Plotting calculations

This equation describes the additional amount of gas adsorbed into the pores due to capillary action. In this case, V is the molar volume of the gas, y its surface tension, R the gas constant, T absolute temperature and r the Kelvin radius. The distribution in the sizes of micropores may be detenninated using the Horvath-Kawazoe method [19]. If the sample has both micropores and mesopores, then the J-plot calculation may be used [20]. The J-plot is obtained by plotting the volume adsorbed against the statistical thickness of adsorbate. This thickness is derived from the surface area of a non-porous sample, and the volume of the liquified gas. 

Make the Br nsted plot, calculate p, and discuss deviations from the line. 

Hammett equation. Consider the reaction39 of a series of benzenediazonium ions with OH" and CIST. Construct Hammett plots, calculate the p s, and discuss their signs and magnitudes. 

Figure 8. One dimensional plot ofx2 as a function of 8Um- The three curves in the figure are exact plots calculated using full dynamical theory (circle) full tensor expansion around 8 Um = -0.0025 A1 (solid line) and line tensor expansion around 8f7m = -0.0025 A1 (cross and solid line).

If we want to increase the margin, we either have to reduce the value ofKc or increase One possibility is to keep = 1.58 min and repeat the Bode plot calculation to find a new Kc which may provide a gain margin of, say, 2 (6 dB), as in the case of using only the proportional controller. To do so, we first need to find the new ultimate gain using the PI controller ... 

From experimental measurements it was found that there was a fairly good linear relation between the SC capacitance and current, the slope being dependent on the electrode material and electrolyte. If the dependence is known, it can be used to make the Ragone plot calculations more accurate. Therefore, the following method was employed in our evaluations. 

Evolving factor analysis can, and should be performed in both forward and backward directions. The forward plot, calculated above and shown in Figure 5-34, indicates the appearance of new components. The backward plots of Figure 5-36 are calculated similarly by determination of the singular values of the set of the last 1, 2, 3,. .. spectra in Y, as seen in the schematic of Figure 5-35. These plots indicate the disappearance of the components. 

Plots of the derivative of the cumulative curves, dV/dP versus pressure or radius, are often useful for the determination of the radius or pressure at which the maximum volume intrudes or extrudes. Figures 11.14a and 11.14b are derivative plots calculated from the cumulative curves in Figures 11.9a and 11.9b. 

Plot calculated kinetic data to obtain information about the order of the kinetic reaction. 

From the slope and the intercept of the r ed versus c plot, calculate the Huggins constant, k. ... 

Fig. 12.5. A crack from an indent in glass (a) z = 0 (b) z = — 3.8 pm (c) z = —5.2 pm ELSAM, 1.5 GHz. The experimental line-scans superimposed on the images can be compared with the plots calculated using two-dimensional theory (eqns (12.2), (12.13), and (12.14)) with elastic constants from Table 6.3 and values of defocus (a) z = 0 (b) z = —4.2 pm (c) z = —6.8 pm. The values of z in the calculations were chosen for best fit the reason for the discrepancy is not known, though no doubt there are the usual uncertainties associated with thermal drift, the measurement of z, and the frequency and pupil function used (Briggs etal. 1990).

Use QRRK theory to calculate kan as a function of pressure for the decomposition of azomethane at T = 563 and 603 K. Parameters needed for the calculation are given in Section 10.4.4, in the discussion of Fig. 10.7. Plot calculated rate constant in units of 1/s versus pressure (in atm) include in the same plot a comparison with experimental data of Ramsperger [326], which can be found in the data file azomethanedata.csv. 

Fig. 8.17. Theoretical IMPS plots calculated for the scheme shown in Fig. 8.16. The upper plot is for the case where ka> kh> kc. The values of these rate constants are given by the corresponding kc. In this case the two higher frequency semicircle condense into one determined...

Fig. 16a d. Fractions of isolated Mj(a), M2(c) units in copolymer of styrene with acrylonitrile M2, and fractions of isolated pairs of these units MjM, (b) and M2M2 (d) Experimental points are compared to the plots calculated via various models terminal (- -), penultimate (----------), CR-1 (---------), CR-2 (--------) [282]... 

Fig. 22a-h. Glass transition temperature versus composition of copolymers methyl methacrylate + styrene (a) styrene + methyl acrylate (b) acrylonitrile + styrene (c) vinyl chloride + methyl acrylate (d) methyl methacrylate + vinyl chloride (e) acrylonitrile + butadiene (f) acrylonitrile + vinyl acetate (g) a-methyl styrene + acrylonitrile (h). Experimental points obtained at low conversions from various publications, are compared to the theoretical plots calculated according to Eqs. (7.1) within the framework of the terminal model [18]... 

Figure 11. Plots of log Z and / v.v. log / for a thiol-hexapeptide-coated mercury drop immersed in 5xlO 3M (a), I.3xlO 2M (b), 3.6xlO 2M (c), and 0.1M (d) KC1, as obtained at -1.000 V over the frequency range from 0.1 to 105 Hz. At frequencies <102 Hz all Bode plots coincide hence, only the experimental points for the lower KC1 concentration were reported. The solid curves are least-squares fits to the simple equivalent circuit of inset (1), which consists of the electrolyte resistance Ra, with in series a RSCS mesh representing the self-assembled monolayer and a further RjiCji mesh representing the diffuse layer. Rs = 0.14 Mfi cm2 C, = 11 pF cm-2 Ra = 4.53 (a), 4.17 (b), 1.27 (c) and 0.87 KO cm2 (d). CW 68 (a), 61 (b), 80 (c) and 84 pF cm 2 (d). Inset (2) shows the reciprocal, 1/Cji, of the experimental diffuse-layer capacitance vs. the l/C fajj = 0) value corresponding to the same KC1 concentration, as calculated on the basis of the Gouy-Chapman (GC) theory. The solid curves are 1 /Ca(OM) vs 1 /C,ii(ctm = 0) plots calculated from the GC theory for different charge densities afo on the metal, whose values are reported on each curve. (Reprinted from Ref.114 with permission from the Am. Chem. Soc.)...

The t-plots calculated from N2 adsorption isotherms for mesoporous materials are shown in Figure 3. The t-plots for calcined 3D SBA-15 prepare at 403 K (without TMB and with TMB) give two straight lines at t = 0.4 0.75 nm and the extrapolation lines go through the origin, indicating that the 3D SBA-15 products have no micropores. For conventional SBA-15 the extrapolation line cuts the y-axis at 0.07 cm /g, suggesting the presence of the micropores. 

In Fig. 15L(a-d) we show four complex-plane-impedance plots calculated for the concentrations of 100, 10, 1 and 0.1 mM, respective-... 

Fig. 3E Semi-logarithmic i/x plots calculated from Eq. 35E for three values of p. Note that the lines are symmetrical around T] = 0 only for p = 0.5, but the lines extrapolated from high over-...

Fig. 51 Tafel plots calculated from the complete rate equation for different values of f. The curves for f = 10 and f = 0 are displaced vertically by 200 mV each, to avoid overlap. There is a transition region, extending over two decades of current density or more, where the slope changes from 2.3RTI2F) for low coverage to 2.3RTIF) for intermeidate values of the coverage.

Figure 51 Relationship between log K( and log Cs for heparin. Solid line refers to the plots calculated assuming that (Ps)d = (Pc)d ...

We consider first the -pH plot corresponding to Eq. (6-64). Figure 6-10 is such a plot calculated with Eq. (6-64) and the typical values TpKa = 4.0, k = 1.10 s , and = 1 x s . Note that this plot is similar to the Fs-pH plot in Fig. 6-9. Differentiating Eq. (6-64) to find dkld li ] and using the identity equation (6-49) gives the first derivative of the /c-pH plot. The second derivative is found with the aid of Eq. (6-51). Imposing the condition k + k" and setting the second derivative equal to zero leads to a simple expression for the pH at the inflection point. 

Make the Br nsted plot, calculate 3, and discuss deviations from the line. These data are for the nucleophilic catalysis of the hydrolysis of p-nitrophenyl acetate by imidazoles and benzimidazoles at pH 8.0. The apparent second-order catalytic rate constants are defined by... 

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