Gibbs equation, generalized

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. 

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 change in the free energy is generally expressed by the Gibbs equation ... 

This is the form of the Gibbs equation for an aqueous solution containing three different ionic species (e.g., R, Na, Cl). Thus, the more general form for solutions containing i number of ionic species would be... 

Gibbs equation, which is perfectly general, may be deduced readily from the potential functions of Gibbs Thermodynamics, p. 221) and Duhem Le Potential Thermodynamique, Paris 1886), or in the following manner. 

Curve 3 in Figure 7.14 applies primarily to amphipathic species. Most long-chain amphipathic molecules are insoluble unless the hydrophobic alkyl part of the molecule is offset by an ionic head or some other suitably polar head such as a polyethylene oxide chain, — (CH2CH20)n—. Like their insoluble counterparts, these substances form an oriented monolayer even at low concentrations. Figure 7.15 shows some actual experimental plots of type 3 for the ether that consists of a dodecyl chain and a hexaethylene oxide chain (n — 6) in the general formula just given. Example 7.4 illustrates the application of the Gibbs equation to these data. 

The general form of the Gibbs equation (dy = -X T d/x,) is fundamental to all adsorption processes. However, experimental verification of the equation derived for simple systems is of interest in view of the postulation which was made concerning the location of the boundary surface. 

Principles of thermodynamics find applications in all branches of engineering and the sciences. Besides that, thermodynamics may present methods and generalized correlations for the estimation of physical and chemical properties when there are no experimental data available. Such estimations are often necessary in the simulation and design of various processes. This chapter briefly covers some of the basic definitions, principles of thermodynamics, entropy production, the Gibbs equation, phase equilibria, equations of state, and thermodynamic potentials. 

A chemical reaction is an irreversible process that produces entropy. The general criterion of irreversibility is d S > 0. Criteria applicable under particular conditions are readily obtained from the Gibbs equation. The changes in thermodynamic potentials for chemical reactions yield the affinity A. All four potentials U, H, A, and G decrease as a chemical reaction proceeds. The rate of reaction, which is the change of the extent of the reaction with time, has the same sign as the affinity. The reaction system is in equilibrium state when the affinity is zero. 

In extended nonequilibrium thermodynamics of polymer solutions, the generalized extended Gibbs equation for a fluid characterized by internal energy U and viscous pressure Pv is... 

In its most general form, at given pressure, the Gibbs equation reads (see 11.2.22.91)... 

There are two keys to this generalization, whereby a rational definition for entropy of nonequilibrium states is obtained. First, a more general second law is postulated, based on the property availability. (The availability at any state of a system reflects the extent to which it could affect any other system.) The second key is the introduction of other integrating factors, in addition to temperature, in order to deduce the fundamental differential property relationship (i.e., Gibbs equation.)... 

Reversibility of Adsorption. Apparently, the data in Figure 10.13 imply that the Gibbs equation (10.2) does not hold for the protein. As we have seen, it is valid for the amphiphile. However, the slopes dll/d In c given in the figure differ only by a factor 2 between the two surfactants, whereas the values of Fm differ by two orders of magnitude. The explanation is not fully clear. Application of the Gibbs equation to polymers is anyway questionable, because it is generally not known what the relation is between concentration (c) and activity (a) of the surfactant. Moreover, proteins and other polymers are virtually always mixtures. 

Polymeric surfactants are generally (far) more surface active, but they give lower surface pressures than most amphiphiles. At the plateau value of the surface excess they are not very tightly packed (most amphiphiles are), but they extend fairly far into the solution. The exchange between solution and interface may be very slow, and the Gibbs equation does not seem to hold. Most amphiphiles can displace polymers from the interface, if present in sufficient concentration, since they give a lower interfacial tension. Mixed surface layers can also be formed. 

The applicability of eq. (11.22) to a successful description of adsorption from a solution was established by Langmuir himself, when he compared his adsorption isotherm to the Gibbs equation and ended up with the Szyszkowski equation as a result. The transition from localized to non-localized adsorption (which can be viewed as the transition from fixed adsorption sites to moving ones) does not, therefore, change general trends in the adsorption in the cases described. One should also keep in mind that the liquid interface is more uniform in terms of energy than the solid interface, which contains active sites with different interaction potentials.4 The latter is probably the reason why... 

The general appearance of this function is shown in Figure 4, and it is seen that for A positive-i.e., for P° > P°-a discontinuity is possible, in that film of finite thickness x° can be in equilibrium with an infinitely thick liquid layer-i.e., bulk liquid-at P = P In combination with the Gibbs equation, we then have ... 

Generally, wastewaters are complex mixtures of solutes, which require theoretical approaches to predict multicomponent adsorption equilibria flxtm pure component adsorption data. The Ideal Adsorbed Solution model (IAS) was first established for a mixed gas adsorption by Myers and Prausnitz [9], and then extended to a multi-solute adsorption from dilute liquid solution by Radke and Prausnitz [10]. The model is based on the fundamental hypothesis that the multicomponent solution has the same spreading pressure s as that of the ideal single solution of the i component, the spreading pressure being the difference between the interfacial tension of the pure solvent and that of the solution containing the solute. This hypothesis is described by the Gibbs equation ... 

In more general constitutive models the Gibbs equations (local equilibrium) are not valid and therefore explicit calculations of entropy are impossible. This seems to correspond to the nonuniqueness of entropy or to irreversibility of processes between nonequilibrium states [see below (1.37) and Rem. 20]. Such are some constitutive models in Sects. 2.1-2.3, but in particular models with long range memory [17, 23, 48]. Even the usefulness of entropy in situations far from equilibrium [11, 101, 114-120] seems questionable, the entropy inequality deduced and used in... 

The general Gibbs equation for the inner energy in the presence of E is obtained by substitution of Eq. (4.7) into (3.26) ... 

For a general case X = grad T/T (thermal force). For a more general case where other thermodynamic variables are involved, we have to make use of Gibbs equation along with laws of conservation of mass, energy and electric charge. 

When one examines the shape of the surface tension-ln C curve for a surfactant, it can be seen that the curve becomes approximately horizontal at some concentration below the cmc. It can be shown that the effectiveness of the adsorption of a surfactant, Ao-omo, can be quantitatively related to the concentration of surfactant at which the Gibbs equation becomes linear, Q, the surface tension attained at Ci, o-i, and the cmc. The relationship has the general form... 

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