Question

Suppose that some dipositive cation, \(M^{2+},\) is able to form a complex ion with a ligand, \(L\), by the following balanced equation: \(M^{2+}+2 L \rightleftharpoons M(\mathrm{~L})_{2}^{2+} .\) The cation also forms a sparingly soluble salt, \(M \mathrm{Cl}_{2}\). In which of the following circumstances would a given quantity of ligand be more able to bring larger quantities of the salt into solution? Explain and justify the calculation involved: (a) \(K_{\text {form }}=1 \times 10^{2}\) and \(K_{\text {sp }}=1 \times 10^{-15}\), (b) \(K_{\text {form }}=1 \times 10^{10}\) and \(K_{\mathrm{sp}}=1 \times 10^{-20}\).

Short answer

The ligand would be more capable of dissolving larger quantities of the salt in case (b) because the complex ion is significantly more stable (higher Kform), and the salt is significantly less soluble (lower Ksp) compared to case (a).

Step-by-step solution

Understanding the Problem

We need to determine in which case the ligand, L, is more capable of dissolving the salt, MCl2, based on the formation constant (Kform) of the complex ion and the solubility product constant (Ksp) of the salt.

Analyze the Relationship between Kform and Ksp

The larger the formation constant (Kform), the more stable the complex ion, meaning the ligand L will be more effective at forming the complex ion over precipitating the salt. Conversely, the smaller the solubility product constant (Ksp), the less soluble the salt is, meaning less of the salt will dissolve to form the cation, M2+.

Comparing Case (a) and Case (b)

For case (a), Kform is 1 x 10^2 and Ksp is 1 x 10^-15, while for case (b), Kform is 1 x 10^10 and Ksp is 1 x 10^-20. A higher Kform and a lower Ksp suggest that more salt will be able to dissolve, given an adequate amount of ligand. Thus, we need to compare the magnitude of increase in Kform and the decrease in Ksp between cases (a) and (b).

Evaluate the Effectiveness of the Ligand

By comparing the values of Kform and Ksp, it is clear that case (b) has a significantly higher Kform, which indicates a stronger tendency to form the complex ion, and a significantly lower Ksp, which indicates a lower solubility of the salt independently. Therefore, in case (b), the ligand will be more effective at solubilizing the salt due to the greatly increased stability of the complex ion and decreased salt solubility.

Key concepts

Formation Constant (Kform)

The formation constant, often denoted as K_form, is a special type of equilibrium constant that measures the stability of a complex ion in solution. When a metal ion associates with one or more ligands to form a complex ion, this association is governed by an equilibrium between the free ions and the complex ion.

In the given exercise, the complex ion is formed through the reaction between a metal cation, M²⁺, and a ligand L, resulting in the equation: M²⁺ + 2L ⇌ ML₂²⁺. The K_form quantifies the extent of this reaction. A higher K_form means the reaction favors the formation of the complex ion; conversely, a lower K_form favors the dissociation back into the cation and ligands.

Students might find it beneficial to think of K_form as the 'stickiness' of the metal cation to the ligand - the higher the constant, the 'stickier' the interaction, leading to more complex ion and less free cation in the solution. This concept is crucial in understanding why certain quantities of ligand can dissolve more of the sparingly soluble salt, as seen in the exercise.

Solubility Product Constant (Ksp)

The solubility product constant, or K_sp, refers to the equilibrium constant for the dissolution of a sparingly soluble ionic compound. It represents the maximum product of the ionic concentrations of the dissolved ions that can exist in a saturated solution before the compound starts precipitating. The lower the value of K_sp, the less soluble the compound is.

In our problem, the salt MCl₂ has a K_sp that indicates how much of it can dissolve in water before it reaches saturation. If we try to dissolve more than what the K_sp allows, excess salt will remain undissolved as a precipitate. It's essential for students to grasp that the K_sp is not about how fast a salt dissolves, but rather how much can dissolve. Understanding the interplay between K_form and K_sp is key to explaining why different conditions affect the salt's solubility in the presence of a ligand.

Ligand Complexation

Ligand complexation is the process by which ligands, which are ions or molecules with lone pair electrons, form a chemical bond to a central metal ion, creating a complex ion. In the exercise, the ligand L is complexing with the metal cation M²⁺ to form ML₂²⁺. The nature and strength of the complexation are influenced by various factors such as the charge, size, and electron configuration of the metal ion, as well as the electronic properties and geometry of the ligand.

Types of Ligands

Ligands can be classified based on their 'denticity' or how many 'teeth' - points of attachment - they have to bind to a metal ion. Monodentate ligands, like L in our exercise, attach at a single point. Polydentate ligands can attach at multiple points, forming very stable complexes. The type and strength of the complex formed have direct implications on the solubility of salts, as stronger complexes can effectively 'pull' metal ions into solution from their salts.

Salt Solubility

Salt solubility is a measure of how much of a salt can dissolve in a given solvent, such as water, to form a solution. Solubility is not just a constant value; it can be influenced by temperature, pressure, the presence of other ions in solution, and, as in the exercise, the formation of complex ions.

For salts like MCl₂ that are sparingly soluble, their solubility can be significantly affected when a ligand capable of complexing with the metal ion is present. The formation of a complex ion increases the overall solubility of the salt by continuously removing the free metal cation, M²⁺, from the solution, thus driving the dissolution process further to re-establish equilibrium. In essence, the addition of a suitable ligand can 'encourage' more of the sparingly soluble salt to go into solution. This is one of the crucial insights that helps students understand the relationship between complex ion formation, ligand complexation, and salt solubility.