Chapter 11
Solubilities of Solids in Liquids

The ability of solids to dissolve in liquids varies enormously; in some cases a solid solute may form a highly concentrated solution in a solvent (e.g., calcium chloride in water) and in other cases the solubility may be barely detectable (e.g., paraffin wax in mercury).

In this chapter we consider some of the thermodynamic principles that govern equilibrium between a solid phase and a liquid phase.

11.1 Thermodynamic Framework

Solubility is a strong function of the intermolecular forces between solute and solvent, and the well-known guide “like dissolves like” is no more than an empirical statement indicating that, in the absence of specific chemical effects, intermolecular forces between chemically similar species lead to a smaller endothermic enthalpy of solution than those between dissimilar species. Since dissolution must be accompanied by a decrease in the Gibbs energy, a low endothermic enthalpy is more favorable than a large one. However, factors other than intermolecular forces between solvent and solute also play a large role in determining the solubility of a solid. To illustrate, consider the solubilities of two isomers, phenanthrene and anthracene, in benzene at 25°C given in Table 11-1. The solubility of phenanthrene is about 25 times larger than that of anthracene even though both solids are chemically similar to each other and to the solvent. The reason for this large difference in solubility follows from something that is all too often overlooked, i.e., that solubility depends not only on the activity coefficient of the solute (that is a function of the intermolecular forces between solute and solvent) but also on the fugacity of the standard state to which that activity coefficient refers and on the fugacity of the pure solid.

Table 11-1 Structures of phenanthrene and anthracene and their solubility in benzene at 25°C

Let the solute be designated by subscript 2. Then the equation of equilibrium is¹

¹ Equation (11-1) is based on the assumption that there is no appreciable solubility of the liquid solvent in the solid phase.

(11-1)

or

(11-2)

where x₂ is the solubility (mole fraction) of the solute in the solvent, γ₂ is the liquid-phase activity coefficient, and the standard-state fugacity to which γ₂ refers. From Eq. (11-2) the solubility is

(11-3)

Thus the solubility depends not only on the activity coefficient but also on the ratio of two fugacities as indicated by Eq. (11-3).

The standard-state fugacity is arbitrary; the only thermodynamic requirement is that it must be at the same temperature as that of the solution. Although other standard states can be used, it is most convenient to define the standard-state fugacity as the fugacity of pure, subcooled liquid at the temperature of the solution and at some specified pressure. This is a hypothetical standard state but it is one whose properties can be calculated with good accuracy provided that the solution temperature is not far removed from the triple point of the solute.

To show the utility of Eq. (11-3), we consider first a very simple case. Assume that the vapor pressures of the pure solid and of the subcooled liquid are not large; in that case we can substitute vapor pressures for fugacities without serious error. This simplifying assumption is an excellent one in the majority of typical cases. Next, let us assume that the chemical natures of the solvent and of the solute (as a subcooled liquid) are similar. In that case we can assume γ₂ = 1 and Eq. (11-3) becomes

(11-4)

The solubility x₂ given by Eq. (11-4) is called the ideal solubility. The significance of Eq. (11-4) can best be seen by referring to a typical pressure-temperature diagram for a pure substance as shown in Fig. 11-1. If the solute is a solid, then the solution temperature is necessarily below the triple-point temperature. The vapor pressure of the solid at T is found from the solid-vapor pressure curve but the vapor pressure of the subcooled liquid must be found by extrapolating the liquid-vapor pressure curve from the triple-point temperature to the solution temperature T. Since the slope of the solid-vapor pressure curve is always larger than that of the extrapolated liquid-vapor pressure curve, it follows from Eq. (11-4) that the solubility of a solid in an ideal solution must always be less than unity, except at the triple-point temperature where it is equal to unity.

Figure 11-1 Extrapolation (dashed line) of liquid vapor pressure on a pressure-temperature diagram for a pure material (schematic).

Equation (11-4) explains why pheeanthrene and anthracene have very different solubilities in benzene: Because of structural differences, the triple-point temperatures of the two solids are significantly different.² As a result, the pure-component fugacity ratios at the same temperature T also differ for the two solutes.

² The normal melting temperatures for phenanthrene and anthracene are, respectively; 100 and 217°C. These are very close to the respective triple-point temperatures.

The extrapolation indicated in Fig. 11-1 is simple when the solution temperature T is not far removed from the triple-point temperature. However, any essentially arbitrary extrapolation involves uncertainty; when the extrapolation is made over a wide temperature range, the uncertainty may be large. It is therefore Important to establish a systematic method for performing the desired extrapolation; this method should be substituted for the arbitrary graphical construction shown in Fig. 11-1. Fortunately, a systematic extrapolation can readily be derived by using a thermodynamic cycle as indicated in Sec. 11.2. This extrapolation does not require the assumption of low pressures but yields an expression that gives the fugacity, rather than the pressure, of the saturated, subcooled liquid.

11.2 Calculation of the Pure-Solute Fugacity Ratio

For the liquid-phase activity coefficient, we define the standard state as the pure, sub-cooled liquid at temperature T under its own saturation pressure. Assuming negligible solubility of the solvent in the solid phase, , the equilibrium equation is

(11-5)

To simplify notation, let

and let

These two fugacities depend only on the properties of the solute (component 2); they are independent of the nature of the solvent. The ratio of these two fugacities can readily be calculated by the thermodynamic cycle indicated in Fig. 11-2. The molar Gibbs energy change for component 2 in going from a to d is related to the fugacities of solid and subcooled liquid by

Figure 11-2 Thermodynamic cycle for calculating the fugacity of a pure subcooled liquid.

(11-6)

where, for simplicity, subscript 2 has been omitted. This Gibbs energy change is also related to the corresponding enthalpy and entropy changes by

(11-7)

The thermodynamic cycle in Fig. 11-2 provides a method to evaluate the enthalpy and entropy changes given in Eq. (11-7); because both enthalpy and entropy are state functions independent of the path, it is permissible to substitute for the path a → d the alternate path a → b → c → d. For the enthalpy change from a to d we have

(11-8)

Equation (11-8) can be rewritten in terms of heat capacity c_p and enthalpy of fusion :³

³ Equations (11-9) and (11-11) neglect the effect of pressure on the properties of solid and subcooled liquid. Unless the pressure is large, this effect is negligible.

(11-9)

where Δc = c_p(solid) and T_t is the triple-point temperature. Similarly, for the entropy change from a to d,

(11-10)

which becomes⁴

⁴ See previous footnote.

(11-11)

At the triple point, the entropy of fusion Δ_fuss is

(11-12)

Substituting Eqs. (11-7), (11-9), (11-11), and (11-12) into Eq. (11-6), and assuming that Δc_p is constant over the temperature range T → T_t, we obtain⁵

⁵ Equation (11-13) assumes that there are no solid-solid phase transition along path a → b in Fig. 11-2. When such transitions are present, Eq. (11-53) must be modified, as discussed, for example, by G. T. Preston et al., 1971, J. Phys. Chem., 75: 2345 and by P. B. Choi and E. McLaughlin, 1983, AIChE J., 29: 150.

(11-13)

Equation (11-13) gives the desired result; it expresses the fugacity of the sub-cooled liquid at temperature T in terms of measurable thermodynamic properties. To illustrate, Fig. 11-3 shows the fugacity ratio for solid and subcooled liquid carbon dioxide.

Two simplifications in Eq. (11-13) are frequently made; these usually introduce only slight error. First, for most substances there is little difference between the triple-point temperature and the normal melting temperature; also, the difference in the enthalpies of fusion at these two temperatures is often negligible. Therefore, in practice, it is common to substitute the normal melting temperature for T_t and to use for Δ_fush the enthalpy of fusion at the melting temperature.⁶ Second, the three terms on the right-hand side of Eq. (11-13) are not of equal importance; the first term is the dominant one and the remaining two, of opposite sign, have a tendency approximately to cancel each other, especially if T and T_t are not far apart. Therefore, in many cases it is sufficient to consider only the term that includes Δ_fush and to neglect the terms that include Δc_p.

⁶ For cases where no experimental enthalpies (or entropies) of fusion are available, they must be estimated. A group-additivity method for estimating enthalpies (and entropies) of fusion of organic compounds is given by Chickos et al., 1991, J. Org. Chem., 56: 927.

Figure 11-3 Fugacity ratio for solid and subcooled liquid carbon dioxide.

11.3 Ideal Solubility

An expression for the ideal solubility of a solid solute in a liquid solvent has already been given by Eq. (11-4), but no clear-cut method was given for finding the saturation pressure of the subcooled liquid. However, this difficulty can be overcome by substituting Eq. (11-13) into Eq. (11-5); if we assume that the solution is ideal, then γ₂ = 1 and we obtain for ideal solubility x₂:

(11-14)

Equation (11-14) provides a reasonable method for estimating solubilities of solids in liquids where the chemical nature of the solute is similar to that of the solvent. For example, in Sec. 11.1 we quoted the experimental solubilities of phenanthrene and anthracene in benzene at 25°C; they are 20.7 and 0.81 mol %, respectively. The corresponding solubilities calculated from Eq. (11-14) are 22.1 and 1.07 mol %.

Equation (11-14) immediately leads to useful conclusions concerning the solubilities of solids in liquids. Strictly, these conclusions apply only to ideal solutions but they are useful guides for other solutions that do not deviate excessively from ideal behavior:

• For a given solid/solvent system, the solubility increases with rising temperature. The rate of increase is approximately proportional to the enthalpy of fusion and, to a first approximation, does not depend on the melting temperature.

• For a given solvent and a fixed temperature, if two solids have similar enthalpies of fusion, the solid with the lower melting temperature has the higher solubility; similarly, if the two solids have nearly the same melting temperature, the one with the lower enthalpy of fusion has the higher solubility.

A typical application of Eq. (11-14) is provided by the solubility data of McLaughlin and Zainal (1959) for nine aromatic hydrocarbons in benzene in the temperature range 30 to 70°C. With the help of Eq. (11-14) these data can be correlated in a simple manner. To a fair approximation, the terms that include Δc_p in Eq. (11-14) may be neglected; also, it is permissible to substitute melting temperatures for triple-point temperatures. Equation (11-14) may then be rewritten

(11-15)

where T_m is the normal melting temperature. For the nine solutes considered here, the entropies of fusion do not vary much; an average value is 54.4 J mol^-1K^-1. Therefore, a semilogarithmic plot of log x₂ versus T_mlT should give a nearly straight line with a slope approximately equal to -(54.4/2.303R) and with intercept log x₂= 0 when T_mlT=1;in this case, R= 8.31451 J mol^-1K^-1. Such a plot is shown by the dashed line in Fig.11-4 and it is evident that this Sine gives a good representation of the experimental data. Thus the ideal-solution assumption is appropriate for these systems. However, for precise work, the assumption of ideality is only an approximation. Solutions of aromatic hydrocarbons in benzene, although the compounds are chemically similar, show slight positive deviations from ideality and therefore the observed solubilities are a little lower than those calculated with Eq. (11-15). The continuous line in Fig. 11-4, determined empirically as the best fit of the data, has a slope equal to-(57.7/2.303/R).

Equation (11-14) gives the ideal solubility of solid 2 in solvent 1. By interchanging subscripts, we may use the same equation to calculate the ideal solubility of solid 1 in solvent 2; by repeating such calculations at different temperatures, we can then obtain the freezing diagram of the binary system as a function of composition, as shown in Fig. 11-5 taken from Prigogine and Defay (1954). In these calculations we assume ideal behavior in the liquid phase and complete immiscibility in the solid phase. The left side of the diagram represents equilibrium between the liquid mixture and solid o-chloronitrobenzene, while the right side represents equilibrium between the liquid mixture and solid p-chloronitrobenzene. At the point of intersection, called the eutectic point, three phases are in equilibrium.

Figure 11-4 Solubility of aromatic solids in benzene.

Figure 11-5 Freezing points for the system o-ch!oronitrobenzene (1)/pchtoronifrobenzene (2).

11.4 Nonideal Solutions

Equation (11-14) assumes ideal behavior but Eqs. (11-5) and (11-13) are general. Whenever there is a significant difference in the nature and size of the solute and solvent molecuies, we may expect that γ₂ is not equal to unity; in nonpolar solutions, where only dispersion forces are important, γ₂ is generally larger than unity (and thus the solubility is less than that corresponding to ideal behavior), but in cases where polar or specific chemical forces are important, the activity coefficients may well be less than unity with correspondingly higher solubilities. Such enhanced solubilities, for example, have been observed for unsaturated hydrocarbons in liquid sulfur dioxide.

An illustration that shows that the ideal-solubility equation [Eq. (11-15)] gives only a crude approximation is provided by the solubility of cholesterol (C₂₇H₄₆O, M = 386.67 g mol^-1, T_m = 421.7 K, Δ_fush = 28.924 kJ mol^-1) in different polar solvents, shown in Fig. 11-6. Cholesterol, a lipid sterol, is an important component of cell membranes. From spectroscopic (infrared and NMR) and dielectric-constant measurements, there is evidence that cholesterol shows self-association (Mercier et al., 1983; G6ralski, 1993, Jadzyn and Hellemans, 1993). It is likely that hydrogen bonds and van der Waals interactions play an important role in the biological functions of cholesterol.

Figure 11-6 Solubility of cholesterol (2) in polar solvents:• Methanol; Ethanol;∎ Acetone. Solid line is the best fit through the experimentai points; the dashed line is the ideal-solubility curve [Eq. (11-15)].

As Fig. 11-6 shows, the experimental solubility curves (Bar et al., 1984) deviate significantly from the ideal-solubility curve as given by Eq. (11-15). In contrast to the previous example concerned with the solubility of aromatic solids in benzene, for cholesterol/polar-solvent systems, all molecular interactions (solute-solute, solute solvent, and solvent-solvent) are relatively complex. For these systems, activity coefficients for cholesterol are far in excess of unity.

In Fig. 11-4 we showed the solubilities of nine aromatic hydrocarbons in benzene and it is apparent that the assumption of ideal-solution behavior is approximately valid. By contrast, Fig. 11-7 shows the solubilities of the same nine hydrocarbons in carbon tetrachloride; the data are again from McLaughlin and Zainal (1960). The dashed line shows the ideal solubility curve using, as before, Δfus_s = 54.4 J mol^-1K^-1; the solid line that best represents the data has a slope -(66.5/2.303/R).

Figure 11-7 Solubility of aromatic solids in carbon tetrachloride.

A comparison of Figs. 11-4 and 11-7 shows that at the same temperature, the solubilities in carbon tetrachloride are lower than those in benzene; in other words, the activity coefficients of the solutes in carbon tetrachloride are larger than those in benzene. If the activity coefficients of the solutes in carbon tetrachloride are represented by the simple empirical relation

(11-16)

then typical values of A are in the range 400 to 1300 J mol^-1.

As for solutions of liquid components, there is no general method for predicting activity coefficients of solid solutes in liquid solvents. For nonpolar solutes and solvents, however, a reasonable estimate can frequently be made with the Scatchard-Hildebrand relation

(11-17)

where is the molar volume of the subcooled liquid, δ2 is the solubility parameter of the subcooled liquid, δ₁ is the solubility parameter of the solvent, and

is the volume fraction of the solvent.

The molar liquid volume and solubility parameter of the solvent can be determined from the thermodynamic properties of the solvent, but it is necessary to use a thermodynamic cycle (as illustrated in Fig. 11-2) to calculate these functions for the subcooled liquid solute.

Let Δ_fusυ stand for the volume change of fusion at the triple-point temperature; that is,

(11-18)

where subscript t refers to the triple-point temperature.

Let be the molar volume of the solid at temperature T of the solution. The molar volume of the subcooled liquid is then given by

(11-19)

where are α^L the volumetric coefficients of expansion of the solid and liquid, respectively.

The energy of vaporization of the subcooled liquid is found in a similar manner. Let Δ_fush stand for the enthalpy of fusion of the solid at the triple-point temperature and let Δ_subh stand for the enthalpy of sublimation of the solid at temperature T. The energy of vaporization of the subcooled liquid is then

(11-20)

where P^s is the saturation pressure of the subcooled liquid and υ^G is the molar volume of the saturated vapor in equilibrium with the solid, all at temperature T.

In Eqs. (11-19) and (11-20) it is often convenient to replace the triple-point temperature with the melting temperature and at moderate pressures this substitution usually introduces insignificant error; in that case, all subscripts t can be replaced by subscript m. Also, if the temperature T is not far removed from the triple-point (or melting) temperature, the last term in Eq. (11-19) may be neglected, and finally, if the saturation pressure of the subcooled liquid is small, as is usually the case, υ^G >> υ^L, and the last term in Eq. (11-20) may be replaced by RT,

The square of the solubility parameter is defined as the ratio of the energy of complete vaporization to the liquid volume;⁷ therefore, if the vapor pressure of the sub-cooled liquid is large, it is necessary to add a vapor-phase correction to the energy of vaporization given by Eq. (11-20). Such a correction, however, is rarely required and for most cases of interest the solubility parameter of the subcooled liquid is given with sufficient accuracy by

⁷ see Sec. 7.2.

(11-21)

where Δ_vapµ2 the energy of vaporization, given by the enthalpy of vaporization minus RT.

To illustrate the applicability of Eq. (11-17), we consider the solubility of white phosphorus in n-heptane at 25°C. The melting point of white phosphorus is 44.2°C; the enthalpy of fusion and the heat capacities of the solid and liquid have been measured by Young and Hildebrand (1942). From Eq. (11-14), the ideal solubility at 25°C is x₂ = 0.942. A much better approximation can be obtained from regular-solution theory as given by Eq. (11-17). From the extrapolated thermal and volumetric properties, the solubility parameter and molar volume of subcooled liquid phosphorus are, respectively, 27 (J cm^-3)^1/2 and 70.4 cm³ mol^-1 at 25°C. The solubility parameter of nheptane is 15.1 (J cm^-3)^1/2 and therefore one can immediately conclude that subcooled phosphorus and heptane form a highly nonideal liquid solution. When and n Eqs. (11-17) and (11-13) are substituted into the fundamental Eq. (11-5), the calculated solubility is x₂=0.022, a result strikingly different from that obtained by assuming liquid-phase ideality. The experimental solubility is x₂= 0.0124.

As discussed in Chap. 7, the regular-solution theory of Scatchard-Hildebrand can be significantly improved when the geometric-mean assumption is not used; in that event, Eq. (11-17) becomes

(11-22)

Preston (1970) has applied Eq. (11-22)⁸ to correlate solubilities of nonpolar solids in nonpolar liquids at low temperatures. To do so, it was first necessary to estimate solubility parameters aad molar volumes for subcooled liquids; these estimates were obtained by extrapolations as shown in Figs. 11-8 and 11-9. For a given binary, parameter l₁₂ must be obtained from some experimental binary datum; it is best to obtain l₁₂ from an experimental solubility. Preston gives l₁₂ parameters for 25 systems at cryogenic conditions.

⁸ This equation was also used by Choi and McLaughlin, 1983, Ind. Eng. Chem. Fund., 22: 46, to correlate solubilities of solid aromatic hydrocarbons in pyridine and in thiophene.

Figure 11-8 Solubility parameters of subcooled liquids.

Figure 11-9 Molar volumes of subcooled liquids.

A more fundamental extension of the Scatchard-Hildebrand equation was introduced by Myers (1965) in his study of the solubility of solid carbon dioxide in liquefied light hydrocarbons. Because carbon dioxide has a large quadrupole moment (see Table 4-2), separate consideration was given to the contributions of dispersion forces and quadrupole forces to the cohesive energy density of subcooled liquid carbon dioxide.⁹ The energy of vaporization was divided into two parts:

⁹ The solubility parameter is the square root of the cohesive-energy density. See Sec. 7.2.

(11-23)

As a result, two cohesive-energy densities can now be computed, corresponding to the two types of intermolecular forces:

(11-24)

(11-25)

where superscript L has been omitted. These cohesive-energy densities for carbon dioxide are shown as a function of temperature in Fig. 11-10.

Figure 11-10 Cohesive-energy density due to dispersion forces and due to quadrupole forces for subcooied liquid carbon dioxide.

The activity coefficient of component 2, the solute, dissolved in a nonpolar solvent, is now written

(11-26)

where C_{2 total} = c_{2 disp} + c_{2 quad}. The last term in the brackets follows from the geometric-mean assumption for the attractive dispersion forces between solute and solvent. If ^C2 quad=0, Eq. (11-26) reduces to Eq.(11-17).

Splitting the cohesive-energy density into a dispersion part and a quadrupole part has an important effect on the calculated solubility of solid carbon dioxide. Although the contribution from quadrupole forces is small, appreciable error is introduced by not separately considering this contribution. As an example, consider the solubility of solid carbon dioxide in a nonpolar (and nonquadrupolar) solvent having the (typical) value c₁ = 251 J cm^-3 for its cohesive-energy density at 160 K. If we do not take account of the presence of quadrupolar forces, we calculate, from Eq. (11-17), a solubility x₂ = 0.067. However, if we do account for quadrupolar forces [Eq. (11-26)], we obtain the significantly different result x₂ = 0.016. In this illustrative calculation we have, for simplicity, assumed that Φ₂ = x₂.

To calculate the cohesive-energy density due to quadrupole forces, Myers derived the relation

(11-27)

where Q_i is the quadrupole moment of species i, υ_i is the molar liquid volume, N_A is Avogadro’s constant, k is Boltzmann’s constant, Tis the absolute temperature, and (3 is a dimensionless constant. If the solvent, component 1, also has a significant quadrupole moment, then an additional term must be added to the bracketed quantity in Eq, (11-26) to account for quadrupole forces between the dissimilar components; further, the geometric-mean term must be modified to include only the dispersion cohesive-energy density of component 1. The bracketed term in Eq. (11-26) then becomes

Using the theory of intermolecular forces, Myers showed that

(11-28)

where β is the same as in Eq. (11-27).

For υ₁₂ Myers used the combining rule

(11-29)

The term c_{12 quad} is frequently negligible but it is important, for example, in carbon dioxide/acetylene mixtures because in this system both solute and solvent have a significant quadrupole moment (see Table 4-2).

In nonpolar systems, the ideal solubility is generally larger than that observed. This result is correctly predicted by the Scatchard-Hildebrand theory [Eq. (11-17)] because γ₂ ≥ 1; as a consequence, this equation says that the ideal solubility is the maximum possible: the larger γ₂, the smaller x₂. However, whenever there is a tendency for the solvent to solvate with the solute, i.e., whenever there are strong specific forces between the dissimilar molecules, the observed solubility may well be larger than the ideal solubility. Enhanced solubility is observed whenever there are negative deviations from Raoult’s law in the liquid solution; such deviations frequently occur in polar systems, especially in systems where hydrogen bonding between solute and solvent is strong. However, even in nonpolar systems, specific solvation forces may, on occasion, be sufficiently strong to result in solubilities above the ideal. For example, Weimer (1965) measured the solubility of hexamethylbenzene (m.p. 165.5°C) in carbon tetrachloride at 25°C and found x₂ = 0.077. Contrary to predictions using the Scatchard-Hildebrand theory, this solubility is larger than the ideal solubility x₂ = 0.062. From spectroscopic and other evidence, we know that carbon tetrachloride forms weak charge-transfer complexes with aromatic hydrocarbons and we also know that the stability of the complex increases with methyl substitution on the benzene ring. For the solutes listed in Fig. 11-6, the complex stability is small, but in the carbon tetrachloride/hexamethylbenzene system the tendency to complex is sufficiently strong to produce an enhanced solubility; in this system, complex formation overshadows the “normal” physical forces that would tend to give a solubility lower than the ideal. Weimer found that in carbon tetrachloride solution the activity coefficient of hexamethylbenzene at saturation (referred to pure subcooled liquid) is γ₂ = 0.79, indicating negative deviation from Raoult’s law.

As briefly indicated in Chap. 6, liquid-phase activity coefficients can sometimes be estimated from a group-contribution method such as UNIFAC. It is therefore possible, in some cases, to use UNIFAC for constructing freezing-point diagrams, as shown by Gmehling et al. (1978). To illustrate, Fig. 11-11 shows a diagram for the system benzene/phenol. A simple eutectic diagram is obtained; the lines calculated with UNIFAC are significantly higher than those calculated assuming ideal solubility. Agreement with experiment (Hatcher and Skirrow, 1917; Tsakalotos and Guye, 1910) is good.

In the liquid phase, it is also possible to calculate the fugacity of the dissolved component using an equation of state. For solute 2, the equation of equilibrium is written

(11.30)

where φ₂ is obtained from an equation of state, valid for the fluid mixture, over the density range from zero density to liquid density, as discussed in Chaps. 3 and 12.

Equation (11-30) has been used, for example, by Soave (1979) for calculating the solubility of solid carbon dioxide in liquefied natural gas.

Figure 11-11 Solid-liquid equilibria for the system benzene/phenol.

From experimental solid-liquid equilibrium (SLE) data we can obtain the binary parameters of a particular liquid-phase activity coefficient model, such as Wilson or UNIQUAC. These parameters may give in turn liquid phase activity coefficients at different system conditions that can be used to predict other equilibria, e.g. vapor-liquid equilibria (VLB). Calculation of VLB from experimental SLE is particularly useful for those systems where it is difficult to obtain experimental VLB data, e.g. due to the high-melting points of the system components and to the large difference of vapor pressures, as in systems containing heterocyclic polynuclear aromatics (naphthalene, acenaphthene, fluorene, phenanthrene, etc.) in organic solvents. To illustrate we now consider VLB at 170°C for the systems (Gupta et al., 1991) acenaphthene¹⁰/tetralin and acenaphthene/cis-decalin, shown in Fig. 11-12. Vapor-liquid equilibria for both systems are predicted using the UNIQUAC equation whose binary interaction parameters a₁₂ and a₂₁ (see Sec. 6.11) were determined from regression of experimental solidacenaphthene solubility data in each solvent from room temperature to the melting-point temperature of the solute.

¹⁰ The melting point of acenaphthene is 93.3°C.

Figure 11-12(a) shows that in the acenaphthene/tetralin system VLB predictions are in good agreement with the experiment. However, this is not always the case; as illustrated in Fig. 11-12(b) for the system acenaphthene/cis-decalin, good agreement with experiment is obtained only if the solid-solubility data and (at least) one vapor liquid equilibrium data point are combined to obtain an improved set of a₁₂ and a₂₁ UNIQUAC binary parameters.

Figure 11-12 Calculated and observed vapor-liquid equilibria at 170°C for: (a) Acenaphthene/tetralin. ——Calculated from UNIQUAC equation and SLE data only. Using also one VLE datum point, the calculated curve is almost indistinguishable from that obtained using SLE data only, (b) Acenaphthene/cis-decalin. Calculated from UNIQUAC equation: ---- SLE data only; —— SLE and one VLE datum. In this case, use of one VLE datum very much improves agreement with experiment.

11.5 Solubility of a Solid in a Mixed Solvent

Scatchard-Hildebrand theory predicts that the solubility of a solid is a maximum in that solvent whose solubility parameter is the same as that of the (liquid) solute; in that event, the activity coefficient of the solute (referred to the subcooied liquid) is equal to unity. Scatchard-Hildebrand theory suggests, therefore, that when a solid solute is dissolved in a mixture of two carefully selected solvents, a plot of solubility versus (solute-free) solvent composition should go through a maximum.

To test this maximum-solubility effect, consider phenaethrene that melts at 100°C; the (liquid) solubility parameter of phenanthrene is 20.3 (J cm^-3)^1/2. When phenanthrene is dissolved in a binary solvent mixture, the effective solubility parameter of the mixed solvent varies with the solvent’s composition. Let phenanthrene be designated by subscript 1 and the two solvents by subscripts 2 and 3. If δ₂ < 20.3 and δ₃ > 20.3, then there is some mixture of solvents 2 and 3 which is the optimum solvent for phenanthrene.

Gordon and Scott (1952) have measured the solubility of phenanthrene in cyclohexane (2), in methylene iodide (3), and in their mixtures. At 25°C, the solubility parameters are δ₂ = 16.8 and δ₃ = 24.4 (J cm^-3)^1/2. Figure 11-13 shows, as expected, that a plot of experimental solubility versus solvent composition (solute-free basis) exhibits a maximum.

Figure 11-13 Solubility of phenanthrene in a mixed solvent containing cydohexane and methylene iodide.

Also shown in Fig. 11-13 are calculations based on regular-solution theory without, however, using the geometric-mean assumption. The activity coefficient of phenanthrene (referred to the subcooled liquid) is given by

(11.31)

where

Parameters l_ij, that give deviations from the geometric mean, are obtained from binary data reported by Gordon and Scott. For the two binaries including phenanthrene, l₁₂ and l₁₃ are obtained from experimental solubilities of phenanthrene in each of the two solvents at 25°C. For the solvent pair, l₂₃ is obtained from binary melting-point data. These parameters are l₁₂ = 0.028, l₂₃ = 0.010, and l₁₃ = 0.0028. For phenanthrene, the enthalpy of fusion is 17991 J mol^-1 and the (liquid) molar volume at 25°C is estimated to be 150 cm³ mol^-1. Molar volumes for cydohexane and methylene iodide are 108 and 80 cm³ mol^-1, respectively.

In these calculations, all parameters were taken as independent of temperature. Agreement between calculated and observed solubilities is nearly quantitative; the somewhat larger deviations at 310 K are probably due to uncertainties in l₂₃ that was found from data at significantly lower temperatures. It is likely that l₂₃ is strongly temperature-dependent near 310 K because cyclohexane and methylene iodide are only partially miscible below the (upper) critical solution temperature near 303 K. The important feature of Fig. 11-13 is that calculations, based on binary data only, correctly predict the maximum solubility that is observed in the ternary system.

A group-contribution method such as UNIFAC may sometimes be used to calculate liquid-phase activity coefficients of solutes in mixed solvents, as discussed by Gmehling et al. (1978). To illustrate, consider the solubility of naphthalene in mixed solvents containing water and alcohol. If naphthalene is designated by subscript 2, the equation of equilibrium is given by Eq. (11-5), again assuming that there is negligible solubility of the other components in solid naphthalene. The fugacity ratio for pure naphthalene is given by Eq. (11-13). UNIFAC is used to calculate activity coefficient γ₂.

Table 11-2 gives calculated and experimental solubilities for naphthalene in aqueous solvents containing methanol, ethanol, propanol, or butanol. The ideal solubilities (γ₂ = 1) are always too large by one order of magnitude. Solubilities calculated with UNIFAC are in semiquantitative agreement with experiment.

Table 11-2 Solubility of naphthalene in alcohol/water systems.

For a solid dissolved in a binary mixed solvent, enhanced solubility (or maximum-solubility effect)¹¹ has been observed for a large variety of systems. As an example, consider the solubilities of two isostructural aromatic compounds 2-acetyl-lnaphthol (1) and l-acetyl-2-naphthol (2), in pure and mixed hydrocarbons and alcohols (Domanska, 1990):

¹¹ However, in other cases, enhanced solubility is not observed; instead, there may exist negative synergetic effect causing a decreased solubility in comparison with those in single solvents. For example, solubility measurements for solid l-benzoyl-2-naphthol show slight negative deviations from additivity (negative synergetic effects) in the hexane+1-butanol mixed solvent system. Negative synergetic effects on solubility have also been found in solubility measurements for octadecanoic, eicosanoic, and docosanoic acids in mixed solvent systems such as 1,4-dioxane+chloroform and tetrahydroforan+chloroform.

Figure 11-14(a) shows the solubility of (1) in cyclohexane, in 1-butanol and in several mixtures of these solvents, at four temperatures. Similarly, Fig. 11-14(b) shows the solubility of (2) in cyclohexane, in 1-butanol and in several mixtures of these solvents. Both isomers have enhanced solubility in the binary solvent mixture, and the deviations from additivity¹² rise with increasing temperature. These deviations are probably due to the decrease of self association of 1-butanol molecules with increasing temperature.

¹² Additivity [given by the broken lines in Fig. (11-14)] means that the solubility is a linear function of solvent composition.

The results presented in Fig. 11-14 indicate that solubilities of compound (2) in the alcohol are much higher than those in cyclohexane. In contrast, (1) exhibits similar solubilities in the alcohol and in the alkane solvents. The large differences observed in the solubilities of these (similar) compounds are attributed to their characteristic behavior in the two solvents. Ultraviolet and infrared absorption spectra of (1) and (2) reveal that both form intramolecular hydrogen bonds. However, for compound (1) this bond is little affected by hydroxylic solvents (such as alcohols), i.e. the long-wavelength bond in its UV spectra is not affected by the solvent, as shown in Fig. 11-15(a). In contrast to (1), UV absorption spectra in (2) reveal that hydroxylic solvents cause the disruption of intramolecular hydrogen bonds in (2), as shown in Fig. 11-15(b).

Figure 11-14 Solubility isotherms for (a) 2-acetyl-1-naphthol (1) and (b) 1-acetyl-2-naphthol (2) in cyclohexane/1-butanol mixed solvent system (Domanska, 1990). —— Smoothed experimental data (symbols); --- Additivity rule.

Figure 11-15 Schematics of ultraviolet absorption spectra for (a) 2-acetyi-1-naphthol (1) and (b) 1-acetyl-2-naphthol (2), in cyctohexane (——) and in 1-butanol (- - - -). Here ε is the extinction coefficient.

In cyclohexane solutions, molecules of isomeric (2) are strongly intramolecular hydrogen bonded and almost planar, but in hydroxylic solvents the intramolecular hydrogen-bond is disrupted due to the formation of intermolecular solute-solvent solvation, as schematically illustrated below:

Moreover, dielectric studies also indicate that the intramolecular hydrogen-bond is relatively stable for (1) but unstable for (2). Therefore, compound (1), that is able to form strong intramolecular hydrogen bonds, behaves quite differently from its isomer that lacks this ability.

Domanska’s studies show once again that understanding of thermodynamic properties (in this case, solubilities) is much enhanced by fundamental physico-chemical measurements that provide information on molecular behavior.

The solubility measurements for solids (1) and (2) can be described by an equation similar to Eq. (11-14), provided that we do not neglect activity coefficient γ₂. To illustrate, consider again the solubility of (1) in the cyclohexane+1-butanol mixed solvent. Domanska (1990) used the Wilson equation to obtain the activity coefficients of each component of the ternary mixture [i.e. Eq.(6-165) with m = 3]. Wilson’s interaction parameters for the binary cyclohexane+1-butanol solvent mixture were obtained from VLB data, whereas those for solute-solvent were regressed from binary SLE data. Figure 11-16 shows the temperature dependence of the solubility of (1) in pure cyclohexane and 1-butanol and also in several of their binary mixtures. As shown, the Wilson equation with temperature-independent parameters can describe well the observed solubilities in the temperature range studied.

11.6 Solid Solutions

In all previous sections we have assumed that whereas the solid has a finite solubility in the liquid solvent, there is no appreciable solubility of the solvent in the solid. As a result, we have been concerned only with the equation of equilibrium for component 2, the solid component. However, there are many situations where components 1 and 2 are miscible not only in the liquid phase but in the solid phase as well; in such cases we must write two equations of equilibrium, one for each component:

(11-32)

(11-33)

Figure 11-16 Temperature as a function of the solubility of 2-acetyl-1-naphthol (1) in: cyclohexane(); 1-butanol (); and cyclohexane+1-butanol mixed-solvent system containing X_1-butanol = 0.1971 () and X_1-butanol = 0.8009(). —— Calculated from Wilson equation.

Introducing activity coefficients we can rewrite these equations:

(11-34)

(11-35)

If system temperature T is above the triple-point temperature of component 1 but below that of component 2, then pure solid 1 and pure liquid 2 are both hypothetical. In Sec. 11.2 we showed how to compute the fugacity of a pure (hypothetical) sub-cooled liquid, and exactly the same principles may be used to calculate the fugacity of a pure superheated solid; the calculation is performed by using a thermodynamic cycle similar to that shown in Fig. 11-2 with temperature T above, rather than below, triple-point temperature T_t. When both components are miscible in both phases, Eqs. (11-34) and (11-35) indicate that information is required for the activity coefficients in both phases. Such information is rarely available. In some systems one may, to a good approximation, assume ideal behavior in both phases but in most systems this is a poor assumption.

When Eqs. (11-34) and (11-35) are applied over a range of temperatures, it is possible to calculate the freezing-point diagram for the binary system. If there is rniscibility in both phases, such a diagram is qualitatively very different from that shown in Fig. 11-5. For example, Fig. 11-17 gives the experimentally determined phase diagram for the system naphthalene/β-naphthol that exhibits approximately ideal behavior in both phases. On the other hand, the system mercuric bromide/mercuric iodide, shown in Fig. 11-18, exhibits appreciable nonideality resulting in the formation of a hylotrope (no change in composition upon melting). The thermodynamics of solid solutions is of importance in metallurgy and many experimental studies have been made of binary and multicomponent metallic mixtures.

Figure 11-17 Phase diagram for the system naphthalene/β-naphthol.

When calculating the solubility of a solid in a liquid, as discussed in Sec. 11.4, we assume that the solid phase is pure, i.e., in Eqs. (11-34) and (11-35). In many cases that is a good assumption, but if it is not, serious error can result. To illustrate, Preston (1970) calculated the solubility of solid argon in liquid nitrogen in the region 64 to 83 K. Using regular-solution theory [Eq. (11-17)] and assuming that solid argon is pure, he obtained the upper line shown in Fig. 11-19. He then repeated the calculation, using Eq. (11-35) coupled with experimental solid-phase composition data, assuming again that the liquid-phase activity coefficient was given by Eq. (11-17) and the solid-phase activity coefficient was unity (ideal solid solution).

Preston then obtained the lower line in Fig. 11-19, in excellent agreement with experimental data (Fedorova, 1938; Long and DiPaolo, 1963; Din et al, 1955). These results show that for the system argon/nitrogen it is necessary to take into account miscibility in the solid phase.

Figure 11-18 Phase diagram for the system HgBr₂/Hgl₂.

Figure 11-19 Solubility of argon in nitrogen.

It is difficult to say a priori whether or not two components are partially or totally miscible in the solid phase. For nonpolar substances the general rule is that solid-phase miscibility is usually negligible provided that the two components differ appreciably in molecular size and shape. However, other factors than size and shape contribute to determine the formation of more than one solid phase. For example, the fluorene/dibenzofuran system, whose solid-liquid phase diagram is shown in Fig. 11-20 (Sediawan et al, 1989), forms solid solutions at any composition, because, as expected, those two molecules have similar molecular sizes and shapes, as illustrated in Table 11-3.

Table 11-3 Molecular structures and physical properties of pure dibenzofuran, dibenzo thiophene, and fluorene.

However, in spite of the components’ similarity in size and shape, in the fluorene/dibenzothiophene system there are two possible mixed-solid phases, as illustrated in Fig. 11-21 (Sediawan et al., 1989). For solid mixtures with fluorene mole fractions smaller than about 0.04, there is one solid phase, α, and for solid mixtures with fluorene mole fractions greater than about 0.18 there is another solid phase, β. In the region with fluorene mole fractions between 0.04 and 0.18, there is a solid immiscibility gap; Table 11-3 indicates that this gap cannot be attributed to differences in size and shape. In this particular case it is likely that the observed solid-phase immiscibility gap is caused by differences in the components’ pure crystal structures: solid phase β has a crystal structure similar to that of pure fluorene, while solid phase a has a crystal structure similar to that of dibenzothiophene. Similarly, the binary dibenzothiophene/dibenzofuran shows a small solid-phase immiscibility gap.

Experimental information on binary systems is always necessary to estimate phase equilibria of multicomponent systems. To illustrate, consider the ternary system benzene/fluorene/dibenzothiophene composed of two polynuclear aromatic solutes in an organic solvent. Domanska et al. (1993) used binary data to obtain the parameters of the UNIQUAC model. Four parameters per binary are required: two describing the liquid phase, obtained from the liquidus curve, and two describing the solid phase, obtained from the solidus curve.¹³ For the ternary system at temperatures in the range 40-60°C, Domanska et al. calculate the compositions of the liquid solution in equilibrium with the solid solution, in good agreement with experiment. Their results suggest that it is possible to predict ternary solid-liquid equilibria using UNIQUAC parameters obtained from binary solid-liquid equilibria.

¹³ However, for the two binary entectic systems (benzeoe/fluorene and benzene/dibenzothiophene), Domanska and coworkers assumed that solid-phase parameters are zero.

Figure 11-20 Experimental solid-liquid phase diagram for the system fluorene/dibenzofuran (Sediawan et al., 1989). —— Smoothed data.

Figure 11-21 Experimental solid-liquid phase diagram for the system fluorene/dibenzothiophene (Sediawan et al., 1989). —— Smoothed data.

Because our knowledge of solid-phase mixtures is so meager, in many typical chemical-engineering calculations for multicomponent systems, it has been customary to assume either that there is complete immiscibility or (more rarely) complete miscibility in the solid phase. (In most cases, it is more realistic to assume no solubility in the solid phase). Unfortunately, calculated phase equilibria are extremely sensitive to this choice of assumption; calculated results depend very strongly on whether we assume complete miscibility or complete immiscibility in the solid phase. An industrially important example is provided by the precipitation of solid waxes from petroleum. For many years it was assumed that waxes (long-chain paraffins) are completely miscible in the solid phase; however, to obtain agreement between calculated and experimental results, elaborate empirical corrections were required. When spectroscopic data showed that long-chain paraffins are, in general, not mutually soluble in the solid phase (unless the chain lengths are similar), subsequent thermodynamic calculations gave much better results, as discussed by Lira-Galeana et al. (1996).

11.7 Solubility of Antibiotics in Mixed Nonaqueous Solvents

An industrial application that requires solid-liquid equilibria is provided by separation and recovery processes for amino acids and antibiotics. For some common bioproducts, the cost in bioseparations may reach ninety percent of the total cost of manufacturing. Design of bioseparation units may require phase equilibria for the bioproducts in the solvents used in the production process. Because amino acids and antibiotics have high melting points, the required phase equilibria are mostly solid-liquid equilibria.

A simple solubility model for amino acids and antibiotics was presented by Gupta and Heidemann (1990) using a modified UNIFAC equation.¹⁴ To illustrate, consider the antibiotic carbomycin-A (C₄₂H₆₇NO₁₆, M = 841.97 g mol^-1, t_m = 214°C). Using available experimental solubilities of the antibiotic in different solvents, and considering the entire antibiotic molecule as a group,¹⁵ Gupta and Heidemann wanted to obtain UNIFAC group-interaction parameters. To do so, they obtained from experimental solubility data the activity coefficient of the solid at saturation, using an equation similar to Eq. (11-15):

¹⁴ B. L. Larsen, P. Rasmussen, and Aa. Fredenstand, 1987, Ind. Eng. Chem. Res., 26: 2274.

¹⁵ For carbomycin-A, the group volume and the group surface area parameters are, respectively, R = 31.6568 and Q = 5.119, as determined by the van der Waals volumes and surface areas of the constituent conventional UNIFAC groups.

(11-36)

For the entropy of fusion they take 56.51 J mol^-1 K^-1. ¹⁶ The optimal UNIFAC parameters, a_ij, were then obtained from regression of the objective function are obtained from Eq. (11-36) with experimental and where is from tne UNIFAC equation. Table 11-4 lists the group-interaction parameters obtained.

¹⁶ S. H. YaSkowsky, 1979, Ind. Eng. Chem. Fundam., 18: 109.

Table 11-4 UNIFAC group-interaction parameters a_IJ(in Kelvin) between carbomycin-A (2) and the standard alkane (CH₂), alcohol (OH), and aromatic (ACH) groups, obtained from the antibiotic’s solubility data in several solvents at 28°C (Gupta and Heidemann, 1990).

Antibiotics are commonly produced from an aqueous fermentation broth as crystalline precipitates by adding solvents wherein the product has a lower solubility. UNIFAC group-interaction parameters can be used to predict solubilities in solvents and in mixed solvents when little or no experimental data are available, thus providing estimates of the required phase equilibria necessary for the design of the separation process. As an illustration, Fig. 11-22 shows calculated solubilities of carbomycin-A in a mixed solvent containing cyclohexane and toluene at 28°C. Calculations were performed with the UNIFAC equation using parameters in Table 11-4. Comparison with experimental data is possible only at the endpoints of the curve, corresponding to the solubility in the pure solvents also at 28°C.

An useful, industrially-oriented way to use calculated solubilities is presented in Fig. 11-23. This figure shows the calculated percent recovery of carbomycin-A that is expected upon diluting a saturated solution in toluene with several solvents. The calculations show that about 90% of the dissolved antibiotic can be recovered as crystalline solid by diluting with n-hexadecane on a one-to-one molar basis.

The results presented in Fig. 11-23 are qualitative only; they require experimental confirmation. The model used in the calculations is based on a limited data base and therefore, the results of Fig. 11-23 can only provide guidance toward a possible separation process. Nevertheless, the results indicate how simple thermodynamic calculations, coupled with limited experimental data, may be useful for preliminary chemical-process design.

Figure 11-22 Solubility of carbomycin-A in mixtures of toluene and cyclohexane at 28°C. —— UNIFAC; Experiment.

Figure 11-23 Predicted (from UNIFAC) recovery of carbomycin-A from saturated solutions in toluene upon dilution with a selected hydrocarbon.

References

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Problems

1. A liquid mixture contains 5 mol % naphthalene and 95 mol % benzene. The mixture is slowly cooled at constant pressure. At what temperature does a solid phase appear? Assume ideal mixing in the liquid phase and total immiscibility in the solid phase. Data are:

2. At 25°C, a solid A is in solution in a liquid solvent B. It is proposed to remove A from the solution by adsorption on a solid adsorbent S that is inert toward B. The adsorption equilibrium constant K is defined by a Langmuir-type expression

where 0 is the fraction of surface sites on S that are occupied by A and a_A the activity of A (a_A = 1 when x_A = 1). At 25°C, K = 130.

What fraction of the surface sites on S is covered by A when x_A is one-half of the saturated mole fraction of A in B at 25°C?

The following data are available for pure A:

= 19700 J mor^-1

T_m = 412 K (melting point)

C_P(liquid) = 33.5 J mol^-1 K^-1

C_P(solid) = 26.4 J mol^-1 K^-1

=0.28 kPa

= 0.050 at 25°C (solubility in B)

3. In a famous paper by Brown and Brady (1952, J. Am. Chem. Soc., 74: 3570) on the basicity of aromatics, the authors comment on Klatt’s measurements at 0°C on the solubilities of aromatics in (acidic) liquid hydrogen fluoride. Klatt finds the solubilities are in the order m-xylene < toluene < benzene. However, there is ample evidence from other equilibrium measurements, from spectra and from kinetic data that the basicity of the aromatic ring increases with methyl substitution. Can you explain the apparent disagreement?

4. Estimate the solubility of naphthalene at 25°C in a mixed solvent consisting of 70 mol % isopentane and 30 mol % carbon tetrachloride.

The following data are available for naphthalene:

Melting point: 80.2°C; enthalpy of fusion: 19008 J mol^-1.

c_p(liquid) - c_p(solid) is sufficiently small to be negligible for these conditions.

Volume of “liquid” naphthalene at 25°Cm³ mol^-1 (extrapolated).

5. The freezing point of benzene is 278.7 K. At 260 K, an equimolar mixture of n-hexane and carbon tetrachloride contains 10 mol % benzene. Calculate the partial pressure of benzene in equilibrium with this mixture.

For benzene: = 30.45 cal g^-1 (260 K) = 0.0125 bar. At 25°C:

6. A liquid mixture of benzene and n-heptane is to be cooled to as low a temperature as possible without precipitation of a solid phase. If the solution contains 10 mol % benzene, estimate what this lowest temperature is.

The data are as follows;

7. The Cu₂O/P₂O₅ system has been investigated by X-ray and thermal techniques (M. Ball, 1968, J. Chem. Sac. (A), Inorg. Phys. Theor., 1113). The following table gives liquidus and solidus temperatures for the system, under argon. Melting point of Cu₂O is 1503 K.

What physicocfaemical conclusions can be drawn from these data?

8. Benzene freezes at 5.5°C; its enthalpy of fusion is 9843 J mol^-1.

(a) At -50°C, would you expect benzene to be more or less soluble in CS₂ than in n-octane? Why? (The freezing points of CS₂ and n-octane are -19.5 and -56.5°C, respectively.)

(b) What composition of a mixed solvent of CS₂ and n-octane is needed such that a solution containing 30 mol % benzene just begins to precipitate at -50°C?

9. The triple point of carbon dioxide is 216.5 K. At 194.3 K, the solubility of carbon dioxide (2) in a solvent (1) is x₂ = 0.25, where x₂ is the liquid-phase mole fraction of carbon dioxide. Solvent (1) is insoluble in solid carbon dioxide. Estimate the partial pressure of carbon dioxide in solvent (1) at 194.3 K when x₂ = 0.05.

The data (all at 194.3 K) are as follows: Saturation pressure of solid carbon dioxide: 0.99 bar; from entfaalpy-of-fusion data, the ratio _{pure 2} = 0.56

10. At 250 K, a nonpolar solid A has a vapor pressure of 35 ton. The melting temperature of A is 300 K where its enthalpy of fusion is 13 kJ mol^-1. Component A is a branched hydrocarbon. At 250 K, a 3 mol % solution of A in carbon tetrachloride has a partial pressure of 5 torr.

What is the partial pressure of A in a 1 mol % solution in n-hexane at 250 K? At 25°C, the following data are available for solubility parameters and molar liquid volumes:

11. A natural gas, containing mostly methane, contains also small amounts of H₂S. It is proposed to remove the H₂S by an absorption process. Consideration is being given to two liquid solvents A and B whose freezing points are, respectively, 20 and 5°C.

As for most gases, the solubility of gaseous H₂S in either A or B rises with decreasing temperature. To minimize solvent requirements, it is therefore advantageous to operate at a low temperature. However, if A is used, the lowest possible temperature is 20°C and if B is used, the lowest possible temperature is 5°C. To operate at a lower temperature, it is proposed to use for the solvent a mixture of A and B.

To test the feasibility of this proposal, estimate the composition in mole percent of a mixed solvent containing A and B that gives the lowest possible temperature for an absorption process. Also estimate this temperature.

For the preliminary-design purposes considered here, assume that liquids A and B are chemically similar and that they are rniscible in all proportions. Also assume that solids A and B are mutually insoluble. The enthalpies of are 8 kJ mol^-1 for A and 12 kJ mol^-1 for B.