The initial slope

One can calculate the initial slope in a K-diagram by use of the system of differential equations according to 

In some cases, a division of zero by zero can happen. Under these circumstances the formalism of BemouIIi-L Hospital has to be applied, which is discussed in detail in the next example. The following reaction mechanism is assumed  

The rate constants of the back reaction are symbolised by kj j = 1 For j = 1 (that means A, is the starting material), one obtains 

Whereas the condition i = 2 gives the same result as above. 

According to Bernoulli and L Hospital both the numerator and denominator are differentiated with respect to t. As soon as the time t approaches zero the numerator becomes zero. Then the denominator is given by 

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Hence, if mass transfer resistance is negligible, Yi > g where is the initial slope of the adsorption isotherm for the more retained species. In our case... 

Note that the initial slope of the adsorption isotherm can be easily obtained from the knowledge of the retention time associated to a small injection performed on a column, as this retention time is given by t = t icj where... 

Using this model, the initial slopes of the adsorption isotherms are given by and... 

Note that knowledge of the initial slopes of the adsorption isotherms gives some constraint to be fullfilled between parameters X, N, and K. In order to fit the adsorption isotherms, frontal analysis has performed with the pure components at 1, 25, 50, 75 and 100 g on the analytical column at 1 ml min k... 

Figure 13 indicates burst kinetics. As discussed before, such biphasic curves indicate the reaction to occur through two steps involving an acylated intermediate. The initial slopes for the presteady state can be taken as the measure of acylation rates, and the slopes of the later straight line for steady-state can be taken as the measure of deacylation rates. 

Table 2 lists the apparent Stern-Volmer quenching constants (Ksv) for APh-x, estimated from the initial slopes of the Stern-Volmer plots, along with the... 

It was found that Zs levels off at dose > 20 krad. The cross-linking process in dioxane occurs at times longer than several milliseconds. Similar results were obtained for PHS in dioxane, however the radiolytic yield of scission, G(S), measured from the initial slope of the plot of Zs vs. D, is different for the two polymers G(S) is about 12 for PBS and about 4 for PHS, in contrast to irradiation of the pure polymers, where for both at 30° G(S) 1080. 

Figure 46. Separation of the overall oxidation curve into its two components a relaxation curve, responsible for the initial slope and the position of the chronoamperometric maximum, and a diffusion curve that controls the overall shape of the chronoampero-gram. (Reprinted from T. F. Otero, H.-J. Grande, and J. Rodriguez, J. Phys. Chem. 101, 3688, 1997, Figs. 1, 3, 6, 7, 13. Copyright 1997. Reprinted with permission from the American Chemical Society.)...

Figure 4.47. Effect of catalyst potential on the desorption activation energy of 02 from Pt/YSZ calculated from the modified Redhead analysis ( ) and from the initial slope of the TPD spectra (O).30 Reprinted with permission from Academic Press.

The initial rate of polymerization was determined from the initial slopes of time-conversion curves (Fig. 1) using f-BuX/Me3 Al/MeCl systems at -40 °C. This... 

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... 

The initial rate of the reaction can be determined from the early linear portion of the plot of [C] vs. n. (Note that there is often abrief adjustment period before the system settles into normal kinetic behavior. This brief adjustment period should not be included in the initial linear period.) This is illustrated in Figure 8.3, where the initial slope is found to be 4.83 ingredients/iteration (ingr/itn). Based on Eq. (8.2), the value of the rate constant k for this system is 4.83/(500)2 2.0 X 10 ... 

To lessen experimental time, the null-point method may be employed by locating the pulse spacing, tnun, for which no magnetization is observed after the 180°-1-90° pulse-sequence. The relaxation rate is then obtained directly by using the relationship / , = 0.69/t n. In this way, a considerable diminution of measuring time is achieved, which is especially desirable in measurements of very low relaxation-rates, or for samples that are not very stable. In addition, estimates of relaxation rates for overlapping resonances can often be achieved. However, as the recovery curves for coupled spin-systems are, more often than not, nonexponential, observation of the null point may violate the initial-slope approximation. Hence, this method is best reserved for preliminary experiments that serve to establish the time scale for spin-lattice relaxation, and for qualitative conclusions. 

This quadratic in Rp is of the form required by the data for styrene-benzoyl peroxide shown in Fig. 14. The first term, corresponding to the intercept, represents the creation of chain ends through transfer with monomer. It occurs to an extent which is independent of the polymerization rate. The second term corresponds to 1/2 according to Eq. (27) it represents the pairs of ends created at the initiation step. Its coefficient is given by the initial slope of the line in Fig. 14. The third term, which accounts for the curvature at higher rates, represents the contribution of chain transfer with benzoyl peroxide. This becomes more prominent at higher rates because of the larger amounts of the initiator which are present. The marked rise in the curves for... 

The initial slope of the /c V8, c plot may be calculated from the parameters F2 and (V )o evaluated in the manner referred to above. This slope is given by RTA which is equal to r2( /c)o. The parameter A 2 is given according to theory by Eq. (75) or (75 ). The quantity (yVvi) i(l —0/T) =J/N occurring in these equations should be independent of the molecular weight, but the factor decreases slowly... 

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