Logarithms reciprocal

The first is to normalize the data, making them suitable for analysis by our most common parametric techniques such as analysis of variance ANOYA. A simple test of whether a selected transformation will yield a distribution of data which satisfies the underlying assumptions for ANOYA is to plot the cumulative distribution of samples on probability paper (that is a commercially available paper which has the probability function scale as one axis). One can then alter the scale of the second axis (that is, the axis other than the one which is on a probability scale) from linear to any other (logarithmic, reciprocal, square root, etc.) and see if a previously curved line indicating a skewed distribution becomes linear to indicate normality. The slope of the transformed line gives us an estimate of the standard deviation. If... 

When comparing the test and reference products, dissolution profiles should be compared using a similarity factor (fz). The similarity factor is a logarithmic reciprocal square root transformation of the sum of squared error and is... 

The similarity factor /2 [137-139] is a logarithmic reciprocal transformation of the sum of squared errors and is a measurement of the similarity in the percentage dissolution between the two curves ... 

In field studies, data are not always normally distributed or parametric, even with relatively high replication (N = 10-16 individuals per site) and the use of data transformation techniques (e.g. logarithmic, reciprocal or arcsine). This could... 

For the formation of very thin films (about 10 run) at low temperature, logarithmic, reciprocal-logarithmic, or cubic time laws cau be deduced. If diffusiou determines the rate of formation of thick and compact scale, a parabolic time law applies ... 

A particular concentration measure of acidity of aqueous solutions is pH which usually is regarded as the common logarithm of the reciprocal of the hydrogen-ion concentration (see Hydrogen-ION activity). More precisely, the potential difference of the hydrogen electrode in normal acid and in normal alkah solution (—0.828 V at 25°C) is divided into 14 equal parts or pH units each pH unit is 0.0591 V. Operationally, pH is defined by pH = pH(soln) + E/K, where E is the emf of the cell ... 

H2O/100 kg of adsorbent. At equilibrium and at a given adsorbed water content, the dew point that can be obtained in the treated fluid is a function only of the adsorbent temperature. The slopes of the isosteres indicate that the capacity of molecular sieves is less temperature sensitive than that of siUca gel or activated alumina. In another type of isostere plot, the natural logarithm of the vapor pressure of water in equiUbrium with the desiccant is plotted against the reciprocal of absolute temperature. The slopes of these isosteres are proportional to the isosteric heats of adsorption of water on the desiccant (see... 

By plotting the natural logarithm (In) of against the reciprocal of the absolute temperature (1/T) at constant stress, as shown in Fig. 17.6, we find that ... 

Figure 29. Graph of the Logarithm of the Water/Methanol Association Equilibrium Constant against the Reciprocal of the Absolute Temperature...

Mathematically, pH is the logarithm (base 10) of the reciprocal of the hydrogen ion concentration. The pH may range from 0 to 14, where 0 is most acidic, 14 most basic, and 7 is neutral. Natural waters usually have a pH between 6.5 and 8.5. 

Table 2.2 gives the pH scale. Note again the reciprocal relationship between [H ] and [OH ]. Also, because the pH scale is based on negative logarithms, low pH values represent the highest H concentrations (and the lowest OH concentrations, as K, specifies). Note also that... 

Metwally et al. [28] also studied the resin-catalyzed hydrolysis of ethyl formate in acetone-water mixtures at different temperatures. The experimental results indicated a linear dependence of the logarithm of rate constant on the reciprocal of the dielectric constant (Fig. 2). The decrease of dielectric constant may lower the concentration of the highly polar transition state and thereby decrease the rate [28]. 

An overview of the superplastic behavior of aluminum alloys to demonstrate the grain-size effect is depicted in Fig. 1, in which the quantitative relation between the logarithm of the optimum strain rate for superplastic flow and the grain size (plotted as the logarithm of reciprocal grain size) is clearly shown [4]. The slope of the curve in Fig. 1 is noted to be about 3. 

The conductivity of a solution containing such molecular ions may be small compared with the value that would result from complete dissociation into atomic ions. In this way, in the absence of neutral molecules, we can have a weak electrolyte. The association constant for (29) has a value that is, of course, the reciprocal of the dissociation constant for the molecular ion (PbCl)+ the logarithms of the two equilibrium constants have the same numerical value, but opposite sign. 

A straight line is produced when the logarithm of a specific reaction rate is plotted against the reciprocal of the absolute temperature. Temperature has a marked influence on the reaction rates, but the range between reactions that are too slow or too fast to measure is really quite narrow. 

Volatilization is also affected by the ventilation rate over the surface of the material, but when this is constant, a straight-line result from the plot of the logarithm of the vapor pressure against the reciprocal of the absolute temperature. 

Thus, whether the changes in the material are due to chemical reactions, volatilization, or diffusion, one can expect a linear relationship between the logarithm of life (i.e., time to failure) and the reciprocal of absolute temperature. But there is no sound basis for extrapolating the effect of changing the concentration of the environmental exposure medium or the physical functions. 

It is possible, in some situations, that two different phenomena which proceed at different rates with different temperature coefficients or activation energies will affect the physical properties. In such complex cases, it is not expect to obtain a linear relation between the logarithm of life and reciprocal absolute temperature. If one obtains a nonlinear curve, however, it may he possible to identify the reaction causing the nonlinearity and correct for it. When one can make such a correction, one obtains a linear relationship. 

Test temperature. Tests should be run at a minimum of three temperatures and preferably four to confirm the linear relation between the logarithm of life and the reciprocal of absolute temperature. Several samples are necessary to plot the results of changes occurring versus time for each testing temperature. The time available and the accuracy of the extrapolation desired determine the lowest test temperature. Usually the most desirable lowest temperature is one that will give results in about 1,000-2,000 h (6-12 weeks). 

Linear least squares treatments of plots of the logarithm of the vapor pressure versus the reciprocal temperature were performed. The second-law enthalpy and entropy of sublimation at the median temperature are proportioned to the slope and... 

To summarize, the surface kinetics (or near surface kinetics) is the limiting step at lower temperature and diffusion is the rate limiting step at higher temperature. It is possible to switch from one rate-limiting step to the other by changing the temperature. This is illustrated in Fig. 2.9, where the Arrhenius plot (logarithm of the deposition rate vs. the reciprocal temperature) is shown for several reactions leading to the deposition of silicon,... 

Theta temperature (Flory temperature or ideal temperature) is the temperature at which, for a given polymer-solvent pair, the polymer exists in its unperturbed dimensions. The theta temperature, , can be determined by colligative property measurements, by determining the second virial coefficient. At theta temperature the second virial coefficient becomes zero. More rapid methods use turbidity and cloud point temperature measurements. In this method, the linearity of the reciprocal cloud point temperature (l/Tcp) against the logarithm of the polymer volume fraction (( )) is observed. Extrapolation to log ( ) = 0 gives the reciprocal theta temperature (Guner and Kara 1998). 

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

Increasing temperature permits greater thermal motion of diffusant and elastomer chains, thereby easing the passage of diffusant, and increasing rates Arrhenius-type expressions apply to the diffusion coefficient applying at each temperature," so that plots of the logarithm of D versus reciprocal temperature (K) are linear. A similar linear relationship also exists for solubUity coefficient s at different temperatures because Q = Ds, the same approach applies to permeation coefficient Q as well. 

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