Multi-Langmuir equation

Alternatively, the isotherm can also be expressed by a multi-Langmuir equation ... 

V = 1.5 mlmin . For the chromatograms shown in Figure 6.29 different values are chosen. The isotherms are described by the multi-Langmuir equation (Equation 2.57). 

Langmuir-Type Relations For systems composed of solutes that individually follow Langmuir isotherms, the traditional multi-component Langmuir equation, obtained via a kinetic derivation, is... 

The Langmuir equation can be easily extended to multi-component adsorption, for example, for a binary mixture of components 1 and 2 ... 

This partial differential equation can be solved analytically for certain types of isotherms by applying the method of characteristics (Helfferich and Klein, 1970 Rhee, Aris, and Amundson, 1970). Following this approach direct solutions are available for isotherms with constant selectivity, such as the multi-Langmuir or the modified competitive multi-Langmuir isotherm (Storti et al., 1993,1995 Mazzotti, Storti, and Morbidelli, 1994,1996,1997). For multi-bi-Langmuir isotherms with nonconstant selectivities a numerical determination is necessary (Gentilini et al., 1998 Migliorini, Mazzotti, and Morbidelli, 2000). 

As an example, the procedure for determining the operating diagram is explained for the EMD53986 system (Appendix A). The adsorption equilibrium of this system can be described by the multi-Langmuir isotherm (Equation 7.91) ... 

We start this book with a chapter (Chapter 2) on the fundamentals of pure component equilibria. Results of this chapter are mainly applicable to ideal solids or surfaces, and rarely applied to real solids. Langmuir equation is the most celebrated equation, and therefore is the cornerstone of all theories of adsorption and is dealt with first. To generalise the fundamental theory for ideal solids, the Gibbs approach is introduced, and from which many fundamental isotherm equations, such as Volmer, Fowler-Guggenheim, Hill-de Boer, Jura-Harkins can be derived. A recent equation introduced by Nitta and co-workers is presented to allow for the multi-site adsorption. We finally close this chapter by presenting the vacancy solution theory of Danner and co-workers. The results of Chapter 2 are used as a basis for the... 

Here we take an example of a solid where the local adsorption isotherm takes the form of Langmuir equation (6.4-3) and the energy distribution has the form of multi-modal distribution ... 

The dependence of the frequency ratios K on the thermodynamic conditions is governed by the properties of the mixture. If the bulk phase is an ideal-gas mixture, it is reasonable to assume that the adsorption fi-equencies are proportional to the partial molar pressmes of gas, P = Pzgj. On the other hand, if no lateral interaction between adsorbed molecules is assumed, desorption rates k are solely the functions of the temperature. In these assumptions, a classical multi-component Langmuir equation is obtained ... 

Engineering theories for multicomponent adsorption can be roughly divided into three categories extensions of the Langmuir equation, the thermodynamic approach (ideal and real adsorbed solution theories, lAST and RAST) by Myers and Prausnitz (1965) and finally the potential adsorption theory, especially as extended to multi-component systems by Shapiro and co-workers (Shapiro and Stenby, 1998 Monsalvo and Shapiro, 2007a, 2009a,b). 

The advantage of Langmuir s approach for multi-component systems is its simplicity and often good performance for relatively simple systems. Addition of new physical mechanisms to improve the model for relaxing some of its assumptions has resulted in cumbersome equations which have not found widespread use. An alternative way to use the Langmuir equation is, in its single adsorption form, in combination with the thermodynamic approaches of lAST and RAST. 

The coupled isotherm equation takes into account the displacement of one component by the other with the term (b, c,j = i). Equation 2.43 is the general form of the multi-component Langmuir isotherm and is also called asymmetric as for each component i specific parameters bt j have to be determined. With symmetric coefficients only a set of bj parameters is taken into account and Eq. 2.43 reduces to Eq. 2.44,... 

Equations 2.45 and 2.46 show that simple multi-component Langmuir isotherms obviously can not explain the decrease in selectivity observed under increased loading factors. The extended form of the modified Langmuir isotherm, however, can represent these phenomena. The modified multi-component Langmuir isotherm is shown in Eq. 2.47 (Charton and Nicoud, 1995). 

The terms qS3i,i represent the maximum loadings for each component. Interaction between the different components is considered by the sum of all Ncom P components. Equation 6.33 is the non-equilibrium form of the multi-component Langmuir isotherm (Eq. 2.44). 

The corresponding equilibrium data for pure components and different mixtures are represented in Fig. 6.21 by filled rhombuses and triangles. Also shown are the results of different isotherm equations (solid and dashed lines) that have been fitted to the experimental data. In this case the modified multi-component Langmuir isotherm (Eq. 2.47)... 

Fig. 6.30 Comparison of the simulated profiles for the modified multi-component Langmuir isotherm and the IAS equation (Fig. 6.26) (Cfeej = 4.4l1 V= lOmlmin-1, Vinj = 120ml, Vc = 54ml for additional data see Appendix B.l).

Antia and Horvith [61] attempted to use these equations to obtain multi-component isotherms in cases in which each individual solute follows single-component Langmuir isotherm behavior, hence... 

Physisorption arises from the van der Waals forces, and these forces also condense gas molecules into their liquid state. Thus, in principle, there is no reason to stop upon completion of a monolayer during physisorption. Indeed, the formation of multi-layers, which are basically liquid in nature, is very common in physisorption experiments. Brunauer, Emmett and Teller developed a theory in 1938 to describe physisorption, where the adsorbate thickness exceeds a monolayer, and this isotherm equation is known by the initials of the authors (B.E.T.). The original derivation of the B.E.T. equation is an extension of Langmuir s treatment of monolayer adsorption from kinetic arguments. Later, in 1946, Hill derived this equation from statistical mechanics. In the B.E.T. isotherm, it is assumed that ... 

The generalized Maxwell-Stefan equation provides a rational basis for the analysis of sorption rate measurements and membrane permeation in multi-component systems. For a binary Langmuir system ... 

Cellulose-cellulase incubation was made at 25 C with 1 mL of 50 mM Na-acetate of pH 5, 0.8 - 62 pM cellulase, 1.8 g/L cellulose and 8 pM BSA in 1.2-mL Pierce ImmimoWare tubes. After 30 min, the supernatant was decanted, filtered (by 0.45 p Millipore Multi Screen-HV), and assayed by PASC hydrolysis activity for free cellulase. Langmuir adsorption equation, l/[E]adsorbed = l/([E]o - [E]) = l/(No tE]) + P/No (35), was then applied to calculate apparent capacity No (in mole/mole) and adsorption constant IC. 

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