Kinetic rate equation, logarithmic

The obvious advantage is that the steady-state solution of an S-system model is accessible analytically. However, while the drastic reduction of complexity can be formally justified by a (logarithmic) expansion of the rate equation, it forsakes the interpretability of the involved parameters. The utilization of basic biochemical interrelations, such as an interpretation of fluxes in terms of a nullspace matrix is no longer possible. Rather, an incorporation of flux-balance constraints would result in complicated and unintuitive dependencies among the kinetic parameters. Furthermore, it must be emphasized that an S-system model does not necessarily result in a reduced number of reactions. Quite on the contrary, the number of reactions r = 2m usually exceeds the value found in typical metabolic networks. 

Most problems associated with approximate kinetics are avoided when Michaelis Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis Menten kinetics are common in biochemical practice almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. 

In order to determine the rate equation for hydrodesulfurization, a semi-logarithmic plot of the total sulfur content with time was made (Figure 2). The plot indicated two independent first-order reactions with greatly different rate constants. This is in agreement with the findings of Gates, et al. (7) and Pitts (3). A procedure similar to that of Pitts (3 ) was used to describe the hydrodesulfurization kinetics. The rate expression is given below ... 

Differential Data Analysis As indicated above, the rates can be obtained either directly from differential CSTR data or by differentiation of integral data. A common way of evaluating the kinetic parameters is by rearrangement of the rate equation, to make it linear in parameters (or some transformation of parameters) where possible. For instance, using the simple nth-order reaction in Eq. (7-165) as an example, taking the natural logarithm of both sides of the equation results in a linear relationship Between the variables In r, 1/T, and In C ... 

The parameters that are plotted versus pH are (1) og(VIK) for each substrate, (2) log(V), (3) pA i (logarithm to the base 10 of the reciprocal of the dissociation constant) for a competitive inhibitor or a substrate not adding last to the enzyme, and (4) pAj or pA , for metal ion activators. It is particularly important to consider the pH variation of VIK and V, the two independent kinetic constants, and not simply to determine the rate at some arbitrary concentration of each substrate. The Michaelis constant is merely the ratio of V and V/K, so its pH profile is a combination of effects on V and V/K. Although we shall discuss the shapes of pH profiles, the reader should remember that graphical plotting is for a preliminary look at the data, and that the data must be fitted to the appropriate rate equation by the least-squares method to obtain reliable estimates of kinetic parameters, pA values, and their standard errors (5). Because pH profiles commonly show decreases of a factor of 10 per pH unit over portions of the pH range, the fits are always made in the log form [i.e., log(V), log(V/A), or pAj versus pH],... 

The isothermal crystallization kinetics is studied mainly with the Avrami equation (logarithmic form in equation 8.2), where < )(f) is the relative crystallinity, n and are the Avrami exponent and crystallization rate constant respectively. 

A very useful method for determining the reasonableness of constants estimated by LH analysis has been advanced by Boudart et al. [M. Boudart, D.E. Mears and M.A. Vannice, Ind. Chim. Beige, 32, 281 (1967)). The method is based on the compensation effect, often noted in the kinetics of catalytic reactions on a aeries of related catalysts, in which there is observed a linear relationship between the logarithm of the pre-exponential factor of the rate equation and the activation energy. Explanations for such behavior abound. Perhaps the most reasonable is... 

These parameters arise from the boundary conditions on Laplace s equation when finite kinetic rates are included. The first is a ratio between the exchange current density and ohmic parameters such as length and conductivity. The second parameter is a dimensionless current level. Values of either or both parameters much greater than unity indicate that ohmic effects dominate and the current distribution resembles the primary case. Low values of both parameters indicate that kinetics limit the process and the current distribution is uniform. Even though a low value of J indicates substantial kinetic effects, the current distribution resembles the primary case at high values of 6 because the linear ohmic dependence on current density dominates the logarithmic overpotential dependence. These... 

Logarithmic Reaction Kinetics Logarithmic reaction kinetics are most often associated with low-temperature service tests, the initial stages of oxidation of certain materials, and with thick scale behavior where internal scale cavities or precipitates interfere with diffusion mechanisms [3]. The rate equation describing logarithmic reaction kinetics (Fig. 3) is... 

As shown in Sect. 2, the fracture envelope of polymer fibres can be explained not only by assuming a critical shear stress as a failure criterion, but also by a critical shear strain. In this section, a simple model for the creep failure is presented that is based on the logarithmic creep curve and on a critical shear strain as the failure criterion. In order to investigate the temperature dependence of the strength, a kinetic model for the formation and rupture of secondary bonds during the extension of the fibre is proposed. This so-called Eyring reduced time (ERT) model yields a relationship between the strength and the load rate as well as an improved lifetime equation. 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

Figure 4. Copper complexation by a pond fulvic acid at pH 8 as a function of the logarithm of [Cu2+]. On the x-axis, complex stability constants and kinetic formation rate constants are given by assuming that the Eigen-Wilkens mechanism is valid at all [M]b/[L]t. The shaded zone represents the range of concentrations that are most often found in natural waters. The + represent experimental data for the complexation of Cu by a soil-derived fulvic acid at various metakligand ratios. An average line, based on equations (26) and (30) is employed to fit the experimental data. Data are from Shuman et al. [2,184]...

Figure 6.2 Graphical method of analyzing the kinetics of a reaction obeying equation 6.16. The logarithm of [B] is plotted against time. The rate constant for the slower process is obtained from the slope of the linear region after the faster process has died out. The rate constant for the faster process is obtained by plotting the logarithm of A (the difference between the value of [B] at a particular time and the value of [B] extrapolated back from the linear portion of the plot) against time for the earlier points. The rate constants for this example are 20 and 2 s 1, respectively.

The continuous decrease with the time of the rate of P h decay at low temperature, which is expressed by the presence in the kinetic curves of several sections approximated by first-order equations with steadily decreasing rate constants, suggests the idea of trying to describe the kinetics of this process in terms of logarithmic kinetics characteristic of electron tunneling reactions in the presence of scatter over the distances between the reagents for different pairs P 1 -A or P+ -Q . In order to do so it is necessary to study the kinetics of the process over a broad enough time interval. Such a study has been carried out in ref. 45 for the process of P decay via reaction (1) in the reaction centres of the PSl of subchloroplasts. Let us consider this work in more detail. 

Equation (2.17), (2.18), or (2.19) indicates that a plot of the negative of the logarithm of [A] or of (a - y) versus time should be a straight line with slope k or k/2.30. As noted earlier (Section IIB, 2b), obtaining such a linear plot from experimental data is a necessary but not sufficient condition for one to conclude that the reaction is kinetically first-order. Even if the kinetic plot using a first-order equation is linear over 90% of the reaction, deviations from the assumed rate expression may be hidden (Bunnett, 1986). When other tests confirm that it is first-order, the rate constant k, is either the negative of the slope [Eq. (2.17) or (2.19)] or 2.30 times the negative of the slope [Eq. (2.18)]. 

In chemical kinetics, the reaction rates are proportional to concentrations or to some power of the concentrations. Phenomenological equations, however, require that the reaction velocities are proportional to the thermodynamic force or affinity. Affinity, in turn, is proportional to the logarithms of concentrations. Consider a monomolecular... 

A plot of the logarithm of the degradation as a function of time for a first-order reaction produces a straight line. Such plots for the warp and weft yarns are shown in Figure 2. The plots are nearly linear and suggest that the degradation is indeed first order. As is well known in kinetics, the rate constant can be estimated from the slope of the line whose equation is given by... 

---