Linear relationships

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

Modem NDT film systems (with Pb screens) are very linear X-ray detectors. This is shown in fig.l for different NDT film systems and a X-ray tube at 160 kV. Note that for histoncal reasons the film response curve is often plotted as film density versus log (radiation dose), which hides this linear relationship. The film density is the difference between the measured optical film density and the fog density Db of the film base. 

Fig. 2 shows the response of a C2 film system on a step wedge (wall thickness range 2. .. 18 mm) exposed with a X-ray tube at 160 kV. For the exposure withy-rays (Irl92 or Co60) corresponding linear relationships are obtained. From this linear relationship it is followed, that the influence of the scattered radiation and the energy dependence of the absorption coefficient can be considered by an effective absorption coefficientPcff in equation (1). 

The same highly linear relationship as shown for X-ray penetration can be obtained for Irl92 or Co60 at higher wall thickness and with a lower eontrast sensitivity. 

Many solids show marked swelling as a result of the uptake of a gas or a liquid. In certain cases involving the adsorption of a vapor by a porous solid, a linear relationship exists between the percentage of linear expansion of Ae solid and the film pressure of the adsorbed material [134, 135]. 

A correlation coefficient of/ = +1 means that there is a completely positive linear relationship. High values on the y-axis are associated with high values on the y-axis. 

Linear regression models a linear relationship between two variables or vectors, x and y Thus, in two dimensions this relationship can be described by a straight line given by tJic equation y = ax + b, where a is the slope of tJie line and b is the intercept of the line on the y-axis. 

Multiple linear regression (MLR) models a linear relationship between a dependent variable and one or more independent variables. 

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

Once the descriptors have been computed, is necessary to decide which ones will be used. This is usually done by computing correlation coelficients. Correlation coelficients are a measure of how closely two values (descriptor and property) are related to one another by a linear relationship. If a descriptor has a correlation coefficient of 1, it describes the property exactly. A correlation coefficient of zero means the descriptor has no relevance. The descriptors with the largest correlation coefficients are used in the curve fit to create a property prediction equation. There is no rigorous way to determine how large a correlation coefficient is acceptable. 

In the case of drug design, it may be desirable to use parabolic functions in place of linear functions. The descriptor for an ideal drug candidate often has an optimum value. Drug activity will decrease when the value is either larger or smaller than optimum. This functional form is described by a parabola, not a linear relationship. 

The relative basicities of aromatic hydrocarbons, as represented by the equilibrium constants for their protonation in mixtures of hydrogen fluoride and boron trifluoride, have been measured. The effects of substituents upon these basicities resemble their effects upon the rates of electrophilic substitutions a linear relationship exists between the logarithms of the relative basicities and the logarithms of the relative rate constants for various substitutions, such as chlorination and... 

Provided that the ratio of activity coefficients is invariant over the range of acidity concerned, a linear relationship with unit slope between logic Aaobs. 2nd +logic % o) i expected. However, there is... 

Intensities of the deformation vibration band near 1600 cm plotted for 2-aminothiazole and other 2-substituted thiazoles versus the Hammett constant give a linear relationship (123). 

On nonpolar columns, the compounds of a homologous series separate as a function of their boiling points, and linear relationships have been established between the logarithms of the retention volumes and the number of carbon atoms in the 2-, 4-, and 5-positions (see Fig. III-l). 

Linear relationships have been established on one hand between the Rf and pAa values of these azaaromatic bases (in the absence of steric hindrance of the ring nitrogen) and on the other hand, between the... 

Here again it is possible to find a linear relationship between the log (k/feo) (ko = methyl) values of 2-alkyl- and 2,4-dialkylthiazoles and between the latter value and Tafts Eg parameter (256). The value of 5 for 2,4-dialkylthiazoles is 1.472 with a correlation coefficient of 0.9994. Thus the sensitivity to substituent effects is more marked than in the case of a single substituent in the 2-position. Furthermore, the 4-position is again more sensitive than the 2-position. 

When we draw a scatter plot of all X versus Y data, we see that some sort of shape can be described by the data points. From the scatter plot we can take a basic guess as to which type of curve will best describe the X—Y relationship. To aid in the decision process, it is helpful to obtain scatter plots of transformed variables. For example, if a scatter plot of log Y versus X shows a linear relationship, the equation has the form of number 6 above, while if log Y versus log X shows a linear relationship, the equation has the form of number 7. To facilitate this we frequently employ special graph paper for which one or both scales are calibrated logarithmically. These are referred to as semilog or log-log graph paper, respectively. 

Equations 10.4 and 10.5, which establish the linear relationship between absorbance and concentration, are known as the Beer-Lambert law, or more commonly, as Beer s law. Calibration curves based on Beer s law are used routinely in quantitative analysis. 

Both the method of continuous variations and the mole-ratio method rely on an extrapolation of absorbance data collected under conditions in which a linear relationship exists between absorbance and the relative amounts of metal and ligand. When a metal-ligand complex is very weak, a plot of absorbance versus Ay or n-J m may be curved, making it impossible to determine the stoichiometry by extrapolation. In this case the slope ratio may be used. 

When possible, a quantitative analysis is best conducted using external standards. Unfortunately, matrix interferences are a frequent problem, particularly when using electrothermal atomization. Eor this reason the method of standard additions is often used. One limitation to this method of standardization, however, is the requirement that there be a linear relationship between absorbance and concentration. 

Shown here is a fiagram obtained for a solution of 100.0-ppm P04 . Determine h, t, T, f. At, and T. What is the sensitivity of this FIA method (assuming a linear relationship between absorbance and concentration) How many samples can be analyzed per hour ... 

A linear relationship exists between the cohesive energy density of an abrasive (10) and the WoodeU wear resistance values occurring between comndum H = 9) and diamond H = 42.5). The cohesive energy density is a measure of the lattice energy per unit volume. 

The Fischer-Tropsch process can be considered as a one-carbon polymerization reaction of a monomer derived from CO. The polymerization affords a distribution of polymer molecular weights that foUows the Anderson-Shulz-Flory model. The distribution is described by a linear relationship between the logarithm of product yield vs carbon number. The objective of much of the development work on the FT synthesis has been to circumvent the theoretical distribution so as to increase the yields of gasoline range hydrocarbons. 

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