Initial slope

Figure 26-65 illustrates that Eq. (26-90) provides a linear approximation to the nonlinear relationship between two-phase specific volume and reciprocal pressure (v vs. P or vs. T ). For single components, me initial slope of the curve is found using me Clapeyron equation to give ... 

The storage capacity of an ANG storage system is always greater than its delivered capacity, usually by about fifteen percent. For some carbons, however, it can be as high as thirty percent because of the large amount of methane which is held by the adsorbent at less than one bar, (0.1 MPa). Carbons which are very microporous, such as polyvinylidene chloride (PVDC) carbons, tend to have very steep initial slopes to their methane isotherm, and as much as thirty percent of their overall uptake occurs at less than 0.1 MPa. These carbons have a high storage capacity but a much lower deliverable VfV. 

Thus, in the purely agglomerative proeess with a typieal value of the parameter a > 1, the number density distribution n L) aeeording to equation 6.11 is linear when plotted in logarithmie eo-ordinates, i.e. In n L) versus logL, thus exhibits an initial slope equal to —5/2, as shown in Figure 6.12. Note If the population density is defined on a unit solid volume, rather than size, basis, then the eorresponding slope is —3/2, see also Jones etal., 1996.)... 

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... 

Consequently, changing the temperature or the strain rate of a TP may have a considerable effect on its observed stress-strain behavior. At lower temperatures or higher strain rates, the stress-strain curve of a TP may exhibit a steeper initial slope and a higher yield stress. In the extreme, the stress-strain curve may show the minor deviation from initial linearity and the lower failure strain characteristic of a brittle material. 

At higher temperatures or lower strain rates, the stress-strain curve of the same material may exhibit a more gradual initial slope and a lower yield stress, as well as the drastic deviation from initial linearity and the higher failure stain characteristic of a ductile material. 

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. 

Fig. 3.—A. Initial Slope Approximation to Determine the Initial, Nonselective, Spin-Lattice Relaxation Rate of H-S of 2,3 S,6-Di-0-isopropylidene-a-D-mannofuranose (2) in Me2SO-d Solution. (Points between 0.01 and l.SS s were selected for tracing the best straight line.) B. The Same as in A for H-1 of a Partially Deuterated Sample of 1,6-Anhydro- -cellobiose Hexaacetate (3). [Note that the relaxation of H-1 is strongly dependent on the choice of I value. An R (ns) value of 0.24s was obtained from the data points 0 t 5s, where a value of 0.18 s was obtained from the terminal decay 5 lOs (see text).]...

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