The Gibbs Isotherm

The Gibbs isotherm assumes that adsorbed layers behave like liquid films, aud that the adsorbed molecules are free to move over the surface. This isotherm cau be derived then using classical thermodynamics using Gibbs free energy equations. This results in an isotherm of the form given in Equation 8.10. This is known as the Harkins-Jura (HJ) equation[90]. For details of derivation please refer to (3). 

Solution The data is for gas solid adsorption so we will concentrate on Langmuir, BET, and Gibbs isotherms for this purpose. 

Langmuir isotherm The equation for the Langmuir isotherm in terms of pressure and volume is given by the following equation 

This equation can be written in linear form as follows 

The isotherms presented earlier are applicable to a single component system. The BET isotherm and the Gibbs isotherm are modified for multiple components only for some particular cases and could not be generalized. On the other hand linear, Langmuir, and Freundlich isotherms can be generalized for multicomponent systems and are given below for component i. 

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The capillary-rise method is used to study the change in surface tension as a function of concentration for aqueous solutions of />butanol and sodium chloride. The data are interpreted in terms of the surface concentration using the Gibbs isotherm. 

In [17] we have calculated (Ss(r)) and shown that the value of (,s(r)) on the wall verifies the contact theorem. The integral of (s(r)) gives the adsorption I. From the calculation of the free energy we have obtained the surface tension and shown that the Gibbs isotherm leads to the same F as the one calculated via the density profile. Thus, as in the case of the homogeneous Yukawa fluid we have a totally self consistent calculation. 

Gas-solid equilibria have been studied for over 200 years, since Fontana showed that activated charcoal adsorbs gases and vapors at room temperature [1]. A considerable amoxmt of theoretical and experimental literature is available. The Gibbs isotherm [2] and the multilayer adsorption theory of Brunauer, Emmett and Teller [3], provide serious theoretical guidelines and support in understanding the results of experimental studies. Although, gas-sohd isotherms are difficult to predict quantitatively [4], this branch of adsorption thermod3mamics is much easier than liquid-solid adsorption because of the relative simplicity of the gas-sohd interface as compared to the liquid-solid interface. The Gibbs equation relates the amoimt of a compoimd adsorbed per unit surface area of a hquid-gas or a hquid-hquid interface and the surface or interfacial tensions [2]. This relationship provides a useful theoretical framework. 

Combined with the Gibbs isotherm (Eq. 3.6) it gives the virial isotherm equation... 

Two serious difficulties are encountered with the Freimdlich isotherm. First, from a fundamental point of view, this isotherm violates the Gibbs isotherm equation and it is not thermodynamically consistent. From a more practical point of... 

The surface tension of aqueous organic solution is lower than that of water. " The Gibbs isotherm predicts surface excess of organic molecules at the gas/ liquid interface, leading to a nonlinear decrease of the surface tension with respect to the bulk mole fraction of organic molecules in the solution. The same effect is expected to occur in the condensed liquid layer of water and alcohol on the substrate surface. The saturated surface concentration of small alcohols reaches molecules/cm, resulting in an... 

Perhaps the most useful general approach is still the ideal adsorbed solution theory (lAST) developed many years ago by Myers and Prausnitz [26]. The spreading pressure for the pure components may be calculated as a function of equilibrium pressure (p,) by integration of the Gibbs isotherm ... 

Let s introduce the concept of surface concentration of surfactant (measured in g mole/m ) [2]. If the environment is a binary solution with the volume molar concentration C of surfactant dissolved in it, then the quantities F and C are related by the Gibbs equation (aka the Gibbs isotherm) ... 

In many practical applications, the wetting liquid in question is a solution, e.g. an aqueous solution containing surface-active components. Then, the possibility of adsorption at all interfaces surrounding the three-phase contact-line (tcl) must be considered. According to the Gibbs isotherm for adsorption at the ij interface ... 

It is now seen that the Gibbs isotherm equation (2.3-13) is very general, and with any proper choice of the equation of state describing the surface phase an... 

Using the equation of state (2.7-3) into the Gibbs isotherm equation (eq. 2.3-13), they obtained ... 

The approach of IAS of Myers and Prausnitz presented in Sections 5.3 and 5.4 is widely used to calculate the multicomponent adsorption isotherm for systems not deviated too far from ideality. For binary systems, the treatment of LeVan and Vermeulen presented below provides a useful solution for the adsorbed phase compositions when the pure component isotherms follow either Langmuir equation or Freundlich equation. These expressions are in the form of series, which converges rapidly. These arise as a result of the analytical expression of the spreading pressure in terms of the gaseous partial pressures and the application of the Gibbs isotherm equation. 

Substituting the above equation into the Gibbs isotherm equation (5.5-2) for pure component systems we have ... 

This means that from the knowledge of the partial pressures Pj and P2 we can solve eq. (5.5-5) and (5.5-7) for the spreading pressure as a function of partial pressures P, and P2. Knowing this, the adsorption isotherm of the species i in the binary mixture can be obtained by applying the Gibbs isotherm eq. (5.5-4), that is ... 

Proceeding in the same manner through the Gibbs isotherm Eq. (3.65) leads to... 

The Gibbs isotherm states that the adsorption of surfactants leads to a decrease of the interfacial tension y. However, the interfacial tension y will not completely vanish as long as the particulate and fluid phase can be distinguished. 

Consider an adsorption isotherm which represents the dependence of surface concentration on the equilibrium pressure, p, of adsorbate, in the gas phase (Fig. 4.1.). The general relation between directly measurable quantities F and p and two-dimensional pressure is given by the Gibbs isotherm [75] ... 

The adsorbed mixture is treated as a two-dimensional phase. From the Gibbs isotherm, one can calculate a spreading pressure for each component based on its pure component isotherm. The basic assumption of the IAS theory is that the spreading pressures are equal for all components at equilibrium. 

Interfacial tension reflects the interfacial free energy of the interface. One of the simplest methods to measure the interfacial tension is the drop volume method, which just measures the volume of the detaching aqueous drop from the tip of a capillary immersed in an organic solution. A reversed situation is also constructed, when the density of the organic solution is higher than the aqueous solution as shown in Figure 2.11. Interfacial tension y is analyzed by the Gibbs isotherm... 

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. 

Discuss the assumption of inert substrate in the derivation of the Gibbs isotherm. How realistic is such an assumption How would you remove this assumption How much improvement do you anticipate if you lifted this assumption ... 

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