Exponential rate equations

The plot of ( versus q which results from Eqn. 9-9 is a polarization curve this polarization crirve is usually divided into two ranges of polarization as shown in Fig. 9-3 one is a range of polarization where a linear rate equation holds near the equilibrium potential (t) - 0) the other is a range of polarization (the Tafel range) where an exponential rate equation applies at potentials away from the equilibrium potential (ti 0). 

Water removal from the trihydrate resulted in a different reactant structure and, providing water was excluded, this was not converted to the monohydrate. Different kinetic behaviour was observed during subsequent decomposition. The temperature of reaction was lower, studied between 509 and 538 K, attributed to decomposition occurring in a less stable structure. In contrast with the monohydrate, ar-time curves for this reactant were fitted by the exponential rate equation when a> 0.05 and = 140 kJ mol". The mechanism proposed was that reaction advanced through the creation of cracks in a strained reactant crystallite assemblage. 

If the same measurement is repeated for different [BJ it should be possible to extractjy by plotting log vs log [BJ. This should be a straight line with slopejy. In a similar manner, can be obtained by varying [CJ. At the same time the assumption that x equals 1 is confirmed. Ideally, a variety of permutations should be tested. Even if xis not 1, and the integrated rate equation is not a simple exponential, a usefiil simplification stiU results from flooding all components except one. 

If C is approximately constant during the initial drydown period, as it is in many closed-system apphcations, then the water concentration decays exponentially with time. The rate equation can be integrated to give the relationship between water concentration and time ... 

FIG. 18 Scaling plot of L t) from the numerical solution of the rate equations for quenches to different equilibrium states [64] from an initial exponential MWD with L = 2. The final mean lengths are given in the legend. The inset shows the original L t) vs t data. Here = (0.33Lqo). ... 

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-... 

The effect of substrate concentration on specific growth rate (/i) in a batch culture is related to the time and p,max the relation is known as the Monod rate equation. The cell density (pcell) increases linearly in the exponential phase. When substrate (S) is depleted, the specific growth rate (/a) decreases. The Monod equation is described in the following equation ... 

It has been suggested that fungi grow in filamentous form at an exponential rate with a constant specific growth rate (ji) until some substrate becomes growth limiting, according to the Monod equation 4 6... 

Equation (6.13), in fact, reflects the physical nature of the electrode process, consisting of the anode (the first term) and cathode (the second term) reactions. At equilibrium potential, E = Eq, the rates of both reactions are equal and the net current is zero, although both anode and cathode currents are nonzero and are equal to the exchange current f. With the variation of the electrode potential, the rate of one of these reactions increases, whereas that of the other decreases. At sufficiently large electrode polarization (i.e., deviation of the electrode potential from Eg), one of these processes dominates (depending on the sign of E - Eg) and the dependence of the net current on the potential is approximately exponential (Tafel equation). 

Exponential rate expressions are also useful in deriving kinetic equations because they can be substituted into differential equations, which can then be integrated. For example, from Scheme 2 the differential equation describing the rate of appearance of unchanged drug in urine may be written ... 

Note that the pre-exponential factors indicate only small entropies of activation in the Eyring form of the rate equations. This is a significant observation which indicates that the decrease of entropy associated with the incorporation of a hydrogen molecule at or prior to the transition state must be compensated for by a dissociation or decrease of coordination number. 

The temperature dependence of many rate constants can be fit over a relatively narrow temperature range by the exponential Arrhenius equation... 

In a graphite atomizer, the atoms will appear according to a kinetic rate equation which will probably contain an exponential function. As the number of atoms in the atom cell increases, so does the rate of removal, until, at the absorption maximum (peak height measurement), the rate of formation equals the rate of removal. Thereafter, removal dominates. 

Malkin s autocatalytic model is an extension of the first-order reaction to account for the rapid rise in reaction rate with conversion. Equation 1.3 does not obey any mechanistic model because it was derived by an empirical approach of fitting the calorimetric data to the rate equation such that the deviations between the experimental data and the predicted data are minimized. The model, however, both gives a good fit to the experimental data and yields a single pre-exponential factor (also called the front factor [64]), k, activation energy, U, and autocatalytic term, b. The value of the front factor k allows a comparison of the efficiency of various initiators in the initial polymerization of caprolactam [62]. On the other hand, the value of the autocatalytic term, b, describes the intensity of the self-acceleration effect during chain growth [62]. 

We may now allow for reactant consumption explicitly by restoring the exponential decay, p = p0e- °. In dimensionless terms this means recognizing that p is a time-dependent parameter, p = /i0e . The governing rate equations are... 

The linear rate equation, eqn. (18), was assumed to hold throughout Sect. 2 because it is the most simple case from a mathematical point of view. Evidently, it is valid in the case of the linear mechanism (Sect. 4.2.1) as it is also in some special cases of a non-linear mechanism (see Table 6 and ref. 6). The kinetic information is contained in the quantity l, to be determined either from the chronoamperogram [eqn. (38), Sect. 2.2.3] or from the chronocoulogram [eqn. (36), Sects. 2.2.2 and 2.2.4], A numerical analysis procedure is generally preferable. The meaning of l is defined in eqn. (34), from which ks is obtained after substituting appropriate values for Dq2 and for (Dq/Dr)1/2 exp (< ) = exp (Z) [so, the potential in this exponential should be referred to the actual standard potential, see Sect. 4.2.3(a)]. 

To test whether the reaction is first order, we simply fit the data to the exponential integrated first-order rate equation (Table 3.1) using a non-linear optimisation procedure and the result is shown in Fig. 3.3. The excellent fit shows that the reaction follows the mathematical model and, therefore, that the process is first order with respect to [N2O5], i.e. the rate law is r = A bsI Os]. The rate constant is also obtained in the fitting procedure, k0bs = (6.10 0.06) x 10 4 s 1. We see that, even with such a low number of experimental points, the statistical error is lower than 1%, which shows that many data points are not needed if... 

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