Adsorption equations

These surface active agents have weaker intermoiecular attractive forces than the solvent, and therefore tend to concentrate in the surface at the expense of the water molecules. The accumulation of adsorbed surface active agent is related to the change in surface tension according to the Gibbs adsorption equation... 

An adsorption equation known as the Frumkin isotherm has the form... 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

It must be remembered that, in general, the constants a and b of the van der Waals equation depend on volume and on temperature. Thus a number of variants are possible, and some of these and the corresponding adsorption isotherms are given in Table XVII-2. All of them lead to rather complex adsorption equations, but the general appearance of the family of isotherms from any one of them is as illustrated in Fig. XVII-11. The dotted line in the figure represents the presumed actual course of that particular isotherm and corresponds to a two-dimensional condensation from gas to liquid. Notice the general similarity to the plots of the Langmuir plus the lateral interaction equation shown in Fig. XVII-4. 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

Bmnauer-Emmett-TeUer (adsorption equation) twice daily t-butyloxycarbonyl biochemical (biological) oxygen demand boiling point becquerel... 

The value of 9 can be estimated on purely theoretical grounds from estimates of the adsorption of surfactant which, in turn, can be estimated from the Gibbs adsorption equation relating adsorption to surface-tension lowering. 

This equation is the mono-layer adsorption isotherm of solvent (B) and is exactly the same as the previously derived Langmuir adsorption equation but somewhat differently expressed. 

Determination of the equilibrium spreading pressure generally requires measurement and integration of the adsorption isotherm for the adhesive vapors on the adherend from zero coverage to saturation, in accord with the Gibbs adsorption equation [20] ... 

The dissolution of passive films is, in the main, controlled by a chemical activation step in contrast to film-free conditions at. Many protective anodic films are oxides and hydroxides whose dissolution depends upon the hydrogen ion concentration, and the rate follows a Freundlich adsorption equation ... 

The Gibbs adsorption equation for the adsorption of an ion / from solution can be written in the form of the thermodynamic equation... 

The adsorption equation shows that a solute may very strongly lower the surface tension of a solvent, but cannot strongly raise it, since although T may reach high values by positive adsorption (in some cases, as with solutions of some aniline dyes, the pure solute appears as a thin skin on the surface), it can never sink below that of the pure solvent by negative adsorption. 

At constant p and T, the Gibbs adsorption equation for an electrode interface leads to the well-known Lippmann equation12 ... 

The identification of bi-layer adsorption of polar solvents on the surface of silica gel arose from some work by Scott and Kucera (5) who measured the adsorption isotherms of the some polar solvents, ethyl acetate, isopropanol and tetrahydrofuran from n-heptane solutions onto silica gel. The authors found that the experimental results for the more polar solvents did not fit the simple mono-layer adsorption equation and, as a consequence, the possibility of bi-layer adsorption on the silica gel surface was examined. 

The adsorption of a component j in a given system depends on temperature T and on the component s concentration, Cyj, in the bulk phase. The overall adsorption equation can be written as Aj =f(T, Cyj). The relation between adsorption and the adsorbate s bulk concentration (or pressure, in the case of gases) at constant temperature is called the adsorption isotherm the relation between adsorption and temperature at constant concentration is called the adsorption isobar. From the shape of the adsorption isotherms, the adsorption behavior can be interpreted. In the case of monolayer adsorption, the isotherms are usually written in the form 9 =f(Cyj). (The subscript j is dropped in what follows.)... 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

The appreciation of the importance of adsorption phenomena at liquid interfaces is probably as old as human history, since it is easily recognized in many facets of everyday life. It is not surprising that liquid interfaces have been a favorite subject of scientific interest since as early as the eighteenth century [3,4], From an experimental point of view, one obvious virtue of the liquid interfaces for studying adsorption phenomena is that we can use surface tension or interfacial tension for thermodynamic analysis of the surface properties. The interfacial tension is related to the adsorbed amount of surface active substances through the Gibbs adsorption equation. 

For this system the Gibbs adsorption equation, Eq. (1), takes the form [2]... 

This relationship is termed the Gibbs adsorption equation. 

In a study of adsorption of organic herbicides by montmorillonite, Bailey and colleagues138 found that none of the compounds conformed to the Langmuir adsorption equation. Of the 23 compounds tested, only a few did not conform well to the Freundlich equation. 

For orthokinetic conditions the ratio of adsorption halftime to coagulation halftime can be calculated by integrating the adsorption Equation (6) and the coagulation Equation (2). 

These excess quantities are independent of the thickness chosen for the interface as long as it incorporates the region where the concentrations are different from those in the bulk that is, it does not matter if one chooses too thick a region (see Problem 1). We cannot refer the surface concentrations of the metal particles M, Mz+, and e to the solution. Nevertheless we will drop the asterisk in their surface concentrations to simplify the writing we will eliminate these quantities later. We can now rewrite the Gibbs adsorption equation in the form ... 

The experimental verification of the adsorption equation is comparatively simple, apart from certain experimental difficulties. Solutions of different known concentrations are prepared and equal amounts of adsorbent are placed in equal volumes of these solutions. Agitation of some sort is generally necessaiy to ensure complete adsorption. When equilibrium has been reached, the concentrations of the various solutions are again determined and represent, of course, the end or equilibrium concentrations, viz., the values of C in the formula. The differences between these and the original concentrations are... 

---