Logarithmic law

The kinetics of reactions in which a new phase is formed may be complicated by the interference of that phase with the ease of access of the reactants to each other. This is the situation in corrosion and tarnishing reactions. Thus in the corrosion of a metal by oxygen the increasingly thick coating of oxide that builds up may offer more and more impedance to the reaction. Typical rate expressions are the logarithmic law,... 

The behavior type may change with temperature range. For instance, oxidation of zinc above 350°C (662°F) obeys the parabolic law, but at lower temperature the product coat seems to develop cracks and the logarithmic law, nw = kt, is obsei ved. 

This function is intermediate between the parallel model and the series model and referred to as the logarithmic law of mixture shown in curve 3. The law of mixture is valid for a composite system when there is no interaction in the interface. However, it is natural to consider that interaction will occur in the interface due to contact between A and B. Then considering the creation of interfacial phase C, different from A and B, the following equation can be presented ... 

The logarithmic law is also observed when the oxide film is an electrical insulator such as AI2O3. The transport of electrons through the oxide is mainly due to a space charge which develops between the metal-oxide interface and the oxide-gas interface. The incorporation of oxygen in the surface of the oxide requires the addition of electrons, and if this occurs by a charging process... 

The course of the curve already indicates that the logarithmic law is not strictly obeyed. By exterpolation we can obtain the point of... 

In contrast to piezoelectric single crystals, such as quartz, the piezoelectricity of PZT ceramics decays with time due to relaxation. Experimentally it is found that on a large time scale (for example, months and years), the aging process can be accurately described by a logarithmic law. For example, the coupling constant k varies with time f as... 

If, contrary to our assumption, electron transfer is slower than Mn+ diffusion and thus controls the oxidation rate, it can be shown that the direct logarithmic law applies ... 

The main features of this investigation are the following. First, a "deactivation process similar to that observed on the pure nickel oxide was found on the modified catalysis as well, with the same logarithmic law to represent its evolution with time. Second, the kinetic equations which were found to fit the data on pure nickel oxide also apply to the modified catalysts. Thus there is a low-temperature mechanism operative between 100° and 180°C. For all the samples assembled in Table II, the activation energies were practically the same, about 2 kcal./mole and essentially equal to the value for pure nickel oxide. This indicates that, for this particular mechanism of the reaction, the added ions and the semiconductivity changes do not affect directly the catalytic process. 

Because logarithms are exponents, we have the following logarithm laws that are derived from the laws of exponents given on page 8. Let A and B be any two numbers. 

The logarithm of a number consists of two parts, called the characteristic and the mantissa. The characteristic is the portion of the log that lies before the decimal point, and the mantissa is the portion that lies after the decimal point. The significance of separating a logarithm into these two parts is evident when you apply the logarithm laws to the logs of numbers such as 2000, and 2, and 0.000002. 

To find the power of a number by means of a log table, you use the logarithm laws cited on page 14, as illustrated in the following problem. 

The galactic redshift could obviously be attributed to the damping of the electromagnetic waves emitted from various galaxies in random motion within a stationary universe. Now, comparison between Hubble relativistic linear law and the logarithmic law that comes out from Maxwell electromagnetic wave equation shows that, in any case, the logarithmic law fits experimental data very well and thus better than linear law. 

Under such conditions, significant changes of the layer-growth mechanism may take place, and this must be reflected on the kinetic dependences observed. Therefore, the number of equations proposed for the mathematical description of those dependences is considerably greater than in the case of systems formed, for example, by two solids. Of these, different forms of linear, parabolic, paralinear, cubic and logarithmic laws are employed most frequently. 

According to U.R. Evans,7 logarithmic equations (direct, reverse and asymptotic) are low-temperature laws. The reverse logarithmic law is valid if the layer-growth process is controlled by the electric potential gradient. This is probably the case with thin films of ionic compounds.6,145 The... 

The addition of positively charged methylviologen ions, which are poorly adsorbed on the colloid under consideration, changes the kinetic dependence of the photobleaching relaxation. The kinetic curves obtained do not obey the logarithmic law (Eq. (2.9)), and are expressed by an exponential dependence on time,... 

However, since (x - 1) = e In x for small exponents, this expression is not very different from the logarithmic law. 

As reported in [97] for a narrow interval of polydispersity the bubble distribution is expressed equally well both by the natural logarithmic law and the gamma-function. De Vries [98] proposed the following mathematical expression of the distribution function, which correlated well with the real distribution... 

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