Isothermal curve, generation

Fig. 6.1 A breakthrough curve generated by the frontal analysis method [31], The analysis represents a high-volume injection of caffeine through a reversed-phase column, at a concentration representative of the linear region of the binding isotherm. Adapted with permission from Elsevier.

Figure 5 demonstrates the kinetic evaluation of these results. On the left-hand side, the initial velocity Vp (intersection points with the ordinate) is plotted versus the concentration of the Al component while the Ti concentration was kept constant. The result is the Al isothermal curve of the second equilibrium reaction generating the active species. It demonstrates in the initial part the demanded sigmoid curve course according to the location of the two successive equilibria. This induction period is enlarged in Fig. 6. It can be clearly seen how sensitively the initial polymerization rate responds to the ratio Al/Ti. With a ratio >1 (i.e., here [AlEtCl2] > 3x10 mol L ), the formation of the active species increases drastically.

The relationship between the concentration of A in the stationary phase (Cs) and the concentration of A in the mobile phase (Cm) with regard to a particular component is measured at constant temperature. For this reason, the curves generated by plotting Cs against Cm are known as partition isotherms (see Fig. 45). 

Simulations typically require millions of displacement, insertion and removal moves in order to equilibrate the system. The result is an adsorption equilibrium between the sorbate molecules in the gas phase and those adsorbed on the zeolite at the specific gas-phase fugacity. This would represent a single point on an adsorption isotherm. The remainder of the isotherm curve can be generated to determine the amount of gas adsorbed at various other pressures . 

For exothermic reactions, > 0, and the P curves generated at each Ys value show that may indeed exceed unity, depending on whether or not the rate increase caused by temperature rise offsets the decrease resulting from the fall in concentration. When // > 1, -Ra)p > -Ra)s> hut this situation may not always be beneficial because the increase in temperature toward the center of the particle may induce deactivation or may promote undesirable reactions with an overall decrease in the selectivity for the desirable product. At high values of the Thiele modulus, t] becomes inversely proportional to 4>s. as in the isothermal case therefore, the reactant is rapidly consumed as it penetrates the particle so that the reaction takes place in a thin shell just underneath the particle surface while the interior is more or less isothermal at a higher temperature. On the other hand, at low values of Os and for highly exothermic reactions, for example, > 0.3, the value of r] is not uniquely defined by the three dimensionless parameters of the analysis. Three possible values of // exist, each representing a different set of conditions at which the rate of heat release balances the rate of heat removal the... 

Very interesting amphiphilic behavior at the air-water interface was observed by van Hest [126] for hydrophobe (focal point) modified poly(propyleneimine) (PPI) dendrons (i.e., poly(styrene)-de (in-PPl (NH2) where n = 2,4,8)(6). These dendritic amphiphiles were essentially a reverse version of the Frdchet examples (see above). Only PS-den(in-PPl-(NH2) with n = 8 and 16 (i.e., G = 3 and 4) exhibited normal pressure-area isotherms. The lower generations (i.e., G = 1 and 2) all displayed isotherm curves indicating they transitioned directly to solid state behavior. 

Two-step isotherms were generated in two cases at pH 8 by isoproturon an L-type curve and at pH 5 by prometryn an S-type curve were formed. Giles suggested for L-type curves that molecules are oriented... 

Figure 15.6 Schematic plot of logarithm of relaxation modulus versus logarithm of time for a viscoelastic polymer isothermal curves are generated at temperatures Tj through T-j. The temperature dependence of the relaxation modulus is represented as log EXh) versus temperature.

Furthermore, the magnitude of the relaxation modulus is a function of temperature to more fuUy characterize the viscoelastic behavior of a polymer, isothermal stress relaxation measurements must be conducted over a range of temperatures. Figure 15.6 is a schematic log ,(f)-versus-log time plot for a polymer that exhibits viscoelastic behavior. Curves generated at a variety of temperatures are included. Key features of this plot are that (1) the magnitude of EXt) decreases with time (corresponding to the decay of stress. Equation 15.1), and (2) the curves are displaced to lower EXt) levels with increasing temperature. 

This is shown graphically by the (Vm, p, T) surface given in Figure 1.3. The three variables p, Vm, and T exist together only on this surface. Cutting through the surface with isothermal (constant T) planes generates hyperbolic curves... 

Since the pressure build up is primarily due to the evolution of CO as MDI is being decomposed to carbodiimide, the thermodynamic relationship PV = nRT may be applied to convert the pressure profiles to plots of moles of CO2 generated vs. time. This is shown for the 225 °C isotherm in Figure 3. The theoretical curve obtained through the application of zero-order kinetics is also shown in this plot and the data seem to be well accommodated by this rate law throughout the majority of the run. 

Differential Scanning Calorimetry. A sample and an inert reference sample are heated separately so that they are thermally balanced, and the difference in energy input to the samples to keep them at the same temperature is recorded. Similarly to DTA analysis, DSC experiments can also be carried out isothermally. Data on heat generation rates within a short period of time are obtained. Experimental curves from DSC runs are similar in shape to DTA curves. The results are more accurate than those from DTA as far as the TMRbaiherm is concerned. 

In order to find points of equal degrees of conversion (or equal Q-values) in Figure 2.18, van Geel [115] developed the method to evaluate kinetic data from the so-called isoconversion lines. A heat generating substance that follows Equation (2-11), when stored under isothermal conditions at different temperatures has generated an equal amount of heat (Q) when the product of t exp(-Ea/RT) has the same value. Thus, for two heat generation/time curves measured at Ti and T2, the same amount of heat (Q) has been generated, and thus the conversion is equal when ... 

If the calorimeter could respond instantaneously to the heat effects associated with the addition of titrant, then the measured curve would coincide with the dashed lines in figure 11.5. The deviation of the data from this ideal behavior corresponds to periods in which the isothermal condition is not observed. When necessary, however, it is possible to use deconvolution techniques to generate the input function represented by the dashed line from the observed experimental curve. 

Figure 4.7 shows the correlation between the viscosity and the ionic mobility based on isothermal runs for this system as monitored by the value of e" (5 kHz). A representative calibration curve relating the FDEMS sensor output to degree of cure is shown in Figure 4.8. Unlike viscosity, separate calibration curves for different temperatures must be generated from the isothermal runs because they are temperature dependent.

Initially, all the hydrocarbon is adsorbed on the core and none is observed at the outlet. Once the core is saturated, hydrocarbon breakthrough is observed. Example breakthrough curves for two temperatures are shown in Fig. 27. The amount of hydrocarbon adsorbed is given by the area above the breakthrough curve (after correction for the residence time of the reactor). By conducting experiments with different hydrocarbon concentrations and at different temperatures, the temperature and concentration dependency of the amount stored can be determined and hence isotherms generated. 

The results of these static measurements can then be used to rate the probable usefulness of different adsorbents. However, the isotherm results from static water solutions do not apply to dynamic column situations in which equilibrium conditions may not occur. A better approach is to generate frontal breakthrough curves that can then be used to estimate the use of different polymers for different solutes dissolved in water. Theoretical and experimental reports (97, 143, 181, 286, 319-321, 537) discuss details about affinity measurements. These details are not included in this review because affinity is discussed only qualitatively in the sections on Theoretical Considerations and Generalized Methodology. These qualitative discussions suggest that neutral polymers such as the styrene-divinylbenzenes are efficient for adsorbing neutral hydrophobic solutes from water solutions but have little affinity for polar and ionic solutes. If the polarity of the polymer is increased to that of the acrylates, the affinity for neutral hydrophobic components will suffer but the more polar solutes will be better adsorbed. In the absence of actual test results under dynamic column flow conditions, the simple likes adsorb likes concept is invoked. 

As already discussed, DFT can be used to predict the capillary condensation and capillary evaporation pressures for pores with homogeneous surface and well-defined geometry. To generate model adsorption isotherms for heterogeneous pores, it is convenient to employ hybrid models based on both DFT data for homogeneous pores and experimental data for flat heterogeneous surfaces [6-9]. Such model adsorption isotherms can be used to calculate PSDs in mesopore [6-9] and micropore [9] ranges. This approach is particularly useful for pores of diameter below 2-3 nm (micropores and narrow mesopores), where an assumption about the common t-curve for pores of different sizes is less accurate, which in turn makes the methods based on such an assumption (even properly calibrated ones) less reliable [18],... 

Figure 8.4 Stress-strain isotherms for PDMS networks reinforced with in situ generated titania particles.39 Each curve is labeled with the wt % of filler introduced, and filled circles locate results used to test for reversibility.

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