Langmuir system, equilibrium

New phenomena compared to nonreactive Langmuir systems are the same as in the binary case - that is, the existence of combined waves due to the occurrence of inflection points of the equilibrium functions y(x) or Y(X) and limitations on feasible product composition due to adsorptivity reversal similar to azeotropic distillation. Nonreactive examples for the latter were treated in Refs. [6 - 8], reactive examples will be discussed in the next section. 

When the system follows Langmuir competitive equilibrium behavior, the coherence condition defines a grid of coherent composition paths to which the system is restricted once the coherence condition is satisfied. Knowing the feed history, i.e., the boxmdary condition, one can use this grid, find the composition routes for the column and predict the column effluent history. 

If the change in concentration over which the uptake curve is measured is large, the effect of nonlinearity of the equilibrium isotherm must be considered. For a Langmuir system... 

FIGURE 9.5. Equilibrium diagram for ternary Langmuir system showing possible coherent composition paths. 

The present discussion of equilibrium theory has been concerned mainly with constant separation factor Langmuir systems and has been restricted to the analysis of the effect of a single step change in feed composition on a previously equilibrated bed. The Langmuir assumption greatly simplifies the analysis since, for such systems, the characteristics are linear and the same for gradual and shock transitions. 

The effect of mass transfer resistance is to broaden the mass transfer zone relative to the profile deduced from equilibrium theory. Where equilibrium theory predicts a shock transition the actual profile will approach constant-pattern form. Since the location of the mass transfer zone and the concentration change over which the transition occurs are not affected by mass transfer resistance, the extension of equilibrium theory is in this case straightforward and requires only the integration of the rate expression, subject to the constant-pattern approximation, to determine the form of the concentration profile. This is in essence the approach of Cooney and Strusi who show that for a Langmuir system with two adsorbable components a simple analytic expression for the concentration profile may be obtained when both mass transfer zones are of constant-pattern form. 

The model of Santacesaria et is an extension of the linear driving force model, with fluid side resistance, for a nonlinear multicomponent Langmuir system. It includes axial dispersion, and the combined effects of pore diffusion and external fluid film resistance are accounted for throu an overall rate coefficient. Intracrystalline diffusional resistance is neglected and equilibrium between the fluid in the macfopores and in the zeolite crystals is... 

Expressions for the steady-state concentration profile have also been derived for some more complex countercurrent systems. The extension to a (plug flow) system in which the equilibrium relationship is of Langmuir rather than linear form (constant separation factor) is given by Pratt. The solution for a linear plug flow system in which the mass transfer rate is controlled by intraparticle diffusion rather than by the linear rate law has been derived by Amundson and Kasten while the asymptotic behavior of a dispersed plug flow Langmuir system has been investigated by Rhee and Amundson. ... 

Thomas (1944) has provided a general analytical solution for a non-linear Langmuir system with a pseudo second-order reaction kinetic law. The results, which are given in graphical form by Hiester and Vermeulen (1952), provide a means of assessing the importance of a mass transfer resistance in any system for which the rate constant and equilibrium parameters are... 

Rate laws have also been observed that correspond to there being two kinds of surface, one adsorbing reactant A and the other reactant B and with the rate proportional to 5a x 5b- For traditional discussions of Langmuir-Hinshelwood rate laws, see Refs. 240-242. Many catalytic systems involve a series of intermediates, and the simplifying assumption of steady-state equilibrium is usually made. See Boudart and co-workers [243-245] for a contemporary discussion of such complexities. 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

For the equilibrium coverage at temperature T, one obtains in this rough Langmuir approximation for the overall reversible systems (y = x), by combining Eqs. (3) and (6),... 

To parameterize a Langmuir model, you determine from experimental measurements not only the equilibrium constant K, but the surface s sorption capacity (or exchange capacity). The latter value is a measure of the number of sorbing sites and is commonly reported in moles or electrical equivalents, per gram of dry sediment. Multiplying the sorption capacity in moles by the mass of sediment in a system gives the mole number of sorbing sites, which is... 

Subsequently, the condition of complete separation has to be coupled with the material balances derived for the nodes of the SMB unit and implemented in the Equilibrium Theory Model for Langmuir-type systems. That leads to the set of mathematical conditions given below, which the flow rate ratios have to fulfil in order to achieve complete separation, in particular ... 

The Langmuir model for competitive adsorption can be used as a common model for predicting adsorption equilibria in multicomponent systems. This was first developed by Butler and Ockrent [77] and is based on the same assumptions as the Langmuir model for single adsorbates. It assumes, as in the case of the Langmuir model, that the rate of adsorption of a species at equilibrium is equal to its desorption rate. This is expressed by Eq. (18) ... 

Of the various equilibrium and non-equilibrium sorption isotherms or sorption characteristics models, the most popular are the Langmuir and Freundlich models. The correct modeling of an adsorbate undergoing both transport and adsorption through a clay soil-solid system necessitates the selection of an adsorption isotherm or characteristic model which best suits the given system. The use of an improper or inappropriate adsorption model will greatly affect the... 

Again, the system consists of M adsorbent molecules fulfilling all the simplifying assumptions listed for the Langmuir model. The sites can be occupied by either one of the two different ligands A and B which are in equilibrium with a reservoir at fixed absolute activities and Xg, respectively. The GPF of the entire macroscopic system is... 

Whilst the conditions of equilibrium for such systems were clearly enunciated by J. Willard Gibbs and Sir J. J. Thomson a great impetus was given to the subject by supplementing the formal thermodynamic treatment with a clearer visualisation of the molecular structure of surfaces by Sir W. B. Hardy and I. Langmuir. 

Based on the Langmuir-Hinshelwood expression derived for a unimolecular reaction system (6) Rate =k Ks (substrate) /[I + Ks (substrate)], Table 3 shows boththe apparent kinetic rate and the substrate concentration were used to fit against the model. Results show that the initial rate is zero-order in substrate and first order in hydrogen concentration. In the case of the Schiff s base hydrogenation, limited aldehyde adsorption on the surface was assumed in this analysis. Table 3 shows a comparison of the adsorption equilibrium and the rate constant used for evaluating the catalytic surface. 

The Langmuir isotherm can be derived from a statistical mechanical point of view. Thus, for the reaction M + Agas Aads, equilibrium is established when the chemical potential on both phases is the same, i.e., pgas = p,ads. The partition function for the adsorbed molecules as a system is given by... 

Assume that typical time scales of both direct and reverse reactions in (3.3.2) are much shorter than any other time scale in the system. Then the reaction (3.3.2) yields the Langmuir s local equilibrium relation between Ci and 0 of the form... 

Adsorption isotherm An equation or distribution on a graph representing the concentration of an adsorbed chemical species as a function of its concentration in an associated aqueous solution. Adsorption isotherms are measured in systems that are at equilibrium and where temperature, pressure, and possibly other conditions are held constant. Isotherms are often described with linear, Freundlich, or Langmuir equations. 

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