Electrolyte solutions, activity coefficient chemical potential

The papers in the second section deal primarily with the liquid phase itself rather than with its equilibrium vapor. They cover effects of electrolytes on mixed solvents with respect to solubilities, solvation and liquid structure, distribution coefficients, chemical potentials, activity coefficients, work functions, heat capacities, heats of solution, volumes of transfer, free energies of transfer, electrical potentials, conductances, ionization constants, electrostatic theory, osmotic coefficients, acidity functions, viscosities, and related properties and behavior. 

The irmer potential is an important property of individual phases. Much more will be said about this property when the interface between two phases is discussed in chapter 8. For the moment, < ) is regarded as a property of phase a which is the same throughout the phase with a value defined with respect to charge-free infinity. On the other hand, in the case of an electrolyte solution, the local electrostatic potential varies from point to point due to the presence of discrete charges on the ions. Thus the electrostatic potential is more positive at a cation and more negative at an anion. These fluctuations occur about the average value, < ) . This can be seen more clearly by writing out the chemical potential of ion i in terms of its concentration c, and activity coefficient y,. Thus, from equation (6.6.1)... 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

In addition to the foregoing, it is customary to include under electrochemistry (I) processes for which the net reaction is physical transfer, e g., concentration cells (2) electrokinetic phenomena, e.g.. electrophoresis. eleclroosmnsis, and streaming potential (3) properties ot electrolytic solutions, if they are determined by electrochemical or other means, e g.. activity coefficients and hydrogen ion concentration (4) processes in which electrical energy is first converted to heal, which in turn causes a chemical reaction that would not occur spontaneously at ordinary temperature. The... 

The reference state of the electrolyte can now be defined in terms of thii equation. We use the infinitely dilute solution of the component in the solvent and let the mean activity coefficient go to unity as the molality or mean molality goes to zero. This definition fixes the standard state of the solute on the basis of Equation (8.184). We find later in this section that it is neither profitable nor convenient to express the chemical potential of the component in terms of its molality and activity. Moreover, we are not able to separate the individual quantities, and /i . Consequently, we arbitrarily define the standard chemical potential of the component by... 

These equations are used whenever we need an expression for the chemical potential of a strong electrolyte in solution. We have based the development only on a binary system. The equations are exactly the same when several strong electrolytes are present as solutes. In such cases the chemical potential of a given solute is a function of the molalities of all solutes through the mean activity coefficients. In general the reference state is defined as the solution in which the molality of all solutes is infinitesimally small. In special cases a mixed solvent consisting of the pure solvent and one or more solutes at a fixed molality may be used. The reference state in such cases is the infinitely dilute solution of all solutes except those whose concentrations are kept constant. Again, when two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be made and clearly stated. 

In the case of an - electrolyte dissociating in solution as Aj,+ Bj, < is+Az+ + z/ Bz where v+z+ = v z to ensure electroneutrality, and the total number of particles formed by each molecule is v = v+ + z/, then the only activity that can be measured is that of the complete species, and the individual ions cannot be assigned meaningful chemical potentials. Under these circumstances, a mean activity coefficient is defined through the equation yv = y++ yvs. Since individual ionic chemical potentials are not measurable, it has become conventional to assign to the chemical potential of the hydrogen ion under standard conditions the value of zero, allowing relative chemical potentials for all other ions to be formulated. 

This free energy change (AUe) represents the electrostatic contribution to the chemical potential of the ion, that is, the electrical woik necessary to charge the ideal solution, and it is responsible for deviations of the solution from ideal behavior. The activity coefficients of single ions are not measurable experimentally [35] for an electrolyte EpHq, the medium activity coefficient is... 

Hence, to analyze the physical significance of the activity coefficient term in Eq. (3.57), it is necessary to compare this equation with Eq. (3.52). It is obvious that when Eq. (3 52) is subtracted from Eq. (3.57), the difference [i.e., /r,- (real) - fij (ideal)] is the chemical-potential change arising from interactions between the solute particles (ions in the case of electrolyte solutions). That is. 

For any imaginary ideal solution of an electrolyte, at any given T and P, in which all activity and osmotic coefficients are unity, we can write for the chemical potential of a solute s. 

Corresponding to each chemical potential there is an activity coefficient defined in terms of equation (20.4). By convention, the activity coefficients of electrolytes are always expressed in terms of the ideal dilute solution as standard reference state, cf. chap. XXI, 3. Thus in the case of an aqueous NaCl solution we may write... 

In the case of pure solids such as Ag and AgCl the chemical potential is identical to the standard chemical potential at 25°C and 1 bar pressure. For solutions, the standard state of the solute is unit activity at the same temperature and pressure. In the case of electrolytes as solutes, the activity is defined on the concentration (molarity) scale, and the standard state is the hypothetical ideal state of unit molarity for which the activity coefficient ye is unity. Under these circumstances, the activity of the solvent, which does not appear explicitly in equation (9.2.9), is also unity to a good approximation when the solvent is water. For gases the standard state is a pressure of 1 bar (10 Pa) at 25°C. In the older literature the standard pressure was 1 atm (101,325 Pa). In data compilations appearing after 1982, the standard state of 1 bar and 25°C is always used for gases [G3]. 

Thermodynamics is used in the analysis of electrochemical cells (1) to predict which electrode reactions occur spontaneously in the anodic and cathodic directions if the two electrodes are in equilibrium with their respective adjacent solutions and are connected to one another via an external wire, and (2) to quantify chemical potentials and activity coefficients in nonideal electrolytic solutions. 

In this chapter we discuss some of the properties of electrolyte solutions. In Sec. 12-1, the chemical potential and activity coefficient of an electrolyte are expressed in terms of the chemical potentials and activity coefficients of its constituent ions. In addition, the zeroth-order approximation to the form of the chemical potential is discussed and the solubility product rule is derived. In Sec. 12-2, deviations from ideality in strong-electrolyte solutions are discussed and the results of the Debye-Hiickel theory are presented. In Sec. 12-3, the thermodynamic treatment of weak-electrolyte solutions is given and use of strong-electrolyte and nonelectrolyte conventions is discussed. 

In principle, the conventions used for nonelectrolyte solutions developed in Chap. 11 could be employed for electrolyte solutions which are subject to the condition of electroneutrality. Agreement with experimental data could be obtained by choosing the molecular weight to be some fraction of the formula weight. However, these conventions generally lead to activity coefficients which are rapidly varying functions of composition. In order to avoid this, we formally define chemical potentials and activity coefficients for ionic components. The definition of chemical potentials for ionic components does not have operational significance since their concentrations cannot be varied independently. 

For the calculation of the chemical potentials of the particular components in an electrolyte solution, it is necessary to define an appropriate standard and reference state. It is convenient to define a standard state where the values of both the concentration and the activity coefficient and hence the activity are unity. For the solvent, there is no problem to use the pure component at the required pressure and temperature as standard state as shown in Chapter 4 ... 

Electrochemistry uses chemical potentials of the type given by Eqs. (91a)-(91c) for single ions. The link to thermodynamics is established by the help of mean mole fractions and mean activity coefficients f . For a binary electrolyte as the solute, Y2 = AjI (z+, valent cation Z, valent anion), for example, Na2S04 (where z+ = +2, z = — 1, = 2, i = 1), the chemical potential/X2(p, T)... 

In order to calculate the equOibrium composition of a system consisting of one or more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is important, This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions, the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions. 

In contrast to most activity coefficients of uncharged species, salt activity coefficients can significantly differ from unity. As a result, they must be always carefully considered in order to make proper calculations, whenever the chemical potential of an electrolyte in solution is involved. 

An electrolyte solution which is not in equilibrium is exposed to generalized forces that are responsible for irreversible processes, such as transport or relaxation processes. A gradient of the chemical potential of the considered ions is the source of such a force, producing a particle flow that leads to diffusion and to electric conductance. Neglecting activity coefficients (dilute solutions) the flow of ion i is given by the relation (with the convection term omitted)... 

In dealing with an electrolyte solute, we can refer to the solute (a substance) as a whole and to the individual charged ions that result from dissociation. We can apply the same general definitions of chemical potential, activity coefficient, and activity to these different... 

The activity coefficients y of electrolytes are defined with respect to ideal solutions. For example, the mean chemical potential for AgCl is written as... 

The linear term, Cm, was dealt with and interpreted in several ways. Empirically, C = 0.l z z was proposed by Davies [13] for fitting activity coefficients of aqueous electrolytes at 25°C up to 0.1 m. Stokes and Robinson [ 14] suggested that the amount of solvent bound by the solvated ions should be deducted from the total amount of solvent in order to represent the entropic part of the chemical potential of the solute appropriately. Therefore, the following expression results for the Unear term ... 

At the Prague Institute of Chemical Technology, F. Jirsa studied anodic oxidation of gold [15] later he published an important paper about silver electrode for a silver-iron battery [16], Jaroslav Chloupek (1899-1975), partly with V. Danes (1907-1980) and B. Danesova, studied the electrode potential in solutions of mixed manganese salts [17], the solubility and activity coefficient of Ag2S04 in some solutions [18], the ions and deviations from the approximation of Debye-Hiickel theory [19], the liquid potentials [20], and the anomalous valency effect of strong electrolytes in aqueous solution [21]. 

The concentration scale of a standard chemical potential and an activity coefficient are specified by additional symbols placed as either the subscript or superscript. For example, the mole fraction scale is specified in Equation 1.3. In this equation, if we want to be precise, should be called the standard chemical potential on the mole fraction concentration scale. Equation 1.3 is usually used for solutions of nonelectrolytes, such as 02(aq), and for solvent (water) in electrolyte solutions. Also, this equation can be used for solid solutions such as metal alloys. For electrolyte solutions, molality is commonly used except (1) electrolyte conductivity and (2) electrochemical kinetics, where molarity is commonly used. 

At finite concentrations this formula needs modifying in two ways. In the first place, diffusion is governed by the osmotic pressure, or chemical potential, gradient (not, strictly, by the concentration gradient), so that the mean activity coefficient of the electrolyte must be taken into account. In the second place, ionic atmosphere effects must be allowed for. In diffusion, unlike conductance, the two ions are moving in the same direction, and the motion causes no disturbance of the symmetries of the ionic atmospheres there is therefore no relaxation effect. There is a small electrophoretic effect, however, the magnitude of which for dilute solutions has been worked out by Onsager, and the most accurate measurements support the extended formula based on these corrections. 

Thermodynamics makes it possible to take into account the deviation from the ideality of the biological media. In the case of media containing ions, these corrections should be taken into account starting from weak concentrations as mentioned previously, the addition of a salt can significantly alter the chemical equilibrium. In the case of media containing non-electrolytes, corrections of activity become significant when the concentration of a solute increases or when the number of solutes increases even if each one has a weak concentration. Figure 1.5 shows all of the relationships between physical-chemical parameters that can be calculated from the activity coefficient and the standard potential. 

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