Equilibrium isotherms rectangular

Nahon 117 from DuPont Chemical Co. was found to be the most effective membrane in terms of kinetics of ion transfer. The equilibrium isotherm showed that the membrane had a high selectivity for trivalent aluminum ions. This resulted in a rectangular isotherm for the Nahon 117 membrane. 

The theory of simple waves applies to large-volume injections, i.e., to the profiles obtained upon injection of rectangular profiles which are so wide that the injection plateau has not been entirely eroded when the band elutes. Then, simplifications of the solution occur because there is a constant state, the concentration plateau. This solution is not valid in overloaded elution chromatography when the injection volume is sufficiently small that the injection plateau has eroded and disappeared by the time the band elutes from the column. It is important to discuss this solution, however, because it takes a finite time for the profile of even a narrow rectangular injection to decay, and the band profile during that period is given by the simple wave solution. Also, this solution is the basis for a method of determination of competitive equilibrium isotherms (Chapter 4, Section 4.2.4). 

K denotes the equilibrium constant, C is in moles dm 3 of solution, and the coefficients A depend upon T but not on C. This isotherm equation approaches Henry s law as C becomes small enough and has already been shown adequate, for a suitable choice and number of coefficients A, to represent experimental isotherms well (4, 6, 20). One to three such coefficients were needed according to the rectangularity of the isotherms. It lends itself to thermodynamic analysis of gas-solid distribution equi-... 

When the equilibrium constant b is large (highly favorable adsorption) the Langmuir isotherm approaches irreversible or rectangular form,... 

If Kd - Cm, this isotherm reduces to the rectangular or irreversible isotherm, Cs Cs max.134 For many natural and synthetic systems, the equilibrium dissociation constant is in the range of 10 6 and 10 n. 

Figure 7 shows breakthrough curves for Cu(I)-Y for an influent containing 500 ppmw thiophene (190 ppmw sulfu ) in n-octane. The saturation capacity was reduced to 1.28 mmol/g, which was about 50% of the amount obtained previously with the 2000 ppmw thiophene feed. This indicates that the equilibrium adsorption isotherm was not rectangular in shape at low concentrations and rather showed a noticeable decrease in adsorbed amount as one decreased the concentration. Despite this, the observed saturated amount was not low, when taking into account that the thiophene concentration was 75% less than the case discussed previously (i.e., 2000 ppmw). 

A typical set of competitive isotherms for two proteins is shown in Figure 4.20 for a given salt concentration [83]. Note that the competitive isotherms exhibit a behavior that is farther removed from a rectangular one than the single-component isotherms. This is because the separation factor, Kf,/Ka, is much smaller than each of these equilibrium constants. 

This value is in agreement with the one derived from band profiles calculated with the equilibrium-dispersive model [9]. The time given by Eq. 16.20 provides useful information regarding the specifications for the experimental conditions under which staircase binary frontal analysis must be carried out to give correct results in the determination of competitive isotherms. The concentration of the intermediate plateau is needed to calculate the integral mass balances of the two components, a critical step in the application of the method (Chapter 4). This does not apply to single-pulse frontal analysis in which series of wide rectangular pulses are injected into the column which is washed of solute between successive pulses. 

A large number of zeolite-gas systems have been investigated with respect to the sorption equilibrium. Some of the results described in the literature are of a more qualitative nature and cannot be evaluated quantitatively. Sorption isotherms are frequently described as rectangular, which means that measurements were made only at relatively high pressures where saturation is approached and that no information about the form of the isotherm was obtained. 

It was found experimentally that water adsorption facilitates the H2S removal by activated carbons, when its value exceeds 10-20 %wt. of the H2S adsorption practically does not depend on H2O content. Taking this into account, and also the absence of any VOCs at the laboratory conditions (lab), the process of HiS adsorption can be described by mass balance and kinetics equations for one component (H2S) with corresponding boundary conditions. In this type of model the rectangular isotherm Eq.(39) is commonly used to describe the equilibrium in a reaction system. 

A multicomponent model of adsorption can be reduced to three main components first, H2S, second, VOC with average properties of found species, and third, H2O. Moreover, H2O adsorption can be described as quasi-equilibrium due to the high concenbation of H2O and the long operation time of the carbon bed. Rectangular type isotherms are used to model the equilibrium of VOC and H2S. The amount of H2S adsorbed depends on available space left after VOC adsorption and it can be described bj equation ... 

The pressure response to the voliune perturbation was recorded with a high-accuracy differential Baratron pressure transducer (MKS 698A11TRC) (Eq. 10) at each frequency over three to five square-wave cycles (256 pressure readings per cycle) after the periodic steady-state had been established. The isotherm describing the equilibrium sorption conditions can be linear or curved. However, the horizontal region of a rectangular isotherm cannot be used as there is no sorption/desorption following the square-wave modulation of the equilibrium volume. 

From the solutions for the fractional uptake in Table 9.2-4, the half time can be obtained by setting the fractional uptake F to one half, and they are tabulated in the third column of Table 9.2-4. For this case of rectangular isotherm, the half time is proportional to the square of particle radius, the maximum adsorptive capacity, and inversely proportional to the bulk concentration. The time it takes to reach equilibrium is half when the bulk concentration is doubled. This is because when the bulk concentration is doubled the driving force for mass transfer is doubled while the adsorptive capacity is remained constant (that is saturation concentration) hence the time to reach saturation will be half. Recall that when the isotherm is linear, the time scale for adsorption is independent of bulk concentration. Hence, for moderately nonlinear isotherm, the time scale would take the following form ... 

When the equilibrium relationship is nonlinear it is generally not possible to determine a general analytic solution for the breakthrough curve. Such solutions have been obtained, however, for a number of special cases of which the irreversible or rectangular isotherm is the simplest. The irreversible isotherm, sketched in Figure 8.14, may be considered as the extreme limit of a favorable type 1 isotherm for which /8 0 and, as such, it represents an important limiting case. 

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