Isotherm derivation, from Gibbs equation

Adsorption isotherms represent a relationship between the adsorbed amount at an interface and the equilibrium activity of an adsorbed particle (also the concentration of a dissolved substance or partial gas pressure) at a constant temperature. The analysis of adsorption isotherms can yield thermodynamic data for the given adsorption system. Theoretical adsorption isotherms derived from statistical and kinetic data, and using the described assumptions (see 3.1), are known only for the gas-solid interface or for dilute solutions of surfactants (Gibbs). Those for the system gas-solid are of a few basic types that can be thermodynamically predicted81. From temperature relations it is possible to calculate adsorption and activation energies or rate constants for individual isotherms. Since there are no theoretically founded equations of adsorption isotherms for dissolved surfactants on solids, the adsorption of gases on solides can be used as a starting point for an interpretation. 

Thermodynamically Consistent Isotherm Models. These models include both the statistical thermodynamic models and the models that can be derived from an assumed equation of state for the adsorbed phase plus the thermodynamics of the adsorbed phase, ie, the Gibbs adsorption isotherm,... 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

This deviation is derived from the isothermal-isobaric form of the Gibbs-Duhem equation and can be applied to isobaric equilibrium data by minimizing the composition and temperature differences between pairs of points. This was done by choosing data paths through the ternary equilibrium diagram so that differences in composition and temperature between pairs of points were minimized. A point was considered to be inconsistent if it deviated more than 0.02 with both of its neighbors. This deviation was chosen since it approached the approximate limit of the deviation caused by analytical inaccuracy. Runs 1, 4, 5, 13, 15, 23, 27, and 28 were determined to be inconsistent on this basis. 

The classical theory of the Gibbs adsorption isotherm is based on the use of an equation of state for the adsorbed phase hence it assumes that this adsorbed phase is a mobile fluid layer covering the adsorbent surface. By contrast, in the statistical thermod)mamic theory of adsorption, developed mainly by Hill [15] and by Fowler and Guggenheim [12], the adsorbed molecules are supposed to be localized and are represented in terms of simplified physical models for which the appropriate partition function may be derived. The classical thermodynamic fimctions are then derived from these partition fimctions, using the usual relationships of statistical thermodynamics. 

The comparison of the empirical Szyszkowski equation (II. 18) with the Gibbs equation (II.5) indicates that Langmuir adsorption isotherm (11.22) is well suited also for the description of adsorption at the air - surfactant solution interface. It is interesting to point out that at the gas - solid interface, for which eq. (11.22) was originally derived various deviations from Langmuirian behavior are often observed. 

As there is no analogue of Butler s equation for ionised surface layers, the procedure used to derive the equation of state has to be based on the Gibbs adsorption equation and a model adsorption isotherm. The isotherm equation can also be derived from the theoretical analysis of the expressions for electrochemical potentials of ions. For the solution of a single ionic surfactant RX, with the addition of inorganic electrolyte XY, starting from Eqs. (2.2) and (2.21) for the electrochemical potentials, one obtains the adsorption isotherm... 

The dependencies of IT on B for various 0 values, calculated from Eqs. (2.118) and (2.119), are shown in Fig. 2.21. It is seen that the phase transition in the monolayer is characterised by a decreased slope of the n (B)-curves, in contrast to the two-dimensional condensation model based on the Gibbs equation (2.114) [119-128] or that derived in [1, 104, 105] employing the Frumkin equation and Maxwell construction. With increasing B, all isotherms exhibiting a... 

We start this book with a chapter (Chapter 2) on the fundamentals of pure component equilibria. Results of this chapter are mainly applicable to ideal solids or surfaces, and rarely applied to real solids. Langmuir equation is the most celebrated equation, and therefore is the cornerstone of all theories of adsorption and is dealt with first. To generalise the fundamental theory for ideal solids, the Gibbs approach is introduced, and from which many fundamental isotherm equations, such as Volmer, Fowler-Guggenheim, Hill-de Boer, Jura-Harkins can be derived. A recent equation introduced by Nitta and co-workers is presented to allow for the multi-site adsorption. We finally close this chapter by presenting the vacancy solution theory of Danner and co-workers. The results of Chapter 2 are used as a basis for the... 

We have addressed the various adsorption isotherm equations derived from the Gibbs fundamental equation. Those equations (Volmer, Fowler-Guggenheim and Hill de Boer) are for monolayer coverage situation. The Gibbs equation, however, can be used to derive equations which are applicable in multilayer adsorption as well. Here we show such application to derive the Harkins-Jura equation for multilayer adsorption. Analogous to monolayer films on liquids, Harkins and Jura (1943) proposed the following equation of state ... 

We see that many isotherm equations (linear, Volmer, Hill-deBoer, Harkins-Jura) can be derived from the generic Gibbs equation (2.3-13). Other equations of state relating the spreading pressure to the surface concentration can also be used, and thence isotherm equations can be obtained. The following table (Table 2.3-1) lists some of the fundamental isotherm equations from a number of equations of state (Ross and Olivier, 1964 Adamson, 1984). 

Table 2.3.1. Isotherm Equations derived from the Gibbs Equation...

In the last chapter, we discussed the description of pure component adsorption equilibrium from the fundamental point of view, for example Langmuir isotherm equation derived from the kinetic approach, and Volmer equation from the Gibbs thermodynamic equation. Practical solids, due to their complex pore and surface structure, rarely conform to the fundamental description, that is very often than not fundamental adsorption isotherm equations such as the classical Langmuir equation do not describe the data well because the basic assumptions made in the Langmuir theory are not readily satisfied. To this end, many semi-empirical approaches have been proposed and the resulting adsorption equations are used with success in describing equilibrium data. This chapter will particularly deal with these approaches. We first present a number of commonly used empirical equations, and will discuss some of these equations in more detail in Chapter 6. 

J. Derivation of Isotherm Equations from the Gibbs Equation 69 and with the Gibbs isotherm [Eq. (3.32)]... 

Since the assumptions from which Eqs. (3.101) and (3.104) are derived cannot be more than rough approximations which may be expected to break down at high sorbate concentrations, an alternative approach based on Eq. (3.105) has been adopted for the correlation and analysis of equilibrium data for strongly adsorbed species such as the xylenes on X and Y zeolites. By integration of the Gibbs equation one may calculate the spreading pressure (or surface potential ) as a function of equilibrium vapor pressure or sorbate concentration, directly from an experimental isotherm [Eq. (3.50)]. It follows from Eqs. (3.90), (3.94), and (3.99) that... 

For fluid interfaces, the Gibbs equation is often used to establish adsorbed amounts of i from the experimentally determined dependency of y on X . This approach is especially useful when little surface area is available so that F cannot be established analytically. In this way, the functionality r,(X ), the adsorption isotherm, can be derived. 

The ELBT program on the CD-ROM permits the correlation of isothermal VLE data of homogeneous binary systems with equations derived from the Redlich-Kister expansion. Vapor-phase imperfection and the variation of the Gibbs energy of the pure liquid components are accounted for through the second molar virial coefficients By and the molar volumes V° under saturation pressures (Chap. 3.5.5). The correlated total vapor pressure P and... 

We have to remember that the isotherm equation was derived based on the assumption that the adsorbate is inert and is not modified upon adsorption. This assumption is not broadly valid and we know that adsorption modifies the surface to a significant extent. However, for all practical purposes, the Gibbs isotherm equation and the isotherms we will derive from it provide practical equations. 

It is seen that the additional (nonnalised) activity coefficients introduced in Eq. (2.10) to establish the consistency between the standard potentials of the pure components and those at infinite dilution, can be incorporated into the constant Kj in Eq. (2.15). Therefore, if a diluted solution with activity coefficients of unity is taken as the standard state, the form of Eqs. (2.13) and (2.14) remains unchanged. The equations (2.14) and (2.15) are the most general relationships from which meiny well-known isotherms for non-ionic surfactants can be obtained. For further derivation it is necessary to express the surface molar fractions, x-, in terms of their Gibbs adsorption values Tj. For this we introduce the degree of surface coverage, i.e. 9j = TjCOj or 0j = TjCO. Here to is the partial molar area averaged over all components or all... 

The derivation of the Gibbs adsorption isotherm from Eqs. (3.23) and (3.24) follows essentially the same logic as the derivation of the Gibbs-Duhem equation. At constant temperature and neglecting the term PdV, Eq. (3.23)... 

For the derivation of Volmer s and Fowler Guggenheim isotherms from the gas phase equations of state and Gibbs adsorption isotherms, we will refer the reader to Ruthven (1984). 

Hydrophilic surfactants adsorb best on aqueous phases, whereas hydrophobic surfactants adsorb best on lipophilic surfaces (oils). Data on adsorption at constant temperature are usually plotted as a function of the surfactant equilibrium concentration plots for solid substrates are termed Langmuir isotherms. From such isotherms the maximum surfactant concentration at the interface (Fmax) can be derived and the maximum area occupied by the surfactant at the interface ( max) can be calculated. In addition, the Gibbs adsorption equation can be extracted. 

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