Langmuir isotherm and

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. 

Sigmoid, the characteristic S-shaped curves defined by functions such as the Langmuir isotherm and logistic function (when plotted on a logarithmic abscissal scale). 

Despite the already discussed oversimplifications built into the Langmuir isotherm and in the resulting LHHW kinetics, it is useful and instructive at this point to examine how a promoter can affect the catalytic kinetics described by the LHHW expressions (2.11) to (2.14). 

FIG. 3 Adsorption isotherm of ONS at the polarized water-DCE interface (square) experimental, (dotted) Langmuir isotherm, and (solid) Frumkin isotherm. (From Ref. 55, cop5right American Chemical Society.)... 

In this chapter, we consider several simple models of ion sorption and exchange that can be applied within the context of a geochemical model. These models include distribution coefficients, Freundlich and Langmuir isotherms, and ion exchange theory. In the following chapter (Chapter 10), we consider surface com-plexation theory, which is more complicated but in some ways more robust than the models presented here. 

When this happens the adsorption phenomenon follows the so-called Langmuir isotherm and the current intensity at the maximum of the forward peak is correlated with the parameter To (that represents the quantity of Ox, in mol m-2, initially adsorbed on the electrode surface) and with the scan rate according to the relationship ... 

Adsorption of NH3 on hematite followed a Langmuir isotherm and had a heat of adsorption of 48.3 kj mol (Blyholder and Richardson, 1962). Infrared spectroscopy indicated that coordination involved the Lewis acid sites on (activated) hematite (Rochester and Topham, 1979 b). 

Let us consider some frequently used nonlinear functions. The Langmuir isotherm and the Michaelis-Menten kinetics are of the form... 

A detailed analysis of the effect of mixed monolayers of 15 and DMPC on the activity of phospholipase A2 was reported by Grainger et al. [53]. Monolayers composed of different ratios of DMPC and either 15 or primarily poly 5 were characterized by Langmuir isotherms and isobars. The phospholipse-A2-mediated hydrolysis of selected monolayer compositions was usefully employed to ascertain the effectiveness of the enzyme. Both 15 and polyl5 were resistant to hydrolysis. The DMPC hydrolysis was sensitive to its molecular environment in a manner that suggests the phase separation of the polyl5 from DMPC. Phospholipase A2 activity is known to be sensitive to the concentration of the hydrolytic products, i.e. the fatty acid and lysophospholipid. The effect of these reaction products of the activity of phospholipase A2 on mixed monolayers of nonpolymerizable lipids is the subject of a series of interesting studies which are beyond the scope of this review. Ahlers et al. reviewed some of this research [54],... 

Infinite fluid volume and liquid diffusion control General case For the case of the Langmuir isotherm and the solid phase initially free of solute, the solution is (Perry and Green, 1999)... 

The following equations constitute the approximate solutions of the fixed-bed model under the constant pattern and plug-flow assumption for the favorable Langmuir isotherm and linear driving forces (Perry and Green, 1999) ... 

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. 

In some cases, adsorption of analyte can be followed by a chemical reaction. The Langmuir-Hinshelwood (LH) and power-law models have been used successfully in describing the kinetics of a broad range of gas-solid reaction systems [105,106]. The LH model, developed to describe interactions between dissimilar adsorbates in the context of heterogeneous catalysis [107], assumes that gas adsorption follows a Langmuir isotherm and that the adsorbates are sufficiently mobile so that they equilibrate with one another on the surface on a time scale that is rapid compared to desorpticm. The power-law model assumes a Fre-undlich adsorption isotherm. Bodi models assume that the surface reaction is first-order with respect to the reactant gas, and that surface coverage asymptotically approaches a mmiolayer widi increasing gas concentration. 

Solution On the basis of the reaction model (equations (10.81) and (10.82)), the impedance can be derived under the assumption that the adsorbate Mg obeys a Langmuir isotherm and that the rate constants of electrochemical reactions are exponentially dependent on potential (e.g., following Tafel s law). Each reaction with index i has a normalized rate constant K, corresponding to its rate constant hi by... 

Approximation of the linear form is not necessary for the Langmuir isotherm, and the first plot of the adsorption data will determine whether or not the model is applicable, and also will allow calculation of the adsorption coefficients. Usually a single model will not be satisfactory for a wide range of adsorbate concentrations but will only serve in narrow range of concentration. At low concentrations, C/C 1, the BET model reduces to a Langmuir model. 

The models described above are termed Langmuir-Hinshelwood-Hougen-Watson (LHHW) models, named after the scientists that contributed a lot to the development of these engineering models. The characteristics of these models are that adsorption follows the Langmuir isotherm, and that reaction takes place between adsorbed species. Sometimes, one distinguishes Eley Rideal models, whereby a molecule reacts directly from the gas phase with a surface complex ... 

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,... 

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).

Electrochemical Measurements Under conditions in which the interactions among adsorbates can be neglected, and assuming that all adsorption sites are strictly identical (Langmuir isotherm), and that the electron transfer rates are very fast, the peak potential, Ep, and peak current density, ip, observed in cyclic voltammetry are given, respectively, by ... 

Figure 3.40 Illustration of the method of isotherm measurements by computation of elution profiles. R-l-indanol on cellulose tribenzoate chiral stationary phase. Mobile phase, n-hexane and 2-propanol (92.5 7.5, v/v). (Left) Calculated (using the bi-Langmuir isotherm) and experimental chromatograms recorded for 46.25 (main figure) and 9.251 mg (insert) of R-l-indanol. The isotherm was determined from the band profile obtained for 46.25 mg. (Right) Bi-Langmuir isotherms obtained by the inverse method (lines) and by frontal analysis (symbols) for the R- and S-l-indanol enantiomers. Cmax indicates the maximum elution concentration. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, /. Chromatogr. A, 35 (2003) 1005 (Figs. 2 and 3).

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