Isotherm systems which obey

Prediction of the breakthrough performance of molecular sieve adsorption columns requires solution of the appropriate mass-transfer rate equation with boundary conditions imposed by the differential fluid phase mass balance. For systems which obey a Langmuir isotherm and for which the controlling resistance to mass transfer is macropore or zeolitic diffusion, the set of nonlinear equations must be solved numerically. Solutions have been obtained for saturation and regeneration of molecular sieve adsorption columns. Predicted breakthrough curves are compared with experimental data for sorption of ethane and ethylene on type A zeolite, and the model satisfactorily describes column performance. Under comparable conditions, column regeneration is slower than saturation. This is a consequence of non-linearities of the system and does not imply any difference in intrinsic rate constants. 

This equation, with a constant value for D, has been shown to provide a satisfactory correlation of experimental diffusivity data for several zeolitic systems (6-8). For a system which obeys the Langmuir isotherm, Equation 5 becomes... 

As we have seen, the Langmuir adsorption isotherm is based on the simplest of assumptions. Systems which obey the equation are often referred to as showing ideal... 

In most adsorption systems the isotherm is favorable for adsorption and therefore unfavorable for desorption. In desorption the mass transfer zone is therefore dispersive, leading to a continuously spreading concentration prOfife (proportionate-pattern behavior) while in adsorptioii the mass transfer zone is compressive, leading to constant-pattern behavior. For example, for a system which obeys the Langmuir isotherm (Eq. (8.6)] I... 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

That is why La is also called the separation factor. Furthermore, it is clear that the constant separation factor aA B in an ion-exchange system means that in practice, this system obeys a Langmuiran equilibrium isotherm, in which La is of course constant. 

Figure 2.11 gives an Illustration of both the Individual, fl(x), and the surface excess, (9-x)(x), Isotherms. Systems obeying the present premises give rise to type (a) curves (fig. 2.8). With Increasing K the maximum Increases and moves to the left. At the same time the individual and excess isotherms remain Identical till higher 0, which is in line with expectation. 

Sorption and diffusion of water vapour in polymers have been studied mainly for development of water vapour barriers. Some of these studies date back to 1944 [41]. Many attempts have been made in order to describe the sorption behavior of water vapor onto solid surfaces of the membrane and its pores. As described by Vieth [41], a deviation from Henry s law was observed in 1944, in the sorption of water by hydrated cellulose membranes. It was postulated that two competing phenomena are responsible for this observation dissolution, which obeys Henry s law, and adsorption, which follows the Langmuir isotherm. With other polymer systems the ability of water molecules and/or polar groups in the polymer matrix to interact with each other has given rise to sorption isotherms which may follow Henry s law, Flory-Huggins or BET types [41]. 

Evidence of a different kind is furnished by the fact that the Gurvitsch rule (p. 113) is often obeyed by systems showing Type I isotherms " the amounts of different adsorptives taken up by a given adsorbent, when expressed as a volume of liquid, agree within a few per cent. The order of agreement is illustrated by the typical examples in Table 4.1 for the adsorption of n-alkanes on ammonium phosphomolybdate, and in Table 4.2 which refers to a variety of adsorptives on a silica gel. It must be admitted, however, that there are cases where considerable deviations from the Gurvitsch mle are found, even though the isotherms are of Type 1. Thus, in Table 4.3 the variation in values of the saturation uptake is far outside... 

Another system obeying Fick s law is one involving the diffusion of small interstitial solute atoms (component 1) among the interstices of a host crystal in the presence of an interstitial-atom concentration gradient. The large solvent atoms (component 2) essentially remain in their substitutional sites and diffuse much more slowly than do the highly mobile solute atoms, which diffuse by the interstitial diffusion mechanism (described in Section 8.1.4). The solvent atoms may therefore be considered to be immobile. The system is isothermal, the diffusion is not network constrained, and a local C-frame coordinate system can be employed as in Section 3.1.3. Equation 2.21 then reduces to... 

In principle the mobility B and therefore the corrected diffusivity D0 are also concentration-dependent, so Eq. (12) does not necessarily predict quantitatively the concentration dependence of D even for a system where the isotherm obeys the Langmuir equation. Nevertheless, the concentration dependence of B is generally modest compared with that of the thermodynamic factor, so a monatonic increase in diffusivity with adsorbed-phase concentration is commonly observed (Fig. 5). Clearly in any attempt to relate transport properties to the physical properties of the system it is important to examine the corrected, diffusivity D0 (or the mobility B) rather than the Fickian diffusivity, which is in fact a product of kinetic and thermodynamic factors. 

Lewin and coworkers [255-260] developed an accessibility system based on equilibrium sorption of bromine, from its water solution at pH below 2 and at room temperature, on the glycosidic oxygens of the cellulose. The size of the bromine molecule, its simple structure, hydrophobicity, nonswelling, and very slow reactivity with cellulose in acidic solutions, contribute to the accuracy and reproducibility of the data obtained. The cellulose (10 g/1) is suspended in aqueous bromine solutions of 0.01-0.02 mol/1 for 1-3 h, depending on the nature of the cellulose, to reach sorption equilibrium. The diffusion coefficients of bromine in cotton and rayon are 4.6 and 0.37 x 10 cm /min, respectively. The sorption was found to strictly obey the Langmuir isotherm, which enables the calculation of the accessibility of the cellulose as follows ... 

Studies on the sorption of some hydrocarbons have shown that above the transition temperature of EBBA (331 K) the isotherms obey Henry s law and the solubility coefficients S can be calculated. The sorption and desorption curves are similar in shape which indicates that these systems follow Fickian sorption. This fact indicates that steady state surface equilibrium is reached and that the diffusion coefficient for hydrocarbons is a function of concentration only. It follows that the membranes containing 60 wt.% of EBBA are homogeneous from the view point of gas permeation at the temperature above transition in EBBA. The permeability coefficients P show a distinct jump in the vicinity of transition temperature from crystal to nematic phase. This phenomenon was observed for hydrocarbon gases, noble gases like He, and for inert gases like N2. 

For many adsorbate/support systems Henry s Law is obeyed and, as a consequence, this procedure is simplified for the experimenter because Line 3 in Figure 3.3 can simply be extrapolated to its zero-pressure intercept to obtain b, which is equal to a, thus eliminating the need to obtain a second isotherm. This latter procedure does not represent the gas uptake at zero pressure, as sometimes mis-stated, rather it represents subtraction of adsorption on the support at saturation coverage on the metal. 

In systems with concentrations that are held constant, the speeds with constant space function - which is the term ( ) of product isothermal conditions. This is true for the volumetric speed of a homogeneous system as well as for the areal speed of a catalytic system. This will be also true in the case of a strictly heterogeneous reaction that obeys the law of < E. Space functions are, however, frequently a function of time, which leads to absolute speeds or to rates that vaiy with time through this space function. 

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