The Gibbs Adsorption Isotherm

Gibbs [2] derived a thermodynamic relationship between the surface or interfacial tension y and the surface excess F (adsorption per unit area). The starting point of this equation is the Gibbs-Duhem equation, as given above [see Eq. (5.1)]. At equilibrium, where the rate of adsorption is equal to the rate of desorption, dG = 0. Hence, at a constant temperature, but in the presence of adsorption. 

Equation (5.3) is the general form for the Gibbs adsorption isotherm. The simplest case of this isotherm is a system of two components in which the solute [Eq. (5.2)1 is the surface-active component - that is, it is adsorbed at the surface of the solvent [Eq. (5.1)]. For such a case. Equation (5.3) may be written as  

Equation (5.9) allows the surface excess (abbreviated as Fi) to be obtained from the variation of surface or interfacial tension with surfactant concentration. Note that 2 C2, since in dilute solutions/2 1. This approximation is valid since most surfactants have a low cmc (usually 10 mol dm ) and adsorption is complete at or just below the cmc. 

Another important point can be made from the y-log C curves. At a concentration just before the break point there is a condition of constant slope, which indicates that saturation adsorption has been reached  

Since y is constant in this region, 02 must also remain constant, which means that the addition of surfactant molecules above the cmc must result in an association to form units (micellar) with low activity. 

The most fundamental equation governing the properties of interphases is the Gibbs adsoi-ption isotherm  

The surface tension, y, is given in units of force per unit length (10 dyne cm = lNm ). It is related to the two-dimensional surface pressure, n, by the simple equation 

We noted earlier that the driving force in chemistry is the decrease in Gibbs energy (cf Eq. (2.1)). Thus, a system will change spontaneously in the direction of decreasing surface tension. This leads to two observations  

1) A pure phase always tends to assume a shape that creates the minimxmi surface area per unit volume. This is why droplets of a liquid are almost spherical (they are completely spherical in the absence of gravity, in a spacecraft orbiting the Earth, for example). 

2) When a solution is in contact with another phase, the composition of the interphase differs from that of the bulk in such a manner as to minimize the total excess surface Gibbs energy of the system. 

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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,... 

The quantitative relationship between the degree of adsorption at a solution iaterface (7), G—L or L—L, and the lowering of the free-surface energy can be deduced by usiag an approximate form of the Gibbs adsorption isotherm (eq. 9), which is appHcable to dilute biaary solutions where the activity coefficient is unity and the radius of curvature of the surface is not too great ... 

Adsorbed-Solution Theoiy The common thennodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is... 

This equation has been criticized on thermodynamic grounds because it does not satisfy the Gibbs adsorption isotherm unless all monolayer capacities n] are equal. 

To satisfy the Gibbs adsorption isotherm for unequal monolayer capacities, exphcit isotherms can be obtained in the form of a series expansion [LeVan and Vermeulen,/. Phy.s. Chem., 85, 3247 (1981)]. A two-term form is... 

When the adsorbed components are electrically charged, then the partial molar Gibbs energy of the charged component depends on the charge of the given phase, and thus the chemical potentials in the above relationships must be replaced by the electrochemical potentials. The Gibbs adsorption isotherm then has the form... 

Equation (4.3.37) can be used to determine the function = T1(c1), which is the adsorption isotherm for the given surface-active substance. Substitution for c1 in the Gibbs adsorption isotherm and integration of the differential equation obtained yields the equation of state for a monomole-cular film = T jt). 

The simplest way to predict the lipid/ water partition coefficient, Kiw, of a drug is based on measurements of the surface pressure, ttd, of the drug as a function of its concentration in the aqueous subphase (Gibbs adsorption isotherm). The Gibbs adsorption isotherm provides the air/water partition coefficient, Kaw, and the cross-sectional area, Ad of the drug and allows calculation of the lipid/water partition coefficient, K]w, according to Eq. (6) [59] ... 

Equation 17.23 has the form of an adsorption isotherm since it relates the amount adsorbed to the corresponding pressure. This is known as the Gibbs Adsorption Isotherm. For it to be useful, an expression is required for T. Assuming an analogy between adsorbed and liquid films, Harkins and Jura(15) have proposed that ... 

The problem has been treated theoretically by the use of the Gibbs adsorption isotherm, which has been used with success in treating the interfaces between liquids and gases (30). One of the most easily measurable properties of a liquid is its surface tension, and changes in this quantity can be determined with great accuracy. The surface tension of a liquid is numerically equal to its surface energy, as also are changes in these quantities. 

Adsorption at liquid surfaces can be monitored using the Gibbs adsorption isotherm since the surface energy, y, of a solution can be readily measured. However, for solid substrates, this is not the case, and the adsorption density has to be measured in some other manner. In the present case, the concentration of adsorbate in solution will be monitored. In place of the Gibbs equation, we can use a simple adsorption model based on the mass action approach. 

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. 

The surface tension data given in Figure 3.5 was obtained for aqueous solutions of a trivalent cationic surfactant (C0RCI3) in both water and in 150 mM NaCl solution. Use the data and the Gibbs adsorption isotherm to obtain estimates of the minimum surface area per molecule adsorbed at the air/water interface. 

Use the Gibbs adsorption isotherm to describe the type of surfactant adsorption occurring at the air/water interface at points A, B, C and D in Figure 3.6. 

It is well known that the surface tension of water decreases when a detergent is added. Detergents are strongly enriched at the surface, which lowers the surface tension. This change of surface tension upon adsorption of substances to the interface, is described by the Gibbs adsorption isotherm. 

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... 

The simplest application of the Gibbs adsorption isotherm is a system of two components, e.g., a solvent 1 and a solute 2. In this case we have... 

The choice of the ideal interface in the Gibbs adsorption isotherm (3.52) for a two-component system is, in a certain view, arbitrary. It is, however, convenient. There are two reasons First, on the right side there are physically measurable quantities (a, 7, T), which are related in a simple way to the interfacial excess. Any other choice of the interface would lead to a more complicated expression. Second, the choice of the interface is intuitively evident, at least for ci > C2. One should, however, keep in mind that different spatial distributions of the solute can lead to the same T. Figure 3.6 shows two examples of the same interfacial excess concentration In the first case the distribution of molecules 2 stretches out beyond the interface, but the concentration is nowhere increased. In the second case, the concentration of the molecules 2 is actually increased. 

For solutions the Gibbs dividing plane is conveniently positioned so that the surface excess of the solvent is zero. Then the Gibbs adsorption isotherm (Eq. 3.52) relates the surface tension to the amount of solute adsorbed at the interface ... 

The interfacial tension decreases with increasing amount of surface potential. The reason is the increased interfacial excess of counterions in the electric double layer. In accordance with the Gibbs adsorption isotherms, the interfacial tension must decrease with increasing interfacial excess. At charged interfaces ions have an effect similarly to surfactants at liquid surfaces. 

For ionic surfactants another effect often dominates and usually salt tends to stabilize emulsions. Reason without salt the distance between surfactants in the interface is large because the molecules electrostatically repel each other. This prevents a high surface excess. The addition of salt reduces this lateral repulsion and more surfactant molecules can adsorb at the interface. Then, according to the Gibbs adsorption isotherm Eq. (3.52), the surface tension is reduced and the emulsion is stabilized. 

If there is still a significant proportion of the amphiphile dissolved in the liquid we talk about Gibbs monolayers. Solubility in water is increased by using molecules with short alkyl chain or a high polarity of the headgroup. In this case T is determined from the reduction of the surface tension according to the Gibbs adsorption isotherm (Eq. 3.52). 

If we compress a surfactant film on water we observe that the surface tension decreases and the surface pressure increases. What is the reason for this decrease in surface tension We can explain it by use of the Gibbs adsorption isotherm (Eq. (3.52)). On compression, the surface excess increases and hence the surface tension has to decrease. This, however, is relatively abstract. A more illustrative explanation is that the surface tension decreases because the highly polar water surface (high surface tension) is more and more converted into a nonpolar hydrocarbon surface (low surface tension). 

Here, 6 is a constant, which depends on the solvent and the amphiphile. We insert this expression into the Gibbs adsorption isotherm (Eq. (3.52)) ... 

With surfactant the surface tension is reduced according to the Gibbs adsorption isotherm Eq. (3.52). To apply Eq. (3.52) we need to know the surface excess ... 

Many different isotherm equations have been described in the literature.15 Even though it has been challenged because it does not agree with the Gibbs adsorption isotherm unless all saturation capacities are identical,16 the competitive Langmuir isotherm is very often used ... 

The correct thermodynamic treatment of adsorption processes is possible only on liquid-gas and liquid-liquid interfaces, where the surface energy or the surface tension of the liquid can precisely be determined. For these systems, the Gibbs adsorption isotherm can be applied. For example on a liquid-liquid interface,... 

On a liquid-gas interface, the partial pressure of the adsorbed gas is substituted in Equation 1.59. On the solid-gas and solid-liquid interfaces, only the excess surface concentration can be measured directly, and not the surface tension. The Gibbs adsorption isotherm is suitable for the calculation of the change of surface tension. 

In equilibrium the adsorbed vapor has the same chemical potential as in the gas phase. Inserting Eq. (14) into Eq. (I3) yields the Gibbs adsorption isotherm... 

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