Extra-thermodynamic assumption

Ion-selective electrodes (ISEs) with ionophore-based membranes allow for quantification of a large number of analytes in various matrixes. Tailoring of the composition of the membranes to comply with the analytical task, requires advanced theory of membrane response. Most of theoretical descriptions include nonrealistic extra-thermodynamic assumptions, in the first place it is assumed that some kind of species strongly predominate in membranes. Ideally, a rigorous theory of ISE response should be based on strict thermodynamics. However, real ISE membranes are too complex. Therefore, known attempts aimed at rigorous thermodynamic description of ISEs proved to be fraritless. 

NB/W interface, which were calculated from the extraction data using an extra thermodynamic assumption. Afterward, a newly developed electrochemical technique (so-called ion-transfer voltammetry) with a polarizable O/W interface was employed to determine... 

The Motomura equation is apparently derived without extra-thermodynamic assumptions, but the other approaches, through the necessary approximations, are based on models for the system. Independent adsorption measurements could reveal any deficiencies in these models and suggest refinements. 

However, if species i is a single ion, the value of AG°(i, R->S) cannot be obtained by purely thermodynamic means. It is necessary to introduce some extra-thermodynamic assumption. Various extra-thermodynamic assumptions have been proposed. Some typical examples are described in (i), (ii) and (iii) below. For practical methods of obtaining the Gibbs energies of transfer for ionic species, see7). 

Various potentiometric indicator electrodes work as sensors for ion solvation. Metal and metal amalgam electrodes, in principle, respond in a thermodynamic way to the solvation energy of the relevant metal ions. Some ion-selective electrodes can also respond almost thermodynamically to the solvation energies of the ions to which they are sensitive. Thus, the main difficulty in the potentiometric study of ion solvation arises from having to compare the potentials in different solvents, even though there is no thermodynamic way of doing it. In order to overcome this difficulty, we have to employ a method based on an extra-thermodynamic assumption. For example, we can use (1) or (2) below ... 

Traditionally, the LJPs between different solvents are estimated indirectly. For example, if AGt°(M+, R->S) is known, we can estimate the LJP ( )) from Eq. (6.13) by measuring the emf (E) of cell (XI). The reliability of the method depends on the reliability of the extra-thermodynamic assumption used in determining AGt°(M+, R->S). Recently the author proposed a new method of estimating the LJP between different solvents, in which the three components are estimated separately and then simply added together. Table 6.9 lists some examples of LJPs esti-... 

Finally, there is an extra-thermodynamic assumption, which one can make about two molecules whose reactivity one wishes to compare. The basic idea is not unfamiliar, since it is inherent in the Bronsted linear free-energy relation. The assumption is that the free-energy difference in the transition states is bracketed by reactants and products. The factor provides a numerical index between zero and unity of the... 

The free energy of solvation of an uncharged species as well as the related enthalpy and entropy quantities are experimentally accessible. The evaluation of the corresponding properties for single ions is only possible with the help of an extra-thermodynamic assumption. This also holds for the free energy of transfer of neutral combinations of ions and the related thermodynamic quantities. 

We return to the complex formation equilibria described in equations 3, 4, 20, 21, and 26. The equilibrium constants as given in these equations are essentially intrinsic constants valid for a (hypothetically) uncharged surface. In many cases we can use these constants as apparent constants to illustrate some of the principal features of the interdependent variables that affect adsorption. Although it is impossible to separate the chemical and electrical contributions to the total energy of interaction with a surface without extra thermodynamic assumptions, it is useful to operationally break down the interaction energy into a chemical and a Coulombic part ... 

Unfortunately, in many nonaqueous solvents there is no completely unambiguous way of determining half-cell potentials. This ambiguity stems from the fact that the free energies of transfer (AGJ ) of individual ions from one solvent to another are not knowable. (It should be noted that no ambiguity necessarily exists if one is content with comparing whole-cell potentials in different solvents. This is true because the free energies of transfer of dissolved salts are knowable.) Some type of extra thermodynamic assumption is usually necessary to compare half-cell potentials measured in one solvent to those measured in some other solvent. Popovich has provided excellent... 

In the study of the interface with two immiscible electrolyte solutions (ITIES), considerable attention has been focused on the estimation of the Galvani potential difference at the water oil interface on the basis of a reasonable extra-thermodynamic assumptions. The discussion of these estimates is often made in terms of the ionic distribution coefficient, which is defined on the basis of equations (8.9.5) and (8.9.6). Generalizing this equation for the ot P interface at which ion i with charge z,- is transferred, one may write... 

The above analysis is easily extended to other half-cell reactions. Much of the data required to do the necessary calculations has been collected for cells involving aqueous solutions [1] and can also be found in thermodynamic tables published by the National Bureau of Standards in Washington [G3]. In practice, standard potentials are always used on the conventional scale because no extra-thermodynamic assumptions are involved in their calculation. Any of these quantities can be converted to the absolute scale by adding the estimate of the absolute potential of the SHE, that is, 4.43 V, to the conventional value of the standard potential. 

The experimental problem in defining the pH of an aqueous solution exists for all p-functions related to ionic activity. The fundamental reason for the problem is that individual ionic activity coefficients carmot be measured experimentally. A variety of extra thermodynamic assumptions are used to circumvent this problem. As might be expected, the exact nature of the assumption depends on the nature of the ion and the way it interacts with the solvent, which in most cases is water. 

In 1977 Koryta et al. [61] reported the Avalues of several common ions at the NB/W interface, which were calculated from the extraction data using an extra thermodynamic assumption. Afterward, a newly developed electrochemical technique (so-called ion-transfer voltammetry) with a polarizable O/W interface was employed to determine AGtr° w for a variety of ions [33,62-71]. In Table 4 the reliable values of AG ° >w are compiled. Regarding the ions whose AGj r° w values are available for both electrochemical and extraction measurements, the electrochemical data, which seem to be more accurate, have been chosen preferentially. For several ions, somewhat different AGf, 0 w values from electrochemical measurements have been reported, as also seen in the database provided by Girault on a website [72], In this study, however, we have carefully chosen reliable values for the respective ions, which were determined under well-defined conditions (reference electrodes, solution compositions, etc.). 

So-called absolute standard molar enthalpies of hydration of a number of individual ions, including Th", have been obtained by using a combination of experimental data and estimates based on the extra-thermodynamic assumptions that... 

Up to this point the treatment is strictly thermodynamic. However, the next steps necessarily demand the use of extra-thermodynamic assumptions. The first of them is the choice of an adsorption isotherm, which in general may be written as Pa = f( , 0). The most commonly adopted isotherm is the Frumkin isotherm, Eq. (1), assuming arbitrarily a linear or a quadratic dependence of upon E. The next necessary assumption concerns the dependence of p upon E, for which the following expression is usually adopted P = Pn,axexp -5( - moU, where b is a constant. Thus if a certain adsorption isotherm px = f( , 0) is selected, the partial derivative of In x with respect to E at constant 0 can be calculated and therefore Eq. (4) after integration yields the relationship = g( ). From this relationship the dependence of y upon E is obtained by integration and the differential capacity C is calculated fromC = dCT /d . 

Partial molal entropy data in ethanol are nearly as sparse as the heat capacity data. The only comprehensive entropy data in this solvent are those of Jakuszewski and Taniewska-Osinska, who report 5 for HCl and several alkali metal halides in ethanol. Ionic entropies have been calculated for the alkali metals from free energies and enthalpies of solvation, but since extra-thermodynamic assumptions were necessary, the meaning of the values is questionable. Ionic entropies in ethanol are somewhat more negative than in methanol and considerably more negative than in water. 

An important quantity required for analysis of ion-solvent interactions and structural properties is the absolute free energy (or enthalpy and entropy) of solvation. Most methods of obtaining these quantities involve some extra-thermodynamic assumption such as the extrapolation of solvation energies versus some function of crystal radii (see sect. 2.11.4). The method based on measurements of volta potential differences avoids the controversy involving the significance of these radii. This method has been used by Frumkin, Klein and Lange, Randles and Parsons et... 

The extra thermodynamic assumptions which have been proposed to split the medium effect for electrolytes, that is for electrically neutral combinations of ions, into values for individual ions are discussed in sect. 2.11.4b. 

Several diverse techniques have been employed for making this division all of them involve extra-thermodynamic assumptions, most of which have been the subject of criticism. The most important single factor leading to confidence in any of the values is the fact that they all give results in a small range ( Sh+ = —1.5 to —6.3 cal mol K ), which fortuitously is not far from the conventional value of = 0. The methods can be divided into three groups, (1) Electrochemical, (2) Born treatment and (3) Correspondence plots. 

Ions in aqueous solutions are characterized by several thermodynamic quantities in addition to the molar volumes, heat capacities and entropies discussed above. These are the molar changes of enthalpy, entropy, and Gibbs energy on the transfer of an ion from its isolated state in the ideal gas to the aqueous solution. They pertain also to the dissolution of an electrolyte in water, since they can be considered as parts in a thermodynamic cycle in which the electrolyte is transferred to the gas phase, dissociates there into its constituent ions, which are then transferred into the solution. Contrary to thought processes, as described in Sect. 2.2., it is impossible to deal experimentally with individual ions but only with entire electrolytes or with such combinations (sums or differences) of ions that are neutral. The assignment of values to individual ions requires the splitting of the electrolyte values by some extra-thermodynamic assumption that cannot be proved or disproved within the framework of thermodynamics. However, for a theoretical estimation of the individual ionic... 

Experimental thermodynamic quantities pertain to entire electrolytes or to combinations (sums or differences) of ions that are neutral. Splitting of the electrolyte values for assignment of values to individual ions requires some extra thermodynamic assumption. 

By using this extra-thermodynamic assumption we get h+ = -6.7 + 0.7 cm moT at 25 °C. Other reference electrolytes could be chosen, but the partial molar volume of the ion results close to the above-cited values. 

The standard molar entropy of hydration of an ion is A 5" = 5 - 5, , the difference between its standard molar entropy in the aqueous solution (Table 2.8) and the standard molar entropy of the isolated ion in the ideal gas phase (Table 2.3). The latter, S°, are calculated from the third law of thermodynamics and spectroscopic data without invoking any extra-thermodynamic assumptions. The former, do involve the assumptions leading to A5 (H+, aq)=-22.2 2J K" mol" for the hydrogen ion (Section 2.3.1.2). With 5°(H% g)=108.9J K" mol", the standard molar entropy of hydration of the hydrogen ion is then A,5"(H ) = -22.2 2-108.9 =-131.1 2 J-K" mor. The standard molar entropies of hydration of ions are shown in Table 4.1, derived from A 5,°° = S -S,° but also obtainable from the conventional values by use of the absolute value of the hydrogen ion. They are related to the effect that ions have on the structure of water according to various approaches. This aspect is fully dealt with in Section 5.1.1.7. 

The individual ionic thermodynamic quantities for hydration, listed in Table 4.1, are based on two extra-thermodynamic assumptions for the enthalpies, they are based on the tetraphenylphosphonium tetraphenylborate (TPTB), A // (Ph4P ) = A //" BPh and for the entropies on the temperature derivative of the electromotive force of... 

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