Plotting yield

A plot of log([A]g j[A — 1) versus log[B], called a Schild plot, yields a straight line of unit slope and intercept of iCg, the latter often expressed on a scale analogous to that for pH, so that = logif (46—48). 

Thus by measuring Vq at several values of c , the log-log plot yields the reaction order. This method also works if the rate equation includes the concentrations of other reactants, provided their concentrations are held constant throughout the series of measurements. 

These equations may be quickly solved from the plot yielding the values of frequency at maximum, minimum, and 0 points, shown in Figures 13-20A and 13-20B. 

The measured potential is thus a linear function of pH an extremely wide (10-14 decades) linear range is obtained, with calibration plots yielding a slope of 59 mV per pH unit. The overall mechanism of the response is complex. The selective response is attributed to the ion-exchange properties of the glass surface, and in particular the replacement of sodium ions associated with the silicate groups in the glass by protons ... 

Thus, In k will vary linearly with 1 fT. This plot yields A from the intercept and Ea from the slope. Equation (7-1) or (7-3) provides a definition of the Arrhenius activation energy, which is expressed as... 

Fig. 5. Log-iog plot illustrating the hardness dependence of density for PE samples crystallized from the melt. The plot yields two straight sections which can be ascribed to two preferential deformation modes a crystal destruction and b compression of amorphous domains...

The slope of such a plot yields the activation energy and the intercept the prefactor. This procedure is represented by the follo ving elegant mathematical expression, which we will often use in the course of this book ... 

Plots of the left-hand side of this equation versus corresponding ratios of [Fe(II)]/ [V(in)] are linear. Such plots appear to pass through the origin so that k2 must be large. The gradients of the plots yield a value for k since k 2l i... 

Thus, in the case of two-step reactions, different methods of determining the exchange CD generally yield different results (in contrast to the case of simple reactions discussed earlier) Extrapolation of the limiting anodic and cathodic sections of the semilogarithmic plots yields values and if, respectively, while the slope of the linear section in an ordinary plot of the polarization curve yields the value of ig. It is typical for multistep reactions that the exchange CD determined by these methods differ. 

The quantum yield for the formation of the cycloaddition product has been found to be temperature dependent, increasing by a factor of approximately three as the temperature is lowered from 65 ( = 0.24) to 5°C ( = 0.69). Photolysis of mixtures of the olefin and f/my-stilbene in the presence of sensitizers yielded no cycloaddition product (42) but rather only m-stilbene. This suggests that the cycloadduct is produced via a singlet reaction. This conclusion is supported by the fact that tetramethylethylene quenches fluorescence from the /rans-stilbene singlet. A plot of l/ (42) vs. 1/[TME] (TME = tetramethylethylene) is linear. The slope of this plot yields rate constants for cycloadduct formation which show a negative temperature dependence. To account for this fact, a reversibly formed exciplex leading to (42) was proposed in the following mechanism<82) ... 

Hydraulic fracturing fluids are solutions of high-molecular-weight polymers whose rheological behavior is non-Newtonian. To describe the flow behavior of these fluids, it is customary to characterize the fluid by the Power Law parameters of Consistency Index (K) and Behavior Index (n). These parameters are obtained experimentally by subjecting the fluid to a series of different shear rates (y) and measuring the resultant shear stresses (t). The slope and Intercept of a log shear rate vs log shear stress plot yield the Behavior Index (n) and Consistency Index (Kv), respectively. Consistency Indices are corrected for the coaxial cylinder viscometers by ... 

The plotted yields are for liquefaction alone. The combined yields including CSD are as follows ... 

The methyl-rhodium complex (2) was detected, at low concentration in neat CH3I solution, by a combination of FTIR and NMR spectroscopy.6 The rate of the reaction (2) —> (3) was measured between 5°C and 35°C. An Arrhenius plot yielded activation parameters of A7/= 63 k.lmol 1 and AS= — 59 JmoC1 K 1 for the methyl migration step. 

In this way it was shown that the opt values derived from data for MCl, MFi, and MFg species gave excellent correlations with the occupation number, q, and that the n -<-75 (ys) peak positions could be well reproduced using Eqs. 5 (7) and 5 (2), with spin-orbit corrections. In all cases the correlations were significantly better when the relativistic terms were included than when they were omitted, and in Table 30 we list the xopt and oPt values derived from the 5 d data for MFg, MFg, and MF6 complexes. In the Table we also show the observed and the calculated band positions using the corrected forms of Eqs. 5 (7) and 5 (2). Once again the xopt vs. q plots yield slopes in excellent agreement with the ( —ri) values deduced from these equations. Finally, in Figs. 16... 

Despite increasing the usable range of the indicators, the MAXCOV plots yielded cusps rather than complete peaks. The estimated base rates were. 085 for unsmoothed and. 083 for smoothed plots. The range of estimates was small (.02). However, this is likely to be an underestimate because it is based on four estimates derived from four aggregated plots averaged over three subanalyses with the same input indicator. This atypical presentation makes the findings difficult to interpret. 

Substitute for kx and k2 and plot yield against T. The plot shows a maximum, Cb/Ca0 = 0.53, at T = 423. q (3) could have been differentiated analytically and the maximum found from the condition that the derivative is zero.-... 

Figure 45 should be compared with the distorted plots shown previously137, 1411 (Fig. 43) for a strongly cationic polyelectrolyte. Normal plots yielding the correct molecular weight are obtained when salt is included KC1 for aqueous solutions and LiCl for methanolic ones in view of the insufficiently great solubility of KC1 in... 

The more common procedure is to perform a linear extrapolation of resistance vs. f1 or/-1/2, the value at infinite frequency being taken as the true resistance. For Nal solutions in acetonitrile it has been found that f l plots yield straight lines at low concentrations, but/-1/2 plots must be used to achieve linearity at higher concentrations 28>. Robinson and Stokes 2> discuss the causes of these variations. 

Fig. 2.19. Ionization efficiency curve of argon plotted on a linear scale (a) and as semilog plot (b). Extrapolation of the linear portion of a gives erroneous IBs, whereas the x-position of the tangent of an empirical critical slope to the sertrilog plot yields accuracies of 0.05 eV. Reproduced from Ref. [25] by permission. American Chemical Society, 1948.

Concerted acid-base catalysed enolizations of a range of simple aldehydes and ketones have been measured in water at 25 °C, using a range of substituted acetic acid-acetate buffers.The buffer plots yield rate constants for acid (A a) and base ( b) catalytic terms in the normal way at low buffer concentrations. Extension up to higher concentrations (as far as [total buffer] = 2 m, typically) yields the third-order term ( ab) via upward curvature of the plots. While ab does not have a simple correlation with either k or b, it does correlate with their product, i.e. 

The discrete factor number of recrystallizations is naturally ordered (see Figure 2.9). It obviously makes sense to plot yield against the number of recrystallizations in the order 1, 2, 3, 4, 5, 6, 7, 8, 9 it would not make sense to plot the yield against a different ordering of the number of recrystallizations, say 3, 7, 5, 8, 2, 1, 4, 9, 6. Other discrete factors are not always as meaningfully ordered, especially if they are... 

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