Laws of logarithms

Then, using the laws of logarithms, we can simplify further ... 

We can split the fraction term in Equation (6.51) by employing the laws of logarithms, to yield... 

Using our calculators, we need to type ln(0.5093) as the numerator rather than a percentage. The minus sign comes from the laws of logarithms. 

It is very common to see this equation cited in an alternative form, with the last term being written as - RT/nF)ln [//(/d - DY- Both forms are correct the difl erence simply illustrates the laws of logarithms. 

By taking the Nemst equation used in Response 9.6, and appreciating that both Ag and AgCI are solid phases, and from the laws of logarithms that In [l/a(Cr)l = — In [a(Cr)], we may rewrite the equation as follows ... 

The major laws of logarithms and the exponential laws from which they are derived are shown in Table 1. 

Using laws of logarithms we can simplify the right-hand side to obtain the following ... 

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 ... 

There are several good reasons to focus on linear models. Theory may indicate that a linear relation is to be expected, e.g. Lambert-Beer s law of the linear relationship between concentration and absorbance. Even when a linear relation does not hold strictly it can be a sufficiently good local approximation. Finally, one may try and find a transformation of the individual variables (e.g. a logarithmic transformation), in order to obtain an acceptable linear model for the transformed variables. Thus, we simplify eq. (36.1) to... 

Listed after the reactions are the corresponding equilibrium quotients. The law of mass action sets the concentration relations of the reactants and products in a reversible chemical reaction. The negative log (logarithm, base 10) of the quotients in Eqs. (3.1)—(3.4) yields the familiar Henderson-Hasselbalch equations, where p represents the operator -log ... 

To construct such a diagram, a set of defect reaction equations is formulated and expressions for the equilibrium constants of each are obtained. The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. To simplify matters, only one defect reaction is considered to be dominant in any particular composition region, this being chosen from knowledge of the chemical attributes of the system under consideration. The simplified equilibrium expressions are then used to construct plots of the logarithm of defect concentration against an experimental variable such as the log (partial pressure) of the components. The procedure is best illustrated by an example. 

We encounter problems when it becomes necessary to take the logarithm of a concentration (which has units), since it contravenes one of the laws of mathematics. To overcome this problem, we implicitly employ a dodge by rewriting the equation as... 

Note how, as a consequence of the laws of arithmetic, we multiply the RT/F term with the logarithm term before adding the value of... 

According to Stevens law the logarithm of the perceived intensity is linearly related to the logarithm of the odour intensity. In the figure this relationship is given for two substances, one with a slope of 1.00 and one with a slope of. 67. As can be seen from the figure, this means that an odour concentration of 100 odour units/m3 is related to very different perceived odour intensities for the two substances. This means that odour concentrations computed in odour units/m3 should not be used as an indication of perceived odour intensity, but can only be used in relative measurements where the effects of measures taken to reduce odour pollution are compared, or in studies where dispersion models are used to find the distance to the source at which threshold is reached. 

Fig. 18.3 Plots of the growth laws of oxidation a) parabolic, b) rectilinear, c) quasi-rectilinear, d) logarithmic (West, 1980, with permission).

The most commonly used form of this Beer-Lambert law involves logarithms to the base 10 ... 

As can be seen from Eq. (25), (nc)-p is proportional to the square of the photon flux nc. It should also be proportional to the product of the maximum absorption cross section ffmax and the cross section oi at wavelength . This relation has been checked experimentally in 48>. The relation between the fluorescence output (wc)F and the excitation power nc for an aqueous solution of rhodamine resulted in a straight line in a double-logarithmic plot with a slope of 2.05 0.1, thus verifying the square law of two-photon absorption. 

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. 

Constant-Stress Layer in Flowing Fluids. In the boundary layer of a fluid flowing over a solid wall. Ihe shear stress varies with distance from Ihe wall bul ii may be considered nearly constant within a small fraction of the layer thickness. The concept is of particular importance in turbulent flow where it leads lo a theoretical derivation of the law of ihe wall," the logarithmic distribution of mean velocity. The constant stress layer is ihe best-known example of the equilibrium flow s near a wall. 

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