Isotherms favorability

The process of adsorption takes place when the concentration of the adsorptive is greater than the equiUbrium value vahd for the given temperature however, desorption requires a fluid concentration of the adsorptive which is smaller than the equilibrium concentratiom An adsorption isotherm favorable for adsorption is unfavorable for desorption and vice versa. Condensation of gases or vapors and solidification or crystalhzation will start when the relative supersaturation becomes > 1. In the case of adsorbents with capillary or very narrow pores, capillary condensation is observed for relative saturations adsorption isotherm vahd for adsorption and desorption can sometimes be ejqrlained, see Fig. 2.4-2. Sohd materials exposed to drying (see Chap. 10) often show such hysteresis behavior which can sometimes be explained by the ciu-vature of the liqttid sttrface in capillaries The radius of this surface is greater in the case of adsorption in comparison to the radius valid for a desorption process, see Fig. 2.4-2. 

Fig. 9.8-5 Concentration bands for a linear isotherm left), an isotherm favorable for desorption center), and an isotherm favorable for adsorption right)...

Figure 7.1.2. Adsorption isotherms (a) two linear isotherms for two species i =1, 2 (b) three isotherms, favorable, linear and unfavorable.

We now turn specifically to the thermodynamics and kinetics of reactions (5. EE) and (5.FF). The criterion for spontaneity in thermodynamics is AG <0 with AG = AH - T AS for an isothermal process. Thus it is both the sign and magnitude of AH and AS and the magnitude of T that determine whether a reaction is thermodynamically favored or not. As usual in thermodynamics, the A s are taken as products minus reactants, so the conclusions apply to the reactions as written. If a reaction is reversed, products and reactants are interchanged and the sign of the AG is reversed also. 

Fig. 11. (a) Equilibrium isotherm and (b) dimensionless equilibrium diagram showiag favorable, linear, and unfavorable isotherms. 

Fig. 12. (a) Development of the physically unreasonable overbanging concentration profile and the corresponding shock profile for adsorption with a favorable isotherm and (b) development of the dispersive (proportionate pattern) concentration profile for adsorption with an unfavorable isotherm (or for... 

Fig. 13. Schematic diagram showing (a) approach to constant pattern behavior for a system with a favorable isotherm and (b) approach to proportionate pattern behavior for a system with an unfavorable isotherm, jy axis cj qlj q,----------------------- < q,-- From ref. 7.

The distance requited to approach the constant pattern limit decreases as the mass transfer resistance decreases and the nonlinearity of the equihbrium isotherm increases. However, when the isotherm is highly favorable, as in many adsorption processes, this distance may be very small, a few centimeters to perhaps a meter. 

Favorable and unfavorable equihbrium isotherms are normally defined, as in Figure 11, with respect to an increase in sorbate concentration. This is, of course, appropriate for an adsorption process, but if one is considering regeneration of a saturated column (desorption), the situation is reversed. An isotherm which is favorable for adsorption is unfavorable for desorption and vice versa. In most adsorption processes the adsorbent is selected to provide a favorable adsorption isotherm, so the adsorption step shows constant pattern behavior and proportionate pattern behavior is encountered in the desorption step. 

Adsorption Isotherms. EquiUbrium dialysis studies indicate around 10 repeat VP units (base moles) are required to form favorable complexes (89,90). This figure can rise to several hundred for methyl orange and other anions depending on stmcture (91,92). 

Historically, isotherms have been classified as favorable (concave downward) or unfavorable (concave upward). These terms refer to the spreading tendencies of transitions in fixed beds. A favorable isotherm gives a compact transition, whereas an unfavorable isotherm leads to a broad one. 

Adsorption with strongly favorable isotherms and ion exchange between strong electrolytes can usually be carried out until most of the stoichiometric capacity of the sorbent has been utilized, corresponding to a thin MTZ. Consequently, the total capacity of the bed is... 

FIG. 16-2 Limiting fixed-bed behavior simple wave for unfavorable isotherm (top), square-root spreading for linear isotherm (middle), and constant pattern for favorable isotherm (bottom). [From LeVan in Rodtigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dotdtecht, The Nethedands, 1989 reptinted withpeimission.]... 

In either equation, /c is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for sohd diffusion and film resistance in series... 

For a favorable isotherm d n lldc f < 0), Eq. (I6-I3I) gives the impossible result that three concentrations can coexist at one point in the bed (see example below). The correct solution is a shock (or abrupt transition) and not a simple wave. Mathematical theoiy has been developed for this case to give weak solutions to consei vation laws. The form of the solution is... 

Using the isotherm to calculate loadings in equilibrium with the feed gives rii = 3.87 mol/kg and ri2 = 1.94 mol/kg. An attempt to find a simple wave solution for this problem fails because of the favorable isotherms (see the next example for the general solution method). To obtain the two shocks, Eq. (16-136) is written... 

With a favorable isotherm and a mass-transfer resistance or axial dispersion, a transition approaches a constant pattern, which is an asymptotic shape beyond which the wave will not spread. The wave is said to be self-sharpening. (If a wave is initially broader than the constant pattern, it will sharpen to approach the constant pattern.) Thus, for an initially uniformly loaded oed, the constant pattern gives the maximum breadth of the MTZ. As bed length is increased, the constant pattern will occupy an increasingly smaller fraction of the bed. (Square-root spreading for a linear isotherm gives this same qualitative result.)... 

Constant pattern solutions for the individual mechanisms and constant separation factor isotherm are given in Table 16-13. The solutions all nave the expected dependence on R—the more favorable the isotherm, the sharper the profile. 

When the adsorption equihbrium is nonlinear, skewed peaks are obtained, even when N is large. For a constant separation-factor isotherm with R < 1 (favorable), the leading edge of the chromatographic peak is steeper than the trailing edge. Wmen R > 1 (unfavorable), the opposite is true. 

Most published studies relate only to isothermal experiments. Hence, in order to make such comparisons we modified our computations to assume isothermal conditions. Figure 11 compares our kinetic model with data by Hui and Hamielec for styrene thermal polymerization at 1A0°C. Figure 12 compares out kinetic model with data by Balke and Hamielec (7) for MMA at 90 C using 0.3 AIBN. Figure 13 compares our kinetic model with data by Lee and Turner ( ) for MMA at 70°C using 2% BPO. Our model compares quite favorably with these published experiments. The percent error was less than S% in most of the ranges of conversions. 

Nanoparticles of the semicondnctor titanium dioxide have also been spread as mono-layers [164]. Nanoparticles of TiOi were formed by the arrested hydrolysis of titanium iso-propoxide. A very small amount of water was mixed with a chloroform/isopropanol solution of titanium isopropoxide with the surfactant hexadecyltrimethylammonium bromide (CTAB) and a catalyst. The particles produced were 1.8-2.2 nm in diameter. The stabilized particles were spread as monolayers. Successive cycles of II-A isotherms exhibited smaller areas for the initial pressnre rise, attributed to dissolution of excess surfactant into the subphase. And BAM observation showed the solid state of the films at 50 mN m was featureless and bright collapse then appeared as a series of stripes across the image. The area per particle determined from the isotherms decreased when sols were subjected to a heat treatment prior to spreading. This effect was believed to arise from a modification to the particle surface that made surfactant adsorption less favorable. 

The equilibrium isotherms were favorable type and the Langmuir equation represents our experimental data very well. 

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