Gibbs equation systems

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. 

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... 

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... 

First, one must determine if this is an exothermic reaction. Gibbs equation states that an exothermic reaction must have a negative value of AH. This means that the heat content of the reactants is greater than the heat content of the products. The difference in heat content between the two states is released during the reaction as the system goes to a lower energy state. The opposite is true of an endothermic reaction, as is shown in Figure 6.1. 

The three isotherms discussed, BET, (H-J based on Gibbs equation) and Polanyi s potential theory involve fundamentally different approaches to the problem. All have been developed for gas-solid systems and none is satisfactory in all cases. Many workers have attempted to improve these and have succeeded for particular systems. Adsorption from gas mixtures may often be represented by a modified form of the single adsorbate equation. The Langmuir equation, for example, has been applied to a mixture of n" components 11). 

When a surfactant is injected into the liquid beneath an insoluble monolayer, surfactant molecules may adsorb at the surface, penetrating between the monolayer molecules. However it is difficult to determine the extent of this penetration. In principle, equilibrium penetration is described by the Gibbs equation, but the practical application of this equation is complicated by the need to evaluate the dependence of the activity of monolayer substance on surface pressure. There have been several approaches to this problem. In this paper, previously published surface pressure-area Isotherms for cholesterol monolayers on solutions of hexadecy1-trimethyl-ammonium bromide have been analysed by three different methods and the results compared. For this system there is no significant difference between the adsorption calculated by the equation of Pethica and that from the procedure of Alexander and Barnes, but analysis by the method of Motomura, et al. gives results which differ considerably. These differences indicate that an independent experimental measurement of the adsorption should be capable of discriminating between the Motomura method and the other two. 

In principle, the penetration or adsorption of surfactant, Tg is given by the Gibbs equation. For a non-ionic monolayer and an ionised surfactant (as in the system examined), this equation is ... 

Equation (46), one form of the Gibbs equation, is an important result because it supplies the connection between the surface excess of solute and the surface tension of an interface. For systems in which y can be determined, this measurement provides a method for evaluating the surface excess. It might be noted that the finite time required to establish equilibrium adsorption is why dynamic methods (e.g., drop detachment) are not favored for the determination of 7 for solutions. At solid interfaces, 7 is not directly measurable however, if the amount of adsorbed material can be determined, this may be related to the reduction of surface free energy through Equation (46). To understand and apply this equation, therefore, it is imperative that the significance of r2 be appreciated. 

Throughout most of this chapter we have been concerned with adsorption at mobile surfaces. In these systems the surface excess may be determined directly from the experimentally accessible surface tension. At solid surfaces this experimental advantage is missing. All we can obtain from the Gibbs equation in reference to adsorption at solid surfaces is a thermodynamic explanation for the driving force underlying adsorption. Whatever information we require about the surface excess must be obtained from other sources. 

This reversible and ideal relationship predicts that the more effective depressants of interfacial tension tend to accumulate in the interface to the exclusion of others. Actually, in many cases the amount of material concentrated at the interface is greater than would be predicted by the Gibbs equation, and the system is not reversible or only sluggishly so. 

PI 1.1 The first of the four fundamental equations of Gibbs equation (11.10) is obtained by combining equation (11.6), the statement of the First Law applied to the system, with equation (11.7), the Second Law statement for a reversible process, again applied to the system, and equation (11.8) that calculates reversible pressure-volume work. Start with equation (11.10) and the defining equations for H, A, and G equations (11.1), (11.2), and (11.3), and derive the other Gibbs equations equations (11.11), (11.12), and (11.13). ... 

From adsorption studies with the help of inverse gas chromatography The interaction between a stationary phase and a well known gas or solvent is measured as in standard gas chromatography. The results are, however, used to characterize the stationary phase. Therefore, let us assume that n moles adsorb on a surface A. The interface excess concentration is T = n/A. With the Gibbs equation for a two-component system (Eq. 3.55) T = — l/RT dy/d wP) we get... 

From the thermodynamic point of view, this is a multiphase system for which, at equilibrium, the Gibbs equation (A.20) must apply at each interface. Because there is no charge transfer in and out of layer (4) (an ideal insulator) the sandwich of the layers (3)/(4)/(5) also represents an ideal capacitor. It follows from the Gibbs equation that this system will reach electrostatic equilibrium when the switch Sw is closed. On the other hand, if the switch Sw remains open, another capacitor (l)/( )/(6) is formed, thus violating the one-capacitor rule. The signifies the undefined nature of such a capacitor. The open switch situation is equivalent to operation without a reference electrode (or a signal return). Acceptable equilibrium electrostatic conditions would be reached only if the second capacitor had a defined and invariable geometry. 

When the system contains two (or more) phases the components of the system interact with these phases in one of two ways. The first is that the component exists in both phases, in which case at equilibrium the chemical potential of that component must be equal in both phases, which follows from the Gibbs equation (A.20). Because the number of moles transferred from phase 1 equals the number of moles received by phase 2(A i = — A 2)... 

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. 

The strict thermodynamic analysis of an interfacial region (also called an -> interphase) [ii] is based on data available from the bulk phases (concentration variables) and the total amount of material involved in the whole system yielding relations expressing the relative surface excess of suitably chosen (charged or not charged) components of the system. In addition, the - Gibbs equation for a polarizable interfacial region contains a factor related to the potential difference between one of the phases (metal) and a suitably chosen - reference electrode immersed in the other phase (solution) and attached to a piece of the same metal that forms one of the phases. 

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. 

The 1 IT and fi/T appear in the thermodynamic conjugates of the extensive variables in the Gibbs equation for the system entropy... 

For example, the amount of adsorbed SRlOl on a water/DCE interface (F) is given by the Gibbs equation F= —(1/2.3RT) dy/d log[SR101]. The F value was then calculated to be 5.0 x 10 mol cm ([SR101] = 1.0 x 10 M). When the interfacial area is assumed to be 1 cm for simplicity, the number of SRIOI molecules adsorbed on the interface (5.0 x 10 mol) is 5000 times higher than that expected to be involved in the excited volume by the evanescent wave (1.0 x 10 mol) without adsorption. Almost the same results with those for SRIOI were obtained for other dye/water/oil systems. At [SRIOI] = 1 x 10 M, therefore, the fluorescence response observed under the TIR conditions is ascribed essentially to that from the interface. 

Surface tensions can also be used to predict the behavior of multicomponent solid systems. Unlike single-component systems, the surface tension is no longer equal to the surface free energy G, but is related to the different component concentrations at the smface by the Gibbs equation ... 

The starting point is, as before, the Gibbs equation [3.4.11, which can be elaborated for the system under consideration. Typically, attention Is now paid to the F d term, expressed as (3.4.2) for the Agl-case. Cross-differentiation between the temperature, the surface charge, or the salt term, leads to useful... 

Let us take by way of example a surfactant of the A Na type, (abbreviated ANa) such as sodium dodecyl sulphate. For cationic surfactants the reasoning is similar. Let the solution also contain dissolved NaCl. For this system the Gibbs equation [4.6.4] becomes... 

Since activation kinetics is affected by the presence of fluctuations in the metastable phase, we will provide a description of the system in terms of the probability distribution function P(y,t). Our task will be then to derive the expression of the activation current, a quantity accessible to experiments in many instances. To this end, our starting point is the expression for the Gibbs equation accounting for entropy variations due to the underlying diffusion process of the probability density... 

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... 

The application of the Gibbs equation to ternary systems can be made only in cross-sections with a constant ratio of the amounts of substances, e.g. in the system A—B—C the pseudo-binary system A/B—C. The Gibbs equation in the ternary system is... 

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