Linearity of the Plot

the Langmuir isotherm is obeyed when p against p/V gives a hnear plot, Freundlich isotherm when a log p against log V is linear, whereas Temkin isotherm gives a linear plot with log p against V plot. 

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For a substantial number of systems the DR plot is indeed a good straight line. In Fig. 4.17, for example, the linearity of the plots extends over a very... 

Fig. 17. An homologous series of alkane probes is generally used because good values for their adsorbed molar areas are available [87]. The linearity of the plot (the alkane line ) lends credence to the procedure.

A clean first-order process may erroneously appear to be a biphasic one, and vice versa. If the distortion to the property-time curve is not so evident as in the example, there might be a smooth rise or fall from reactant to product. The linearity of the plot of In (Y, - Kcc) versus time depends on the end point reading Yr.. One must be cautious, however, in ascribing a mildly curved plot of In Y, - W) versus time to a biphasic pattern. Were the observed value of Yx off by a small amount, a simple adjustment could give a straight-line plot indicative of first-order kinetics. Of course, if Tec is adjusted to force linearity, one must surely ask whether the revised value of Yx represents a reasonable extrapolation of the data, lest the proper but more complex reaction pattern be concealed. 

This olefin was chosen because its slow bromination rate allows to make accurate measurements of transient CTC s with a conventional spectrophotometer. Curve d is a difference spectrum between a solution of Bi2 plus a hundred fold excess of olefin (curve c) and those of the single reagents (curves a and b), and represents the tail of the expected CT band. From the linearity of the plots of the difference absorbance agains the olefin concentration (Fig. 2) it was possible to evaluate a limiting value of Kf < 0.1 M l. On the other hand, a AH = - 0.9 kcal moTl was obtained from a plot (Fig. 3) of the in of the products KfScx. obtained from the first plot, against 1/T, assuming ecj to be temperature independent (ref. 3). 

Mathematical methods for determining the gas holdup tine are based on the linearity of the plot of adjusted retention time against carbon number for a homologous series of compounds. Large errors in this case can arise from the anomalous behavior of early members of the homologous series (deviation from linearity in the above relationship). The accuracy with which the gas holdup time is determined by using only well retained members of a homologous series can be compromised by instability in the column temperature and carrier gas flow rate [353,357]. The most accurate estimates... 

The linearity of the plot confirms the assumed rate equation. The correlation is... 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

Convergence or divergence in the linearity of the plots of retention time versus the logarithm of the number of residues, i.e. /c iex i versus In d, can be employed to characterize the regular ion-exchange behavior of polypeptides in terms of the role of polypeptide chain length... 

The linearity of the plot of 77sp/c versus c and the dependence of the slope of the linear plot on [77], which is the intercept, has been described by a purely empirical equation by Huggins. 

The linearity of the plot f (C), for concentrations between 5.10 -6.10 m (Figure 6 ), demonstrates the kinetic order 1 with respect to the active centers. The rate constant of... 

The solution is EtOH/O.lM [n-Bu,fN C10 k ferrocene and illumination is at 632.8 nm, 50 W/cm2. The strict linearity of the plots shows that oxidation of solution ferrocene is mass transport-limited for all w used and for each ferrocene concentration used. 

The increase in double bond character is assumed to increase the intrinsic barrier for reaction at the a-carbon atom. As this increase is greatest for the thermodynamically least stable (CF3-substituted) carbocation, changes in thermodynamic driving force and intrinsic barrier oppose each other. The constancy of the values of kn2o thus reflects a change in intrinsic barrier overriding the second and third terms in the Marcus expression of Equation (20). This is a more radical effect than the lesser variation preserving the linearity of the plots for the reaction families in Fig. 3 (p. 77), for which only the third term is overridden. 

Hence linearity of the plot of log10 kobs versus y/l/ + fl) does not, in itself, necessarily give the correct interpretation of what is happening in this reaction. 

The linearity of the plot of number-average molecular weight versus percentage conversion indicates that the amount of transfer reactions was low throughout the reaction. The increase in molecular weight was proportional to the degree of monomer conversion. The same characteristics have been observed for ROP of L-LA initiated by other cyclic tin alkoxides [83]. 

The linearity of the plot shows that the propagation was first order with respect to lactide monomer. The absence of an induction period indicates that the initiator was reactive from the beginning and that no rearrangement of initiator aggregates was necessary to form the active species. The nature of the metal, alkoxide groups, solvent, and temperature does not generally influence the first order in monomer [71,104]. 

The number of branches per starting PVAc molecule versus the PS/PVAc ratio in the reaction mixture is plotted in Fig. 22. The good linearity of the plot leads to the conclusion that the PS prepolymers which carry non-deactivated acyl chloride groups all undergo the coupling reaction with PVAC/NH2, irrespective of the molar mass of the PS molecule, at least in the molar mass range studied. 

Fig. 1.2. Effect of silylation on the linearity of the plot of peak height ratio versus amount of solute. (Reproduced from ref. 2 by courtesy of K. Hammarstrand and the publisher.)...

The electrostatic model for the micellar effect on the hydrolysis of phosphate monoesters is also consistent with the results of inhibition studies (Bunton et al., 1968, 1970). The CTAB catalyzed hydrolysis of the dinitrophenyl phosphate dianions was found to be inhibited by low concentrations of a number of salts (Fig. 9). Simple electrolytes such as sodium chloride, sodium phosphate, and disodium tetraborate had little effect on the micellar catalysis, but salts with bulky organic anions such as sodium p-toluenesulfonate and sodium salts of aryl carboxylic and phosphoric acids dramatically inhibited the micelle catalysis by CTAB. From equation 14 and Fig. 10, the inhibitor constants, K, were calculated (Bunton et al., 1968) and are given in Table 9. The linearity of the plots in Fig. 10 justifies the assumption that the inhibition is competitive and that incorporation of an inhibitor molecule in a micelle prevents incorporation of the substrate (see Section III). Comparison of the value of for phenyl phosphate and the values of K for 2,4-and 2,6-dinitrophenyl phosphates suggests that nitro groups assist the... 

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