The Gibbs Adsorption Equation

The combined expression for the first and second laws for a system that 

The internal energy associated with the interface is the difference between the internal energy of the system and that of the a and (3 phases, as determined by the position of the Gibbs dividing surface, 

Differentiation of Equation (7.15) for an infinitesimal change in state of the system yields 

Dividing through Equation (7.18) hy the interfacial area, A, and rearranging yields 

In order to define unambiguously the state of a bulk phase, a, it is necessary to specify the values of several variables associated with that state. Those variables include the temperature, T , its volume, y , and its composition, rt-. Alternatively, the system pressure, P , may be fixed. In terms of the Helmholtz free energy F, the system can be specified by 

For a two-phase system, a similar equation can be written for the second or 13 phase. At equilibrium, the two phases will have the same temperature, T the same pressure, P and the same chemical potential, /x, for all components. The complicating factor in a system of two phases in contact is that the presence of the a-f3 interface that may be considered to be a third phase, (T, and which will make a separate contribution to the overall energy of the system. The total energy, then, will be given by 

In bulk thermodynamics, one can derive the Gibbs-Duhem equation by integration of Equation (9.5) while holding the intensive properties T, P, and (Xi constant to give 

Similarly, the interfacial phase yields the Gibbs adsorption equation 

For a two-component Uquid-vapor system where the Gibbs dividing surface is defined so that the surface excess concentration of the solvent is zero (F = 0), the summation in Equation (9.12) is no longer necessary and a simple relationship between the surface tension of the liquid phase, o, and the surface excess concentration of solute i, F is obtained. It is therefore possible to employ experimentally accessible quantities such as surface tension and chemical potential to calculate the surface excess concentration of a solute species and to use that information to make other indirect observations about the system and its components. 

One of the most important equations in surface thermodynamics is that which links changes of surface tension to adsorption processes, The derivation of this equation is given by Appendix III. 

The equation is most commonly applied to the case in which the temperature is constant, when it is known as the Gibbs adsorption isotherm, and has the form, for a binary mixture, 

Since in any stable phase the chemical potential increases with increase in concentration, it follows that if component 2 is adsorbed at the interface (F)1 positive) then the surface tension decreases with increase in concentration of component 2. 

Substances that are strongly adsorbed at an interface and hence cause a substantial lowering of the surface tension are called surface-active agents or surfactants. The difference between the surface tension (cr) of a solution of a surface-active agent at a mole fraction of x[, and that of the pure solvent (erf) is given simply by integrating equation (5.2) or (5.4)  

An important application of the Gibbs adsorption equation is to the calculation of the relative adsorption from measurements of the variation of surface tension with concentration  

An interface in real systems is an area of small but non-zero thickness (of about 10 molecular diameters). The adsorprion, as defined in Equation 4.7a, is the number of moles at an interface per area and has units of concentration (molar) per area. The number of moles at the interface and thus the adsorption itself can be either positive or negative (for compounds avoiding the interface). 

Using some algebra and concepts from thermodynamics (see Appendix 4.3), the Gibbs adsorprion equation can be derived  

Equation 4.7a provides the relationship between adsorption (which carmot be easUy measured directly) and the change of surface tension with concentration, dy/dCi (which is rather easily obtained from experiments). This equation is derived for the case of a single adsorbing solute, e.g. a non-ionic surfactant. For ionic surfactants (Chapter 5) we have two species that adsorb at the interface and we should use 

There are a couple of interesting points First, we cannot determine absolute adsorptions using the Gibbs equation, only relative ones, typically of the solute with respect to the solvent, e.g. of a surfactant assuming no solvent present at the interface. In addition, when the hquid phase is non-ideal, we should use activities (acrivily coefficient x concentration) instead of the concentration in Equation 4.7a, but typically we assume ideal dilute solutions and thus only Equation 4.7a,b will be used in our apphearions. 

There are many methods that have been used for experimental verification of the Gibbs adsorprion equation. McBain, using very thin films (0.05 mm) from surfactant solutions, measured the concentration of the interfacial phase. This and other methods have provided excellent agreement with the Gibbs adsorprion equation (Shaw, 1992). 

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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 ... 

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 ... 

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

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). 

In the previous sections we have noted that the hypothesis of a unimolecular Gibbs layer for solutions of liquids of markedly different internal pressures together with the equation of Gibbs leads to values for molecular areas and thicknesses which are not at all unreasonably different from those determined by means of X-ray measurements, or from a study of insoluble substances on the surface of water, but cannot be said to be identical within the limits of experiment. In one respect, however, such soluble films differ from the insoluble films which we shall have occasion to examine in the next chapter the surface tension of solutions which according to the Gibbs adsorption equation... 

Ionic surfactants are electrolytes dissociated in water, forming an electrical double layer consisting of counterions and co-ions at the interface. The Gouy-Chapman theory is used to model the double layer. In conjunction with the Gibbs adsorption equation and the equations of state, the theory allows the surfactant adsorption and the related interfacial properties to be determined [9,10] (The Gibbs adsorption model is certainly simpler than the Butler-Lucassen-Reynders model for this case.). 

Assuming that there are N components adsorbed at the interface, the Gibbs adsorption equation at the constant temperature gives... 

Calculate the surface energies of each of these liquids and plot a graph of y for the CTAB solutions as a function of logio(conc.). Use your results and the Gibbs adsorption equation (see later) to estimate the minimum surface area per CTAB molecule adsorbed at the air-water interface. 

Surface Adsorption. From Fig.l and Fig.2 we can calculate the total surface adsorption (17 ) of the RDH-surfac-tant mixture by applying the Gibbs adsorption equation(7). In the case of a mixed aqueous solution with a constant ionic strength, the equation is written as... 

Hence the surface adsorption of surfactant 1 and 2, and their surface mole fractions can be obtained from the surface (interfacial) tension-concentration relationships (Fig.1 and fig.2) by applying the Gibbs adsorption equation. 

Information on the chemical potentials of components in a solution of biopolymers can serve as a guide to trends in surface activity of the biopolymers at fluid interfaces (air-water, oil-water). In the thermodynamic context we need look no further than the Gibbs adsorption equation,... 

In line with the Gibbs adsorption equation (equation 3.33 in chapter 3), the presence of thermodynamically unfavourable interactions causes an increase in protein surface activity at the planar oil-water interface (or air-water interface). As illustrated in Figure 7.5 for the case of legumin adsorption at the n-decane-water interface (Antipova et al., 1997), there is observed to be an increase in the rate of protein adsorption, and also in the value of the steady-state interfacial pressure n. (For the definition of this latter quantity, the reader is referred to the footnote on p. 96.)... 

This is the Gibbs adsorption equation that relates y to the number of moles and the chemical potentials of the components in the interface. 

We have conceptually arrived at the same point as we did in the argument leading to Equation 4 for Equation 11 is valid for arbitrary choice of multipliers x and y, and each such choice corresponds to a different Gibbs convention. The choice which leads to the Gibbs adsorption equation is that which makes the coefficients of dP and dm (conventionally defined to be the solvent) vanish ... 

While the derivation of these quantities seems at first a bit of a mathematical card trick without a real application, interfacial scientists do utilize these equations on a day-to-day basis. The Gibbs adsorption equation that relates interfacial tension to the interfacial coverage is a perfect example of a thermodynamic relationship that can be obtained from... 

Takahashi et al.67) prepared ionene-tetrahydrofuran-ionene (ITI) triblock copolymers and investigated their surface activities. Surface tension-concentration curves for salt-free aqueous solutions of ITI showed that the critical micelle concentration (CMC) decreased with increasing mole fraction of tetrahydrofuran units in the copolymer. This behavior is due to an increase in hydrophobicity. The adsorbance and the thickness of the adsorbed layer for various ITI at the air-water interface were measured by ellipsometry. The adsorbance was also estimated from the Gibbs adsorption equation extended to aqueous polyelectrolyte solutions. The measured and calculated adsorbances were of the same order of magnitude. The thickness of the adsorbed layer was almost equal to the contour length of the ionene blocks. The intramolecular electrostatic repulsion between charged groups in the ionene blocks is probably responsible for the full extension of the... 

Surfactants are compounds that exhibit surface activity, or more generally, interfacial activity, and migrate to the interface when placed in solution. This migration results in lowering the solution surface tension (interfacial tension) as compared to the surface tension of the pure solvent. Thermodynamically, adsorption of a surfactant is deLned by the Gibbs adsorption equation ... 

The Gibbs adsorption equation enables the extent of adsorption at a liquid surface to be estimated from surface tension data. 

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