Equilibrium isotherm Langmuir

Hiester and Vermeulin [39] extended the model of Thomas by deriving a breakthrough curve for a slow diffusion rate-limiting step with a nonlinear Langmuir equilibrium isotherm. The model was further adapted by Arnold and Blanch [40] to describe affinity chromatographic experiments ... 

The Langmuir equilibrium isotherm model can be extended to multicomponent systems [7,8]. However, when several components are simultaneously present in the solution, these compounds interfere. The amount of each of them that is adsorbed at equilibrium is smaller than if this compoimd were alone. Although... 

We will assume a Langmuir equilibrium isotherm. The general equation of the characteristic line that begins in a point of this boundary condition, i.e., from the point X = Q,t = Tjtfi, (0 < / < 1) is... 

It is evident from Eq. (8.3) that the equilibrium. factor is in general concentration dependent. However, in the special case of a Langmuir equilibrium isotherm... 

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

The chromatographic resolution of bi-naphthol enantiomers was considered for simulation purposes [18]. The chiral stationary phase is 3,5-dinitrobenzoyl phenyl-glycine bonded to silica gel and a mixture of 72 28 (v/v) heptane/isopropanol was used as eluent. The adsorption equilibrium isotherms, measured at 25 °C, are of bi-Langmuir type and were proposed by the Separex group ... 

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 ... 

Table 9-3. Operating conditions for complete separation under Equilibrium Theory. Langmuir adsorption isotherms (see Fig. 9-8).

The competitive adsorption isotherms were determined experimentally for the separation of chiral epoxide enantiomers at 25 °C by the adsorption-desorption method [37]. A mass balance allows the knowledge of the concentration of each component retained in the particle, q, in equilibrium with the feed concentration, < In fact includes both the adsorbed phase concentration and the concentration in the fluid inside pores. This overall retained concentration is used to be consistent with the models presented for the SMB simulations based on homogeneous particles. The bed porosity was taken as = 0.4 since the total porosity was measured as Ej = 0.67 and the particle porosity of microcrystalline cellulose triacetate is p = 0.45 [38]. This procedure provides one point of the adsorption isotherm for each component (Cp q. The determination of the complete isotherm will require a set of experiments using different feed concentrations. To support the measured isotherms, a dynamic method of frontal chromatography is implemented based on the analysis of the response curves to a step change in feed concentration (adsorption) followed by the desorption of the column with pure eluent. It is well known that often the selectivity factor decreases with the increase of the concentration of chiral species and therefore the linear -i- Langmuir competitive isotherm was used ... 

Assuming adsorption to behave according to the Langmuir adsorption isotherm, we get Eq. (1.22b). Both the rate constant of proton activation and the equilibrium constant of adsorption K q depend on cavity details. 

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

The Langmuir adsorption isotherm is easy to derive. Again we assume that the catalyst contains equivalent adsorption sites, and that the adsorbed molecules do not interact. If the adsorbed molecules are in equilibrium with the gas phase, we may write the reaction equation as... 

Figure 2.10. Langmuir adsorption isotherm for associative adsorption as a plot of coverage versus pressure for three values of the equilibrium constant, K.

In this assumption, the singlet oxygen adsorption on a surface may be considered a quasistationary process and described by the Langmuir Adsorption Isotherm. Then the equilibrium constant is... 

This is the important Hill-Langmuir equation. A. V. Hill was the first (in 1909) to apply the law of mass action to the relationship between ligand concentration and receptor occupancy at equilibrium and to the rate at which this equilibrium is approached. The physical chemist I. Langmuir showed a few years later that a similar equation (the Langmuir adsorption isotherm) applies to the adsorption of gases at a surface (e g., of a metal or of charcoal). 

Figure 8.8 Langmuir adsorption isotherm At adsorption equilibrium, with raA = rdA,...

The multicomponent Langmuir adsorption isotherm given in Eq. (7) is the simplest model for the description of non-linear, multicomponent, adsorption equilibrium. At high concentration, the model predicts saturation of the stationary phase and overload of the chromatographic column. At low concentration (high dilution) the behavior can be correctly described by the non-competitive linear adsorption isotherm ... 

In 1938, Brunauer, Emmett and Teller(12) and Emmett and de Witt(13) developed what is now known as the BET theory. As in the case in Langmuir s isotherm, the theory is based on the concept of an adsorbed molecule which is not free to move over the surface, and which exerts no lateral forces on adjacent molecules of adsorbate. The BET theory does, however, allow different numbers of adsorbed layers to build up on different parts of the surface, although it assumes that the net amount of surface which is empty or which is associated with a monolayer, bilayer and so on is constant for any particular equilibrium condition. Monolayers are created by adsorption on to empty surface and by desorption from bilayers. Monolayers are lost both through desorption and through the adsorption of additional layers. The rate of adsorption is proportional to the frequency with which molecules strike the surface and the area of that surface. From the kinetic theory of gases, the frequency is proportional to the pressure of the molecules and hence ... 

For the Langmuir adsorption isotherm it is assumed that this reaction (Eq. 1) has a fixed free energy of adsorption equal to AG°, which is not dependent on the extent of adsorption and not affected by interaction among sites. In addition, each site is assumed to be capable of binding at most one molecule of adsorbate. If Q is the maximum number of moles of a pollutant adsorbed per mass adsorbent when the surface sites are saturated with an adsorbate (i.e., a full monolayer), and q is the number of moles of adsorbate per mass adsorbent at equilibrium, then according to the law of mass action Eq. (2) follows ... 

This identity of formulation means that in normal cases correct results for the velocity equation will be obtained by assuming that adsorption equilibrium is established and the adsorbed molecules react at a rate proportional to their surface concentration when they possess the critical energy. Then, for the surface concentration 07 of every reactant the Langmuir adsorption isotherm holds ... 

This is called a Langmuir adsorption isotherm for a species A, and the function 9a( Pa) isi shown in Figure 7-23. The KjS are the adsorption-desorption equilibrium constants for species A and B. By historical convention we call these the adsorption isotherms. Before i we proceed let us note that this is a true thermodynamic equilibrium relation so that... 

EXAMPLE 7.5 Use of Langmuir Adsorption Isotherm. The moles of solute B adsorbed per gram of solid C were determined by measuring concentration changes in the solution. The accompanying results report the adsorption versus the concentration of the equilibrium solution ... 

Furthermore, in 3.3 we turn to reactive binary ion-exchange. An equilibrium binding reaction (adsorption) with a Langmuir-type isotherm is considered. Formation of sharp propagating concentration fronts is studied via an unconventional asymptotic procedure [1]. 

The Langmuir adsorption isotherm describes the equilibrium between a single-component gas A and adsorbed species A(s) at a surface [237]. The expression relates the fraction of the surface 6a covered by adsorbed species as a function of the partial pressure pa exposed to the surface. The usual form of the Langmuir adsorption isotherm is... 

---