Simple Adsorption Isotherms

N is the number of ligand molecules (or moles) adsorbed and P is the partial pressure 

Our goal is to find a molecular model for the adsorption process that reproduces the same functional dependence of A on P as in (2.8.1) and thereby to provide a molecular interpretation of the quantities M and K in this equation. Later, we shall generalize this simple isotherm in various directions. 

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The relationship between the activity of adsorbed ions fiQO in Eqn. 5-21 and the adsorption coverage 6i is known as an adsorption isotherm. Equations 5-23 and 5-24 show simple adsorption isotherms ... 

The Davies and Jones derivation makes some fundamental assumptions concerning the surface concentrations of the lattice ions and the BCF theory is only applicable to very small supersaturations. Thus, both theories have limitations which affect the interpretation of the results of growth experiments. Nielsen [27] has attempted to examine in detail how the parabolic dependence can be explained in terms of the density of kinks on a growth spiral and the adsorption and integration of lattice ions. One of the factors, a = S — 1, comes from the density of kinks on the spiral [eqns. (4) and (68)] and the other factor is proportional to the net flux per kink of ions from the solution into the lattice. Nielsen found it necessary to assume that the adsorption of equivalent amounts of constituent ions occurred and that the surface adsorption layer is in equilibrium with the solution. Rather than eqn. (145), Nielsen expresses the concentration in the adsorption layer in the form of a simple adsorption isotherm equation... 

Kinniburgh D. G., J. A. Barker, and M. Whitfield. 1983. A comparison of some simple adsorption isotherms for describing divalent cation adsorption by ferrihydrite. J. Coll. Interface Sci. 95 370-384. 

The values of am can be found from the measurement of the amount of adsorbed gas and an adsorption isotherm (the relationship between the partial pressure of the adsorbate and the amount adsorbed at equilibrium at constant temperature). A simple adsorption isotherm is given by the Langmuir equation... 

Surfactant adsorption close to the cmc may appear Langmuirian, but this does not automatically imply a simple orientation. For example, rearrangement from a horizontal to a vertical orientation or electrostatic interaction and counterion binding may be masked by simple adsorption isotherms. It is essential, therefore, to combine the adsorption isotherms with other techniques such as microcalorimetry and various spectroscopic methods in order to obtain a full picture of surfactant adsorption. 

To evaluate the adsorption of chelates and organic ligands, such as surfactants, a simple adsorption isotherm model may be used. The perhaps most straightforward model has been developed by Langmuir. Assuming that the surface is basic and the adsorbate is acidic in the Bronsted or the Lewis sense, we obtain for the displacement of the solvent liquid (L) molecules by the adsorbate (A) ... 

In the case of multilayer formation, it is useful to express the coverage degree by 6 = V/Vm where denotes the volume of the adsorbed substance in a monolayer. Adsorption and desorption can be described as well as the kinetic and equilibrium process. Equilibrium is described by the general condition /( , p, T) = 0. The process is described either at a constant pressure (adsorption isobar) or constant temperature (adsorption isotherm). The most simple adsorption isotherm after Freundlich and Langmuir is based on the equilibrium... 

Simple adsorption isotherm studies are widely used to assess the amount of coating that is needed to fully treat a filler surface. The isotherms may be determined by a number of techniques for example following either the reduction in solution concentration (gravimetrically, spectroscopically, titrimetrically, etc.), or the bnild np on the filler after isolation and washing. Reversibility of adsorption may also be stndied. 

The problem for the diffusion-controlled adsorption of a mixture of surfactants at a small deviation from equilibrium can be solved analytically by using the Laplace transform the resulting long expressions can be found in Ref. 73. The results are compared with experimental data for different alkanoates at the air-liquid interface [73]. Simple adsorption isotherms for mixtures of surfactants are reported in Refs. 31 and 44 surface equations of state applicable to mixtures of surfactant molecules of different size can be found in Res. 27 and 74. 

Examples of mono-layer adsorption isotherms obtained for chloroform and butyl chloride are shown in Figure 5. The adsorption isotherms of the more polar solvents, ethyl acetate, isopropanol and tetrahydro-furan from -heptane solutions on silica gel were examined by Scott and Kucera [4]. Somewhat surprisingly, it was found that the experimental results for the more polar solvents did not fit the simple mono-layer... 

For determining the adsorption isotherm, the equilibrium concentrations of bound and free template must be reliably measured within a large concentration interval. Since the binding sites are part of a solid, this experiment is relatively simple and can be carried out in a batch equilibrium rebinding experiment or by frontal analysis. 

For the case of a binary system with linear adsorption isotherms, very simple formulas can be derived to evaluate the better TMB flow rates [19, 20]. For the linear case, the net fluxes constraints are reduced to only four inequalities, which are assumed to be satisfied by the same margin /3 (/3 > 1) and so ... 

As noted in Chapter 1, the most simple and theoretically sound model for drug-receptor interaction is the Langmuir adsorption isotherm. Other models, based on receptor behavior (see Chapter 3), are available. One feature of all of these models (with the exception of some instances of the... 

Such a model should be as simple as possible, without however missing any of the underlying thermodynamic and physicochemical factors which cause electrochemical promotion. In particular it will be shown that even the use of Langmuir-type adsorption isotherms, appropriately modified due to the application of potential (or equivalently by the presense of promoters) suffice to describe all the experimentally observed rules G1 to G7 as well as practically all other observations regarding electrochemical promotion including the effect of potential on heats of adsorption as well as on kinetics and reaction orders. 

Bi-layer adsorption is not uncommon and the development of the bi-layer adsorption isotherm equation is a simple extension of that used for the mono-layer equation. The Langmuir equation for bi-layer adsorption is as follows ... 

Volume overload can be treated in a simple way by the plate theory (8,9). In contrast, the theory of mass overload is complicated (10-12) and requires a considerable amount of basic physical chemical data, such as the adsorption isotherms of the solutes, before it can be applied to a practical problem. Volume overload is useful where the solutes of interest are relatively insoluble in the mobile phase and thus, to apply a sample of sufficient size onto the column, a large sample volume is necessary. If the sample is very soluble in the mobile phase then mass overload might be appropriate. 

The principle underlying surface area measurements is simple physisorb an inert gas such as argon or nitrogen and determine how many molecules are needed to form a complete monolayer. As, for example, the N2 molecule occupies 0.162 nm at 77 K, the total surface area follows directly. Although this sounds straightforward, in practice molecules may adsorb beyond the monolayer to form multilayers. In addition, the molecules may condense in small pores. In fact, the narrower the pores, the easier N2 will condense in them. This phenomenon of capillary pore condensation, as described by the Kelvin equation, can be used to determine the types of pores and their size distribution inside a system. But first we need to know more about adsorption isotherms of physisorbed species. Thus, we will derive the isotherm of Brunauer Emmett and Teller, usually called BET isotherm. 

Adequate description of many catalysts will require a large number of bits of data since they are usually rather complicated materials rather than simple chemicals. Attempts at tMs were just beginning by ICC 1, but now, one expects authors to give specific surface areas and some details of the porosity of their catalysts. Automation of the former tedious point by point measiirement of the N2 adsorption isotherm has greatly facilitated this. 

Heterogeneous chemical reactions in which adsorbed species participate are not pure chemical reactions, as the surface concentrations of these substances depend on the electrode potential (see Section 4.3.3), and thus the reaction rates are also functions of the potential. Formulation of the relationship between the current density in the stationary state and the concentrations of the adsorbing species in solution is very simple for a linear adsorption isotherm. Assume that the adsorbed substance B undergoes an... 

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