13.7: The Colligative Properties of Strong Electrolyte Solutions
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Learning Objectives
- To understand the factors that determine the solubility of ionic compounds.
The solubility product of an ionic compound describes the concentrations of ions in equilibrium with a solid, but what happens if some of the cations become associated with anions rather than being completely surrounded by solvent? Then predictions of the total solubility of the compound based on the assumption that the solute exists solely as discrete ions would differ substantially from the actual solubility, as would predictions of ionic concentrations. In general, four situations explain why the solubility of a compound may be other than expected: ion pair formation, the incomplete dissociation of molecular solutes, the formation of complex ions, and changes in pH.
Ion-Pair Formation
An ion pair consists of a cation and an anion that are in intimate contact in solution, rather than separated by solvent (Figure \(\PageIndex{1}\)). The ions in an ion pair are held together by the same attractive electrostatic force in ionic solids. As a result, the ions in an ion pair migrate as a single unit, whose net charge is the sum of the charges on the ions. In many ways, we can view an ion pair as a species intermediate between the ionic solid (in which each ion participates in many cation–anion interactions that hold the ions in a rigid array) and the completely dissociated ions in solution (where each is fully surrounded by water molecules and free to migrate independently).
As illustrated for calcium sulfate in the following equation, a second equilibrium must be included to describe the solubility of salts that form ion pairs:
\[\mathrm{CaSO_4(s)}\rightleftharpoons \underbrace{\mathrm{Ca^{2+}}\cdot \mathrm{SO_4^{2-}(aq)}}_ {\textrm{ion pair}} \rightleftharpoons \mathrm{Ca^{2+}(aq)}+\mathrm{SO_4^{2-}(aq)} \label{17.5.3.1}\]
The ion pair is represented by the symbols of the individual ions separated by a dot, which indicates that they are associated in solution. The formation of an ion pair is a dynamic process, just like any other equilibrium, so a particular ion pair may exist only briefly before dissociating into the free ions, each of which may later associate briefly with other ions.
Ion-pair formation can have a major effect on the measured solubility of a salt. For example, the measured Ksp for calcium sulfate is 4.93 × 10−5 at 25°C. The solubility of CaSO4 should be 7.02 × 10−3 M if the only equilibrium involved were as follows:
\[CaSO_{4(s)} \rightleftharpoons Ca^{2+}_{(aq)} + SO^{2−}_{4(aq)} \label{17.5.2}\]
In fact, the experimentally measured solubility of calcium sulfate at 25°C is 1.6 × 10−2 M, almost twice the value predicted from its Ksp. The reason for the discrepancy is that the concentration of ion pairs in a saturated CaSO4 solution is almost as high as the concentration of the hydrated ions. Recall that the magnitude of attractive electrostatic interactions is greatest for small, highly charged ions. Hence ion pair formation is most important for salts that contain M2+ and M3+ ions, such as Ca2+ and La3+, and is relatively unimportant for salts that contain monopositive cations, except for the smallest, Li+. We therefore expect a saturated solution of CaSO4 to contain a high concentration of ion pairs and its solubility to be greater than predicted from its Ksp.
The formation of ion pairs increases the solubility of a salt.
Incomplete Dissociation
A molecular solute may also be more soluble than predicted by the measured concentrations of ions in solution due to incomplete dissociation. This is particularly common with weak organic acids. Although strong acids (HA) dissociate completely into their constituent ions (H+ and A−) in water, weak acids such as carboxylic acids do not (Ka = 1.5 × 10−5). However, the molecular (undissociated) form of a weak acid (HA) is often quite soluble in water; for example, acetic acid (CH3CO2H) is completely miscible with water. Many carboxylic acids, however, have only limited solubility in water, such as benzoic acid (C6H5CO2H), with Ka = 6.25 × 10−5. Just as with calcium sulfate, we need to include an additional equilibrium to describe the solubility of benzoic acid:
\[ C_6H_5CO_2H_{(s)} \rightleftharpoons C_6H_5CO_2H_{(aq)} \rightleftharpoons C_6H_5CO^−_{2(aq)} + H^+_{(aq)} \label{17.5.3}\]
In a case like this, measuring only the concentration of the ions grossly underestimates the total concentration of the organic acid in solution. In the case of benzoic acid, for example, the pH of a saturated solution at 25°C is 2.85, corresponding to [H+] = [C6H5CO2−] = 1.4 × 10−3 M. The total concentration of benzoic acid in the solution, however, is 2.8 × 10−2 M. Thus approximately 95% of the benzoic acid in solution is in the form of hydrated neutral molecules—\(C_6H_5CO_2H_{(aq)}\)—and only about 5% is present as the dissociated ions (Figure \(\PageIndex{2}\)).
Although ion pairs, such as Ca2+·SO42−, and undissociated electrolytes, such as C6H5CO2H, are both electrically neutral, there is a major difference in the forces responsible for their formation. Simple electrostatic attractive forces between the cation and the anion hold the ion pair together, whereas a polar covalent O−H bond holds together the undissociated electrolyte.
Incomplete dissociation of a molecular solute that is miscible with water can increase the solubility of the solute.
Complex Ion Formation
Previously, you learned that metal ions in aqueous solution are hydrated—that is, surrounded by a shell of usually four or six water molecules. A hydrated ion is one kind of a complex ion (or, simply, complex), a species formed between a central metal ion and one or more surrounding ligands, molecules or ions that contain at least one lone pair of electrons, such as the [Al(H2O)6]3+ ion.
A complex ion forms from a metal ion and a ligand because of a Lewis acid–base interaction. The positively charged metal ion acts as a Lewis acid, and the ligand, with one or more lone pairs of electrons, acts as a Lewis base. Small, highly charged metal ions, such as Cu2+ or Ru3+, have the greatest tendency to act as Lewis acids, and consequently, they have the greatest tendency to form complex ions.
As an example of the formation of complex ions, consider the addition of ammonia to an aqueous solution of the hydrated Cu2+ ion {[Cu(H2O)6]2+}. Because it is a stronger base than H2O, ammonia replaces the water molecules in the hydrated ion to form the [Cu(NH3)4(H2O)2]2+ ion. Formation of the [Cu(NH3)4(H2O)2]2+ complex is accompanied by a dramatic color change, as shown in Figure \(\PageIndex{1}\). The solution changes from the light blue of [Cu(H2O)6]2+ to the blue-violet characteristic of the [Cu(NH3)4(H2O)2]2+ ion.
The Formation Constant
The replacement of water molecules from [Cu(H2O)6]2+ by ammonia occurs in sequential steps. Omitting the water molecules bound to Cu2+ for simplicity, we can write the equilibrium reactions as follows:
\\ \mathrm{[Cu(NH_3)]^{2+}_{(aq)}}+\mathrm{NH_{3(aq)}}&\rightleftharpoons\mathrm{[Cu(NH_3)_2]^{2+}_{(aq)}}\hspace{3mm}K_2
\\ \mathrm{[Cu(NH_3)_2]^{2+}_{(aq)}}+\mathrm{NH_{3(aq)}}&\rightleftharpoons\mathrm{[Cu(NH_3)_3]^{2+}_{(aq)}}\hspace{3mm}K_3
\\ \mathrm{[Cu(NH_3)_3]^{2+}_{(aq)}}+\mathrm{NH_{3(aq)}}&\rightleftharpoons \mathrm{[Cu(NH_3)_4]^{2+}_{(aq)}}\hspace{3mm}K_4 \end{align} \label{17.3.1}\]
The sum of the stepwise reactions is the overall equation for the formation of the complex ion: The hydrated Cu2+ ion contains six H2O ligands, but the complex ion that is produced contains only four \(NH_3\) ligands, not six.
\[Cu^{2+}_{(aq)} + 4NH_{3(aq)} \rightleftharpoons [Cu(NH_3)_4]^{2+}_{(aq)} \label{17.3.2}\]
The equilibrium constant for the formation of the complex ion from the hydrated ion is called the formation constant (Kf). The equilibrium constant expression for Kf has the same general form as any other equilibrium constant expression. In this case, the expression is as follows:
\[K_\textrm f=\dfrac{\left[[\mathrm{Cu(NH_3)_4}]^{2+}\right]}{[\mathrm{Cu^{2+}}][\mathrm{NH_3}]^4}=2.1\times10^{13}=K_1K_2K_3K_4\label{17.3.3}\]
The formation constant (Kf) has the same general form as any other equilibrium constant expression.
Water, a pure liquid, does not appear explicitly in the equilibrium constant expression, and the hydrated Cu2+(aq) ion is represented as Cu2+ for simplicity. As for any equilibrium, the larger the value of the equilibrium constant (in this case, Kf), the more stable the product. With Kf = 2.1 × 1013, the [Cu(NH3)4(H2O)2]2+ complex ion is very stable. The formation constants for some common complex ions are listed in Table \(\PageIndex{1}\).
| Complex Ion | Equilibrium Equation | Kf | |
|---|---|---|---|
| *Reported values are overall formation constants.
Source: Data from Lange’s Handbook of Chemistry, 15th ed. (1999). |
|||
| Ammonia Complexes | [Ag(NH3)2]+ | Ag+ + 2NH3 ⇌ [Ag(NH3)2]+ | 1.1 × 107 |
| [Cu(NH3)4]2+ | Cu2+ + 4NH3 ⇌ [Cu(NH3)4]2+ | 2.1 × 1013 | |
| [Ni(NH3)6]2+ | Ni2+ + 6NH3 ⇌ [Ni(NH3)6]2+ | 5.5 × 108 | |
| Cyanide Complexes | [Ag(CN)2]− | Ag+ + 2CN− ⇌ [Ag(CN)2]− | 1.1 × 1018 |
| [Ni(CN)4]2− | Ni2+ + 4CN− ⇌ [Ni(CN)4]2− | 2.2 × 1031 | |
| [Fe(CN)6]3− | Fe3+ + 6CN− ⇌ [Fe(CN)6]3− | 1 × 1042 | |
| Hydroxide Complexes | [Zn(OH)4]2− | Zn2+ + 4OH− ⇌ [Zn(OH)4]2− | 4.6 × 1017 |
| [Cr(OH)4]− | Cr3+ + 4OH− ⇌ [Cr(OH)4]− | 8.0 × 1029 | |
| Halide Complexes | [HgCl4]2− | Hg2+ + 4Cl− ⇌ [HgCl4]2− | 1.2 × 1015 |
| [CdI4]2− | Cd2+ + 4I ⇌ [CdI4]2− | 2.6 × 105 | |
| [AlF6]3− | Al3+ + 6F− ⇌ [AlF6]3− | 6.9 × 1019 | |
| Other Complexes | [Ag(S2O3)2]3− | Ag+ + 2S2O32− ⇌ [Ag(S2O3)2]3− | 2.9 × 1013 |
| [Fe(C2O4)3]3− | Fe3+ + 3C2O42− ⇌ [Fe(C2O4)3]3− | 2.0 × 1020 | |