Chapter 9
Electrolyte Solutions

Many industrial and natural processes require information on phase equilibria of electrolyte solutions. Some examples are: partitioning processes in biochemical systems; precipitation and crystallization processes in geothermal-energy systems or drilling muds; desalination of water; water-pollution control; salting-in and salting-out effects in extraction and distillation; production of natural gas from high-pressure aquifiers where natural gas is in equilibrium with brines; food processing; and production of fertilizers.

Extensive modifications of traditional activity-coefficient models described in Chap. 6 are required to describe phase equilibria in systems containing electrolytes.

The literature is rich in theoretical and experimental studies of electrolyte solutions. Unfortunately, much of it is confusing, primarily because authors often are not clear in their definitions of thermodynamic functions and because there are no universally accepted standards of notation. One source of confusion concerns standard states but another, often not appreciated, follows from the choice of variables. In mixtures of nonelectrolytes, we use the Lewis-Randall system where the variables are temperature, pressure and concentrations of all components, usually expressed by mole fractions. In electrolyte solutions, especially dilute solutions, the variables may be temperature, concentrations of all solute species, and osmotic pressure. A variety of units is available for solute concentrations; a common one is molality, defined as moles of solute per kg of solvent (not per kg of solution). This unit is suitable for dilute solutions but not for highly concentrated solutions because it tends to infinity as the solvent-to-solute ratio tends to zero.

Extension of a thermodynamic framework for nonelectrolytes to contain also electrolytes is not a trivial task. It is a common misconception that such extension is only a small detail, a little perturbation, like adding a short tail to a big dog. Not so. Extension to include also electrolytes requires concepts and constraints (e.g. electro-neutrality) that can be mastered only with patient and devoted study.

This chapter can only provide an overview. The best thorough discussion of electrolyte-solution thermodynamics is given in the book by Robinson and Stokes (1970). However, this fine book is somewhat out of date. A good supplement, reviewing more recent work, is the monograph edited by Pitzer (1991) and the book by Barthel et al. (1998).

The following sections present the thermodynamic basis for describing activities of components in electrolyte solutions, some of the theoretical and semi-empirical models that have been developed for electrolyte solutions and, finally, a few applications of these models to practical phase-equilibrium calculations.

9.1 Activity Coefficient of a Nonvolatile Solute in Solution and Osmotic Coefficient for the Solvent

Before taking into account dissociation of electrolytes into ions, we first discuss the thermodynamics of solutions containing a nonvolatile solute in a volatile solvent. This discussion is relevant because most electrolytes (salts) are essentially nonvolatile at normal temperatures.

As shown in Chap. 6, for a component i at some temperature, pressure, and composition, activity a_i and activity coefficient γ_i are related to chemical potential μ_i by

(9-1)

where is the chemical potential of i in some conveniently defined standard state and ξ is some convenient measure of concentration.

In Chap. 6, for liquid mixtures of volatile nonelectrolytes, we define by choosing as standard state pure liquid i at system temperature and pressure. Mole fractions (or volume fractions) are typically used for ξ.

For a mixture containing a nonvolatile solute dissolved in a solvent, we use Eq. (9-1) for the solvent with the conventional definition of However, for a nonvolatile solute, pure liquid i at system temperature and pressure is often not a convenient standard state because for most cases, at normal temperatures and pressures, a pure nonvolatile solute cannot exist as a liquid.¹

¹ However, for some applications it may be useful to define the standard state of the nonvolatile solute as that of the pure liquid at system temperature and pressure, even when that standard state is hypothetical. When the system temperature is not very far below the melting point of the nonvolatile component, it is possible to estimate the chemical potential (or fugacity) of the subcooled liquid by systematic extrapolation as shown in Chap, 11.

For the dissolved solute, the chemical potential is written

(9-2)

where , the standard state chemical potential of i, is independent of composition but depends on temperature, pressure, and the nature of solute and solvent. A convenient choice for is the chemical potential of i in a hypothetical ideal solution of i in the solvent, at system temperature and pressure, and at unit concentration ξ_i = 1. In this ideal solution, γ_i = 1 for all compositions. In the real solution, γ_i → 1 as ξ_i → 0.

It is a common misconception to say that the standard state for the solute is the solute at system temperature and pressure and at infinite dilution. That is not correct; at infinite dilution, the chemical potential of the solute is -∞. The standard-state chemical potential for solute i must be at some fixed (non-zero) concentration. That concentration is unit concentration. Unit concentration is used because its logarithm is zero.

For solutions of polymers, the common composition scale is volume fraction but for solutions of other nonvolatile solutes, there are three commonly-used composition scales: molarity (moles of solute per liter of solution, c_i); molality (moles of solute per kg of solvent, m_i); and mole fraction, x_i.

When we set ξ_i = c_i, Eq. (9-2) becomes

(9-3)

where is the molarity activity coefficient. The standard state is the hypothetical, ideal, 1-molar solution of solute i in solvent j. Because ideal behavior is approached as the real solution becomes infinitely dilute, →1 as c_i → 0.

It is often more convenient to use the molality scale because it does not require density data. When we set ξ_i = m_i, Eq. (9-2) becomes

(9-4)²

² For a solution containing a solvent and a solute i, the conversion between molarity (c_i.) and molarity (m_i) is given by c_i = d m_i/(1 + 0.001 M_i m_i.) where d is the mass density of the solution and M_i. is the molecular mass of solute i. However, in practice this conversion is often made by assuming that c_i ≈ d m_i a good approximation for dilute solutions.

where is the molality activity coefficient. The standard state is the hypothetical, ideal, 1-molal solution of solute i in solvent j. In the real solution, → 1 as m_i → 0³.

³ Because it is aesthetically preferable to use activity coefficients that are nondimensional, and because m_i has dimensions of mol kg^-1, we could write where m_o is unit molality, i.e., 1 mol kg^-1. For simplicity, we always use Eq. (9-4) that omits m_o. The same applies to Eq. (9-3) where we omit unit molarity c_o.

Molality is an inconvenient scale for concentrated solutions because m_i → ∞ when we reach the pure solute. For concentrated solutions, the mole fraction is often a more convenient scale. When ξ_i = x_i, Eq. (9-2) becomes

(9-5)

where is the unsymmetric⁴ mole-fraction activity coefficient. The standard state is the hypothetical, ideal solution where x_i = 1. In the real solution, → 1 as x_i → 0.

⁴ The convention presented in Sec. 6-4 adds an asterisk (*) to the unsymmetrically normalized activity coefficients. In this chapter, the activity coefficients of the solute are always normalized in this way. However, to clarify notation, instead of we use here or or , depending on the variable used for solute concentration.

To illustrate Eq. (9-4), Fig. 9-1 shows solute activity plotted against solute molality. The hypothetical ideal solution is shown by a straight line that goes through the coordinates (0,0) and (1,1). The standard-state activity is given by A corresponding to unit molality. If we arbitrarily choose a molality of 1.5, the activity of the solute in the real solution is given by C, while that in the hypothetical ideal solution is given by B. In the hypothetical ideal solution, activity is equal to molality because, in that ideal solution, , for all solute concentrations. Therefore, the activity coefficient of the real solution, = a_i / m_i, is given by the ratio

Figure 9-1 Schematic representation of the activity of a nondissociating solute as a function of its molality. Point A shows the standard state. When the solute’s molality is 1.5, its activity coefficient is given by .

In Eqs. (9-3), (9-4), and (9-5), , or , or . is the chemical potential of the solute in its thermodynamic standard state, that is, the state from which we measure or calculate changes in state properties, such as chemical potentials. A standard state need not be physically realizable, but it must be well-defined.⁵

⁵ A mixture at infinite dilution is not appropriate for a standard state because, in the limit of infinite dilution, the chemical potential of the solute approaches -∞.

For the solvent, we use the pure liquid at system temperature and pressure as the standard state.

In a given solution, the chemical potential of the solvent and that of the solute are unaffected by the choice of the composition scale but the activity and the activity coefficient are affected not only by the choice of standard state but also by the choice of concentration scale. We can readily derive relationships to convert activity coefficients for one concentration scale to those for another. In a binary mixture of a nondissociating solute 2 in solvent s, these relations are:

(9-6)

(9-7)

(9-8)

where d is the mass density (g cm^-3) of the solution; d_s is the mass density (g cm^-3) of the pure solvent; M₂ is the molar mass of the solute and M_s is the molar mass of the solvent.

To obtain the activity of the solvent, we use the Gibbs-Duhem equation that interrelates the activity of the solute and that of the solvent. At constant temperature and pressure, the Gibbs-Duhem equation gives

(9-9)

The osmotic pressure π of the solution is ⁶

⁶ Osmotic pressure is discussed in Sec. 4.11.

(9-10)

where υ_s is the molar volume of the solvent.

The osmotic coefficient is the ratio π(real)/π(ideal). To find π(real) and π(ideal), we substitute Eq. (9-9) into Eq. (9-10); for the ideal solution, the second term on the right-hand side of Eq. (9-9) is zero.

The osmotic coefficient is

(9-11)

Superscript (m) for and a_s (ideal) indicates that these quantities depend on the choice of concentration scale.⁷ However, no superscript is needed for a_s (real) because this quantity is independent of concentration scale.

⁷ If the concentration scale is molality, a_i(ideal) = m_i; if the concentration scale is molarity, a_i(ideal) = c_i

For representing experimental data of dilute solutions, osmotic coefficients are preferred because they are more sensitive to composition than activity coefficients γ_s. Experimentally, is often obtained from vapor-pressure measurements where, at modest pressures, the activity is

(9-12)

Here P_s is the partial pressure of the solvent and is the vapor pressure of the pure solvent, both at system temperature T. Equations (9-10) and (9-12) give a useful expression to replace the general equation [Eq. (9-11)] for the osmotic coefficient:

(9-11a)

9.2 Solution of an Electrolyte. Electroneutrality

The equations of Sec. 9.1 define the activity and the activity coefficient of a nondissociating solute. However, in a solution of an electrolyte, the solute dissociates into cations and anions. Cations and anions are not independent components because of electroneutrality. For example, when one mole of a strong electrolyte⁸ like NaCl is dissolved in one kilogram of water, we have a one-molal solution of NaCl that is fully dissociated to a one-molal solution of positively-charged sodium ions and a one-molal solution of negatively-charged chloride ions.

⁸ A strong electrolyte (e.g. NaCl in water) is an electrolyte that in solution is completely dissociated into its constituent ions. By contrast, a weak electrolyte (e.g. acetic acid in water) is only partially dissociated.

Ordinary termodynamic measurements give properties not of individual ionic species, but of the neutral electrolytes formed by cations and anions. In a solution of an electrolyte, electroneutrality imposes the condition that the number of moles of the individual ionic species cannot be varied independently. In aqueous NaCl there are three species but only two (not three) components.

When dissolved in a high-dielectric-constant solvent like water, an electrically neutral electrolyte M_V+X_V– is dissociated into v₊ positive ions (cations) each with a charge z₊, and v_– negative ions (anions) of charge z_–. Charges are given in normalized units where z₊ = 1 for a proton. Electrolytic dissociation is represented by

(9-13)

Electroneutrality requires that

(9-14)

For example, in the dissociation

we have v₊ = 2, z₊ = 1, v_– = 1, and z_– = –2.

Equation (9-13) expresses a chemical equilibrium. The criterion for chemical equilibrium is

(9-15)

Using the molality scale for activity coefficients [Eq. (9-4)], substitution in Eq. (9-15) gives the chemical potential of the electrolyte:

(9-16)

where we have dropped superscript (m) on activity coefficients γ₊ and γ_- and where

(9-17)

In Eq. (9-17), is the chemical potential of ion M (with charge z₊) in a hypothetical ideal solution where the molality of ion M is unity. A similar definition holds for

Equation (9-16) can be written in a more efficient form:

(9-16a)

where, for convenience, we have dropped the subscripts on MX.

The mean ionic molality, m_±, and the mean ionic activity coefficient, γ_±, are defined by

(9-18)

(9-19)

where ν = ν₊ + ν_-. After substitution, Eq. (9-16a) becomes

(9-20)

with

(9-21)

where a_± is the mean ionic activity.

For strong electrolytes, where ionization is essentially complete, m₊ = ν₊ m_MX and m_- = ν_-m_MX.⁹ The mean ionic activity coefficient is

⁹ m_MX is the molality of the electrolyte as determined from the preparation of the solution, ignoring dissociation.

(9-22)¹⁰

¹⁰ Similarly, using the molarity concentration scale, this equation becomes where . Here c_MX is the molarity of the salt. For the mole fraction scale, . Here x_MX is the mole fraction of the electrolyte, ignoring dissociation. However, this definition is arbitrary. Many applications assume complete dissociation of the electrolyte; in these applications, mole fraction may be defined by Eq. (9-26).

For example, the mean molality and mean ionic activity coefficient for NaCl, a 11 electrolyte, are . Similar relations hold for 2-2 or 3-3 electrolytes. For a 2-1 (or 1-2) electrolyte (e.g. CaCl₂), and where m_MX is the molality of the electrolyte. Table 9-1 gives the mean molality for salts with specified stoichiometry.

Table 9-1 Mean ionic molality (m_±) for several electrolytes.

Figure 9-2 shows activity coefficient as a function of concentration (molality) for a few electrolytes in water at 25°C. By definition, is unity at zero molality for all electrolytes. In dilute solutions, decreases rapidly with rising concentration; the steepness of this initial drop varies with the type of electrolyte. However, for a given valence type, Fig. 9-2 shows that for low molalities (say, to about 0.01), is essentially independent of the chemical nature of the constituent ions. The theoretical basis for this observation is supplied by the Debye-Hückel theory, as briefly discussed in Sec. 9.7. For most electrolytes, curves similar to those in Fig. 9-2 show a minimum at intermediate concentrations. At high concentrations, may be much larger than unity.

Figure 9-2 Mean ionic activity coefficients (Robinson and Stokes, 1970) for some electrolytes in aqueous solution at 25°C.

Equation (9-19) indicates that the measurable quantity is determined by individual ion activity coefficients γ₊ and γ_- that, usually are not independently measurable.

Activity data for electrolytes are obtained from measurements of the potentials of electrochemical cells, or from solubility and colligative-property measurements. An important source of experimental data is vapor-pressure measurements.¹¹ The most accurate relative-vapor-pressure method is the isopiestic method discussed elsewhere (Robinson and Stokes, 1970; Rard and Platford, 1991).

¹¹ Vapor-pressure measurements give the activity of the solvent. To obtain the activity of the solute, we use the Gibbs-Duhem equation that relates the activity of the solvent to that of the solute. See Sec. 9.4.

For a nondissociating solute, the standard state is at point A in Fig. 9-1. For a dissociating solute (i.e. an electrolyte), the standard state is obtained from a procedure similar to that for a nondissociating solute shown in Fig. 9-1, provided that the appropriate quantities are plotted.

To illustrate, consider a 2-1 electrolyte, such as calcium chloride (ν = 3). Because m_± = , the standard state is determined from a plot of a_± as a function of , as shown in Fig. 9-3 for aqueous CaCl₂ solutions at 25°C. As Fig. 9-3 shows, the standard state for a 2-1 electrolyte is a hypothetical ideal dilute solution with unit molality. In that standard state, the mean ionic activity coefficient is 1. For a 2-1 electrolyte, upon selecting the plotted quantities indicated in Fig. 9-3, we assure that a_±/m_± → 1 as m_± → 0. In the hypothetical ideal solution, a_±/m_± remains unity for all m_±.

Figure 9-3 Mean ionic activity of aqueous CaCl₂ at 25°C as a function of its molality (Robinson and Stokes, 1970). Point A shows the standard state. The (ideal solution) straight line goes through coordinates (0, 0) and (1, 4^1/3). When mol kg^-1, = 1 mol kg^-1

9.3 Osmotic Coefficient in an Electrolyte Solution

An electrolyte MX is dissolved and completely dissociated in solvent s. Using the molality scale, the chemical potential of the solvent μ_s is

(9-23)

Here is the chemical potential of pure solvent s at system temperature T and pressure is the osmotic coefficient.

If one molecule of salt MX dissociates into v ions, the ideal-solution activity of the solvent is

(9-24)

For the real solution, substitution of Eq. (9-12) into Eq. (9-23) gives

(9-25)

A superscript (m) is written for and for a_s (ideal) because these quantities depend on the choice of concentration scale. However, no superscript is needed for a_s because that quantity is independent of concentration scale.

If we use the mole fraction scale, we write for the chemical potential of the solvent, μ_s

(9-23a)

where γ_s → 1 x_s → 1 and the standard state is the pare liquid solvent at the temperature and pressure of the solution.

Assuming complete dissociation of an electrolyte, for aqueous solutions the mole fraction of the electrolyte, x_MX, is defined

(9-26)¹²

¹² Equation (9-26) is readily extended to the case where the aqueous solution contains two (or more) salts, provided that these salts do not have a common ion. In general, it is better to define mole fractions of ions rather than mole fractions of salts.

Here n_MX is the number of moles of electrolyte per kg of water, n_w is the number of moles of water per kg of water (n_w = 1000/M_w = 55.51 mol, where M_w is the molar mass of water), m_MX is the molality of the salt and ν = ν₊+ ν_-. According to this definition, x_w + νx_MX = 1; the salt mole fraction can never equal unity because, for pure salt, it becomes 1/ν.

In dilute and moderately concentrated electrolyte solutions, the activity coefficient of the solvent, γ_s, is close to unity, as illustrated in Table 9-2 for aqueous sulfuric acid at 25°C. For the calculations in Table 9-2, mole fractions of water, x_w were obtained from x_w = n_w/(n_w + vn_MX) = 55.51/(55.51 + νm_MX).¹³

¹³ An equivalent definition for

Table 9-2 Activity coefficients (γ), and rational (g) and molal osmotic () coefficients of aqueous sulfuric acid solutions at 25°C.*

Consider a 1-molal solution of sulfuric acid at 25°C. Table 9-2 shows that water has an activity coefficient γ_w = 1.014. This number does not suggest the strong departure from ideal behavior indicated by the mean ionic activity coefficient of the solute, = 0.1316. Deviations from ideal behavior are more effectively characterized by the osmotic coefficient of the solvent, i.e. be the rational osmotic coefficient, g, or the more-often-used practical (or molality) osmotic coefficient, . The former is related to the chemical potential through

(9-27)

with g → 1 as x_s 1. Here x_s is calculated assuming total dissociation of the electrolyte [Eq. (9-26)].

Comparing with Eq. (9-23a), we obtain

(9-28)

Like the molality osmotic coefficient the rational osmotic coefficient g is also defined by the ratio π(real)π(ideal). However, and g are not identical because π(ideal) using the molality scale is not equal to π(ideal) using the mole-fraction scale.

For the mole fraction scale,

(9-29)

where the definition of mole fraction x_s is given by

(9-26a)

π(real) is found from Eq. (9-10) and experimental data for a_s. The molality osmotic coefficient and the rational osmotic coefficient are related by

(9-30)

Compared to γ_s (mole fraction scale), osmotic coefficients are much more sensitive to deviations from ideality, i.e., differs from unity by a much larger amount than does γ_s, as shown in Table 9-2. For example, for a 1-molal aqueous solution of sulfuric acid at 25°C, while γ_s = 1.014, g = 0.745 and = 0.721. The definition of osmotic coefficient assures that it goes to unity at infinite dilution; it recognizes that van’t Hoff’s equation and Raoult’s law for the solvent become exact at infinite dilution.

9.4 Relation of Osmotic Coefficient to Mean Ionic Activity Coefficient

The mean ionic activity coefficient refers to the solute while the osmotic coefficient refers to the solvent. They are inter-related through the Gibbs-Duhem equation.

For a binary system at constant temperature and pressure, the Gibbs-Duhem equation is

(9-31)

where x_i is the mole fraction of component i, calculated according to Eq. (9-26).

We rewrite Eq. (9-31) in the form

where, for convenience, we have dropped subscript MX on m.

Substituting Eq. (9-21), we obtain

We note that d In m_± = d In m because In m_± and In m differ only by a constant. From Eq. (9-22), coupled with Eq. (9-25), we obtain

(9-32)

In the very dilute solution, as m → 0; therefore, Eq. (9-32) can be integrated to give

(9-33)

For an electrolyte solution, the integrand ( – l)/m diverges to –∞ as m →. 0 This problem is easily solved by changing the independent variable from m to m^1/2. Then the equation above becomes

(9-34)

In Eq. (9-34), the integrand now approaches a finite value as m → 0 where the Debye-Hückel coefficient for the osmotic coefficient.¹⁴

¹⁴ can be calculated from the ionic charges and the solvent’s dielectric constant, as shown in Sec. 9.7 [Eq. (9-49a)].

From cryoscopic, osmotic-pressure or vapor-pressure measurements, can be determined as a function of m; Eq. (9-34) can then be used to calculate . However, data must be available from (nearly) m = 0 and must be spaced closely enough to permit accurate evaluation of the integral in Eq. (9-34).

Alternatively, we may assume a given analytical form to extrapolate the experimental curve (m) to infinite dilution. The limiting law of Debye and Hückel (Sec. 9.7) provides a theoretical basis for this extrapolation. We cannot here go into details. The essential result is that the experimental activity of the solvent gives osmotic coefficient according to (Eq. 9-25.). The activity coefficient of the solute is then calculated from (Eq. 9-34.). However, for these calculations, it is necessary to determine the activity of the solvent to a very high degree of accuracy, especially in the dilute region.

9.5 Temperature and Pressure Dependence of the Mean Ionic Activity Coefficient

Activity-coefficient data for electrolyte solutions are mostly available at or near 25°C and 1 atmosphere. However, some important applications require data for wide ranges of temperature and pressure. For example, geothermal solutions may exist at temperatures to 600°C and pressures to several kilobars. It is therefore of interest to inquire what thermodynamics can tell us about the temperature and pressure dependence of activity coefficients.

The temperature dependence of the activity coefficient for a nonelectrolyte solute 2 is given by (Eq. 6-34.).

A similar expression holds for an electrolyte MX:

(9-35)

where ν is the number of ions formed by one molecule of the electrolyte and

(9-35a)

Here is the excess partial molal enthalpy of the dissolved electrolyte, relative to that at infinite dilution. In Eq. (9-35a), is the partial molal enthalpy of solute i in an infinitely dilute solution. The partial molal enthalpy in the standard state is the same as that at infinite dilution.

Similar to Eq. (6-36a), at constant temperature and composition, the effect of pressure on the mean ionic activity coefficient for an electrolyte MX is given by

(9-36)

where

(9-36a)

In Eq. (9-36a), is the partial molal volume of the electrolyte at the concentration of the solution and is the partial molal volume of the electrolyte at infinite dilution, equal to that in the standard state.

Unfortunately, (Eqs. 9-35.) and (9-36) are of little practical use because experimental partial molal enthalpies and partial molal volumes are rarely available. Further, these quantities depend not only on composition but also on temperature and, to a lesser extent, on pressure.

For some electrolyte solutions, semi-empirical methods have been developed to estimate the effect of temperature and pressure on mean ionic activity coefficients (Millero, 1977).

9.6 Excess Properties of Electrolyte Solutions

As discussed in Sec. 6.2, excess functions are thermodynamic properties of solutions in excess of those of an ideal solution at the same temperature, pressure, and composition. Care must be taken to define ideal precisely.

Consider a binary electrolyte solution containing m moles of (totally dissociated) salt MX in 1 kg of solvent s. The number of moles of solvent is n_s = 1000 / M_s, where M_s is the molar mass of the solvent; if the solvent is water, n_s = 55.51 mol. At constant pressure and temperature, the total Gibbs energy G of this solution is

(9-37)

where the chemical potential of the salt, μ_MX is given by (Eq. 9-20.) and the chemical potential of the solvent, μ_s, is given by (Eq. 9-23.). Substitution of (Eqs. 9-21.) and (9-22) in the expression for μ_MX, (Eq. 9-24.) in the expression for μ_s, gives

(9-37a)

where, for convenience, we dropped subscript MX from molality m of the electrolyte.

For an ideal solution, , and the Gibbs energy is

(9-38)

Therefore, the total unsymmetric excess Gibbs energy of the solution containing m moles of salt in 1 kg of solvent is given by

(9-39)¹⁵

¹⁵ The excess Gibbs energy is unsymmetric because the activity coefficients of solute and solvent are not normalized in the same way. see Sec. 6.4. Note that G^E* → 0 as m → 0.

This solution contains n_s moles of solvent and n_MX moles of completely dissociated electrolyte. Using Eq. (6.2-9) and (Eq. 9-39.) gives the partial excess Gibbs energy of the solvent (s) and that of the solute (MX):

(9-40)

(9-41)

Equations (9-40) and (9-41) can be rewritten in the form

(9-42)

(9-43)

where, as before, ν is the total number of ions formed from the dissociation of the salt.

For a binary solution containing 1 kg of solvent, the excess enthalpy of an electrolyte solution can be obtained using the Gibbs-Helmholtz relation [(Eq. 6-12.)],

(9-44)

The last term in (Eq. 9-44.) is related to the partial molal enthalpy of the electrolyte, given by (Eq. 9-35.).

To determine the excess enthalpy of an electrolyte solution, we need highly accurate data for the osmotic and mean ionic activity coefficients as a function of temperature.¹⁶ More reliable results may be obtained from heat-of-dilution flow calorimetry and heat-capacity measurements (Messikomer and Wood, 1975; Picker et al., 1971).

¹⁶ Because the Gibbs-Duhem equation relates to γ_±, we need not measure both. Accurate measurements for either one are sufficient.

9.7 Debye-Hückel Limiting Law

At a fixed concentration of ions, electrolytes containing ions with multiple charges have a stronger effect on the activity coefficients of ions than electrolytes containing only singly-charged ions. To express this dependence it is useful to introduce the (molal) ionic strength of the solution, I, defined by

(9-45)

where z_i is the charge on ion i and m_i is its concentration expressed by molality. The summation extends over all ionic species in the solution. For example, for seawater, whose composition is shown in Table 9-3, I = 0.72 mol kg^-1.

Table 9-3 Concentrations of major ions in oceanic seawater (Ciegg and Whitfield, 1991).

According to its definition, the ionic strength of a 1-1 electrolyte is equal to its molality, m_MX; that for a 1-2 electrolyte (e.g. Na₂SO₄) is 3m_MX, and that for a 2-2 electrolyte (e.g. ZnSO₄) is 4m_Mx.

The activity coefficient of an electrolyte depends strongly on its concentration. As summarized in Chap. 4, charged particles interact with coulombic forces: for two ions with charges z₊ and z_– at a separation r, the force is proportional to (z₊z_–)/4πε_oε_rr², where ε_r is the dielectric constant of the solvent and ε_o is the vacuum permittivity. Thus the potential energy of interaction varies inversely with the first power of r, it therefore has a much longer range than other intermolecular forces that depend on higher powers of r^–1. Contrary to what is observed with nonelectrolytes (where short-range forces are dominant), solutions of electrolytes depend on both long-range electrostatic attractions and repulsions and on short-range interactions between ions and between ions and solvent molecules. Moreover, since ε_r > 1, the potential is reduced from its value in vacuo. For example, for water at 25°C, ε_r = 78.41; at a given separation distance, the Coulomb potential is reduced from its value in vacuo by almost two orders of magnitude. It is therefore not surprising that water is a better solvent for electrolytes than say, benzene. The dielectric constant of water is very much larger than that of benzene (for benzene at 25°C, ε_r = 2.27).

At infinite dilution, the distribution of ions in solution can be considered completely random because the ions are too far apart to exert any significant influence on each other. In this case, the mean ionic activity coefficient of the electrolyte is unity. However, for dilute (not infinitely dilute) solutions, where the ions are no longer “blind” to each other, Coulombic forces become important; in the neighborhood of a negative ion, the local concentration of positive ions is slightly higher than that for the bulk solution.

A slightly positive atmosphere around anion i, and a slightly negative atmosphere around cation j, produce a decrease in attraction between i and j, as illustrated in Fig. 9-4 for an aqueous solution at 25°C of ions Na⁺ and Cl^–17. This decrease due to preferential spatial distribution of ions gives a shielding effect. To account for shielding, the theory of Debye-Hückel shows that r^-1 in Coulomb’s potential should be multiplied by a “damping factor”,

(9-46)

¹⁷ As shown in Sec. 4.2, for isolated Cl^- and Na⁺ ions in contact (r = 0.276 nm), Γ_ij = -8.36 x 10^–19 J. In water at 25°C, this value is reduced to Γ_ij = -0.106 x 10^–19 J.

where K^-1 is the shielding length commonly called the Debye length. The Debye length is a characteristic distance of interaction; it plays an important role in the Debye-Hückel theory. When k^-1 is very large, rk is small and the exponential in (Eq. 9-46.) is close to unity; in that event, we recover the original Coulomb potential that, lest we forget, is for two isolated charges in a continuous medium characterized by ε_r. The substitution indicated by (Eq. 9-46.) extends Coulomb’s potential to the case where charges i and j are no longer the only charged particles in the liquid solution. As Fig. 9-4 shows, when K^-1 is small, the shielded potential is weaker than the unshielded potential, even for short distances.

Figure 9-4 Coulomb potential for Cl^– and Na⁺ as a function of their (center-to-center) separation distance: in vacuo, ε_r = 1; in an infinitely-dilute aqueous solution at 25°C, ε_r = 78.41, K^-1 = ∞; shielded Coulomb potential in water at 25°C₁ ε_r = 78.41 and k^-1 = 3.05 nm. Debye length 3.05 nm is that of a 0.01 mol kg^-1 aqueous NaCl solution at 25°C.

The Debye length is defined as

(9-47)

where ε_o is the vacuum permittivity (ε_o = 8.85419 x 10^-12C² N^-1 m^-2), εr is the relative permittivity or dielectric constant, d_s is the solvent density (kg m^-3), N_A is Avogadro’s constant, e is the electronic charge (e = 1.60218 × 10^-19C), and I is the ionic strength given by Eq. (9-45).

Equation (9-47) tells us that the Debye length decreases with rising concentration; the higher the concentration of ions (ionic strength), the more effective the shield. For example, for a 1-molal aqueous solution of an 1-1 electrolyte at 25°C, K^-1 = 0.30 nm, giving strong shielding. If the molality is reduced to 0.001 mol kg^-1, K^-1 = 9.64 nm, giving almost no shielding.

Because ionic strength depends on the charges of the ions, even a low concentration of highly charged ions may form an effective shield. Also, because Debye length increases with rising dielectric constant, at a fixed ionic strength, shielding in water is less than that in most other solvents whose dielectric constants are below that of water.

Because of long-range Coulombic forces, electrolyte solutions are nonideal even at low electrolyte concentration because electrostatic attractive and repulsive forces between ions are significant even at high distances of separation.

Using well-established concepts from classical electrostatics, Debye and Hückel derived a simple expression (Robinson and Stokes, 1970; Newman, 1991) for the activity coefficient γ_i of an ion with charge z_i in a dilute solution of ionic strength I:

(9-48)

where k is the inverse of the Debye length [defined by Eq. (9-47)].

For dilute aqueous solutions near ambient temperature, there is no significant difference between molality and molarity. Switching to molality units for γ and I, substitution of constants in (Eq. 9-48) gives

(9-48a)

where constant A_γ is given by

(9-49)

Equations (9-48) and (9-48a) give the activity coefficients of ions, not of electrolytes in an electrically-neutral solution. However, the quantity that is usually measured experimentally is the mean ionic activity coefficient, . For electrolyte is defined by Eq. (9-19). Upon substitution of Eq. (9-48a) in Eq. (9-19) and introduction of the electroneutrality condition expressed by Eq. (9-14), we obtain

(9-50)

where |z₊z_–| is the absolute value of the product of the charges.

A similar derivation yields for the osmotic coefficient

(9-51)

where the Debye-Hückel constant A is directly related to constant A_γ given by Eq. (9-49):

(9-49a)

Equation (9-50) is the Debye-Hückel limiting law, useful for interpreting the properties of electrolyte solutions. It is an exact limiting law at low concentrations in the same sense that the virial equation of state, truncated after the second virial coefficient, is an exact limiting law for the compressibility factor of a gas at low pressure.

If the solvent is water at 25°C and atmospheric pressure, A_γ = 1.174 kg^1/2 mol^-1/2 (ε_r = 78.41;¹⁸ d_s = 0.997 g cm^-3). Converting to base 10 logarithms, Eq. (9-50) gives

¹⁸ From the 1997 international standard (D. P. Fernández, A. R. H. Goodwin, E. W. Lemmon, J. M. H. Levelt-Sengers, R. C. Williams, 1997, J. Phys. Chem. Ref, Data, 26: 1125),

(9-50a)

where I is in mol kg^-1.

For very dilute solutions, Eq. (9-50a) is in good agreement with experimental data, as shown in Fig. 9-5. This figure also shows that the Debye-Hückel limiting law always predicts negative deviations from ideal-dilute behavior. At low concentrations, depends on the valence but not on the chemical nature of the electrolyte.

Figure 9-5 Mean ionic activity coefficients as a function of concentration for some aqueous strong electrolytes (Robinson and Stokes, 1970), Full lines show experimental data. Dashed lines are calculated from the Debye-Hückel limiting law [Eq. (9-50a)].

The Debye-Hückel equation is applicable only to solutions at very low concentrations (typically, for ionic strengths to 0.01 mol kg^-1). Several factors account for deviations from the Debye-Hückel law at high concentrations; these include ion-ion repulsion due to the finite sizes of the ions and interactions arising from forces other than electrostatic forces (e.g. dispersion forces). An important deviation from the Debye-Hückel law follows from strong and specific ion-solvent solvation forces that invalidate the assumption that the solvent is a dielectric continuum. For realistic applications, it is necessary to derive expressions that apply to more concentrated solutions. For example, the ionic strength of seawater is about 0.7 mol kg^-1. For industrial or geothermal applications, the ionic strength may be much larger. For such solutions, the limiting law of Debye-Hückel is insufficient.

For concentrated electrolyte solutions, several semi-empirical corrections to the Debye-Hückel limiting law have been proposed (Zemaitis et al., 1986). A common expression for aqueous solutions with I to 0.1 mol kg^-1 is

(9-52)

For ionic strengths to 1 mol kg_-1, a linear term is added, yielding

(9-53)¹⁹

¹⁹ Equation (9-53) [and (Eq. 9-52)] may include in the denominator the term aI^1/2 (instead of I^1/2), where a is a parameter that reflects the finite size of ions. In practice, however, parameter a is an adjustable parameter or, for simplicity, it is set equal to unity.

where b is an adjustable parameter. Table 9-4 compares experimental activity coefficients for aqueous solutions of sodium chloride at 25°C with those calculated from (Eqs. 9-50.), (9-52), and (9-53).

Table 9-4 Mean-ionic activity coefficients of aqueous sodium chloride at 25°C (Robinson and Stokes, 1970).

Research toward a fundamental theory for concentrated electrolyte solutions is an active topic pursued by physical chemists using modern statistical mechanics. As discussed elsewhere (Mazo and Mou, 1991; Ohtaki and Yamatera, 1992), much progress has been made but, for engineering applications, it is often more useful to utilize a semi-theoretical model. Section 9.10 introduces a few (mostly) empirical models for concentrated electrolyte solutions.

9.8 Weak Electrolytes

The thermodynamic relations presented in the preceding sections are for strong electrolytes, i.e., salts that completely dissociate in the solvent (usually water).

Weak electrolytes are compounds (such as acetic acid) that are only partially dissociated in aqueous solutions. At equilibrium, in addition to the ions, there exists a significant concentration of the molecular (undissociated) electrolyte. The dissociation constant of the weak electrolyte (that depends only on temperature) relates the concentration of the undissociated electrolyte to the concentrations of the ions formed by partial dissociation. This relation, however, also requires activity coefficients for the ions and for the undissociated electrolyte.

Consider electrolyte M_v+X_v- that dissociates according to

The dissociation (or ionization) equilibrium constant is

(9-54)

where m_MX and are, respectively, the molality and the activity coefficient of the molecular (undissociated) part of the electrolyte. For strong electrolytes, m_MX is zero by definition. An equivalent definition of a strong electrolyte is to say it is a solute whose dissociation constant K is infinite.

To illustrate, consider the dissociation constant²⁰ of acetic acid (HAc) into H⁺ and acetate (Ac^-) ions,

²⁰ Dissociation (ionization) constants can be obtained from spectroscopic data. Classically, they are obtained from conductivity or electromotive-force measurements. See Robinson and Stokes (1970), Chap. 12.

To calculate vapor-liquid equilibria in aqueous solutions containing a weak electrolyte, it is necessary to know the equilibrium constant in addition to activity coefficients for the aqueous solutes and for water. It is also necessary to know fugacity coefficients for all volatile components that exist in the vapor phase. At normal conditions, we can neglect the concentration of ions in the vapor phase. An application is shown in Sec. 9.17.

9.9 Salting-out and Salting-in of Volatile Solutes

When an appreciable amount of salt dissolves in a liquid, it significantly affects that liquid’s vapor pressure. Further, the dissolved salt affects the solubility of a gas (or liquid) in that solvent and finally, if the solvent is a mixture of two (or more) volatile components, the dissolved salt influences the composition of the vapor in equilibrium with the solvent mixture.

The solubility of a gas in a salt solution is usually less than that in salt-free water; this solubility decrease is called salting-out. A simplistic but incomplete explanation of salting-out follows from a consideration of hydration forces. Ions (especially cations) like to form complexes with water (hydration), thereby leaving less “free” water available to dissolve the gas. (This explanation, however, is seriously oversimplified because it neglects the subtle effect of ions on water structure). The salting-out influence of an ion usually rises with increasing ionic charge and decreasing ionic radius.²¹

²¹ Salts with large polarizable ions (usually anions) tend to salt in, i.e. to increase the solubility of the gas.

To illustrate, Fig. 9-6 shows experimental results at 80°C for the solubility of carbon dioxide in salt-free water and in aqueous solutions containing sodium sulfate or ammonium sulfate or both. Figure 9-6 shows that when 2 moles of sodium sulfate are added to 1 kg of water at 80°C, the pressure needed to dissolve 0.2 moles of carbon dioxide increases from 17 to 51 bar. Similarly, at the same temperature and at 40 bar, 1 kg of salt-free water dissolves 0.43 moles CO₂ while a 2-molal aqueous sodium sulfate solution containing 1 kg of water dissolves only 0.16 moles CO₂. The same figure also shows that, at the same concentration, sodium ions cause a larger salting-out effect than the larger ammonium ions.

Figure 9-6 Solubility of carbon dioxide in aqueous solutions of sodium sulfate and ammonium sulfate at 80°C (Rumpf and Maurer, 1993). For the salt-containing solutions the ionic strength is 6 mol kg^-1.

Salting-out effects are often described by an empirical equation proposed many years ago by Setchenov (1889). For a simple derivation of the Setchenov equation, consider a three-phase system consisting of a gas phase and two aqueous phases, as schematically shown in Fig. 9-7. One aqueous phase (') contains no salt. The other aqueous phase contains salt with molality m_MX. We assume that temperature T is sufficiently low so that water is essentially nonvolatile, i.e. the gas phase contains only solute i.

Figure 9-7 Three-phase system consisting of a gaseous solute in equilibrium with two aqueous liquid phases: phase’ contains only water; phase “contains water and a salt with molality m_MX. It is assumed that water is nonvolatile.

For solute i, at equilibrium,

As discussed earlier, this equation of equilibrium can be written in the form

where and are, respectively, the molalities of solute i in the liquid phase’ (no salt) and in the liquid phase” (with salt).

Now assume that the difference between the standard Gibbs energy change of solution of a solute into a solvent with salt and that into a solution without salt is expressed as a power series of the salt concentration; i.e. assume

where k_MX is a constant characteristic of the salt.

As an approximation, consider only the first term in the series. For low salt concentrations, the Setchenov equation is then obtained:

(9-55)

where and are, respectively, the solubilities (expressed in molality units)²² of the gas in salt-free solvent (water) and in the aqueous salt solution; k_MX is the salting parameter, and m_MX is the molality of the salt in the aqueous solution. Strictly, Setchenov’s constant, k_MX, is independent of m_MX only in the limit of infinite dilution of the electrolyte; Eq. (9-55) is therefore a limiting relation. Constant k_MX depends on the salt, the solute, and the temperature. Table 9-5 presents Setchenov constants for some common gases in aqueous salt solutions at 25°C.

²² Any consistent set of units may be used for gas solubility. However, constant k_MX depends on the units chosen.

Table 9-5 Molality Setchenov constants for volatile solutes in aqueous electrolyte solutions at 25°C (Krishnan and Friedman, 1974).

Constant k_MX can be either positive or negative. If it is positive, the solubility of the gas decreases with rising salt concentration (the gas is salted-out). If it is negative, the solubility of the gas increases with rising salt concentration (the gas is salted-in). Figure 9-6²³ shows that at 80°C, carbon dioxide is salted-out significantly by sodium sulfate, less so by ammonium sulfate. Table 9-5 shows that at 25°C, methane, ethane, and butane are salted-in by tetramethylammonium bromide.

²³ Setchenov’s equation cannot be applied to the results shown in Fig. 9-6 because the high salt concentration indicated in Fig. 9-6 requires higher terms in Setchenov’s power series in electrolyte concentration.

Several empirical models have been proposed for estimation of Setchenov constants. The model of Schumpe (1993) is the most general because it can also be applied to mixed electrolyte solutions. In this model for gaseous solutes, Setchenov constants are estimated from a set of ion-specific and gas-specific parameters that have been evaluated for 45 ions and 22 gases by non-linear regression of solubility data for salt solutions (Schumpe et al, 1995).

Lang (1996) showed that the Setchenov equation [Eq. (9-55)] and the Schumpe model can also be applied to aqueous solutions containing ionogenic²⁴ organic compounds, such as amino acids, zwitterion peptides, proteins, and bases. To illustrate, Fig. 9-8 shows that the linear relation [Eq. (9-55)] holds; on a semi-log plot, the relative solubility of oxygen, (m’/m”)_O2, is a linear function of the concentration of the organic solute. It is useful to predict the salt effect of an organic solute on the solubility of oxygen, e.g., in aerobic fermentation processes, where the oxygen concentration is a key parameter for optimal control or in medicine, where the solubility of oxygen in blood may change due to dissolved solutes.

²⁴ Organic compounds that are ionogenic show their ionic behavior only if dissolved in aqueous solution.

A dissolved salt can also have a large effect on the composition of a vapor in equilibrium with an aqueous solution of a volatile liquid (Furter and Cook, 1967; Furter, 1976, 1977). When the dissolved salt solvates preferentially with the molecules of one component (Ohe, 1976, 1991), the salt can have a selective effect on the volatilities of the two liquids, and hence on the composition of the equilibrium vapor, although no salt is present in the vapor phase. For example, a preference for solvation with the less volatile component would result in an increase in the relative volatility of the more volatile component and therefore would enhance separation by distillation. Addition of a soluble salt to a liquid phase of a system may provide a convenient technique for extractive or azeotropic-distillation operations. However, industrial applications of this technique are often hindered by difficulties in salt recovery from the remaining liquid phase and by corrosive properties of salt solutions.

The effect of salt on vapor-liquid equilibria can be described by a Setchenov-type equation proposed by Furter and coworkers (Johnson and Furter, 1960). For a single salt in a binary mixed-solvent at fixed (salt-free) composition, Furter’s equation is

(9-56)

Figure 9-8 Setchenov plot of oxygen in aqueous solutions of some bio-organic compounds at body temperature(37°C).²⁵ HAc, acetic acid; Ammediol, 2-amino-2-methyl-1, 3-propan-diol; THIS, Tris(hydroxymethyl)amino-methane; Gly, glycine; TRIS.HAc, Tris(hydroxymethyl)amino-rnethane acetate; Gly-Na, glycine-sodium salt; L-Lys.HCl, L-lysine-monohydrochloride; L-Glu-Na, L-glutamic acid-monosodium salt; L-Glu-Na₂, L-glutamic acid-disodium salt. (Lang, 1996).

²⁵ In Fig. 9-8 the molarity scale is used. To convert from molarity to molality we need the mass density of the solution. See Footnote 2.

where α and α° are the relative volatilities²⁶ with and without salt, respectively; is the salt-effect parameter (which remains constant for moderate salt concentration), and x_MX is the mole fraction of the salt in the liquid phase, given by moles salt/(moles salt + moles solvent).

²⁶ In a binary solution, relative volatility of i is defined by α_i = y₁ (1 - x_i) / × ₁ (1 - y_i), where y_i; and x_i are, respectively, the mole fraction in the vapor phase and in the liquid phase. Since the salt is nonvolatile, the vapor phase contains only the two volatile species, whereas the liquid phase contains all three components. However, to facilitate direct comparison, the definition of α used here uses liquid compositions on a salt-free basis.

To illustrate, Fig. 9-9 shows smoothed experimental data (Burns and Furter, 1976) for the effects of KBr and R₄NBr (where R = H, CH₃, C₂H₅, n-C₃H₇, and nC₄H₉) salts in the ethanol/water system at fixed (salt-free) liquid composition (x_ethanol = 0.206).Figure 9-9 shows the large variety of salt effects for vapor-liquid equilibria in the ethanol/water system: they range from a large salting-out effect (KBr) to a large salting-in effect [(n-C₄H₉)₄NBr].Figure 9-9 shows that (Eq. 9-56.) provides a reasonable method for representing the experimental data.

Figure 9-9 Salt effects in ethanol/water system at constant liquid composition x_ethanol = 0.206 (salt-free basis).

Another example, shown in Fig. 9-10, concerns vapor-liquid-equilibrium data at 1 bar for acetic acid/water and for acetone/methanol salt-free mixtures; and for saturated mixtures with inorganic salts. Figure 9-10(a) shows that calcium chloride and barium chloride lower the relative volatility of water. In the absence of salt, water (boiling point 100°C) is more volatile than acetic acid (boiling point 118°C). However, in a solution saturated with CaCl₂ when w_water > 0.12, relative volatility α for acetic acid exceeds unity while a for water is less than unity. Barium chloride has a similar effect but not until w_water > 0.66. Here w is weight fraction on a salt-free basis. Figure 9-10(a) indicates that both CaCl₂ and BaCl₂ induce an azeotrope in this system but at different conditions, respectively, 0.12 weight fraction water and 122.2°C, and 0.66 weight fraction water and 102.3°C. The opposite effect is also possible, as illustrated in Fig. 9-10(b) for acetone/methanol mixtures at 1 bar. The salt-free system has an azeotrope at 55.4°C and a weight fraction for acetone of 0.87. Addition of Nal breaks the azeotrope and produces a salting-out effect on acetone. These examples show how salts may be useful for creating favorable conditions for separation by distillation. However, salts are not often used for that purpose because solids-handling is not convenient and because dissolved salts tend to be corrosive.

A salt dissolved in a mixed solvent may affect the boiling point, the mutual solubilities of the two liquid components, and the equilibrium vapor-phase composition. Generally, the non-dissociated molecules or ions (or both) of dissolved salt tend to attract preferentially one type of solvent molecule, as Fig. 9-10 illustrates. Usually, the molecules of the more polar component are preferentially attracted by the electrostatic field of the ions. In that event, the vapor composition is enriched by the less polar solvent, wherein the salt is less soluble.

Figure 9-10 A salt may invert relative volatility or eliminate an azeotrope. Vapor-liquid equiiibrium at 1 bar for: (a) Acetic acid/water system with no salt (- - -), saturated (about 1.8 mol kg^-1) with BaCl₂ (o), and saturated (about 7.5 mol kg^-1) with CaCl₂ (⇓) (Ramalho et al., 1964).(b)² Acetone/methanol system with no salt (- - -), and saturated (about 3.8 mol kg^-1) with Nal (Δ) (lliuta and Thyrion, 1995).

9.10 Models for Concentrated Ionic Solutions

When ion concentrations are low, the average distance between ions is large; in that case, only long-range electrostatic forces are important. As ion concentration rises, ions begin to interact also with hard-core repulsive forces (leading to excluded-volume effects) and with short-range (van der Waals) attraction forces.

While much effort has been expended by physical chemists toward a fundamental theory for concentrated electrolyte solutions, for most applications it is necessary to resort to semi-empirical models (Rafal et al, 1994). A number of models has been developed based on different assumptions. Typical models are based on perturbation theory (Henderson et al., 1986), or equations of state (Jin and Donohue, 1988, 1988a, 1991), or on solvation concepts leading to chemical theories (Stokes and Robinson, 1973; Schönert, 1986).

To correlate activity coefficients in aqueous electrolyte solutions, semi-empirical models can be divided into three broad categories:

Physical models. Deviations from the Debye-Hückel limiting law are attributed to physical interactions between ions, as, e.g., in the Pitzer model (Pitzer, 1973, Pitzer and Mayorga, 1973, 1974). Here physical interactions refer primarily to excluded-volume and van der Waals-attraction effects.

Chemical models. Nonideal behavior of an electrolyte solution is attributed to chemical reactions that lead to the formation of semi-stable chemical species, particularly solvated ions; the solvation model of Robinson and Stokes (1973) is a typical example.

Local-composition models. The local-composition concept²⁷ is used to account for nonrandomness. These models are special cases of physical models. The effect of van der Waals interactions between ions is expressed not as a function of bulk composition but as a function of local composition. These models use the NRTL equation, or the Wilson equation, or the UNIQUAC equation for that part of the activity coefficient which is due to short-range forces; examples are provided by the models of Chen et al. (1982, 1986), Mock et al (1986), Liu et al. (1989), Sander et al. (1986), Macedo et al., (1990), and Vera et al. (1980, 1988).

²⁷ see Sec. 6.11.

In all these models, the key adjustable parameters for the excess Gibbs energy are determined by regression of experimental data for the binary mixture.

We cannot here provide a comprehensive discussion of a very large subject with a huge literature. We provide only a modest introduction.

9.11 Fundamental Models

With few exceptions, theoretical descriptions of electrolyte solutions have been based on the primitive model where the solvent is a dielectric continuum (characterized by its dielectric constant) and the ions are considered to be charged hard spheres. In this model, there are no explicit terms for solvent-solvent and ion-solvent interactions; it is assumed that these interactions are taken into account through the dielectric constant in the ion-ion interaction terms. This crude approximation is satisfactory for dilute solutions or else for solutions of particles (e.g. colloids) that are much larger than the solvent molecules. Nevertheless, it is a bad approximation for concentrated ionic solutions where the size of the solute ion is comparable to that of the solvent molecule.

Henderson et al. (1986) developed a non-primitive model where a perturbation expansion is applied to a mixture of dipolar hard spheres (the solvent) and charged hard spheres (the ions) of the same diameter. However, for most practical applications the perturbation expansion converges too slowly.

Using perturbation theory, Jin and Donohue (1988, 1988a, 1991) derived a four-parameter equation of state for aqueous solutions containing strong²⁸ or volatile weak electrolytes, including multisalt systems. In the model, Jin and Donohue calculate short-range interactions using the perturbed anisotropic-chain-rfieory (PACT) and long-range coulombic (charge-charge) interactions between ions from a perturbation expansion. The adjustable parameters used are the sizes of the ions.

²⁸ For a review of modeling thermodynamic properties of aqueous strong electrolyte solutions see J. R. Loehe and M. D. Donohue, 1997, AIChE J., 43: 180.

The mean spherical approximation (MSA) (Blum, 1980) has also been used to correlate activity coefficients in aqueous electrolyte solutions (see, e.g., Gering et al., 1989). This approximation uses the primitive model of aqueous electrolyte solutions but it takes serious account of the finite sizes of the charged particles (ions). The MSA model reduces to the Debye-Hückel theory for point charges in a very dilute electrolyte solution. Results based on the MSA model are analytical. The MSA yields reasonable activity coefficients of ions to moderate concentrations. It has been applied to describe vapor-liquid equilibria of mixed-solvent electrolyte systems (Wu and Lee, 1992) and combined with an equation of state to gas-electrolyte solution equilibria at high pressures (Harvey and Prausnitz, 1989). Results are very sensitive to ionic diameters. To achieve agreement with experiment, these diameters may depend on electrolyte concentration, reflecting the effect of hydration. That effect, however, is more conveniently taken into account by a chemical theory (Stokes and Robinson, 1973; Schonert, 1986; Zerres and Prausnitz, 1994).

Using powerful computers, molecular simulation provides a promising method for describing the properties of electrolyte solutions without using the primitive-model assumption.

9.12 Semi-Empirical Models

Numerous semi-empirical models have been developed for representing equilibrium properties of electrolyte solutions (for a review see, e.g., Grigera, 1992; Renon, 1986; Maurer, 1983). These models correct the Debye-Hückel theory through additional terms that account for ion-ion interactions and, at high concentrations, for incomplete dissociation that in modern terminology is called ion pairing.

In these semi-empirical models, it is customary to assume that the molar excess Gibbs energy of electrolyte solutions is the sum of two contributions, one arising from the long-range (LR) coulombic forces (represented by the Debye-Hückel theory or its extension) and the other from short-range forces (SR):

(9-57)

Following the relations given in Sec. 9.3, (Eq. 9-57.) gives two contributions to the unsymmetric mean ionic (mole-fraction-based) activity coefficient:²⁹

²⁹ Mole-fraction based activity coefficients, can be converted to mean rnolai activity coefficients, , using an expression similar to the one presented in Sec. 9.1: = /(1 + 0.001M_s vm), where v = v₊ + v_-. Here, mole fraction x is defined by (Eq. 9-26.).

To obtain In from a model of the excess Gibbs energy, we first calculate ln and ln , where and are, respectively, the activity coefficients of the cations and of the anions, and then use (Eq. 9-19.):

Although almost all semi-empirical models use a Debye-Hückel-type term for the long-range contribution, several choices are available to account for the short-range contribution. Possible choices include local-composition expressions (UNIQUAC, NRTL, Wilson) and van Laar or Margules equations. Most models assume complete dissociation of the electrolytes; using at least two adjustable binary parameters, most models are reasonably successful for dilute and semi-concentrated solutions, to about 6 molal.

Long-range forces between ions dominate at dilute electrolyte concentrations, while short-range forces between all species dominate at high electrolyte concentrations. The models of Cruz and Renon (1978), and Chen (Chen et al., 1982, 1986), use an expression for the excess Gibbs energy that includes a contribution derived from the NRTL equation, whereas the model of Liu et al. (1989) uses the Wilson equation. In the model of Pitzer (Pitzer, 1973; Pitzer and Mayorga, 1973, 1974) g^E* is given by a virial series in molality of the solute.

9.13 Models Based on the Local-Composition Concept

Cruz and Renon and Chen et al, postulate that the local composition of cations around cations is zero and similarly, that the local composition of anions around anions is zero. However, Cruz and Renon also assume that, for a completely dissociated electrolyte, the ions are always completely solvated by solvent molecules. In the more realistic local-composition model of Chen, all ions are surrounded by solvent molecules only in very dilute electrolyte solutions. At higher solute concentrations, the ions are partially surrounded by solvent molecules and partially by other ions of opposite charge. For the long-range contribution, Cruz and Renon use an expression obtained from the Debye-Hückel theory. For the short-range contribution, they use the NRTL model. Cruz and Renon show that their model can represent with very good accuracy the osmotic coefficients of partially or completely dissociated electrolytes using four adjustable binary parameters.

The NRTL model of Chen et al. uses (Eq. 9-57.) with a Debye-Hückel long-range term and a short-range interaction term of the NRTL form.

Chen makes two assumptions to define local composition:

Like-ion repulsion assumption. Due to large repulsive forces between ions whose charge is of the same sign, the region immediately surrounding a cation does not contain other cations; similarly, the region surrounding an anion does not contain other anions.

Local electroneutrality assumption. The distribution of cations and anions around a central solvent molecule is such that the net local ionic charge is zero.

With these assumptions, Chen derived an expression for the short-range contribution to the excess Gibbs energy that includes (see Sec. 6.15) two adjustable parameters τ_MX,s and τ_s,MX, that are, respectively, the salt-solvent and solvent-salt interaction parameters for a binary pair of a single completely dissociated electrolyte in solution. Chen’s model reproduces well the mean ionic activity coefficients of aqueous single electrolytes to a molality of six, as illustrated in Fig. 9-11 for aqueous KOH at 25°C. For multisalt systems, this model requires binary parameters for solvent-salt pairs (τ_s,MX), obtained from data correlation (e.g. from osmotic or meanionic activity-coefficient data) of the corresponding binary solvent-salt systems. However, binary salt-salt energy parameters (that have significant effect on the nonideality of the ternary systems) are estimated using binary data for salt solubility in water or ternary solvent/salt(1)/salt(2) activity-coefficient data.

Figure 9-11 Mean ionic activity coefficients for aqueous KOH at 25°C. —— Experiment (Zemaitis et al., 1986); ----- Chen’s model.

Similar to Chen’s model, Haghtalab and Vera (1988) use the NRTL equation [as modified by Panayiotou and Vera (1980)] for short-range interactions, whereas long-range Coulombic interactions are obtained from the Debye-Hiickel theory. This model provides good representation of experimental data for the entire range of electrolyte concentration, from the dilute region to saturation, using only two adjustable parameters. However, attempts to use it for multisalt solutions have not been successful.

Liu et al. (1989) propose that in Eq.(9-57) obtained from an extended Debye-Hückel theory, and the second term of Eq. (9-57) is obtained from a local-composition expression of the Wilson type. However, contrary to other models, Liu does not assume additivity of long-range and short-range contributions; in Liu’s model, the effect of long-range electrostatic forces on local composition is taken into account. The Debye-Hiickel term accounts for the contributions due to electrostatic forces between each central ion and all the ions outside the first coordination shell; the local-composition term accounts for the contributions from both the short-range interaction forces of all kinds and the long-range electrostatic forces between each central ion and all ions inside the first coordination shell. While the parameters in Chen’s model are salt-specific, in Liu’s model they are ion-specific.³⁰ For a single-electrolyte solution, Liu’s model uses only one adjustable energy parameter; other fitted parameters are common to all electrolyte systems containing the same cation and/or the same anion. Extension to multicornponent electrolyte solutions (Liu et al., 1989a) follows without additional assumptions, i.e., the parameters for a multicornponent system are obtained from data correlation of its constituent binary systems; no higher-order parameters are required. Liu’s model successfully fits data for a variety of concentrated electrolyte solutions. For example, Liu and Gren (1991) use Liu’s model to describe vapor-liquid equilibria for the hydrogen-chloride/water system over the temperature range 0-110°C and salt concentrations to 21.55 M. For the calculated results shown in Fig. 9-12, Liu used four ion-specific parameters (the radius of H⁺ and the interaction energy parameters given in Table 9-6. The anion radius r^Cl- was 1.81 Å.

³⁰ Parameters for 9 cations and 8 anions are presented by Y. Liu and U. Gren, 1991, Chem, Eng. Sci, 46:1815.

Table 9-6 Parameters obtained by Liu and Grén (1991) from fitting data converted from partial pressure data for the HCl/H₂O system in the temperature range 0-110°C and for HCl concentrations to 21.55 M. Temperature T is in Kelvin.*

Figure 9-12 Partial pressures of HCl (with pressure in bar) in the HCl/H₂O system as a function of the mean ionic mole fraction x_± of HCl [defined by Eq. (9-26)]. O Experiment; —— Liu’s model with parameters in Table 9-6.

Sander et al (1986) proposed a method to correlate and predict salt effects in vapor-liquid equilibria for water+cosolvent mixtures. This model combines a term of the Debye-Hückel type with a modified UNIQUAC equation with concentration-dependent parameters. As in the model of Liu, the parameters are ion-specific; no ternary parameters are required. However, a large number of parameters are needed: 7 for a salt/solvent system and 14 for a salt/cosolvent/system. Sander’s model, as well as that by Mock et al. (1986), suffers from improper combination of Lewis-Randall and McMillan-Mayer formalisms (see Sec. 9.16). This deficiency was corrected in Sander’s model by Macedo et al. (1990).

9.14 The Model of Pitzer

Since about 1973, the ion-interaction model of Pitzer has achieved wide acceptance; it has been applied successfully to several geochemical systems (e.g., equilibria of multicomponent brines with solid phases, solubilities of atmospheric gases in seawater) and to systems of interest in chemical industry. Availability of a substantial model-parameter database and of pertinent computer programs³¹ makes Pitzer’s model a versatile and accessible method for describing thermodynamic properties of electrolyte solutions.

³¹ See, e.g., Chemical Modeling of Aqueous Systems, 1990, Vol. II, Chaps. 8, 9 and 10, (D. C. Melchior and R. L. Bassett, Eds., A.C.S. Symp. Ser. 416, Washington: A.C.S.).

The Debye-Hückel theory is valid only at very low electrolyte concentrations. To extend Eq. (9-51) to higher concentrations, terms may be added on an ad-hoc basis to take into account short-range forces and thereby improve agreement with experiment. A common form of an extended Debye-Hückel equation gives the mean ionic activity coefficients,, as a sum of an electrostatic (Debye-Hückel type) term and a virial expansion in electrolyte concentration. Such extended Debye-Hückel equations are useful for representing experimental data.

Pitzer presented a virial expansion that provides very good representation of the properties of electrolyte solutions. Although Pitzer’s equation has a theoretical basis, his final equation is at least partly empirical.

For an electrolyte solution containing w_s kilograms of solvent, with molalities m_i, m_j,…, of solute species i,j,…, Pitzer assumed that the excess Gibbs energy is given by

(9-58)³²

³² The mole-fraction scale could also be used (and it is sometimes used for highly concentrated electrolyte solutions) but molality is the most common concentration scale in the electrolyte literature.

Function f(I) depends on ionic strength I, temperature and solvent properties; it represents long-range electrostatic forces and includes the Debye-Hückel limiting law.

λ_ij(I) represents the short-range interaction between two solute particles in the solvent; the ionic-strength dependence of λ_ij facilitates rapid convergence in the virial expansion.

Λ_ijk terms account for three-body ion interactions; they are important only at high salt concentrations. Coefficients λ(I) and Λ are analogous to second and third virial coefficients because they represent the effects of short-range forces between, respectively, two and three ions. For highly concentrated solutions, fourth or even higher-order interactions may be required in Eq. (9-58).

Pitzer assumed that the λ. and Λ matrices are symmetric, i.e., λ_ij = λ_ji, and Λ_ijk = Λ_ikj = Λ_jik, etc. Functions f(I) and λ(I) were established by Pitzer; they are presented elsewhere (Pitzer, 1973, 1973a, 1974; 1991a).

Applying Eq. (9-42) and (9-43) to Eq. (9-58) we obtain expressions for the activity coefficient and for the osmotic coefficient. For a binary (i.e. single electrolyte) solution, they are:

(9-59)

(9-60)

From a systematic analysis using reliable experimental osmotic and activity coefficient data for 1-1, 2-1, 1-2, 3-1, and 4-1 electrolytes, Pitzer found that the best general agreement was obtained when terms f, B_MX, and C_MX have the form

(9-61)

(9-62)

(9-63)

(9-64)

(9-65)³³

³³ Pitzer gives somewhat different expressions for for 2-2 electrolytes.

(9-66)

In Eqs. (9-61) and (9-64), is the Debye-Hückel constant for the osmotic coefficient given by Eq. (9-49a) (for water at 25°C, = 0.392 kg^1/2 mol^-1/2); b is a universal parameter equal to 1.2 kg^1/2 mol^-1/2; and α is another universal parameter equal to 2.0 kg^1/2 mol^-1/2 for most electrolytes (2-2 salts are an exception).

Adjustable binary parameters are specific for each salt; they have been obtained from least-square fittings to experimental osmotic and activity-coefficient data for aqueous electrolytes at room temperature. A list of these parameters is given elsewhere (Clegg and Whitfield, 1991; Zemaitis et al., 1986; Pitzer, 1991a; Pitzer, 1995). Parameters depend on triple-ion interactions; they are important only at high concentrations (usually higher than 2 mol kg^-1). All parameters (except α and β) are temperature-dependent.

Equations (9-59) and (9-60) give good agreement with experimental data for salt concentrations to about 6 mol kg^-1. Therefore, unless higher terms are added, Pitzer’s model cannot be applied to highly concentrated electrolyte solutions, e.g. to the limit of saturation for a highly soluble salt. However, using binary parameters regressed over a wide range of electrolyte concentration, significant improvements can be achieved at high concentrations at the cost of less accurate results at low concentrations. To illustrate, Table 9-7 gives standard deviations of In for several electrolytes at 25°C calculated from Pitzer’s model with binary parameters obtained from data over different ranges of salt concentration. Parameters from Pitzer and Mayorga (1973, 1974) were obtained from regression of experimental data over a limited range of concentration (up to the lower concentration in the second column of Table 9-7); those of Kim and Frederick (1988, 1988a) were regressed over a range of concentration extended to saturation (higher concentration in the second column of Table 9-7).

Table 9-7 Deviations for estimated versus experimental mean ionic activity coefficients () at 25°C. Calculations from Pitzer’s equation with binary parameters obtained at different conditions: Pitzer and Mayorga from data regression to the lower concentration shown for each electrolyte; Kim and Frederick from data regression to the higher concentration shown for each electrolyte.

Table 9-7 shows that the binary parameters of Kim and Frederick give generally better agreement with experiment over wide ranges of concentration. While the fit of Pitzer and Mayorga gives accurate results at low concentrations, it gives only fair agreement with experiment at high concentrations.

Pitzer’s equation has been applied to many aqueous electrolyte solutions, including aqueous mixed-electrolyte solutions. For mixed electrolytes, Pitzer uses additional terms in Eq. (9-58) that require additional interaction parameters, θ_ij, and ψ_ijk obtained from experimental data for aqueous mixed-electrolyte solutions with a common ion.³⁴

³⁴ Expressions for activity coefficients and osmotic coefficients of multi-electrolyte solutions are presented in App. I together with model parameters and their temperature derivatives for several common aqueous electrolytes.

However, for a multi-electrolyte solution, the principal contributions to G^E* usually come from the single-electrolyte parameters; parameters θ_ij, and ψ_ijk have only a small effect, as illustrated in Figure 9-13. This figure shows contributions to the activity coefficient of NaCl in a multi-electrolyte aqueous mixture containing the same electrolytes as those in seawater (see Table 9-3). Contributions are from the extended Debye-Hückel term [Eq. (9-61)]; from binary (parameters from mixed-salt terms (parameters θ_ij, and ψ_ijk) as a function of ionic strength (Clegg and Whitfield, 1991).

Figure 9-13 Mean ionic activity coefficient of NaCl, , in water at 25°C as a function of ionic strength: (a) Extended Debye-Hückei term [Eq. (9-61)]; (b) Pitzer equation for a multi-electrolyte system with electrolytes as in seawater [Eq. (9-58)], with , but omitting mixed-salt parameters θ_ij, and ψ_ijk; (c) All terms including mixed-salt parameters θ_ij, and ψ_ijk. For comparison, the solid line (d) gives experimental of NaCl (in the range 3 ≤/≤ 6 mol kg^-1) in aqueous NaCl at the same ionic strength as that of the multi-electrolyte system. Line (d), based on data for the single-solute solution, closely approximates line (c) for seawater.

As Fig. 9-13 shows, the like-sign interactions have little effect on of NaCl; even at I = 6 mol kg^-1, they account for only 3.5% of the total. This result suggests that, particularly at low concentrations, mixed-salt terms may be omitted without introducing significant error. This omission greatly simplifies Pitzer’s model. Because NaCl constitutes about 90 mol % of the salts in seawater, of NaCl in an aqueous solution of NaCl is similar to for the multi-electrolyte solution at the same ionic strength.

It is an important characteristic of the Pitzer model that all parameters can be evaluated from measurements for single electrolytes and common-ion two-salt solutions. No new parameters appear for more complex mixtures. Therefore, with Pitzermodel parameters and with solubility products of salts available from the experimental solubility in single-salt solutions, Pitzer’s model can be used to predict solid solubilities in mixed-salt systems. For a solid electrolyte M_v+ X_v- ·nH₂O the solubility equilibrium in water is

(9-67)

The concentrations of cationic species M^z+ and anionic species X^z- in the liquid phase are determined by the solubility product. If the activity of the pure solid electrolyte is taken as unity, the solubility product is defined as

(9-68)

where a_i, m_i, and represent, respectively, the activity, molality and activity coefficient of the aqueous ion i, and is the mean ionic activity coefficient defined by Eq. (9-19). The activity of water, a_H2O. is related to the osmotic coefficient . The Gibbs-Duhem equation relates the composition dependence of (or a_H2O) to various solute γ_i or γ_±. Because these activity and osmotic coefficients can be calculated with the Pitzer model, it is possible to predict solid-salt solubilities, provided K_sp is known.

The solubility product, K_sp, can be calculated if the standard-state Gibbs energy of the solid and aqueous species are available at the temperature of interest. The standard state for the aqueous ions and electrolytes is the ideal, molal solution at fixed pressure and temperature. For the solid and solvent, the standard state is the pure phase at the pressure and temperature of interest. At reference temperature T_r = 298.15 K and standard pressure, K_sp can be calculated from tabulated standard-state values of , i.e., the Gibbs energies of formation of the various species:

(9-69)

with

where, for convenience, we have dropped the superscripts on M and X and the subscripts on MX.³⁵

³⁵ Here, MX stands for a salt that may or may not be hydrated. If the solid salt is not hydrated, n = 0 in Eq. (9-67).

Using the Gibbs-Helmholtz equation, the temperature dependence of K_sp is

(9-70)

where is obtained from tabulated standard-state values for the enthalpy of formation.

In Eq. (9-70), . If the temperature dependence of (the standard-state heat capacity) is not known or if the difference between T and T_r is not large, may be assumed constant. In that case, Eq. (9-70) simplifies to

(9-70a)

At typical saturation pressures, the effect of pressure on the solubility of salts is small and can be neglected. At high pressures, however, this effect can be significant and has to be taken into account (Pitzer et al, 1984).

Figures 9-14 and 9-15 present two examples comparing experimental and calculated solubilities (expressed in molalities) of two solid salts in an aqueous ternary mixture.

The system NaCl/KCl in Fig. 9-14 is simple with no intermediate (i.e. hydrated) solid phase but the system NaCl/Na₂SO₄ exhibits an intermediate solid phase due to the formation of hydrate Na₂SO₄·10H₂O. In both examples the two salts have a common ion thereby simplifying the calculations.

Equations (9-68) and (9-70) are applied for each salt. After K_sp has been calculated from Eq. (9-70), the molality of one of the non-common ions is fixed and Eq. (9-68) is solved for the molality of the other. An iterative method is required because the three terms on the right side of Eq. (9-68) depend on both molalities. Even when there is no hydrated phase, it is not possible to solve for the molality without iteration. Repetition of this iterative process for various assumed values of one molality gives the curves shown in Figs. 9-14 and 9-15.³⁶ The intersection of the two curves gives the fixed-point composition where the two solids are in equilibrium with the aqueous solution.

³⁶ Appendix I gives model parameters and thermodynamic properties of various pertinent species for calculations leading to Figs. 9-14 and 9-15.

Figure 9-14 Experimental (Linke and Seidell, 1958, 1965) and calculated solubilities in the ternary mixture NaCl/KCl/H₂0 at several temperatures. Intersections of isothermal curves represent calculated ternary invariant points, where three phases are in equilibrium: solid NaCl, solid KCl, and aqueous solution.

Figure 9-15 Experimental and calculated solubilities in the ternary mixture NaCl/Na₂SO₄/H₂O at several temperatures. Hydrate Na₂SO₄·10H₂O is stable only at temperatures below about 38°C. Symbols: experiment (Linke and Seidell, 1958, 1965); —— Pitzer model. Intersections of isothermal curves represent calculated ternary invariant points where three phases are in equilibrium: aqueous solution, solid NaCl, and solid Na₂SO₄ or solid hydrate; or aqueous solution, solid Na₂S0₄ and solid hydrate.

For both examples, agreement with experiment is good, especially considering the large temperature range covered. As expected, the solubilities increase with rising temperature. For the aqueous system NaCl/Na₂SO₄ at 15°C the solid phases are NaCl and the hydrate Na₂SO₄•10H₂O. With rising temperature, the hydrate becomes unstable relative to unhydrated Na₂SO₄ as indicated by the results at 25 and 30°C. At temperatures in excess of 38°C, the hydrate disappears and the only solid phases are NaCl and Na₂SO₄.

If the two salts do not have a common ion or, if there are more than two salts, the calculations become much more complex and require simultaneous solution of several equations.

Another example of application of Pitzer’s model to multi-salt mixtures is provided by studies of mineral solubilities in brines by Weare and collaborators (Harvie et al, 1980, 1982, 1987; Weare, 1987). To illustrate Weare’s results, Fig. 9-16 compares experimental with calculated solubilities of gypsum (CaSO₄•2H₂O) in Na₂SO₄/NaCl solutions (Harvie and Weare, 1980, 1987).

Figure 9-16 Gypsum (CaSO₄•2H₂O) solubilities in aqueous Na₂SO₄/NaCl solutions at 25°C. • Experiment; ___ Pitzers model (Harvie et al., 1982).

Because the dissolution of gypsum in brines has a significant effect on ionic strength (and hence on activity coefficients), the solubility must be calculated by an iterative procedure. Weare (Harvie et al., 1982) proposes a convenient method based on a Gibbs-energy-minimization routine.

In Fig. 9-16, the abscissa is the total ionic strength of added salt (3m_Na2SO4 + m_NaCl). For each curve, the molality ratio of Na₂SO₄ to NaCl is fixed. Figure 9-16 shows how solutions of different compositions affect gypsum solubilities. Calculated gypsum solubilities are in good agreement with experimental data at all ionic strengths. However, for this ternary system the model requires 12 single-electrolyte parameters and 5 mixed-electrolyte parameters (θ_ij and ψ_ijk). The large number of parameters required in Pitzer’s model for multi-salt mixtures is its major disadvantage.

9.15 The “Chemical” Hydration Model of Robinson and Stokes

There is much physicochemical evidence to support the idea that when a strong electrolyte dissolves in water, the ions are solvated; water molecules are bound to the ions forming stoichiometric complexes. In some cases the stability of the complex may be sufficiently high to produce a solid hydrated salt. In many aqueous systems, cations are more extensively hydrated than anions.

In a manner similar to that for chemical theories described in Chap. 7, it is possible to relate the activity of the water to the equilibrium constant (or constants) which characterize hydration equilibria. A particularly successful example for a single-solvent solution is provided by the work of Stokes and Robinson (1973) which was later extended (with modifications) to a binary-solvent solution by Zerres (1994). We give here only a summary of the method developed by Stokes and Robinson. We omit the detailed derivation that is clearly presented in the original article.

Consider an aqueous solution of a strong electrolyte, containing n_w moles of free (i.e. not hydrated) water, n₀ moles of anhydrous cations, n₁ moles of singly-hydrated cations, n_i moles of i-hydrated cations and n_A moles of anions, at temperature T and total volume V. We neglect hydration of anions. This solution was prepared by adding c moles of anhydrous salt to water to give a final volume of 1 liter.

We assume stepwise hydration where a hydrate containing i-1 molecules of water per cation can add one molecule of water to form a hydrate containing i molecules of water per cation.

The hydration equilibrium is

hydrate(i - 1) + water hydrate i

and the corresponding equilibrium constant K_i is

(9-71)

where a stands for activity. The lowest possible value for i is one when the ion is anhydrous. The maximum possible value is somewhat arbitrary. Physicochemical data suggest that it is (about) 5 for univalent cations and (about) 9 for divalent cations.

For a fixed i, K_i depends only on temperature. In general, however, K_i also depends on i. To reduce the number of adjustable parameters, Stokes and Robinson suggest that

At constant temperature, this assumption leaves two adjustable parameters, K and k. Stokes and Robinson also assume that in a mixture of hydrated cations and anhydrous anions, there is a Flory-Huggins-type contribution to the entropy of mixing because the various dissolved species do not all have the same size.³⁷ The molar volumes of the hydrates are assumed linear functions of i according to

³⁷ The Flory-Huggins contribution to the entropy of mixing is small compared to other contributions; little error is incurred by omitting it, as was done by Zerres (1994).

(9-72)

where υ₀ is the molar volume of anhydrous cation and υ_w is the molar volume of water.

Cations are electrically charged, regardless of hydration. Stokes and Robinson obtained an expression for the activity based on the theory of ion-ion interactions given by Waisman and Lebowitz (1970) which, in effect, is a higher-order Debye-Hückel theory; here “higher-order” means that the theory is applicable to higher salt concentrations.

The activity of water is given by

(9-73)

where

Here h, the (concentration-dependent) average hydration number, is found from a known (but complicated) function of the maximum hydration number, K, k, a_w and Y where

In Y = c(υ_h – ν υ_w)

The activity-coefficient factor Y arises from the Flory-Huggins contribution to the entropy of mixing. Parameter ν is the total number of ions produced by one molecule of undissociated salt (ν = ν₊ + ν_). If we neglect the Flory-Huggins contribution, Y = 1.

In Eq. (9-70), κ^-1 is the Debye length; d is the average diameter of the anhydrous cation and anion; N_A is Avogadro’s constant; and function S(t) is given by the Waisman-Lebowitz theory:

(9-74)

Because activity a_w depends on h [Eq. (9-73)], and because h depends on a_w, it is necessary to use an iteration method to find a_w.

Volumetric data give υ₀, υ_A, and υ_w. Crystallographic data give diameter d. The Debye length is found from Eq. (9-47). The important adjustable binary parameters are K and k.

The chemical model of Stokes and Robinson gives very good results for both dilute and concentrated electrolyte solutions. To illustrate, Fig. 9-17 shows calculated and observed (molality) osmotic coefficients for water³⁸ in three aqueous systems: NaOH and LiBr [Fig. (9-17a)]; and CaCl₂ [Fig. (9-17b)]

³⁸ The quantity In a_w is related to the molality osmotic coefficient in Eq. (9-25).

Figure 9-17 Osmotic coefficients for aqueous electrolyte solutions at 25°C: (a) NaOH and LiBr; (b) CaCl₂. o Experiment (Hamer and Wu, 1972; Staples and Nuttall, 1977); ____ Stokes and Robinson chemical theory using parameters in Table 9-8.

Table 9-8 gives parameters for several binary aqueous systems. Results are not sensitive to ion diameter d and to the maximum hydration number, provided that reasonable values are used. However, results are sensitive to adjustable binary parameters K and k. For all systems shown in Table 9-8, calculated results are in excellent agreement with experiment over the entire concentration range, from zero molality to the indicated maximum molality. At high molality, these solutions show very strong deviations from ideal behavior (for the LiBr system, γ±^(m) = 485 at m_LiBr = 20 mol kg^-1).

Table 9-8 Parameters in the Stokes-Robinson model and maximum molality of fit. Aqueous solutions at 25°C. For all solutes, d = 0.4 nm.

The law of mass action suggests that, when water is present in excess, cations tend to hydrate with large hydration numbers. However, as the ratio of water to electrolyte declines, hydration numbers decrease. To illustrate, Fig. 9-18 shows calculated distributions of hydrate stoichiometries for the lithium chloride system at 25°C for three concentrations. At m_LiCl = 1 mol kg^-1, most of the hydrates have a stoichiometry of 5 molecules of water per lithium ion. At m_LiCl = 20 mol kg^-1, there are only a few hydrates with hydrate number equal to 5; most hydrates have stoichiometries of 2 or 3 molecules of water per lithium ion.

Figure 9-18 Distribution of hydration numbers for aqueous lithium chloride at 1, 10, and 20 mol kg^-1.

Lu et al. (1993, 1996) have developed a model that combines chemical (solvation) equilibria with physical interaction represented by a local-composition equation.

The model is for the excess Gibbs energy in mixed-electrolyte aqueous solutions. Solvation equilibria are applied to calculate the “true” concentrations of solvated and nonsolvated ions from the overall concentrations of dissolved electrolytes. Deviation from ideal mixing is taken into account by combining the Debye-Hückel law with the UNIQUAC model. Model parameters are obtained from data for single-electrolyte aqueous systems coupled with an empirical universal temperature dependence for some of the model parameters. No additional parameters are required for describing aqueous solutions of mixed electrolytes. The model accurately predicts activity coefficients at temperatures to 573 K and at concentrations up to the solubility limit in aqueous solutions containing K⁺, Na⁺, NH⁺₄, SO^2-₄, C1^-, and NO^-₃.

9.16 Conversion from McMillan-Mayer to Lewis-Randall Formalisms

In the semi-empirical models briefly described in Sec. 9.12, the excess Gibbs energy G^E* is written as the sum of two contributions: first, a long-range contribution described by a Debye-Hückel-type electrostatic expression and second, a short-range contribution from another model, such as NRTL, Wilson, or UNIQUAC, or a virial expansion, or an expression based on a chemical theory that accounts for ion hydra-don. However, this simple addition of two contributions gives rise to an inconsistency because it combines an electrostatic contribution derived from the framework of Mac-Millan and Mayer with a contribution due to short-range forces calculated in the customary framework of Lewis and Randall.

In the Lewis-Randall framework, the independent variables are temperature T, pressure P, and the mole numbers of all species n_i. However, the loag-range interactions, represented by the Debye-Hückel expression, are obtained from electrolyte solution theories based on a framework where the solvent is considered to be a continuous medium containing discrete solute particles. The solvent is not a component but only a background. The electrostatic contribution to G^E* is calculated in the McMillan-Mayer framework, where the independent variables are temperature T, volume V, the mole numbers of the solute species n_k, and the chemical potential of the solvent μ_s. To maintain consistency, it is necessary to convert the activity coefficients from the McMillan-Mayer to the Lewis-Randall framework.

Thermodynamic properties obtained from the McMillan-Mayer formalism are not identical to those obtained from the Lewis-Randall formalisms. In a series of papers, Friedman (1972) has thoroughly discussed this issue and presented extensively the formulas for the conversion of thermodynamic variables from one formalism to the other. The same subject has been discussed also by Pailthorpe et al. (1984), Cardoso and O’Connell (1987), Wu and Lee (1992), Cabezas and O’Connel! (1993), Haynes and Newman (1998), and Barthel et al. (1998).

Fortunately, for ordinary electrolyte solutions, the inconsistency has no appreciable effect (Cardoso and O’Connell, 1987), on the thermodynamics of single-solvent systems. However, when the empirical models described in previous sections are extended to multi-solvent systems, there is an inconsistency that requires attention, as discussed by Cardoso and O’Connel!.

9.17 Phase Equilibria in Aqueous Solutions of Volatile Electrolytes

Waste streams from chemical plants, as well as stack gases from power plants, may contain volatile components (e.g. ammonia, hydrogen sulfide, carbon dioxide, and sulfur dioxide) that ionize, in part, in aqueous solution. Design of operations to remove volatile weak electrolytes from aqueous solutions requires representation of pertinent vapor-liquid equilibria.

In aqueous solution, volatile electrolytes exist in ionic and molecular (undissociated) form, as briefly discussed in Sec. 9.8. At ordinary temperatures and pressures, only the molecular form exists in the vapor. Calculation of vapor-liquid equilibria requires simultaneous solution of phase-equilibrium equations (for the molecular species), chemical-equilibrium equations for the liquid phase, and material balances.

A molecular-thermodynamic framework proposed by Edwards et al. (1978) has been successfully used for calculating vapor-liquid equilibria in aqueous solutions containing one or more weak volatile electrolytes for temperatures from 0 to 200°C and for total ionic strengths to more than 6 molal (Bieling et al., 1989; Kurz et al., 1995). Figure 9-19 gives a schematic representation of the system under consideration.

Figure 9-19 Schematic representation of vapor-liquid equilibria for an aqueous solution containing a single volatile weak electrolyte.

At a fixed temperature and pressure, the weak electrolyte is distributed between the vapor phase and the aqueous phase. For some solute i at low concentration of i, the (vertical) phase equilibrium is primarily determined by Henry’s constant, H_i, and the molality of molecular (not ionic) solute i. This molality, in turn, is strongly influenced by the (horizontal) chemical dissociation equilibrium, characterized by the dissociation (or ionization) constant, K.³⁹ At higher concentration of i, Henry’s constant H_i must be multiplied by an activity coefficient that depends on the concentrations of all solute species and on temperature.

³⁹ At moderate pressures, dissociation of the electrolyte in the vapor phase is appreciable only at very high temperatures and therefore is neglected here.

First consider the single-solute case; the multi-solute case is described later. Since ions are not volatile, the (vertical) phase equilibrium is governed only by the molecular (undissociated) concentration of the electrolyte in the liquid phase. Vapor-liquid equilibria for the solvent (water) and for the (molecular electrolyte) solute are described by

(9-75)

(9-76)

where f is fugacity, subscript w refers to water and subscript i to electrolyte. Equation (9-75) is rewritten (see Sec. 3.3),

(9-77)

where aw is the activity of water, is the saturation pressure of pure water, and is the fugacity coefficient of pure water at saturation, all at system temperature. Mole fraction y is for the vapor phase.

For the electrolyte, phase equilibrium considers only molecular electrolyte (designated with subscript M); therefore, Eq. (9-76) is rewritten

(9-78)

In Eq. (9-78), y_MX,MP is the partial pressure of the molecular weak electrolyte and is the vapor-phase fugacity coefficient; m_MX,M is the molality of the electrolyte in molecular form, is the activity coefficient and H_MX,M is Henry’s constant for the molecular solute.⁴⁰

⁴⁰ Henry’s constant H depends on temperature. When pressure is high, there is also an effect of pressure on H according to the rigorous thermodynamic equation ∂In H_i/∂p = /RT, where i stands for solute and is the partial molar volume of i in the solvent at infinite dilution. Because solute in Eq. (9-78) is the molecular species (MX,M), here refers to the partial molar volume of molecular i at infinite dilution.

From a mass balance for the weak electrolyte in the liquid phase, we relate the overall electrolyte concentration (stoichiometric), m_MX, to that in molecular form, m_MX,M, and in chemically reacted form. For example, when NH₃ is dissolved in water,

(9-79)

In addition, the bulk electroneutrality condition of the liquid phase relates the concentration of cations to those of anions:

(9-80)

Finally, using Eq. (9-54) the (horizontal) chemical equilibrium relates the molecular concentration of the undissociated electrolyte and the ionic concentrations of the weak electrolyte; for example, for the reaction MX v₊M^z+ + v_-X^z-,

(9-81)

where , the mean ionic activity coefficient, is defined by Eq. (9-19).

For very dilute solutions, it is reasonable to set all activity coefficients equal to unity. The important parameters are Henry’s constant, H, and dissociation constant, K. Activity coefficients are important only for higher concentrations.

To solve the vapor-liquid equilibrium equations, we need to evaluate , , and H_MX,M. These quantities are used to solve the phase equilibrium equations given by Eqs. (9-77) and (9-79), coupled with Eq. (9-79), (9-80), and (9-81). However, to find from Eq. (9-81), we need information concerning . Further, we need vapor-phase equation-of-state data to find .

For aqueous solutions of sulfur dioxide, Fig. 9-20 compares experiment with results calculated using the method of Edwards. Dissociation of sulfur dioxide in the liquid phase was taken into account through the chemical equilibria

Each of these chemical equilibria is characterized by a chemical-equilibrium constant, defined according to Eq. (9-81). Vapor-phase fugacity coefficients were calculated using the method of Nakamura et al. (1976). Henry’s constants as a function of temperature were obtained from binary-data reduction. Activity coefficients for the electrolyte, , and the activity of water, a_w, were obtained from Edward’s extension of Pitzer’s model [Eq. (9-58)]. However, because SO₂ is a weak electrolyte with a low dissociation constant, the concentration of ions is so small that Eq. (9-59) reduces to

(9-82)

and for water

(9-83)

Combining Eqs. (9-78) and (9-82), the equation for the phase equilibrium of SO₂ is

(9-84)

where stands for Henry’s constant of molecular SO₂ in water at infinite dilution, i.e. when the total pressure is equal to the vapor-pressure of water . An independent estimate for is required.

Figure 9-20 Solubility of sulfur dioxide in water at 10°C. Comparison with experiment of the calculated (Edwards et al., 1978) results considering (——) or neglecting (---) ionization of SO₂ in water.

Plotting the left-hand side of Eq. (9-84) as a function of the molecular concentration of SO₂, the slope gives the molecule-molecule interaction parameter, β⁽⁰⁾_so2so2 and the intercept gives Henry’s constant,

Figure 9-20 shows that calculations from Edwards’ model are in good agreement with experiment. Figure 9-20 also illustrates the importance of considering weak-electrolyte dissociation in vapor-liquid equilibrium calculations. The broken curve presents results calculated with the (erroneous) assumption that sulfur dioxide is a non-dissociating solute in an ideal dilute solution where (in the asymmetric convention) γ SO₂=1. The solid curve gives calculated results considering dissociation, as explained above. Large differences are observed at low concentrations where most of the solute is in ionic rather than in neutral molecular form.

The thermodynamic framework described above for a single-solute system can be extended to multisolute systems. The necessary parameters are obtained primarily from binary-data reduction but at high salt concentration, some ternary data are required.

Figure 9-21 compares calculated and experimental results (Rumpf et al., 1993a) for a two-solute system: ammonia and sulfur dioxide in water, from 40 to 100°C at two overall molalities of ammonia (3.2 or 6.1 mol kg^-1 of water) and at pressures to 22 bar.

Figure 9-21 Experimental and calculated results for the simultaneous solubilities of ammonia and sulfur dioxide in water at 100°C. Experiment: ο, •; Calculated from mode! of Edwards and Maurer: m_NH3 = 3.2 mol kg^-1; — m_NH3 = 8.1 mol kg^-1

This example is qualitatively different from the previous one because there is strong chemical interaction between the two solutes: acidic sulfur dioxide and basic ammonia. In this example, in addition to water, the liquid phase contains molecular ammonia, molecular sulfur dioxide, ammonium ion, hydrogen ion, hydroxyl ion, sulfite ion, and bisulfite ion, as indicated in Fig. 9-22. As for single-solute systems, Rumpf et al. (1993) wrote chemical-equilibrium expressions for each of the equilibria indicated in Fig. 9-22; applied mass and charge balances, and phase-equilibrium equations similar to Eq. (9-77) for water, and to Eq. (9-78) for each solute (NH3 and SO2). The calculations required four temperature-dependent equilibrium constants; activity coefficients of all species present in the liquid phase; Henry’s constants for each volatile solute at infinite dilution in water; the vapor pressure, molar volume, dielectric constant of water; partial molar volumes of the dissolved gases; and information on vapor-phase aonideality (obtained from the virial equation). Excepting activity coefficients, all these requirements were obtained from available experimental data. For seven dissolved species (see Fig. 9-22), activity coefficients were calculated from Pitzer’s model. If all Pitzer’s terms are included, the seven species dissolved in the liquid phase would require 56 binary parameters β⁽⁰⁾_ij and β⁽¹⁾_ij, and 84 ternary parameters >A_ijk Rumpf made many reasonable approximations to reduce the total number of adjustable parameters to 13. Most of these were obtained from reduction of experimental single-solute data; Rumpf required only 5 ternary parameters obtained from reduction of experimental ternary data. The temperature dependence of the binary interaction parameters was taken into account, whereas it was neglected for the ternary parameters.

Figure 9-22 Vapor-liquid equilibria and chemical equilibria in the ammonia/sulfur dioxide/water system.

The work of Rumpf and Maurer shows that it is not a simple matter to describe phase equilibria in an aqueous system containing two volatile weak electrolytes over an appreciable range of temperature and solute concentrations, especially when there is strong chemical interaction between the two solutes. The calculations, while not trivial, can be performed with a suitable computer program. However, these calculations require an extensive database that can only be supplied by painstaking experimental measurements.

Rumpf et al. (1993, 1994; Bieling et al., 1995) have also measured the solubilities of carbon dioxide and ammonia in aqueous solutions containing salts such as sodium sulfate, ammonium sulfate and ammonium chloride. Experimental results in the temperature range 313 K to 433 K and in the pressure range to 100 bar could again be correlated well with the ion-interaction model of Pitzer.

Coal-gasification and sweetening of natural gases often require removal of acid gases such as carbon dioxide and hydrogen sulfide from gaseous fuels. Such removal is best accomplished by absorption with aqueous alkanolamine solutions. Proper design of absorption equipment requires information on vapor-liquid equilibria, caloric effects and also on the kinetics of mass transfer and of chemical reactions. Because of chemical reactions and strong deviation from ideality in the liquid phase, it is not simple to model the thermodynamic behavior of aqueous mixtures containing alkanolamines and sour gases.

Numerous models (for a review see, e.g., Kohl and Riesenfeld, 1985) have been proposed to describe vapor-liquid equilibria for such systems. Useful models include that of Mather (Deshmukh and Mather, 1981, Xu et al, 1992) and Kuranov et al. (1996). The latter, based on Pitzer’s model, correlates the solubility of carbon dioxide in aqueous solutions containing N-methyldiethanolamine (MDEA).

Silkenbaumer et al. (1998) used a similar correlation for the solubility of carbon dioxide in aqueous solutions containing 2-amino-2-methyl-l-propanol (AMP) and the alkanolamines MDEA and AMP. Due to chemical reactions in the liquid phase, carbon dioxide dissolves in both neutral and (non-volatile) ionic forms. Figure 9-23 shows good agreement for calculated and experimental results for the solubility of carbon dioxide in 2.4 molal aqueous AMP solutions at three temperatures. At low loading a (moles of CO2/moles of AMP), the total pressure of the solution is essentially the vapor pressure of water because all carbon dioxide is absorbed “chemically”, i.e., in its non-volatile form. With increasing a, the total pressure rises, in particular when all AMP is “consumed” (at α = 1) and more carbon dioxide can only be dissolved physically.

Figure 9-23 Solubility of carbon dioxide in aqueous solutions of 2-amino-2-methyl-1 propanol (AMP). Experiment: A m_AMP = 2.44 mol kg^-1 at 353.15 K; • m_AMP = 2.45 mol kg^-1 at 333.15 K; • m_AMP = 2.43 mol kg^-1 at 313.15 K. — Pitzer model. (Silkenbäumer et al., 1998).

The good agreement between calculation and experiment is achieved only by taking into account all the chemical reactions possible in the liquid phase. In the system CO₂/AMP/H₂O, in addition to the solvent (water), 8 species are present: CO₂, RNH₂, RNH⁺₃, RECOCT^-, HCO^-₃, CO^-23, H⁺, and OH.^{- 41}

⁴¹ Here R denotes the HOCH2C(CH3)2-group in AMP (R-NH₂).

The model calculations require, at any temperature, the equilibrium constants of the pertinent chemical reactions, the activities a_i of all species present in the liquid phase, Henry’s constant for the solubility of carbon dioxide in pure water, the vapor pressure, dielectric constant and molar volume of pure water, the partial molar volume of carbon dioxide at infinite dilution in water, and also information on vapor-phase non-ideality.

With equilibrium constants and thermodynamic properties available in the literature, Silkenbaumer calculated activity coefficients of molecular and ionic species from Pitzer’s model,⁴² Using phase-equilibrium equations similar to Eq. (9-77) for water and to Eq. (9-78) for the solute carbon dioxide, the calculation procedure is similar to that of Rumpf et al. (1993, 1994). Making reasonable approximations, Silkenbaumer reduced the total number of adjustable model parameters to 6 binary and 2 ternary parameters. For the binary parameters, a dependence on temperature was taken into account, but for ternary parameters, it was neglected.

⁴² The equations for multi-electrolyte solutions are given in App. I.

Figure 9-24 shows predicted molalities of the major molecular species present in the liquid phase as a function of the overall molality of carbon dioxide for a 2.43 molal AMP aqueous solution at 313.15 K. (Results for H⁺, OH^- and the carbamate ion RNHCOO^- are not shown because their molalities remain small compared to those of the other species).

Figure 9-24 Predicated molalities (Pitzer’s model) of major species present in aqueous 2.43 molal AMP solutions at 313.15 K as a function of the overall molality of CO₂ (Silkenbäumar et al., 1998).

As expected, adding carbon dioxide to an AMP-solution reduces the amount of neutral AMP, thereby producing mainly RNH⁺₃, HCO^-₃, and CO^2-₃. As long as some AMP remains, the molality of molecular carbon dioxide is very small in comparison to the overall amount dissolved. Although it is small, it is very important because it is the concentration of the molecular carbon dioxide, m_CO2.M, tnat mainly determines the total pressure. Only after the molality of neutral AMP approaches zero, does the molality of molecular carbon dioxide increase, along with a strong increase in total pressure.

Silkenbäumer et al. (1998) also demonstrated that the solubility of carbon dioxide can be predicted reliably in solutions containing two alkanolamines, MDEA and AMP, using parameters of the Pitzer model obtained from reduction of experimental data for aqueous solutions containing only one of these alkanolamines.

To illustrate, Fig. 9-25 shows the solubility of carbon dioxide in solutions containing either one or both alkanolamines with about the same total molality. Predicted pressures for mixed-amine solutions agree well with experiment. Because AMP is a stronger base than MDEA, at a given total pressure the AMP-solution shows higher carbon dioxide loading than the MDEA-solution. Loadings for the two-amine solution fall in between.

Figure 9-25 Solubility of carbon dioxide at 313.15 K in aqueous alkanoiamine solutions. Results for the single-amine and double-amine solutions. Experiment:• m_MDEA = 2.63 mol kg^-1 (N-methyldiethanolamine); ∇ m_MDEA = 1.28 mol Kg^-1 + m_AMP = 1.27 mol kg^-1; • m_AMP = 2.43 mol kg^-1 (2-amino-2-methylpropanol). — Pitzer’s model. (Silkenbäumer et al., 1998).

The work described above shows that is possible but not easy to describe phase equilibria of aqueous systems containing weak electrolytes and other solutes that react with those electrolytes. The ion-interaction model of Pitzer is suitable for such calculations; however, the large number of adjustable parameters requires an extensive data base that can be established only by carefully performed experiments.

9.18 Protein Partitioning in Aqueous Two-Phase Systems

Separation of biologically active materials is an important operation in biotechnology. One useful separation process is provided by liquid-liquid extraction using an aqueous two-phase polymer system formed when two water-soluble polymers (e.g. polyethyleneglycol (PEG) and dextran) are dissolved in excess water. This aqueous two-phase system contains mainly water, with the first polymer predominating in one phase and the second polymer predominating in the other phase as shown schematically in Fig. 9-26

Figure 9-26 Schematics of a two-phase aqueous system HpO+PEG+dextran for extraction, (a) Phase’: aqueous PEG-rich phase containing only a small amount of dextran; phase”: aqueous dextran-rich phase containing only a small amount of PEG; (b) Calculated phase diagram. Tie-line length is given by (), where A and B represent the equilibrium compositions of phases’ and”, respectively.

Due to their high water content, both equilibrium phases provide a suitable environment for biomacromolecules. When, for example, a mixture of proteins is added to a two-phase aqueous system, each type of protein partitions uniquely between the phases. Therefore, separation can be achieved with an extraction process as indicated in Fig. 9-27 To prevent denaturation of the biomacromolecules and to maintain pH control, small amounts of (buffer) salts may be added. A useful feature of such systems is that the partitioning of biornacromolecoles between the two phases can be altered by changing the solution pH, ionic strength or the type of salt (electrolyte) added.

Figure 9-27 Separation of a mixture of proteins i,j,k,… by extraction: partition coefficient K_i. = (concentration of protein in phase’)/(concentration of protein in phase”).

Because the compositions of the two aqueous phases are not the same, a salt partitions unequally between the two aqueous polymer phases. The difference in salt concentration establishes an electric-potential difference between the two phases; that difference can greatly affect the partitioning of charged biomacromolecules like proteins. Because the net surface charge of a protein depends on pH, a change in the pH of the solution can result in a significant change in the partitioning behavior of proteins.

While the electric-potential effect is often dominant, partitioning of biomacromolecules also depends on the properties of the phase-forming polymers.

There are many reports on the feasibility of aqueous two-phase systems for extractive biotechnical separations (Kula, 1979; Albertsson, 1986; Fisher and Sutherland, 1989; Prausnitz, 1989). For engineering design, quantitative information is required for thermodynamic properties, especially phase equilibria. Therefore, extensive experimental and theoretical studies have been directed at two-phase equilibria of such systems with and without partitioning biomacromolecules. Some representative articles are those by Curtis et al. (1998), Foster (1994), and Rothstein (1994). While details of these studies cannot be presented here, they all have the same objective: to provide a molecular-thermodynamic model for correlating and, perhaps, predicting the partition coefficients of biomacromolecules in aqueous two-phase systems.

The first task is to calculate the liquid-liquid phase diagram formed by water and the two water-soluble polymers in absence of salt or biomacromolecules. This calculation is achieved by expressing the chemical potential of all three components through an osmotic virial expansion⁴³ in the polymer concentrations; the coefficients in that expansion are best obtained from low-angle laser-light-scattering data (King et al, 1988).

⁴³ The osmotic virial expansion is discussed in Sec. 4.11.

For equilibrium between’ phase”,

(9-85)

where μ is the chemical potential; subscript 1 refers to water and subscripts 2 and 3 refer to the water-soluble polymers. An osmotic virial expansion truncated after the second term gives chemical potentials μ2 and μ3:

(9-86)

(9-87)

where m_i is the molality of solute i, b_ij is a constant characterizing the interaction between a molecule of polymer i and a molecule of polymer j in the aqueous solvent, and is the standard-state chemical potential of component i (hypothetical state of ideal solution at unit molality). An expression for the chemical potential μ₁ of water is obtained from the Gibbs-Duhem equation,

(9-88)

where n_i is the number of moles of component i. With Eq. (9-86) for μ₂ and Eq. (9-87) for μ₃, the chemical potential μ₁ is

(9-89)

where M₁ is the molar mass of water. In Eq. (9-89) the standard state is pure water at system temperature T.

Interaction parameters b₂₂, b₃₃, and b₂₃ are directly related to osmotic second virial coefficients B^*₂₂, B^*₃₃, and B^*₂₃ by

(9-90)

(9-91)

(9-92)

where M_i is the molar mass.

With virial coefficients obtained from low-angle laser-light-scattering measurements, it is possible to generate a reliable phase diagram, as shown schematically in Fig. 9-26(b).

Consider now a protein component (subscript 4) distributed between the two aqueous phases. The distribution coefficient K is defined by

(9-93)

Depending on pH, the protein may be electrically charged and therefore the presence of ions (salts) must be taken into account. When all proteins in the system are dilute, the distribution coefficient for a particular protein is given by (Haynes et al., 1993),

(9-94)

Here F is Faraday’s constant, the electric potential, z the electric charge and β the chemical activity coefficient, i.e., the coefficient in the absence of electrostatic effects caused by unequal distribution of ions between phase’ and phase”.

Activity coefficients are found from the osmotic virial expansion with coefficients B^*₂₂, B^*₃₃, B^*₂₃, B^*₂₄, and B^*₃₄; additional terms for protein-salt and polymer-salt interactions are obtained from osmometric data (Haynes et al., 1989).

The electric-potential difference Δ = ”-’ between the two phases arises as the result of the addition of a salt that fully dissociates into v₊ cations of charge z₊ and v_- anions of charge z_- but that does not partition equally between the two phases. A direct relation between Δ and measurable equilibrium properties of the two-phase system can be established through application of quasi-electrostatic potential theory. Applying quasi-electrostatic potential theory gives the relation (Haynes et al., 1991)

(9-95)

where γ₊ and γ_- are the activity coefficients of the cation and anion, respectively. The importance of Eq. (9-95) can be seen more clearly by applying it to the description of a two-phase system at equilibrium containing a 1:1 electrolyte, i.e., z₊/z_- = -1 and z₊ - z_- = 2. In this case, Eq. (9-95) reduces to

(9-95a)

where K_s is the partition coefficient of the salt and, as usual,

(9-96)

Here, γ_± is the mean ionic activity coefficient for the neutral salt and v = v₊ + v_-. Mean ionic activity coefficients are tabulated for most strong electrolytes in water at 25°C; for those salts or temperatures where such data are unavailable, γ_± can often be estimated using Pitzer’s ion-interaction model discussed in Sec. 9.14. The last equality in Eq. (9-95a) holds because, at equilibrium,

(9-97)

where m_s is the molality of the salt. Equation (9-95a), provides a means for directly calculating the electric-potential difference Δ from the equilibrium properties of a two-phase system that then can be used for calculating K₄ with Eq. (9-94).

In typical cases, the potential difference is small, perhaps a few millivolts. Nevertheless, that small potential difference can have a large effect. In some cases, the effect of the potential difference on K₄ is dominant, much more important than that of the chemical activity coefficients.

Figure 9-28 compares calculated and experimental partition coefficients for three proteins: albumin, chymotrypsin and lysozyme (Haynes et al, 1991). The horizontal axis of Fig. 9-28 is the tie-line length [see Fig. (9-26)] that provides a measure of how different phase’ is from phase”; when the tie-line length is zero, the two phases are identical.

The molecular-thermodynamic analysis shows that when the protein is charged, the influence of the electric potential is often decisive. This analysis, therefore, suggests that enhanced partition coefficients for a protein could be obtained by raising the asymmetry of partition of the salt. One method for doing so is provided by adding to the two-phase system a very small amount of a-cyclodextrin and a salt whose anion is strongly absorbed by that cyclodextrin. Figure 9-29 shows the dramatic effect on the partition coefficient of chymotrypsin when the salt is KI (Haynes et al., 1991). Because the anion (iodide) is bound by a-cyclodextrin and because a-cyclodextrin is predominantly in the aqueous dextran-rich phase, the salt KI partitions toward that phase. The more asymmetric the partitioning of salt, the larger Δ.

Figure 9-28 Predicted and experimental partition coefficients for a dilute protein mixture in an aqueous two-phase system containing PEG 3350, dextran T-70, and 50 mM KGl (overall) at pH = 7.5 and 25°C. d: diameter (Ä); z: electric charge; • : albumin (d= 78; z- -8); : chymotrypsin (d= 34; z= 2); Δ: iysozyme (d= 22; z = 7). Lines are calculated using second osmotic virial coefficients obtained from single-phase light-scattering measurements.

Figure 9-29 Effect of electric-potentia! difference Δ on the partition coefficient for chymotrypsin in PEG 3350/dextran T-70/water. z = 5; pH = 7.3; 25°C. • : 1 mM Kl; • : 1 mM Ki + 1 mM a-cyclodextrin. Tie-line iength is not affected by α-cyclodextrin.

We have here a striking example to indicate how molecular thermodynamics can benefit process design. Thermodynamic analysis showed the unexpectedly large influence of salt partitioning on the partition coefficient of a charged protein. Once that influence was identified, the sharpness of separation could be significantly improved by enhancing the electric-potential difference between phase’ and phase” through the addition of a suitably chosen entrainer for “pulling” a salt preferentially into one of the two aqueous phases.

9.19 Summary

The thermodynamics of solutions containing electrolytes is not a simple extension of the thermodynamics of solutions containing nonelectrolytes. Electrolyte solutions require a much more elaborate framework because, in a solvent of high dielectric constant (e.g. water), an electrolyte splits into two or more ions. Therefore, a binary solution, e.g. sodium chloride in water, is in some sense a ternary solution containing water, a cation and an anion. But in another sense, it is not a ternary solution because the concentrations of cation and anion are not two independent variables; if one is fixed, so is the other because of material balance and the requirement of electroneutrality. Because ionization and the constraint of electroneutrality must be taken into account in the thermodynamics of electrolyte solutions, but not in the thermodynamics of nonelectrolyte solutions, the framework required for electrolyte solutions is necessarily much more elaborate.

For strong electrolytes (salts), ionization is usually complete but for weak electrolytes (e.g. acetic acid), ionization is only partial. In that event, to obtain a useful thermodynamic description, it is necessary to consider chemical equilibria in addition to one or more material balances and to the constraint of electroneutrality.

To keep this chapter from excessive length, attention is given primarily to solutions containing only one solvent and one solute. Only brief attention is given to multicomponent systems.

Electrolyte-solution thermodynamics often uses concentration scales that are different from those used in nonelectrolyte-solution thermodynamics. In the latter, the most common scales are mole fractions or volume fractions that have a desirable characteristic: they vary from zero to unity. For electrolyte solutions, the most popular concentration scale is molality (moles of solute per kg of solvent). Molality, however, can go from zero to infinity. Nevertheless, molality is often used because, for most applications, attention is restricted to dilute or moderately concentrated solutions where molality may go as high as 10 or 20 but, with rare exceptions, no higher.

At ordinary temperatures, most strong electrolytes (salts) are solids, not liquids. Therefore, in most cases, activity coefficients in electrolyte solutions are normalized according to the unsymmetric convention where the activity coefficients of both solute and solvent go to unity at infinite dilution. This normalization requires that the stan dard state of the solute be defined not as the pure solute at solution temperature but instead, as an ideal dilute solution at some fixed concentration at solution temperature. That fixed concentration is conveniently chosen as unit concentration. It is a common error to confuse that standard state with the ideal dilute solution at infinite dilution. The ideal dilute solution at infinite dilution cannot be used as the standard state because the chemical potential of a solute at infinite dilution is minus infinity. If molality is the concentration scale, the standard state for the solute is the ideal dilute solution at system temperature when the solute molality is unity. While the correct standard state for the solute is hypothetical, its properties are experimentally accessible.

In nonelectrolyte solutions of liquid components, no distinction need be made between solute and solvent; they are described with the symmetric convention, as discussed in Chap. 6. For nonelectrolyte solutions of ordinary liquids, the reference system is one that (essentially) obeys Raoult’s law and therefore, activity coefficients indicate deviations from ideality as given by Raoult’s law. However, for solutions of electrolytes, activity coefficients indicate deviations from ideality as given by Henry’s law.

In a solution of electrolytes, it is useful to think of electrolyte activity as analogous to pressure in the virial equation for gases: in both cases, solute molecules are dissolved in a medium. For a gas, the medium is a vacuum. For ions, the medium is a liquid solvent, typically water. In a gas, as in a solution of ions, we are concerned with interactions between solute molecules that “swim” in a medium.

By contrast, when we consider solutions of nonelectrolytes with activity coefficients normalized symmetrically, we are concerned with interactions between solute and solvent molecules. Therefore, the excess Gibbs energy for a solution of nonelectrolyte liquids is necessarily different from that for a solution of electrolyte (ions) in a solvent.

However, the analogy of an electrolyte solution to a gas has a serious deficiency. At ordinary temperatures, a typical gas consists of molecules with no net electric charge. However, ions, by definition, are electrically charged. Intermolecular forces between charged particles are much different from those between uncharged particles, as indicated in Chap. 4. Interactions between uncharged molecules are short-range, whereas those between charged molecules are long-range. Therefore, for nonelectrolyte gases, the ideal-gas law provides a fair approximation even when the concentration (density) reaches say, 5 or 10% of the maximum possible concentration. However, for a solution of ions, deviations from ideality become appreciable at less than 1% of the maximum possible concentration. When two ions are say, 5 diameters apart, there is a strong interaction between them. However, when two uncharged molecules are 5 diameters apart, one molecule hardly knows that the other one is there.

This chapter presents some equations for activity coefficients of ions dissolved in a solvent. However, because of electroneutrality, cations and anions always appear together; while theoretical or semi-empirical equations can be written separately for anions or cations, in a real experiment, it is generally not possible to measure ionic activity coefficients separately, unless we use special techniques, often based on questionable assumptions. In a real situation, we usually measure a mean ionic activity coefficient; for a 1-1 electrolyte, this mean activity coefficient is related to the individual ionic activity coefficients by γ_± = (γ₊γ_-)^1/2.

In a medium of fixed density, temperature and dielectric constant, at very high dilution, when ions are (on the average) far apart, γ₊ and γ_- are independent of the chemical nature of the ions; they depend only on the ionic valences and on the ionic concentration as expressed by the ionic strength. This result from the Debye-Huckel theory is physically reasonable because at large distances of separation, the only important forces between ions are those due to charge-charge interactions; all other intermolecular forces (induction, dispersion, repulsion due to overlap of ionic diameters) are then insignificant. It is therefore not surprising that, for example, in water at 25°C, the mean ionic activity coefficient of 0.01 molar potassium chloride is the same as that of 0.01 molar sodium nitrate.

As the ionic concentration rises, short-range intermolecular forces also become important; indeed, at high ionic concentrations (typically 1 molar and beyond), short-range forces become dominant relative to long-range forces. Several methods have been proposed for taking short-range forces into account; most of these do no more than add terms to the Debye-Hiickel expression in such a manner that the added terms tend to disappear at very low ionic concentration where the Debye-HiSckel expression must be recovered in the limit. One effective method, initiated many years ago by Guggenheim, uses a power series in electrolyte concentration. This method, systematically developed by Pitzer, gives excellent results and can be extended in a logical manner to multi-salt solutions; however, it requires a large number of parameters that can only be obtained from an extensive data base. Other methods, with fewer parameters, are based on one of the local-concentration models discussed in Chapters 6 and 7. Accuracy is generally not as good as that obtained using Pitzer’s equation and – depending on model details – extension to multisolute solutions may require some doubtful assumptions.

As in the thermodynamics of nonelectrolytes, it is also possible to describe the effect of short-range interactions through a “chemical” theory that considers ions to be solvated (if water is the solvent, we say hydrated); as ion concentration rises, thereby increasing short-range ion-ion interactions, the extent of solvation declines. The chemical theory appears to be particularly useful for describing properties of electrolyte solutions over the entire concentration range when the electrolyte is highly soluble in the solvent.

In addition to the few semi-empirical methods mentioned in this chapter, the scientific literature is rich in theoretical methods based, for example, on the integral-equation theory of fluids, the theory of fluctuations and Monte-Carlo simulations. These theoretical studies are not discussed here in part, because they are beyond the scope of this book and in part, because, as yet, they have only limited utility for typical practical calculations in the applied chemical sciences.

While chemists, chemical engineers and related professionals have given some attention to mixtures containing two (or more) electrolytes in one solvent, little fundamental attention has been given to the thermodynamics of solutions containing one (or more) electrolytes in two (or more) solvents. The literature reports many experimental vapor-liquid-equilibrium (VLE) data for two miscible solvents saturated with an electrolyte but there is little fundamental thermodynamic analysis. More difficult is the case of liquid-liquid equilibria for two partially miscible solvents saturated with an electrolyte because in that case (unlike for VLE) the electrolyte is present in both fluid phases.

Finally, there is a fundamental theoretical problem for reconciling a fundamental inconsistency between the thermodynamic framework used for nonelectrolyte solutions (Lewis-Randall framework) with that used for electrolyte solutions (McMillan-Mayer framework). In principle, this problem has been solved by Friedman but his papers do not make easy reading. Studies by O’Connell and others indicate that, for many cases, the consequences of this theoretical problem are of little importance for single-solvent systems.

This chapter on electrolyte solutions has provided only a short introduction. Electrolyte solutions are of interest in many fields: electrochemistry, geology, biochemistry, metallurgy, electrical engineering, material science, physiology, and more. The technical literature is vast. The main point of this summary is a reminder: the thermodynamics of electrolyte solutions is not a minor extension of the thermodynamics of nonelectrolyte solutions; it is a science of its own. It has much overlap with nonelectrolyte-solution thermodynamics and there are many similarities but the differences are far from trivial.

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Problems

1. As determined from emf measurements, the solubility product constant at 25°C of AgCl is K_sp = 1.72xl0^-10 (molal units).

(a) Find the solubility (in mol kg^-1) of AgCl in pure water.

(b) If sufficient NaCI is added to the system to form a 0.01 molal solution of NaCl, what is the solubility of AgCl in this new solution?

(c) What is the solubility of AgCl in a 0.01 molal solution of NaNO₃?

2. At 25°C, the solubility of PbI₂ in water is 1.66xl0^-3 mol kg^-1 of water. At the same temperature, what is the solubility of PbI₂ in an aqueous 0.01 molal solution of KI?

3. At 25°C, the solubility of PbI₂ in water is 1.66xl0^-3 mol kg^-1, in a 0.01 molal NaCl aqueous solution is 1.86xl0^-3 mol kg^-1, and in a 0.01 molal KI aqueous solution is 2.80xl0^-4 mol kg^-1. Explain. For these dilute solutions, use the Debye-Hiickel limiting law.

4. Acetic acid is a weak electrolyte. Determine the fraction ionized for a 10^-33 molal aqueous solution at 25°C.

At 25°C, the equilibrium constant is K -1.758x10^-53.

5. Calculate the Debye length of 0.001 M and 0.1 M NaCl solutions at 25°C in:

(a) Water (ε_r = 78.4).

(b) Methanol(ε_r = 31.5)

6. Consider seawater with 3.5 weight % NaCl at 25°C. The density of pure water at 25°C is 0.997 g cm^-3.

(a) Calculate the molal osmotic coefficient.

(b) Compute the osmotic coefficient and the osmotic pressure. Compare your result with those listed in Perry (Chemical Engineers Handbook) for osmotic pressures of aqueous sodium chloride solutions at 25°C: 27.12 atm for m_NaCl = 0.60 mol kg^-1 and 0.80 atm for m_NaCl = 0.80 mol kg^-1.

To obtain γ+ use Bromley’s model:

where, for NaCl aqueous solutions at 25°C, A_γ = 1.174 kg^1/2 mol^-1/2 and B = 0.0574 kg^1/2 mol^-1/2.

7. For a 0.12 molal K₂SO₄ solution at 25°C, the experimental mean ionic activity coefficient γ^(m)_± is 0.40. Estimate the equilibrium pressure of water above 0.33 molal solution of K₂SO₄ at 25°C.

The vapor pressure of pure water is 0.0317 bar at 25°C. For a moderately concentrated electrolyte solution, the extended Debye-Hckel equation is:

For water at 25°C, A_y = 1.174 kg¹¹² mol^-1/2 and B = 0.33 kg^1/2 mol^-1/2 Ź. Parameters a and b are specific to K₂SO₄, with a = 4.0 Å.

When the activity coefficient is given by the above equation, the osmotic coefficient for water, as obtained from the Gibbs-Duhem relation, is:

where σ(y) is the function:

8. An osmometer at 25 °C has two chambers separated by a semi-permeable membrane. One chamber contains 1 M aqueous sodium chloride. The other chamber contains an aqueous solution of bovine serum albumin (BSA) and sodium chloride at pH 7.4; BSA concentration is 44.6 g L^-l and sodium chloride concentration is 1 M. The measured osmotic pressure is 224 mmH₂O.

What is the osmotic second virial coefficient of BSA in this solution? The molar mass of BSA is 66,000 g mol^-1. The semi-permeable membrane has a cut-off at molecular weight 10,000, At pH 7,4, the electric charge on BSA is -20, Indicate all simplifying assumptions.

9. Using ion-selective electrodes, Khoshkbarchi and Vera (1996, AlChE J., 42: 249) measured activity coefficients of individual ions in aqueous sodium bromide solutions at 25°C, which then used to obtain mean ionic activity coefficients. These were correlated with a truncated Pitzer equation,

where A_x = 8.766 is the Debye-Hflckel constant, ρ = 9 and I_x is the ionic strength expressed in terms of mole fractions. For aqueous solutions of NaBr to 5 molal, they obtained B_± = 124.598.

For NaBr solutions at 25°C with compositions between m_NaBr = 0 and 5 mol kg^-1:

(a) Calculate the activity coefficients of water.

(b) Plot γ± as a function of m_NaBr using Pitzer equation and Debye-Hflckel equation.

(c) Obtain the osmotic pressures from van’t Hoff equation and determine the range of its validity, i.e., the range of composition where the effect of solution non-ideality can be neglected.

At 25°C, the mass density of pure water is 0.997 g cm^-3 and those of NaBr aqueous solutions are given by d(g cm^-3) = 0.997 + 0.0670m_NaBr.

10. The polar species AB dissociates in water according to the reaction

The equilibrium constant for this reaction at 25°C (with molality as the concentration unit) is

where a_i = m_i γ_i is the activity of species i. The Henry’s constant (based on molality) for molecular AB in water at 25°C is 30 bar. What is the total solubility of AB in water at 25°C and 50 bar? Ignore the vaporization of water, and state clearly any other assumptions you make.

The following additional information is available at 25°C:

Second virial coefficient of AB: B = -200 cm³ mol^-1.

Partial molar volume of AB infinite dilute in water, ^∞_AB = 80 cm³ mol^-1.

Dielectric constant, ε_r = 78.41.

Electron charge, e = 1.602xl0^-19 C.

11. Consider two reactions in dilute aqueous solution at 25°C:

CO[(NH₃)₅Br]²⁺ + OH^- Products (I)

[Cr(NH₂CONH₂)₆]³⁺ + H₂O Products (II)

For reaction I and for reaction II, using the (Eyring) theory of absolute reaction rates, calculate the effect on the reaction rate constant k produced by adding an inert salt (e.g., NaCl) to the aqueous solution, that is, by increasing the ionic strength. Assume that the limiting Debye-Hückel relation is valid.

If the molality of NaCl is 0.01 (and if the molalities of the charged reactants are negligibly small), calculate the change in k. For each reaction, does k increase or decrease upon addition of NaCl?

12. Derive Eqs. (9-59) and (9-60) for a single electrolyte solution from, respectively, Eqs. (1-13) and (1-10).