Isotherm slope

Fig. 10. UOP Sorbex operation with linear isotherms. Slope of slope of = K2. Conditions for separation > 1 2, L jS > ifj,...

Figure 5. Isotherms for Cd(II) adsorption onto orAl Oj. Isotherm slopes are approximately 0.7.

The latter relationship is readily derived from a mass balance on the contaminant entering the column. This wave speed is referred to as the stoichiometric wave speed. It applies strictly only to one component. It will be recognized that manipulation of Eq. (9.11) with similar substitutions yields the wave speed result for bulk separations where the derivative of v with respect to the spatial variable does not drop out and where two or more components may adsorb. The result has a very similar form but of course depends on not one isotherm slope but on two or more and very significantly the wave speed depends on the relative mole fractions of the adsorbing species and the initial loadings in the column. 

EXAMPLE 9.3 Analysis of Adsorption Data Using the Langmuir Isotherm. Slope and intercept values for the linearized plots of the data in Figure 9.7 are as follows ... 

The limitation of Equation (3) is that the isotherm slope dq/dc is a function of solution concentration therefore it may not be a constant over a range of concentration normally encountered in kinetic tests. It is also noted that Deand Djarea function of solid concentration q, while Dp is independent of adsorbate concentration. 

The adsorption analysis can be performed using the experimental sensor effect data from gravimetric QCM measurements. The frequency shift, which is proportional to the partial pressure of the analyte, is correlated to the number of specific incorporation sites in the linear range of the isotherm slope. Additional evidence for specific interactions between the analyte and the polymer matrix can be demonstrated using infrared spectroscopic analysis [19]. NMR [4,7] can also provide information about imprinting effects. 

In summary with this system it has been possible to observe the transition from the osmotic to a salted and then to a collapsed brush. One might suspect that these transitions are also reflected in the pressure/area isotherms. However, we have not yet been able to relate the breaks in the isotherm slopes with parameters deduced from X-ray reflectivity. This would require an even higher precision of data analysis, and precision in density determination better than 1% is difficult to achieve. We realize that the transition at nc is accompanied by a change in relaxation times as expected for a... 

Equation 6.46 predicts the propagation velocity w of an arbitrary concentration c inside the column depending on the isotherm slope. 

According to equilibrium theory, elution of the small disturbance depends on the isotherm slope at the plateau concentration. Because the perturbation peak is almost Gaussian, the time at the peak maximum respectively minimum can be taken to calculate simply the isotherm slope by Eq. 6.49 ... 

As the overall aim of parameter determination is the (simulation-based) prediction of the process behavior the final decision about the suitability of the isotherm equation can only be made by comparing experimental and theoretical elution profiles (Section 6.6). Depending on the desired accuracy, this may involve iteration loops for the selection of isotherm equations or in some cases even the methods (Fig. 6.16). In this context it should be remembered that, according to the model equations (e.g. Eq. 6.47), the migration velocity and thus the position of the profiles is a function of the isotherm slope. Therefore, it is most important for the reliability of process simulation that the slopes of the measured and calculated elution profile fit to each other. If the deviations are unacceptable, another isotherm equation should be tested. 

In section II the minimum value for the corresponding flow rate ratio m is fixed to the maximal possible initial isotherm slope of the less retained component, which... 

Henry s law constant (Ki), which is a function of temperature only. The amount adsorbed monotonically increases with increasing pressure beyond the Henry s law region with a progressively decreasing isotherm slope and finally the amount adsorbed asymptotically approaches the saturation adsorption capacity (m ) of the adsorbate. Figs. 3A and 3B show examples of Type I isotherms for adsorption of pure N2 and pure O2, respectively, on various zeolites at The... 

The values of K and s can be measured independently in separate batchwise experiments. In fact, the isotherm slopes for oxygen and nitrogen with zeolite 5A at 45°C, as shown in Figure 14.24, are 4.51 and 8.24, respectively. The void fraction was determined to be 0.478, and the bulk density was... 

Fig. 10. UOP Sorbex operation with linear isotherms. Slope of = Klm, slope of (D = K2. Conditions for separation Kl > K2i Lzj S > Kly k2 < l2/s < k19 k2 Ll — L2 = F where K = adsorption coefficient, L = net liquid flow rate, S = net solids flow rate, and...

Although similar methods of activation give similar isothermal slopes, the converse does not necessarily follow, and one should not conclude that two carbons were produced by the same method because the isothermal slopes are found to be parallel under a single set of conditions. Thus, the isotherms for molasses color with carbons L and M were found to be almost identical at 25° C, but were dissimilar when the adsorption was conducted at 95° C (Table 9 5). Here, although two carbons have a similar slope at 25° C, the same type of surface is not present on both carbons, as is revealed by the different response to temperature. From a practical point of view, the different response to temperature emphasizes the fact that industrial studies of adsorptive power should be conducted at the temperature to be employed in the process. 

Fig. 10 2. Hypothetical isotherm to illustrate elution of fungicide (from carbons A and B that have dilferent isotherm slopes)...

Fitting Equation 1 to a real fluid requires not only knowledge of the (p, P, T) behavior of the fluid, but also of the isothermal slope dP/dp and of the coexistence envelope. Therefore it was decided expedient to use data which had been fitted already, so that detailed dP/dp data was already available. Obviously, however, the fit of Equation 1 will reflect any error contained in such prefitted data. 

It is seen from these dependencies that, with the increase of m, the onset of surface tension decrease (or the surface pressure increase) corresponds to lower surfactant concentrations at the same time, the decrease of the isotherm slope at high concentrations takes place. The increase of surface activity with the increase of m for low pressure in the framework of reorientation model can be qualitatively explained by strong increase of the molar area of C EO, molecule in the state 1 (coj), while the decrease of the isotherm slope (and also the decrease of surface activity) at high concentrations can be ascribed to slight increase of the CO2 value. Therefore, the Intersection of the isotherms which is observed for the oxyethylated alcohols with different m values is the consequence of the fact that these two molar areas are increased with the increase of m, but the rate of this increase is different for CO2 and coj values. 

In an ideal analytical chromatogram, that is, in the linear range of the adsorption isotherms, the retention time is a function of the isotherm slope, quantified by the Henry constant Hi (Figure 2.6a), according to the basic equation of chromatography (Equation 2.16) (Guiochon et al. 2006) ... 

To illustrate the strength of the results that can be obtained applying the ideal model, we will consider below just the case of single-component elution of a component i. Using an established equilibrium the time derivative of the loading can be expressed by the isotherm slope and the corresponding liquid phase time derivative ... 

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