Thermodynamic approach

This approach is based on classical thermodynamics and statistical mechanics the latter makes the link to the microscopic picture of the adsorption processes. From the point of view of thermodynamics, adsorption can be treated like a chemical reaction. It will be shown that adsorption can proceed with or without a change of the number of gaseous molecules, so that the thermodynamic equilibrium constant expressed in partial pressures may or may not equal the constant expressed in concentrations. Notice that the standard values of some quantities accepted here when deriving formulae for the adsorption characteristics are not the standard states commonly used in chemical thermodynamics. In particular, it concerns the concentrations. 

Let ca in cm-2 be the equilibrium number concentration of the tracer entities in the adsorption layer, and cg in cm-3 the concentration in the gaseous phase. Experimental studies have shown that a range of small ca values (in the sense of surface coverage much less than unity) always exists, over which the dimensional ratio of the concentrations fca, the distribution coefficient 

To analyze the thermodynamics of adsorption, it is necessary to know some basic characteristics of the microscopic profile of the adsorption energy across the surface, as well as the characteristics of the state of the adsorbate. Otherwise, we must make 

Another idealized model is the localized adsorption, a rationale for the Langmuir isotherm. Now the surface is supposed to consist solely of identical distinct adsorption sites with the number concentration Ca in cm-2 the latter has a definite value for each particular adsorbent-adsorbate system. Let the fractional surface coverage be 6 = ca/Ca. In such a model, proportional to cg is ca/Ca(l - 6), the surface concentration divided by the concentration of free adsorption sites. Hence, strictly speaking, the corresponding dimensional constant fc c, now in cm3, is  

the initial straight part of the adsorption isotherm does not contain any information about the nature of the adsorbed state — the experiments allow obtaining only fca evaluated from the primary data through Eq. 5.3. 

This is adequate for signals related to a two-state transition and obtained at a heating rate as low as to practically achieve thermodynamic equilibrium in any point of the temperature range across the DSC peak, viz.. 

This expression corresponds to a skewed peak, the first term of which can be often replaced with a gaussian function across the respective temperature range the second term can be reliably reproduced with a sigmoidal function centred on the same gaussian mean. 

We denote AG(i) as the free enthalpy of formation of a species of a degree of polymerization of i. Equations (17.18) can be rewritten in the language of thermodynamics as 

Meethong, H.-Y. S. Huang, W. C. Carter, and Y.-M. Chiang, lectrochenL Solid-State Lett. 10, A134 (2007). 

Casa-Cabanas, L. Laffont, S. Levasseur, P. Carlach, S. Hamelet, 

Martin, A. Yamada, G. Kobayashi, S. Nishimura, R. Kanno, D. Guyomard, and N. Dupre. Eieetroehem. Solid.State Lett. 11, A12 (2008). 

Yamada, H. Koizumi, S. Nishimura, N. Sonoyama, R. Kanno, M. Yonemura, T. Nakamura, and Y. Kobayashi, Nature Mater.. 5, 357 (2006). 

We now address the thermod5mamic experimental evidences for the solid solution outside of the miscibility gap. Microcalorimetry can directly measure entropy changes vs. x at fixed temperature T during the electrochemical reaction. Observed heat flow P is related to entropy change A5 by P = -ITAS/nF, where / is the input current and F is the Faraday constant The classical entropy of mixing 5rf(x) can be calculated for a fully disordered lattice of lithium atoms and vacant sites, and heat flow varies linearty with = kBln[l/x], 

Electrochem. Solid-State Lett 4, A170 [2001). 

Delmas, A. Nadiri, J. L. Soubeyroux, Solid State Ionics 28-30, 419 (1988). 

Yamada, H. Koizumi, N. Sonoyama,R Kanno,Electrochem. Solid-State Lett,8,A409 (2005). 

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In calculations of pore size from the Type IV isotherm by use of the Kelvin equation, the region of the isotherm involved is the hysteresis loop, since it is here that capillary condensation is occurring. Consequently there are two values of relative pressure for a given uptake, and the question presents itself as to what is the significance of each of the two values of r which would result from insertion of the two different values of relative pressure into Equation (3.20). Any answer to this question calls for a discussion of the origin of hysteresis, and this must be based on actual models of pore shape, since a purely thermodynamic approach cannot account for two positions of apparent equilibrium. 

Chang and his co-workers, adopting a statistical-mechanical rather than thermodynamic approach, made calculations for five different... 

R. A. Greenkorn, L. B. Koppel, and S. Raghavan, "Heat Exchanger Network Synthesis—A Thermodynamic Approach," 71 stAlChE Meeting, Miami, Fla., 1978. 

T. Umeda, T. Harada, and K. Shiroko, "A Thermodynamic Approach to the Synthesis of Heat Integration Systems in Chemical Processes," Proceedings of the 12th Symposium on Computer Applications in Chemical Engineering, Montreaux, Swit2edand, 1979, p. 487. 

Enzymatic Catalysis. Enzymes are biological catalysts. They increase the rate of a chemical reaction without undergoing permanent change and without affecting the reaction equiUbrium. The thermodynamic approach to the study of a chemical reaction calculates the equiUbrium concentrations using the thermodynamic properties of the substrates and products. This approach gives no information about the rate at which the equiUbrium is reached. The kinetic approach is concerned with the reaction rates and the factors that determine these, eg, pH, temperature, and presence of a catalyst. Therefore, the kinetic approach is essentially an experimental investigation. 

The holistic thermodynamic approach based on material (charge, concentration and electron) balances is a firm and valuable tool for a choice of the best a priori conditions of chemical analyses performed in electrolytic systems. Such an approach has been already presented in a series of papers issued in recent years, see [1-4] and references cited therein. In this communication, the approach will be exemplified with electrolytic systems, with special emphasis put on the complex systems where all particular types (acid-base, redox, complexation and precipitation) of chemical equilibria occur in parallel and/or sequentially. All attainable physicochemical knowledge can be involved in calculations and none simplifying assumptions are needed. All analytical prescriptions can be followed. The approach enables all possible (from thermodynamic viewpoint) reactions to be included and all effects resulting from activation barrier(s) and incomplete set of equilibrium data presumed can be tested. The problems involved are presented on some examples of analytical systems considered lately, concerning potentiometric titrations in complex titrand + titrant systems. All calculations were done with use of iterative computer programs MATLAB and DELPHI. 

The full explanation of solute retention on silica, bonded phases or for that matter in liquid/liquid systems is still elusive and controversial. The thermodynamic approach... 

Umeda, T., Itoh, J., and Shiroko, K. (1979). A thermodynamic approach to the synthesis of heat integration systems in chemical processes. Comp. Chem. Eng. 3, 273-282. 

The thermodynamic method has limitations. Since the method ignores the intermediate stages, it cannot be used to determine shock-wave parameters. Furthermore, a shock wave is an irreversible thermodynamic process this fact complicates matters if these energy losses are to be fully included in the analysis. Nevertheless, the thermodynamic approach is a very attractive way to obtain an estimate of explosion energy because it is very easy and can be applied to a wide range of explosions. Therefore, this method has been applied by practically every worker in the field. 

Transition elements, for which variable valency is energetically feasible, frequently show non-stoichiometric behaviour (variable composition) in their oxides, sulfides and related binary compounds. For small deviations from stoichiometry a thermodynamic approach is instructive, but for larger deviations structural considerations supervene, and the possibility of thermodynamically unstable but kinetically isolable phases must be considered. These ideas will be expanded in the following paragraphs but more detailed treatment must be sought elsewhere. " ... 

In this example of the corrosion of zinc in a reducing acid of pH = 4, the corrosion product is Zn (aq.), but at higher pHs the thermodynamically stable phase will be Zn(OH)j and the equilibrium activity of Zn will be governed by the solubility product of Zn(OH)j and the pH of the solution at still higher pHs ZnOj-anions will become the stable phase and both Zn and Zn(OH)2 will become unstable. However, a similar thermodynamic approach may be adopted to that shown in this example. 

The data given in Tables 1.9 and 1.10 have been based on the assumption that metal cations are the sole species formed, but at higher pH values oxides, hydrated oxides or hydroxides may be formed, and the relevant half reactions will be of the form shown in equations 2(a) and 2(b) (Table 1.7). In these circumstances the a + will be governed by the solubility product of the solid compound and the pH of the solution. At higher pH values the solid compound may become unstable with respect to metal anions (equations 3(a) and 3(b), Table 1.7), and metals like aluminium, zinc, tin and lead, which form amphoteric oxides, corrode in alkaline solutions. It is evident, therefore, that the equilibrium between a metal and an aqueous solution is far more complex than that illustrated in Tables 1.9 and 1.10. Nevertheless, as will be discussed subsequently, a similar thermodynamic approach is possible. 

Vermilyea" has adopted a thermodynamic approach to pitting, and considers that the critical pitting potential is the potential at which the metal salt of the aggressive ion (e.g. AICI3) is in equilibrium with metal oxide (e.g. AljOj). On the basis of this theory the critical pitting potential should decrease by 0-059V per decade increase in chloride ion concentration. Vermilyea s theory successfully predicts the values of the critical potentials for Al, Mg, Fe and Ni, but in the case of Zr, Ti and Ta there are large discrepancies. 

As an example of the way in which these data could be used, the temperatures at which carbides separate from an 18/8 stainless steel are calculated for carbon contents of 01, 0-01, 0 001 and 0-0001 wt /o. These calculations, which of necessity involve several approximations due to our present lack of knowledge, demonstrate the value of the thermodynamic approach to problems involving the precipitation of phases which may have a pronounced effect on the corrosion behaviour of the alloy (see Section 3.3). 

Although important contributions in the use of electrical measurements in testing have been made by numerous workers it is appropriate here to refer to the work of Stern and his co-workerswho have developed the important concept of linear polarisation, which led to a rapid electrochemical method for determining corrosion rates, both in the laboratory and in plant. Pourbaix and his co-workers on the basis of a purely thermodynamic approach to corrosion constructed potential-pH diagrams for the majority of metal-HjO systems, and by means of a combined thermodynamic and kinetic approach developed a method of predicting the conditions under which a metal will (a) corrode uniformly, (b) pit, (c) passivate or (d) remain immune. Laboratory tests for crevice corrosion and pitting, in which electrochemical measurements are used, are discussed later. 

The following features of the thermodynamic approach to chemical reactions should be noted ... 

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

Viscoelasticity of metal This subject provides an introduction on the viscoelasticity of metals that has no bearing or relationship with viscoelastic properties of plastic materials. The aim is to have the reader recognize that the complex thermodynamic foundations of the theory of viscoplasticity exist with metals. There have been developments in the thermodynamic approach to combined treatment of rheologic and plastic phenomena and to construct a thermodynamic theory non-linear viscoplastic material that may be used to describe the behavior of metals under dynamic loads. 

Equation (13) is valid for r/Nlp < 0.25 (Fig. 3). At much higher extension ratios, the force must increase indefinitely since the molecule is almost straightened out. The thermodynamic approach to the problem of coil stretching for a freely-jointed chain was considered by Treloar [32], who obtained the following expression for the stress-strain relationship when the two chain ends are kept a distance r apart ... 

Naslain, R., Thebault, J., Hagenmuller, P., and Bernard, C., The Thermodynamic Approach to Boron CVD Based on the Minimization of the Total Gibbs Free Energy, J. Less Common Metals, 67(1) 85-100 (1979)... 

The reactant mixture may be so nonideal that Equation (7.28) is inadequate. The rigorous thermodynamic approach is to replace the concentrations in Equation (7.28) with chemical activities. This leads to the thermodynamic equilibrium constant. 

The most fundamental thermodynamic approach of Rudakov (6) applies to all condensed systems. The actual linear relationship is argued to exist between enthalpy (AH) and entropy (AS) of intermolecular interaction, as reflected in an approximately linear relationship between the total enthalpy and entropy. Special attention has been given to hydrophobic interaction (89, 90) in water solutions, which makes the isokinetic behavior more pronounced and markedly changes its slope. 

The fact that mbber shows mbber elasticity was discovered more than 100 years earlier than professor H. Staudinger s proposal. The memory effect acquired by vulcanization, so-called Gough-Joule effect, and its thermodynamic explanation were the great achievements in the nineteenth century. As seen in many textbooks, this thermodynamic approach was the easiest one to gain consistency between ever-performed experiments and theory. In fact, thermodynamics of mbbery material can be treated in parallel with thermodynamics of gas. One could show experimentally that... 

Equilibrium data are thus necessary to estimate compositions of both extract and raffinate when the time of extraction is sufficiently long. Phase equilibria have been studied for many ternary systems and the data can be found in the open literature. However, the position of the envelope can be strongly affected by other components of the feed. Furthermore, the envelope line and the tie lines are a function of temperature. Therefore, they should be determined experimentally. The other shapes of the equilibrium line can be found in literature. Equilibria in multi-component mixtures cannot be presented in planar graphs. To deal with such systems lumping of consolutes has been done to describe the system as pseudo-ternary. This can, however, lead to considerable errors in the estimation of the composition of the phases. A more rigorous thermodynamic approach is needed to regress the experimental data on equilibria in these systems. 

Saturated hydrocarbons are stable. Only cycloalkanes with a tight ring are unstable. Alkenes and alkynes have a strong endothermic character, especially the first homologues and polyunsaturated conjugated hydrocarbons. This is also true for aromatic compounds, but this thermodynamic approach does not show up their real stability very well. Apart from a few special cases, the decomposition of unsaturated hydrocarbons requires extreme conditions, which are only encountered in the chemical industry. 

The thermodynamic approach followed by Raevsky considers the property P to be based on contributions from three main intermolecular interactions steric, electrostatic and H-bonding [38] ... 

In this chapter, we will give a general description of electrochemical interfaces representing thermodynamically closed systems constrained by the presence of a hnite voltage between electrode and electrolyte, which will then be taken as the basis for extending the ab initio atomistic thermodynamics approach [Kaxiras et ah, 1987 Scheffler and Dabrowski, 1988 Qian et al., 1988 Reuter and Scheffler, 2002] to electrochemical systems. This will enable us to qualitatively and quantitatively investigate and predict the structures and stabilities of full electrochemical systems or single electrode/electrolyte interfaces as a function of temperature, activi-ties/pressures, and external electrode potential. 

Now having specified the bulk electrode, the bulk electrolyte, and the interface between them, our aim in this section is to quantify the atomistic structure of the interface and derive an expression that allows us to evaluate its stabUity. Based on (5.5), we wUl extend the ab initio atomistic thermodynamics approach to electrochemical systems. 

On the basis of (5.19), in this section we will evaluate the importance of different contributions to the overall interfacial free energy by applying the extended ab initio atomistic thermodynamics approach to two examples. 

Before we can apply the extended ab initio atomistic thermodynamics approach to the oxygen-covered surface or the surface/bulk oxide, we have to investigate the structure of the bulk electrode. 

We have also discussed two applications of the extended ab initio atomistic thermodynamics approach. The first example is the potential-induced lifting of Au(lOO) surface reconstmction, where we have focused on the electronic effects arising from the potential-dependent surface excess charge. We have found that these are already sufficient to cause lifting of the Au(lOO) surface reconstruction, but contributions from specific electrolyte ion adsorption might also play a role. With the second example, the electro-oxidation of a platinum electrode, we have discussed a system where specific adsorption on the surface changes the surface structure and composition as the electrode potential is varied. 

Although the extended ab initio atomistic thermodynamics approach provides an exact expression for the interfacial stability, the formalism requires self-consistent modeling of the entire electrochemical system, or electrode/electrolyte interface, exceeding presently available computational capabilities. Therefore, certain assumptions had to be made that reduce the effort to the calculation of the electrode surface only. Even with this simplified approach, which has been applied to the two examples discussed in this chapter, the qualitative behavior can be reproduced. 

Unfortunately, the above thermodynamic approach has only been followed for two systems, namely As-Pt(lll) and Bi-Pt(lll) [Blais et al., 2001, 2002]. Table 7.3 summarizes the main results. Thermodynamic data about the bulk formation of As(OH)s and Bi(OH)2 are not available for comparison. The only data available is the standard enthalpy for the bulk formation of Bi(OH)3 (A/7 = 711.3kJ/mol). 

Garcia-Araez N, Climent V, Herrero E, Feliu J, Lipkowski J. 2006. Thermodynamic approach to the double layer capacity of a Pt(lll) electrode in perchloric acid solutions. Electrochim Acta 51 3787. 

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