Gibbs adsorption isotherm derivation

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,... 

Derivation of the Gibbs adsorption isotherm. Determination of the adsorption of surfactants at liquid interfaces. Laboratory project to determine the surface area of the common adsorbent, powdered activated charcoal. 

Beside the theoretically derived Gibbs adsorption isotherm, a large number of models have been developed that empirically describe a relationship between the interfacial coverage, the surface tension, and the surfactant concentration in the bulk phase. These adsorption isotherms are known under the names of the authors that first described them—i.e., the Fangmuir, Frumkin, or Volmer isotherms. A complete mathematical description of these isotherms is beyond the scope of this unit and the reader is encouraged to consult the appropriate literature instead (e.g., Dukhin et al., 1995). 

The Gibbs adsorption isotherm is a relationship between the surface tension and the excess interfacial concentrations. To derive it we start with Eqs. (3.27) and (3.28). Differentiation of... 

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. 

The Gibbs adsorption isotherm is derived in many standard textbooks of physical chemistry and surface chemistry. We shall not repeat this derivation here. Rather, we show how this isotherm is modified when it is applied in electrochemistry. 

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 use of an equation of state to derive isotherms is based on the assumption that the adsorbed layer can be treated as a two-dimensional phase. In this case, the fimdamental equations in classical thermodynamics can be applied. Thus, at constant temperature, the Gibbs adsorption isotherm becomes... 

Under these circumstances the distribution coefficient for MiXi is zero, and that for M2X2, infinity. It follows that the Galvani potential difference is not defined for this interface. However, the thermodynamics of the interface can be derived on the basis of the Gibbs adsorption isotherm. 

The isotherm equations discussed so far are based on simpHfied mechanistic models for the adsorbed phase. In an alternative approach, pioneered by Willard Gibbs [12], the adsorbed phase is regarded simply as a fluid held within the force field of the adsorbent and characterized by an equation of state. The Gibbs adsorption isotherm, which is derived in a manner similar to the derivation of the Gibbs-Duhem equation, may be written ... 

The derivative d In c /dT can be calculated for each adsorption isotherm in Table 1 and then the integration in Eq. (6) can be carried out analytically (17). The expressions for J thus obtained are also listed in Table 1. The integration of the Gibbs adsorption isotherm, Eq. (2), along with Eq. (6), yields (17) ... 

The reduced spreading pressure xp can be derived from the Gibbs adsorption isotherm. With the condition... 

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)... 

In the final part of considerations about early adsorption science it should be stated that only the most important conceptions and equations of adsorption isotherms have been discussed. However, the isotherms including the lateral interactions between molecules in the surface monolayer as well as the equations concerning mobile and mobile-localized adsorption have been omitted. These equations can be derived in a simple way by assuming that molecules in the surface phase produce the surface film whose behaviour is described by the so-called surface equation of state. This equation is a two-dimensional analogue of the three-dimensional equation of state and relates the surface pressure (spreading pressure) of the film to the adsorption. This adsorption can be expressed by the Gibbs adsorption isotherm [26]. Consequently, it is possible to... 

The thermodynamics of adsorption is particularly important for analyzing the catalytic reactions as weU as the surface characterization of the catalysts. In this section, first, we will derive the Gibbs adsorption isotherm, and then we will use the Gibbs adsorption isotherm to derive the more useful isotherms such as Langmuir s and Fowler Guggenheim isotherms. 

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). 

The shapes of the charge-potential and the capacity-potential curves follow from the Gibbs adsorption isotherm immediately. The shape of the surface-excess-potential curve is more difficult to derive. This will be the subject to be discussed next and in order to do that we shall consider the general adsorption isotherm... 

The dependence of interfacial tension upon applied potential may be derived by application of the Gibbs adsorption isotherm to the system of... 

Downer et al. [29] attempted to remove divalent metal ions by an addition of EDTA in amounts sufficient to chelate the contaminants, but below the concentration at which EDTA affects the surface tension. However, adsorption isotherms derived from surface tension with a Gibbs prefactor of 2 did not agree with those obtained from neutron reflection data. A better agreement was found when using a prefactor of 1.7, consistent with about 30% dissociation of counterions. 

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. 

Historically, the first derivation (by Gibbs) of an adsorption isotherm was that for the hquid—vapor interface. This derivation is presented next to put in place the nomenclature used in adsorption on both hquid—vapor and soUd—hquid interfaces. A derivation (by Langmuir) for adsorption at the sohd—hquid interface is presented after that for the hquid—vapor interface. Adsorption at the liquid—vapor interface is... 

Under electrochemical conditions and T, P = constant, adsorption isotherms can be derived using standard statistical considerations to calculate the Gibbs energy of the adsorbate in the interphase and the equilibrium condition for the electrochemical potentials of the adsorbed species i in the electrolyte and in the adsorbed state (eq. (8.15) in Section 8.2). A model for the statistical considerations consists of a 2D lattice of arbitrary geometry with Ns adsorption sites per unit area. In the case of a 1/1 adsorption, each adsorbed particle can occupy only one adsorption site so that the maximal number of adsorbed particles per unit area in the compact monolayer is determined by A ax = Ng. Then, this model corresponds to the simple Ising model. The number of adsorbed particles, A ads< and the number of unoccupied adsorption sites, No, per unit area are given by... 

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. 

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