Adsorption equations for

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

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

Substitution of these relations into the expression of Ax,(= x - x,) and simple algebraic transformations leads to the usual adsorption equation for the reduced surface excess... 

From thermodynamics, the lowering of surface free energy due to surfactant adsorption is given by the Gibbs adsorption equation for a binary, isothermal system containing excess electrolyte ... 

Ikeda,S. (1977) On the Gibbs adsorption equation for electrolyte solutions. Bull. Chem. Soc. Japan, 50(6), 1403-08. 

For a neutral gas, the adsorption (which is proportional to the free energy of the system, see Eq. (8)) can be calculated easily. For charged particles one should account in the Gibbs adsorption equation for the adsorption of all particles of the system (including those responsible for the charging of the surface [23]). Therefore, one should first identify the mechanism of formation of the double layer. In this case,... 

The thermodynamics of the electrochemical interface is based on the Gibbs adsorption equation. For a plane electrode in contact with an ionic conductor, under equilibrium conditions, the Gibbs equation is [1]... 

The reason for the lowering of y when using two surfactant molecules can be understood from consideration of the Gibbs adsorption equation for multicomponent systems [9]. For two components, sa (surfactant) and co (cosurfactant). Equation (15.4) becomes. 

Some molecules are dissociated upon adsorption, and this type of adsorption is called dissociative adsorption. The adsorption reaction can be shown as [(XY)gas + 2 Surface <-> Xads-Surface + Yads-Surface], and the Langmuir adsorption expression must be modified because two sites on the adsorbent are consumed per adsorbate molecule. The probability of desorption is also different. When these differences are considered, the Langmuir adsorption equation for the dissociative adsorption becomes... 

Refined methods for combining the Kelvin relationship with adsorption equations for obtaining pore size distribution have recently been suggested (Barrett and Joyner, 2 Shull, Elkin, and Roess, 57 Wheeler, 64, 64a). With some experience, however, simple visual inspection of the isotherms proper provides a reasonable impression of the relative pore size distributions. It must be borne in mind when considering pore structure and particularly the structure of high area gels that the shape of the ultimate particles and thus the shape of the pores is not known. It remains to be established whether the particles are platelets, fibers, spheres, or complex combinations of many structures. 

The main problem in the thermodynamic theory of penetration is to determine the dependence of the adsorption of a soluble surfactant on its bulk concentration for any given (constant) adsorption of the insoluble surfactant (surface concentration), and the onset of the surface pressure jump in mixed monolayers, caused by the adsorption of a soluble surfactant in the presence of the insoluble component. There exist several main theoretical approaches to the description of the penetration thermodynamics. One is based on the Gibbs adsorption equation for multicomponent monolayers [143-146], Another approach, initially proposed by Pethica... 

In the Gibbs adsorption equation for multicomponent surface layers (2.22) the value of p for soluble components can refer both to the bulk and to the surface layer (as equilibrium exists), and for the insoluble components to the surface layer only. For systems with one insoluble and one soluble component, denoted by subscripts 1 and 2, respectively, and the assumption that the area per mole of the insoluble component 1 is Aj = l/F, Eq. (2.22) can be rewritten as... 

To obtain an equation for a heterogeneous solid, the adsorption equation for a micropore volume element dW is ... 

Write the Gibbs adsorption equation for a three-component system in which species (1) is the solvent, species (2) is the primary surface-active solute, and species (3) is an impurity in (2) present at a constant percent. It is found that in the dilute solution region a plot of surface tension versus the concentration of (2) goes through a minimum. Provide an... 

Show that the Gibbs adsorption equation for constant temperature is satisfied by ideal surface solution theory as discussed in Section 8. Note that T, = (nfla), for / = 1, 2. 

It may be, however, that by selecting the most important aspects, useful patterns can be delineated. It would appear reasonable to start with those substances having the highest ion-exchange capacity, to study solution-solid equilibria for the most common ions, to carry out measurements over a wide range of compositions, and to evaluate the extent to which results can be correlated by conventional adsorption equations, for example, those describing ion exchange. 

It is thus demonstrated that the factor 2 in the Gibbs adsorption equation for a strong monovalent ionic surfactant vanishes when changing the ionic strength of the solution from low to high. In other words, at high ionic strength the ionic surfactant tends to behave as a nonionic surfactant. 

Mixture of Anionic Surfactants onto Alumina. Most EOR surfactants are mixtures of isomers, but these mixtures are too complex for application of basic theory. In contrast, the effectiveness of ideal solution theory in explaining region II adsorption for binary mixtures of anionic surfactants has been demonstrated [5Jj. These controlled isomeric mixtures allow application of the ideal solution theory. The application of this theory utilized a reduced adsorption equation for mixtures of anionic surfactants [52]. The parameters for this reduced... 

In (2.12) n x) refers to the polymer segment concentration profile near a single wall whereas in (2.11) n x) is the profile between two walls. Expression (2.10) is the extension of the Gibbs adsorption equation for a single surface to the case of two surfaces at finite separation [6-8]. Integration of (2.10) gives... 

Applying exactly the same line of reasoning as for the derivation of the extended Gibbs adsorption equation for two flat plates, see (2.9), we now obtain... 

Reference has already been made to the unusual behavior, e.g., lowering of surface tension (57,58,75), development of surface viscosity (74), and so forth, observed in mixed solutions of polycations and anionic surfactants. Buckingham et al. (75) arrived at the following form of the Gibbs adsorption equation for their system (poly-L-lysine/SDS) ... 

Sircar, S., and Gupta, R., Semi-empirical adsorption equation for single component gas-solid equilibria, AlChE J, 27(5), 806-812(1981). 

Ziolkowska, D., and Garbacz, J.K., Adaptation of single gas adsorption equations for the description of adsorption from non-aqueous liquid solutions of iodine onto active carbons, Adsorpt. Sci. Technol., 15(3), 155-164 (1997). 

Rudzinski, W., et al., Simple adsorption equation for adsorption of nonionic surfactants on hydrophilic surfaces of silica, Adsorpt. Sci. Technol., 2(4), 207-218 (1985). 

The linear differential equation given by Eq. (8) uniquely characterizes the interfaces using the surface tension and the various excess quantities. Applying the Young-Schwartz theorem, the Gibbs adsorption equation for a solid/fluid interface s = S fl can be deduced directly ... 

The Gibbs adsorption equation for a simple 1 1 ionic surfactant, such as NaSDS (SD, Na). 

Using the Gibbs adsorption equation for a multicomponent solution [this is Eq. (1) with c, = c, J, one can derive the long time asymptotics of the surface tension... 

Explain the difference between the Gibbs adsorption equation for non-ionic and ionic surfactants. 

This ehapter is devoted largely to the derivation of adsorption equations for the adsorption of solute at a solution—metal oxide (S—MO) interfaee. The derivations are based on rigorous Gibbs—Lewis thermodynamies [1], employing both mass aetion and mass balanee equations. The thermodynamie mass aetion/mass balanee (TMAB) approaeh is relatively new, appearing first in the literature in the mid-1990s [2-4]. Many of the subtle but important distinetions between TMAB and older adsorption models will be examined in Seetion VII. 

Adsorption at the nonaqueous— metal oxide interface will not be examined in this chapter. However, based on diseussions in this chapter and Chapter 7, adsorption equations for the nonwater solvent— metal oxide interfaces can be derived by using similar methods. 

A sufficient number of equations have now been derived to make possible the derivation of explieit adsorption equations for proton adsorption. This ean best be seen by eolleeting the relevant equations into a table in order to examine the number of equations and unknowns which have been introduced so far (see Table 1). 

In a similar fashion, adsorption equations for the mole fraction of the other surface sites can be derived from Eqs (9), (18) and (20) to give... 

---