Gibbss adsorption equation

The adsorption of an adsorbent of a solution /3 on an adsorbate a is formally described below. The adsorbent is atoms or molecules and the solution a liquid or a 

By subtracting the internal energy of the two homogeneous phases adjacent to the dividing surface from equation (6.76) the internal energy of the dividing surface is obtained  

Differentiation of eq. (6.77), when combined with eq. (6.14), gives the Gibbs-Duhem equation in internal energy for a system where the surface energy is not negligible  

Reorganization of eq. (6.78) and using eq. (6.3) leads to the Gibbs adsorption equation  

For solid-gas and liquid-gas interfaces, where /3 is a gas phase (eq. 6.79) can be further simplified if the adsorbate contains only two components, A and B, since changes in the chemical potential of the two components of the adsorbate due to a change in mole fraction are related by eq. (1.92) as 

The Gibbs adsorption equation has been a subject of many investigations in the literature (Chattoraj and Birdi, 1984 Fainerman et al., 2002). Gibbs considered that the interfacial region is inhomogeneous and difficult to define, and he therefore also considered a more simplified case in which the interfacial region is assumed to be a mathematical plane. 

Gibbs defined a quantity, surface excess fn, of the ith component as follows  

In an exactly similar manner, one can define the respective surface excess internal energy, E,, and entropy, by the following mathematical relationships (Birdi, 1989 Chattoraj and Birdi, 1984)  

Here E and S are the total energy and entropy, respectively, of the system as a whole for the actual liquid system. The energy and entropy terms for a and 6 phases are denoted by the respective superscripts. The excess (x) quantities thus refer to the surface molecules in an adsorbed state. At constant T and p, for a two-component system (say water(l) + alcohol(2)), the classical Gibbs adsorption equation has been derived (Adamson and Gast, 1997 Chattoraj and Birdi, 1984)  

The chemical potential P2 is related to the activity of alcohol by the equation 

In deriving the Gibbs adsorption equation we need to define an interface or interfacial area. As the interface in real systems is an area of small but non-zero 

The starting point is the thermodynamic definition of the surface tension based on the Gibbs energy  

We recall the definitions of the Gibbs energy in relation to enthalpy and internal energy  

The change of Gibbs energy at the surface ( r) should also include the effect of surface tension and so the above equation is written as  

At equUibria the chemical potentials of the compounds in aU phases including the surface phase are equal. 

Adsorption equilibrium of hydrated ions at the interface of metal electrodes is represented by the Gibbs adsorption equation as in Eqn. 5-17  

For the adsorption of chloride ions on the interface of metallic electrode in aqueous potassium chloride solution, the Gibbs adsorption equation is written as in Eqn. 5-18  

Other than [/ , that is, W, F°, and G°. It is further noted that the summation in Equations 3.84 and 3.85 includes all components except the one (usually the solvent) for which T is set zero by the Gibbs convention on positioning the interface. 

Under isothermal conditions, which is often the case in practice. 

For an ideal gas or solution, for which is given by Equation 3.30, Equation 3.87 

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. 

In the case of competitive adsorption, that is, the simultaneous adsorption of two or more components from a mixture, it follows from cross-differentiation in Equation 3.85 that 

It is important to remember that an equilibrium is established between the surfactant molecules at the surface or interface and those remaining in the bulk of the solution. This equilibrium is expressed in terms of the Gibbs equation. In developing this expression it is necessary to imagine a definite boundary between the bulk of the solution and the interfacial layer (see Fig. 6.3). The real system containing the interfacial layer is then compared with this reference system, in which 

We con treat the thermodynamics of the surface layer in a similar way to the bulk of the solution. The energy change, dU, accompanying on infinitesimal, reversible change in the system is given by 

For an open system (one in which there is transfer of material between phases) equation (6.2) must be written 

When applying equation (6.3) to the surface layer, the work is that required to increase the area of the surface by an infinitesimal amount, dA, at constant 7, P and n. This work is done against the surface tension and is given by equation (6.1) as dw = y dA. 

If the energy, entropy and number of moles of component are allowed to increase from zero to some finite value, equation (6.4) becomes 

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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... 

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. 

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. 

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 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 ... 

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. 

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 ... 

Gibbs adsorption equation phys chem A formula for a system involving a solvent and a solute, according to which there Is an excess surface concentration of solute if the solute decreases the surface tension, and a deficient surface concentration of solute if the solute increases the surface tension. gibz ad sorp shan i.kwa-zhon Gibbs adsorption isotherm physchem An equation for the surface pressure of surface [< ... 

The role of the cosurfactant in reducing the interfacial tension can be understood from application of the Gibbs adsorption equation in the form (14). 

Gibbs adsorption equation is an expression of the natural phenomenon that surface forces can give rise to concentration gradients at Interfaces. Such concentration gradient at a membrane-solution Interface constitutes preferential sorption of one of the constituents of the solution at the interface. By letting the preferentially sorbed Interfacial fluid under the Influence of surface forces, flow out under pressure through suitably created pores in an appropriate membrane material, a new and versatile physicochemical separation process unfolds itself. That was how "reverse osmosis" was conceived in 1956. 

To analyze these data, the well-known Gibbs adsorption equation (Chattoraj and Birdi, 1984 Birdi, 1989) has to be used. A liquid column containing i number of components is shown in Figure 3.14, according to the Gibbs treatment of two bulk phases, that is, a and (3, separated by the interfacial region AA BB. ... 

At constant T and p, for a two-component system (say, water(l) + alcohol(2)), we thus obtain the classical Gibbs adsorption equation as... 

The Gibbs adsorption equation is a relation about the solvent and a solute (or many solutes). The solute is present either as excess (if there is an excess surface concentration) if the solute decreases the y, or as a deficient solute concentration (if the surface tension is increased by the addition of the solute). 

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