Logarithmic rate equation

Surface reaction mechanisms include adsorption, desorption, surface nucleation, polynucleation, mononucleation and ion exchange reaction. The dependencies of amounts of precipitate and solution composition on time are different for each mechanism. For example, linear, exponential and logarithmic rate equations are established for volume diffusion, polynuclear growth and spiral growth, respectively. 

The treatment given here follows that of Freeman and Carroll [531], which was also considered by others [534,569]. Again, the logarithmic form of the basic rate equation is used and the reaction order expressed in the form f(a) = fc(l — a)" so that, for incremental differences in (da/dT), (1 — a) and T 1, one can write... 

Writing the rate equation in the logarithmic form for an nth order of reaction, we have... 

The variation of a rate constant with temperature is described by the Arrhenius equation. According to its logarithmic form (Equation ), a plot of in k vs. 1 / Z, with temperature expressed in keIvins, shou id be a straight line. 

In addition to its constraints on the concentration dependent portions of the rate expression thermodynamics requires that the activation energies of the forward and reverse reactions be related to the enthalpy change accompanying reaction. In generalized logarithmic form equation 5.1.69 can be written as... 

Since some adulteration of raw data occurs when they are transformed mathematically, by differentiation or taking logarithms or reciprocals or otherwise, it is better from a statistical point of view to change the rate equation to read in terms of total pressure, rather than to change the data to partial pressures or concentrations. Such a transformation is worked out for a... 

Equation (8.24) is the integrated first-order rate equation. Being a logarithm, the left-hand side of Equation (8.24) is dimensionless, so the right-hand side must also be dimensionless. Accordingly, the rate constant k will have the units of s-1 when the time is expressed in terms of the SI unit of time, the second. 

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. 

The approximation of biochemical rate equations by linear-logarithmic (lin-log) equations [318] seeks to avoid several drawbacks of the power-law formalism. Using the lin-log framework, all reaction rates are described by their dependencies on logarithmic concentrations, based on deviations from a... 

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. 

When the logarithmic form of the burning rate equation given by Eq. (3.73) is differentiated with respect to the initial temperature of the energetic material at a constant pressure, the following is derived ... 

For a given cup and bob combination, the term n" represents the slope of a logarithmic plot of torque t versus rotational speed N at the particular value of N in question. Since the derivative in the last term of Eq. (48) is normally small, one may usually take as the final shear-rate equation ... 

The complete rate equation (E) reduces to the following approximation on substituting, simplifying, and returning to Briggsian logarithms ... 

Taking logarithms of Equation 3.14, we obtain Equation 3.15- another relationship between the rate at any instant, the rate constant, and the instantaneous concentrations of reactants ... 

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 ... 

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 ... 

Time and temperature correlation. Figure 10 shows that the data are not fitted by a linear rate equation and the plots of Fig. 11 of the weight gain vs. the logarithm of the time show smooth curves of increasing slope. Thus neither a linear nor a logarithmic equation applies to the decay of oxidation rates with time. 

The integrated form of this equation yields the familiar linear first-order rate equation in which In stands for the logarithm to the base e, and the subscripts o and t refer to the initial value of DP and to the value at any time, t, respectively ... 

In such a case there is a direct equality between the activation-energy difference and the standard heat of the reaction as well as between the logarithm of the ratio of the frequency factors and the standard entropy change. This gives a rationale for the nomenclature activation energy, which is used to designate the constant E in the rate equation. 

A second rate method found in most texts is based on the rate equation 3.10 in its logarithmic form ... 

Here k is the rate constant, and Kj is the equihbrium constant for an exchange reaction between protons and the metal M, at the surface. Oelkers argues that when the term Kfa +la Y is small, significant M, remains in the surface leached layer, and the rate equation simplifies in that the denominator becomes unity. For such a case, the logarithm of the far-from-equilibrium rate becomes linearly related to the logarithm of the activity of the aqueous species M, and is dependent only upon pH and activity of M,. Oelkers (2001b) has used this simplified rate equation to describe dissolution of basalt glass... 

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 ... 

Because the cost of a heat exchanger depends on its size, and because its size will depend on the heat-transfer rate, a rate equation must be introduced. The rate equation is given by Equation 4.4.3. The logarithmic-mean temperature difference in Equation 4.4.3 is given by Equation 4.4.4. Because perfect countercurrent flow can never be achieved in an actual heat exchanger, the logarithmic-mean temperature difference correction factor, F, is needed. For simplicity, Equation 4.10, discussed earlier, is expressed as Equation 4.4.5, which states that F depends only on the terminal temperatures, once a particular heat exchanger is selected. 

Vyazovkin and Lesnikovich [42] have emphasized that the majority of NIK methods involve linearization of the appropriate rate equation, usually through a logarithmic transformation which distorts the Gaussian distribution of errors. Thus non-linear methods are preferable [89]. Militky and Sest [90] and Madarasz et al. [91] have outlined routine procedures for non-linear regression analysis of equation (5.5) above by transforming the relationship ... 

DIFFERENTIAL METHOD Replacing l/K and with their numerical values and taking logarithms of the rate equation (B) yields... 

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