreases (is inhibited) as the concentration of that substance increases. 

The rate of change of C has been given already as Equation (8.42). Equations (8.42) and (8.43) show why the derivation of integrated rate equations can be difficult for consecutive reactions while we can readily write an expression for the rate of forming C, the rate expression requires a knowledge of [B], which first increases, then decreases. The problem is that [B] is itself a function of time. 

This expression can be integrated for different forms of the rate equation. 

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