Langmuir 43, isotherm

Geochemical models, as with the approach, are commonly formulated with a variant of the Freundlich isotherm based on a chemical reaction, like Reaction 9.1. In this approach, known as the reaction Freundlich model or the activity Freundlich model, the extent of sorption by the reaction can be expressed, 

we represent the Freundlich coefficient as K f (mol g-1), as a reminder that we are not working with Freundlich isotherms in the strictly traditional sense. Rearranging gives 

To cast the model in general form, we begin with the basis shown in Equation 9.5 and write each sorption reaction in the form of Equation 9.7. The mass action equation corresponding to the reaction for each sorbed species Aq is 

The Langmuir isotherm (or Langmuir model) provides an improvement over the K( and Freundlich approaches by maintaining a mass balance on the sorbing sites (Stumm and Morgan, 1996). The model, for this reason, does not predict that species can sorb indefinitely, since the number of sites available is limited. When the calculation carries reactions for the sorption of more than one aqueous species, furthermore, it accounts for competition such a calculation is known as a competitive Langmuir model. 

In the Langmuir model, a species sorbs and desorbs according to a reaction such as, 

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Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

Adsorption isotherms in the micropore region may start off looking like one of the high BET c-value curves of Fig. XVII-10, but will then level off much like a Langmuir isotherm (Fig. XVII-3) as the pores fill and the surface area available for further adsorption greatly diminishes. The BET-type equation for adsorption limited to n layers (Eq. XVII-65) will sometimes fit this type of behavior. Currently, however, more use is made of the Dubinin-Raduschkevich or DR equation. Tliis is Eq. XVII-75, but now put in the form... 

The form of the functions f depends on Che particular isotherm used for example the Langmuir isotherm gives the familiar relation... 

In order to test the Langmuir isotherm against experimental data. Equation (4.1) may be rewritten in the form... 

Eig. 4. The Bmnaner classification of isotherms (I V). Langmuir Isotherm. Type I isotherms are commonly represented by the ideal Langmuir model ... 

Markham andBenton. This model (34) is known as the extended Langmuir isotherm equation for two components, i and j. 

For adsorbed hydrocarbons, the adsorption—desorption process can be thought of as a reaction and the adsorption isotherm as a description of the reaction at equihbtium. For the Langmuir isotherm,... 

Isotherms. When a fibei is immersed, in a dyebath, dye moves fiom the external phase into the fibei. Initially the late is quick but with time this slows and eventually an equiUbrium is reached between the concentration of dye in the fiber and the concentration of dye in the dyebath. For a given initial dyebath concentration of a dye under given dyebath conditions, eg, temperature, pH, and conductivity, there is an equiUbrium concentration of dye in fiber, Dj and dye in the dyebath external solution, D. Three models describe this relationship simple partition isotherm, Freundhch isotherm, and Langmuir isotherm. 

In synthetic fibers the number of ionic groups or dye sites is relatively small, and may have been introduced dehberately to make the base polymer dyeable. The restrictions on dye absorption are therefore very great the dye molecule must find an available specific site from among the limited number of sites in the fiber. This situation follows a Langmuir isotherm, where the reciprocal of dye in fiber 1 /DF is direcdy proportional to the reciprocal of dye in the dyebath 1 /HT. A plot of 1/against 1/H therefore gives a straight line. 

Langmuir isotherms are typically found with ionic synthetic fibers and ionic dyes, eg, dyeing polyacrylonitrile with modified basic dyes, and on hydrophilic fibers in situations when the number of sites becomes very low. This may arise when the internal pH is such that only a small number of sites ionise. 

The classical isotherm for a homogeneous flat surface, and most popular of all nonhnear isotherms, is the Langmuir isotherm... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

The Langmuir isotherm, Eq. (16-13), corresponds to the constant separation rac tor isotherm with... 

Example 4 Application of Isotherms Thomas [Ann. N.Y. Acad. Sci., 49, 161 (1948)] provides the following Langmuir isotherm for the adsorption of anthracene from cyclohexane onto alumina ... 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, nl =. 3 mol/kg, Kipi = K2P2 = 1. Use the ideal adsorbed-solution theory to determine ni and n. Substituting the pure component Langmuir isotherm... 

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

For a Langmuir isotherm with neghgible solute accumulation in the particle pores, the solution for an infinite fluid volume ... 

For noncoustaut diffusivity, a numerical solution of the conseiwa-tion equations is generally required. In molecular sieve zeohtes, when equilibrium is described by the Langmuir isotherm, the concentration dependence of the intracrystalline diffusivity can often be approximated by Eq. (16-72). The relevant rate equation is ... 

In LC, at very low concentrations of moderator in the mobile phase, the solvent distributes itself between the two phases in much the same way as the solute. However, as the dilution is not infinite, the adsorption isotherm is not linear and takes the form of the Langmuir isotherm. 

Equation (3) is one form of the Langmuir isotherm and it should be noted that, when (Cm) tends to zero,... 

Volume overload results from too large a volume of sample being placed on the column, and this effect will be discussed later. It will be seen that volume overload does not, in itself, produce asymmetric peaks unless accompanied by mass overload, but it does broaden the peak. Mass overload, however, frequently results in a nonlinear adsorption isotherm. However, the isotherm is quite different from the Langmuir isotherm and is caused by an entirely different phenomenon. 

Brunauer further developed the Langmuir isotherm expression to include multilayer adsorption ... 

Inserted in Eq. (3), this gives the Langmuir isotherm P = Zi ,exp( u ABr)= - exp(-KoABr,, from which we get the isosteric heat of adsorption... 

A. Gentilini, C. Migliorini, M. Mazzotti and M. Morbidelli, Optimal operation of simulated moving-bed units for non-linear cliromatograpliic separ ations. II. Bi-Langmuir isotherm , 7. Chromatogr. 805 37-44 (1998). 

However, when complexation experiments are performed with both d- and L-enan-tiomers (Fig. 5-18), this leads to selectivity values between 1.4 and 1.9. It was shown that complexation by enantioselective micelles can effectively be described using straightforward multicomponent Langmuir isotherms [74]. 

Fig. 5-18. Selectivity for D,L-phenylalanine, fitted with a multicomponent Langmuir isotherm [73],...

The bi-Langmuir model (Equation (2)) or tri-Langmuir model, the sum of two or three Langmuir isotherms, correspond to models that assume the adsorbent surface... 

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 ... 

Gentilini A., Migliorini C., Mazzotti M., Morbidelli M. (1998) Optimal Operation of Simulated Moving-Bed Units for Non-Linear Chromatographie Separations. II. Bi-Langmuir Isotherm, J. Chromatogr. A 805 37-44. 

The knowledge of these adsorption isotherms allows quantification of the respective affinity for the stationary phase with respect to the different solutes. Many different isotherm equations have been described in the literature, and experimental methods allowing their determination are reviewed by [58]. As a first approximation, modified competitive Langmuir isotherms can often he used ... 

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