The Gibbs Equations

Equation (1.107) relates the total change in internal energy to the sum of the products of intensive variables T. P, F, lib if/, and the changes in extensive properties (capacities) of dS, dV, dl, dNb and de. The Bronsted work principle states that the overall work A W performed by a system is the sum of the contributions due to the difference of extensive properties AK across a difference of conjugated potentials A). 1-. 2 

Equation (1.107) is more useful if it is integrated with the Pfaffian form however, this is not a straightforward step, since intensive properties are functions of all the independent variables of the system. The Euler relation for 

Comparing Eq. (1.109) with Eq. (1.107) yields the definitions of intensive properties for the partial differentials 

The chemical potential fi indicates that the internal energy is a potential for the chemical work (or mass action) juy/ Y,. and it is the driving force for a chemical reaction. The chemical potential cannot be measured directly, and its absolute values are related to a reference state. However, the change of chemical potential is of common interest. 

By introducing the definitions given in Eq. (1.110) into Eq. (1.107), we obtain the integrated form of the Gibbs equation 

With the preceding introduction to the handling of surface excess quantities, we now proceed to the derivation of the third fundamental equation of surface chemistry (the Laplace and Kelvin equations, Eqs. II-7 and III-18, are the other two), known as the Gibbs equation. 

If one now allows the energy, entropy, and amounts to increase from zero to some finite value, keeping T, A (area), and the n constant, Eq. III-76 becomes 

Equation III-77 is generally valid and may now be differentiated in the usual manner to give 

Moreover, since F and are defined relative to an arbitrarily chosen dividing surface, it is possible in principle to place that surface so that F = 0 (this is discussed in more detail below), so that 

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The type of behavior shown by the ethanol-water system reaches an extreme in the case of higher-molecular-weight solutes of the polar-nonpolar type, such as, soaps and detergents [91]. As illustrated in Fig. Ul-9e, the decrease in surface tension now takes place at very low concentrations sometimes showing a point of abrupt change in slope in a y/C plot [92]. The surface tension becomes essentially constant beyond a certain concentration identified with micelle formation (see Section XIII-5). The lines in Fig. III-9e are fits to Eq. III-57. The authors combined this analysis with the Gibbs equation (Section III-SB) to obtain the surface excess of surfactant and an alcohol cosurfactant. 

The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

As an example, Tajima and co-workers [108] used labeling to obtain the adsorption of sodium dodecyl sulfate at the solution-air interface. The results, illustrated in Fig. Ill-12, agreed very well with the Gibbs equation in the form... 

Fig. in-12. Verification of the Gibbs equation by the radioactive trace method. Observed (o) and calculated (line) values for for aqueous sodium dodecyl sulfate solutions. (From Ref. 108.)... 

Smith [113] studied the adsorption of n-pentane on mercury, determining both the surface tension change and the ellipsometric film thickness as a function of the equilibrium pentane pressure. F could then be calculated from the Gibbs equation in the form of Eq. ni-106, and from t. The agreement was excellent. Ellipsometry has also been used to determine the surface compositions of solutions [114,115], as well polymer adsorption at the solution-air interface [116]. 

Although Gibbs published his monumental treatise on heterogeneous equilibrium in 187S, his work was not generally appreciated until the turn of the century, and it was not until many years later that the field of surface chemistry developed to the point that experimental applications of the Gibbs equation became important. 

If the surface tension of a liquid is lowered by the addition of a solute, then, by the Gibbs equation, the solute must be adsorbed at the interface. This adsorption may amount to enough to correspond to a monomolecular layer of solute on the surface. For example, the limiting value of in Fig. Ill-12 gives an area per molecule of 52.0 A, which is about that expected for a close-packed... 

Fig. Ill-IS. Surface tension data for aqueous alcohol illustration of the use of the Gibbs equation. (1) -butyl (2) -amyl (3) -hexyl (4) -heptyl (5) -octyl. (Data from Ref. 126).

Adsorption may occur from the vapor phase rather than from the solution phase. Thus Fig. Ill-16 shows the surface tension lowering when water was exposed for various hydrocarbon vapors is the saturation pressure, that is, the vapor pressure of the pure liquid hydrocarbon. The activity of the hydrocarbon is given by its vapor pressure, and the Gibbs equation takes the form... 

The data could be expressed equally well in terms of F versus P, or in the form of the conventional adsorption isotherm plot, as shown in Fig. Ill-18. The appearance of these isotherms is discussed in Section X-6A. The Gibbs equation thus provides a connection between adsorption isotherms and two-dimensional equations of state. For example, Eq. III-57 corresponds to the adsorption isotherm... 

Two alternative means around the difficulty have been used. One, due to Pethica [267] (but see also Alexander and Barnes [268]), is as follows. The Gibbs equation, Eq. III-80, for a three-component system at constant temperature and locating the dividing surface so that Fi is zero becomes... 

One way of doing this makes use of the Gibbs equation (Eq. III-81) in the form (see also Section V-7A)... 

The treatments that are concerned in more detail with the nature of the adsorbed layer make use of the general thermodynamic framework of the derivation of the Gibbs equation (Section III-5B) but differ in the handling of the electrochemical potential and the surface excess of the ionic species [114-117]. The derivation given here is after that of Grahame and Whitney [117]. Equation III-76 gives the combined first- and second-law statements for the surface excess quantities... 

The surface excess per square centimeter F is just n/E, where n is the moles adsorbed per gram and E is the specific surface area. By means of the Gibbs equation (111-80), one can write the relationship... 

According to Eq. X-12, straightforward application of the Gibbs equation gives... 

Adsorption The separation achieved depends in part on the selectivity of adsorption at the bubble surface. At equihbrium, the adsorption of dissolved material follows the Gibbs equation (Gibbs, Collected Works, Longmans Green, New York, 1928). 

The expression for (dU/dp)T is also easily obtained. Again we start with the Gibbs equation involving dU ... 

Dividing the Gibbs equation by dp and specifying constant T gives... 

So far our discussion of chemical thermodynamics has been limited to systems in which the chemical composition does not change. We have dealt with pure substances, often in molar quantities, but always with a fixed number of moles, n. The Gibbs equations... 

The Gibbs equation applied to each component relates the change in chemical potential to the change in temperature and pressure. 

Chapter 3 starts with the laws, derives the Gibbs equations, and from them, develops the fundamental differential thermodynamic relationships. In some ways, this chapter can be thought of as the core of the book, since the extensions and applications in all the chapters that follow begin with these relationships. Examples are included in this chapter to demonstrate the usefulness and nature of these relationships. 

The Gibbs equation allows the amount of surfactant adsorbed at the interface to be calculated from the interfacial tension values measured with different concentrations of surfactant, but at constant counterion concentration. The amount adsorbed can be converted to the area of a surfactant molecule. The co-areas at the air-water interface are in the range of 4.4-5.9 nm2/molecule [56,57]. A comparison of these values with those from molecular models indicates that all four surfactants are oriented normally to the interface with the carbon chain outstretched and closely packed. The co-areas at the oil-water interface are greater (heptane-water, 4.9-6.6 nm2/molecule benzene-water, 5.9-7.5 nm2/molecule). This relatively small increase of about 10% for the heptane-water and about 30% for the benzene-water interface means that the orientation at the oil-water interface is the same as at the air-water interface, but the a-sulfo fatty acid ester films are more expanded [56]. 

Solubility occurs where the free energy of mixing, AG , is negative. This value is related to the enthalpy of mixing, AH, and the entropy of mixing, AiSjjj, by the Gibbs equation ... 

When the Gibbs equation is used for an electrode-electrolyte interface, the charged species (electrons, ions) are characterized by their electrochemical potentials, while the interface is regarded as electroneutral that is, the surface density, 2, of excess charges in the metal caused by positive or negative adsorption of electrons ... 

The Gibbs equation for metal-electrolyte interfaces is of the form... 

The values for A.S s and AHs are substituted into the Gibbs equation to give... 

Each individual particle has a closed shape, and thus a curved surface. Obviously, this should be taken into account in the Gibbs equation [11] ... 

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