By graphical method

This equation relates the (instantaneous) copolymer composition with the monomer feed of M and M2. Values for and are usually determined by graphical methods (9,10). Today, with the prevalence of powerful desktop computers, numerical minimisa tion methods are often used (11—14). 

Cycles and feedback loops are analyzed by graphical methods. 

To evaluate r for other reactions, we can obtain p for the reaction by measuring kx values for m -substituted compounds only, and then measure kx for p-substituted compounds where the values of graphical methods. Thus for the base-catalysed hydrolysis of p-substituted phenoxytriethylsilanes... 

NUMBER OF STAGES FOR COUNTERCURRENT WASHING BY GRAPHICAL METHODS... 

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 ... 

BOD may also be calculated by graphical method (Hach, 1982). These graphical calculations have the following advantages over the single point calculation outlined above ... 

Frantisek Hrncirik, Properties of Hydrocarbons C1-C8 Determined by Graphic Methods, SNTL, Prague, 1967. 

Evaluating grinding efficiency by graphical methods. Eng. Mining J., 126 934-938. 

It is believed that the technician may have made a dilution factor-of-10 error in one of the concentration measurements, What do you think How do your answers compare using regression (POLYMATH or other software) compare with those obtained by graphical methods ... 

Process design by graphical method based on residue curve map ... 

In former days, before electronic caelulations came into existance all calibrations and evaluations had to be carried out manually by graphical methods Linearized solutions had been used instead of nonlinear regression. Lineweaver and Burk 73) derived the linearized equation (19). It was introduced for calibration purposes to TLC by Kufner and Schlegel74). Kaiserf,3), Hulpke and Stegh 64) also used calibration techniques with reciprocal transformations of R and m without reference to the Michaelis-Menten transformation. 

Patterns of hexagonal crystals can also be indexed by graphical methods, since the hexagonal unit cell, like the tetragonal, is characterized by two variable parameters, a and c. The plane-spacing equation is... 

Substances with these low-symmetry structures yield powder patterns which are almost impossible to index by graphical methods, although the patterns of some orthorhombic crystals have been indexed by a combination of graphical and analytical methods. The essential difficulty is the large number of variable parameters involved. In the orthorhombic system there are three such parameters (flf, b, c), in the monoclinic four (a, b, c, P), and in the triclinic six a, b, c, oc, p, y). If the structure is known, patterns of substances in these crystal systems can be indexed by comparison of the observed sin 0 values with those calculated for all possible values of hkl. 

Integrated forms of the simple mass-action rate functions produce linear equations that are easily tested by graphical methods. The advantage of piu ameter optimization methods is that the computer programs can be writ-Icn lo generate statistics for a more quantitative estimation of goodness-of-I ii rather than the visual estimation that graphical methods provide. 

For crystals the lower limit of the integral is zero from the considerations just outlined. Equation (57) may be integrated if a relation between Cp and T is known. The available analytical relations are. however, complicated and of limited validity. Fortunately values of S may be obtained from measurements of heat capacity at different temperatures by graphical methods, A convenient method is one proposed by Lewis and Gibson17 which consists in plotting values of Cp/T against T and determining the area of the enclosed plot. Such a plot is shown in Fig. 7 for tine estimation of the entropy of metallic silver, from the work of Eucken, Clusius and Woitinek.18 Below the lowest... 

Tray Efficiencies and Real Trays by Graphical Methods... 

The general case of this expression cannot be given in closed form, but it is nevertheless possible to obtain the value of da/de for the cases for which M has been calculated as a function of e Y (= e2 Y/R) by graphical methods, and then plot d as a function of e. This has been done in Fig. 21 for the model with = 6. The curves obtained show, as expected from the basic assumptions for the model, a decreasing sensitivity to rate as the rate of extension increases. At very low rates, the curve appears to become almost parallel to the strain-axis this is only an illusion because the gradient of the stress-strain curve at such slow rates is determined by the relative values of m and /, and the lengths p and N. The particular case selected has a creep rate only slightly smaller than the strain rate at this level of stress. 

In the calculation of protolytic equilibria, the ionic product of water, equations for dissociation constants of acids and bases, equations of analytical concentrations and equations of electroneutrality or proton balance are taken for the starting point. Due to the difficulty of the numerical calculation of pH these systems are generally solved by graphical methods. 

The formalism of a numerical rank analysis can be substituted by graphical methods. Even though this graphical approach is formally equivalent to the numerical rank analysis, diagrams frequently are more descriptive and result in more information. 

Guidance on safe welding procedures by graphical methods... 

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