Thermodynamics the Gibbs equation

The Gibbs convention presented in the previous section can be applied to extensive thermodynamic functions. So, the total internal energy of a system composed of 

By using eqs 2.1, 2.4 and 2.5 for each phase, it is easy to show that a similar relation exists for U in which -yA replaces -pV, that is 

the Gibbs-Duhem relation that expresses the relation between the intensive surface excess thermodynamic variables reads 

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Based on the extended irreversible thermodynamics, the Gibbs equation for a simple single-component fluid in the presence of a viscous pressure tensor P (up to the second order in P ) is... 

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. 

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... 

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... 

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. Thermodynamically from Eq. (4.1) the Gibbs Equation... 

Any reaction or process for which the enthalpy change is large and negative, such that the AH term greatly dominates the TAS term in the Gibbs equation (AG = AH TAS), and acts thereby as the thermodynamic driving force for that reaction or process. 

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. 

Until now, we have focused our attention on those adsorption isotherms that show a saturation limit, an effect usually associated with monolayer coverage. We have seen two ways of arriving at equations that describe such adsorption from the two-dimensional equation of state via the Gibbs equation or from the partition function via statistical thermodynamics. Before we turn our attention to multilayer adsorption, we introduce a third method for the derivation of isotherms, a kinetic approach, since this is the approach adopted in the derivation of the multilayer, BET adsorption isotherm discussed in Section 9.5. We introduce this approach using the Langmuir isotherm as this would be useful in appreciating the common features of (and the differences between) the Langmuir and BET isotherms. 

The impact of the eluent temperature is shown in Figure 12.20. Increasing temperature tends to increase retention. This effect is usually not observed in liquid chromatography, in accordance with the thermodynamics of the retention phenomenon described by the Gibbs equation. However, temperature has an... 

A more comprehensive introduction is Ref. [399], We restrict ourselves to uncharged species and dilute solutions (not binary mixtures). The important subject of polymer adsorption is described in Ref. [400], Adsorption of surfactants is discussed in Ref. [401], Adsorption of ions and formation of surface charges was treated in Chapter 5. In dilute solutions there is no problem in positioning the Gibbs dividing plane, and the analytical surface access is equal to the thermodynamic one, as occurs in the Gibbs equation. For a thorough introduction into this important field of interface science see Ref. [8],... 

This interface is also known as the perm-selective interface (Fig. 6.1a). It is found in ion-selective sensors, such as ion-selective electrodes and ion-selective field-effect transistors. It is the site of the Nernst potential, which we now derive from the thermodynamic point of view. Because the zero-current axis in Fig. 5.1 represents the electrochemical cell at equilibrium, the partitioning of charged species between the two phases is described by the Gibbs equation (A.20), from which it follows that the electrochemical potential of the species i in the sample phase (S) and in the electrode phase (m) must be equal. 

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. 

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. 

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. 

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

No more than the slopes do they contain the electrosorption Gibbs energy. The surface pressure is experimentally accessible by Integration of the Gibbs equation dy = -Xjr d/i. The fact that some components are dubbed "charge determining" does not matter provided the sum Is taken over electroneutral components. This Is our choice, not that of thermodynamics which Is model-free. In practice this means that the r.h.s. of. for instance (3.12.4b( Is Integrated from the pristine surface at the p.z.c. to the final condition, determined by the final values of a°. cr and r. The surface pressure is a function of state, therefore... 

Scientists of a retiring disposition who make important discoveries, but pubHsh them in journals of limited circulation, tend to be overlooked in their own day and receive appreciation only after they have died, when their innovations have been rediscovered. The work of Willard Gibbs (1839-1903) at Yale University on chemical thermodynamics and statistical mechanics, published in the Transactions of the Connecticut Academy of Sciences, was approved by Clerk Maxwell in 1875, but little known to others in Europe at the time. Yet the Gibbs Free Energy function and the Gibbs equations became standard in chemical thermodynamics student courses from the 1920s onwards. Chemists who work on non-standard topics receive... 

As the Gibbs equation remains our primary tool, let us start with item (v). Some time ago this issue has given rise to a lively discussion in the literature but the issue is now resolved. Originally the question was whether for a fully dissociated ionic surfactant such as A Na or C Br" the adsorption term in the Gibbs equation (T dju ) should be written as RTF d In c or 2RTF d In c if the solution is ideal. We shall now analyze the problem thermodynamically. 

The thermodynamics of the electrochemical interface is based on the Gibbs adsorption equation. For a plane electrode in contact with an ionic conductor, under equilibrium conditions, the Gibbs equation is [1]... 

From classical thermodynamics ( ,6.) it is known that a form of the Gibbs equation can be integrated to give E = TS - pV +... 

Statistical theories of thermodynamics yield many correct and practical results. For example, they yield the canonical and grand canonical distributions for petit and grand systems, respectively these distributions, which were proposed by Gibbs, have been shown by innumerable comparisons with experiments to describe accurately the properties of quasistable states. Again, they predict the equality of temperatures of systems in mutual stable equilibrium, the Maxwell relations, and the Gibbs equation. 

Adsorption. Some substances tend to adsorb onto an interface, thereby lowering the interfacial tension the amount by which it is lowered is called the surface pressure. The Gibbs equation gives the relation between three variables surface pressure, surface excess (i.e., the excess amount of surfactant in the interface per unit area), and concentration—or, more precisely, thermodynamic activity—of the surfactant in solution. This relation only holds for thermodynamic equilibrium, and the interfacial tension in the Gibbs equation is thus an equilibrium property. Nevertheless, also under nonequilibrium conditions, a tension can be measured at a liquid interface. 

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