Solution of a single electrolyte

In a solution of a single electrolyte solute that is not necessarily symmetrical, the ion molalities are related to the overall solute molality by 

From the additivity rule for the Gibbs energy, we have 

The mean ionic activity coefficient y is defined in general by 

Thus y is a geometric average of y+ and y weighted by the numbers of the cations and anions in the solute formula unit. With a substitution from Eq. 10.3.7, Eq. 10.3.6 becomes 

---

For a solution of a single electrolyte, the relationship between the mean activity coefficient and the osmotic coefficient is given by the equation... 

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

In a solution of a single electrolyte, here called a "pure solution", subscripts 1 and 2 are used to designate cations and anions respectively. When more than two ions are present, then... 

It has been proposed to define a reduced temperature Tr for a solution of a single electrolyte as the ratio of kgT to the work required to separate a contact +- ion pair, and the reduced density pr as the fraction of the space occupied by the ions. (M+ ) The principal feature on the Tr,pr corresponding states diagram is a coexistence curve for two phases, with an upper critical point as for the liquid-vapor equilibrium of a simple fluid, but with a markedly lower reduced temperature at the critical point than for a simple fluid (with the corresponding definition of the reduced temperature, i.e. the ratio of kjjT to the work required to separate a van der Waals pair.) In the case of a plasma, an ionic fluid without a solvent, the coexistence curve is for the liquid-vapor equilibrium, while for solutions it corresponds to two solution phases of different concentrations in equilibrium. Some non-aqueous solutions are known which do unmix to form two liquid phases of slightly different concentrations. While no examples in aqueous solution are known, the corresponding... 

The value of depends in general on the nature of functions x that describe the concentration distribution c,- and thus also on quantity t,-. The simplest situation occurs when there are solutions of a single electrolyte at different concentrations Ci and C2 on the two sides of the liquid junction. For the transport number,... 

The following text is only intended to provide the reader with a brief outline of the Pitzer method. This approach consists of the development of an explicit function relating the ion interaction coelScient to the ionic strength and the addition of a third virial coefficient to Eq. (6.1). For the solution of a single electrolyte MX, the activity coefficient may be expressed by Eq. (6.29) [15] ... 

A solution of a single electrolyte in a solvent contains four identifiable species the solvent, undissociated electrolyte, anions, and cations. Therefore, it might seem appropriate, following Eqs. 8.1-12 and 8.1-13, to write the Gibbs energy of the solution as... 

For example, the solution molecular weight of NaCl dissolved in water is 29. Thus, if we choose the arbitrary formula weight of NaCl to be 58, V will be 2. In a solution of a single electrolyte or a mixture of electrolytes with no common ion... 

Thermodynamic Excess Functions. Solutions of a Single Electrolyte... 

A crucial test was made in the context of dissolution and precipitation potential of KI in water and DMF. In one case f/+ < U while in the other case > U. Accordingly, one would expect difference in the sign of the potential in the two cases. Experimental results agree with this prediction [26]. Similarly, in solution of mixed electrolytes, the order of mobilities of cations and anions can be reversed as compared to the case of solution of a single electrolyte [26]. Effort has also been made to have a quantitative test of the theory. There is agreement between experimental and theoretically computed values of potentials as regards sign and order [29]. 

Secondary thermodynamic tables are unfortunately much more sparse. For example, while the deviations from perfect-gas behaviour have been studied for many pure gases, relatively little information is yet available even for binary gas mixtures, let alone for the usually multi-component mixtures relevant to chemical reactions in gases. Again, while the activity coefficient of the solute or the osmotic coefficient of the solvent has been measured over useful molality ranges for many solutions of a single electrolyte, little such information is yet available for the mixed electrolyte solutions relevant to chemical reactions in solutions. 

In principle the activity coefficients yb of solute substances B in a solution can be directly determined from the results of measurements at ven temperature of the pressure and the compositions of the liquid (or solid) solution and of the coexisting gas phase. In practice, this method fails unless the solutes have volatilities comparable with that of the solvent. The method therefore usually fails for electrolyte solutions, for which measurements of ye in practice, much more important than for nonelectrolyte solutions. Three practical methods are available. If the osmotic coefficient of the solvent has been measured over a sufficient range of molalities, the activity coefficients /b can be calculated the method is outlined below under the sub-heading Solvent. The ratio yj/ys of the activity coefficients of a solute B in two solutions, each saturated with respect to solid B in the same solvent but with different molalities of other solutes, is equal to the ratio m lm of the molalities (solubilities expressed as molalities) of B in the saturated solutions. If a justifiable extrapolation to Ssms 0 can be made, then the separate ys s can be found. The method is especially useful when B is a sparingly soluble salt and the solubility is measured in the presence of varying molalities of other more soluble salts. Finally, the activity coefficient of an electrolyte can sometimes be obtained from e.m.f. measurements on galvanic cells. The measurement of activity coefficients and analysis of the results both for solutions of a single electrolyte and for solutions of two or more electrolytes will be dealt with in a subsequent volume. Unfortunately, few activity coefficients have been measured in the usually multi-solute solutions relevant to chemical reactions in solution. 

It can also be of interest to define the transport numbers t and, which are the fraction of the current carried by the cation and anion respectively, i.e, for a solution of a single electrolyte ... 

Because of the electroneutrality condition, the individual ion activities and activity coefficients cannot be measured without additional extrather-modynamic assumptions (Section 1.3). Thus, mean quantities are defined for dissolved electrolytes, for all concentration scales. E.g., for a solution of a single strong binary electrolyte as... 

The first approximate calculation was carried out by Debye and Hiickel and later by Onsager, who obtained the following relationship for the relative strength of the relaxation field AE/E in a very dilute solution of a single uni-univalent electrolyte... 

If this equation is employed for a solution of a single valence-symmetrical electrolyte (z+ = z = z), then... 

Consider the system shown in Fig. 6.3. The ion-exchanger membrane separates solutions of a single, completely dissociated, uni-univalent electrolyte. Two pistons can be employed to form a pressure difference between the two compartments. The two electrodes W and W2 are... 

Optical techniques, in particular interferometry, may be used to measure a nonzero concentration of the reactant at the electrode. However, such measurements are restricted to (a) dilute solutions, because refraction occurs in addition to interference (B4a), and (b) solutions in which only the concentration of the reacting species varies, that is, to solutions of a single salt. If the solution contains two electrolytes with dissimilar concentration profiles in the diffusion layer, then a second independent measurement is needed to establish the reactant concentration at the electrode. Interferometric methods are considered in detail by Muller (M14). 

The ideality of the solvent in aqueous electrolyte solutions is commonly tabulated in terms of the osmotic coefficient 0 (e.g., Pitzer and Brewer, 1961, p. 321 Denbigh, 1971, p. 288), which assumes a value of unity in an ideal dilute solution under standard conditions. By analogy to a solution of a single salt, the water activity can be determined from the osmotic coefficient and the stoichiometric ionic strength Is according to,... 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

For an ideal solution, Jq = I and is unity. Then Eq. (9) is consistent with Eq. (10 11), since the total molality of all solute species is vm for a completely dissociated solute of molality m. For ionic solutions, the Debye-Hiickel theory predicts a value of yo different from unity and therefore a deviation of g from unity. A treatment of this aspect of the Debye-Hiickel theory is beyond the scope of this book, and we shall merely state the result. The osmotic coefficient g at 0°C for dilute solutions of a single strong electrolyte in water is given by... 

Solute and Solvent Activity Calculations. For the purposes of this study, the derivations necessary to the calculation of the solute and solvent activities will begin with the equation for the prediction of the excess free energy of a single electrolyte solution based on the work of Friedman (9). 

Once a complete set of interfacial tension data have been assembled as a function of electrode potential and solution composition, they may be differentiated to obtain the relative surface excess. For example, if the solution contains a single electrolyte and the reference electrode is reversible to the anion in the... 

Numerous models predict the activity coefficient of individual ions in solution. The one by Debye and Hiickel [8] considers only electrostatic (columbic) interactions between cations and anions in a dilute solution of a single, completely dissociated salt. It is assumed that ion-ion interactions (as opposed to other phenomena such as ion-solvent interactions, ion solvation effects, and variations in the solvent dielectric constant with salt concentration) cause the ion activity coefficients to deviate from 1.0. From a practical point, only the Debye-Hiickel activity coefficient relationship is needed, along with some knowledge of the theory s shortcomings, which restrict its application. For a dilute electrolytic solution containing a binary salt (i.e., a salt with one type each of cation and anion species), the ion activity coefficient from Debye-Hiickel theory is given by... 

Using an argument based on the Gibbs-Duhem equation, Stokes and Robinson (1) derived the following equation for a solution of a single aqueous electrolyte (A), presumed to be fully dissociated ... 

This fraction ti is the transport, transference, or migration number of the given ion in the particular solution. For the simplest case of a solution containing a single electrolyte yielding two ions the transport numbers of the two ions, t+ and are... 

As there is no analogue of Butler s equation for ionised surface layers, the procedure used to derive the equation of state has to be based on the Gibbs adsorption equation and a model adsorption isotherm. The isotherm equation can also be derived from the theoretical analysis of the expressions for electrochemical potentials of ions. For the solution of a single ionic surfactant RX, with the addition of inorganic electrolyte XY, starting from Eqs. (2.2) and (2.21) for the electrochemical potentials, one obtains the adsorption isotherm... 

Shortly after Debye and Hiickel had presented their momentous work on the free energy of electrolyte solutions, Onsager derived theoretically, the empirical equation proposed by Kohlrausch to represent the molar conductance of an electrolyte solution. For solutions of a single symmetrical electrolyte this equation is given by... 

Equations (1.17) and (1.18) become uncoupled (this is also the case in a solution of a single binary electrolyte ). 

---