Problem 106 - Chemistry The Central Science

Question

The solubility product for \(\mathrm{Zn}(\mathrm{OH})_{2}\) is \(3.0 \times 10^{-16} .\) The formation constant for the hydroxo complex, \(\mathrm{Zn}(\mathrm{OH})_{4}^{2-},\) is \(4.6 \times 10^{17}\). What concentration of \(\mathrm{OH}^{-}\) is required to dissolve 0.015 mol of \(\mathrm{Zn}(\mathrm{OH})_{2}\) in a liter of solution?

Step-by-Step Solution

Answer

The concentration of \(\mathrm{OH}^-\) required to dissolve 0.015 mol of \(\mathrm{Zn}(\mathrm{OH})_{2}\) in a liter of solution is approximately \(4.81 \times 10^{-3} \ \text{M}\).

Step 1: Write the solubility product expression for \(\mathrm{Zn}(\mathrm{OH})_{2}\)

The solubility product expression for \(\mathrm{Zn}(\mathrm{OH})_{2}\) is given by: \[K_{s} = [\mathrm{Zn^{2+}}][\mathrm{OH}^-]^2\] where \(K_s = 3.0 \times 10^{-16}\).

Step 2: Write the formation constant expression for the hydroxo complex \(\mathrm{Zn}(\mathrm{OH})_{4}^{2-}\)

The formation constant expression for the hydroxo complex \(\mathrm{Zn}(\mathrm{OH})_{4}^{2-}\) is given by: \[K_f = \frac{[\mathrm{Zn}(\mathrm{OH})_{4}^{2-}]}{[\mathrm{Zn^{2+}}][\mathrm{OH}^-]^4}\] where \(K_f = 4.6 \times 10^{17}\).

Step 3: Relate the solubility product and formation constant expressions

From the solubility product expression, we can write the concentration of \(\mathrm{Zn^{2+}}\) as: \[[\mathrm{Zn^{2+}}] = \frac{K_s}{[\mathrm{OH}^-]^2}\] Substitute this expression for the concentration of \(\mathrm{Zn^{2+}}\) into the formation constant expression and solve for the concentration of \(\mathrm{Zn}(\mathrm{OH})_{4}^{2-}\): \[[\mathrm{Zn}(\mathrm{OH})_{4}^{2-}] = K_f [\mathrm{Zn^{2+}}][\mathrm{OH}^-]^4 = K_f \frac{K_s}{[\mathrm{OH}^-]^2}[\mathrm{OH}^-]^4\]

Step 4: Calculate the concentration of \(\mathrm{Zn}(\mathrm{OH})_{4}^{2-}\) required to dissolve 0.015 mol of \(\mathrm{Zn}(\mathrm{OH})_{2}\) in a liter of solution

To dissolve 0.015 mol of \(\mathrm{Zn}(\mathrm{OH})_{2}\) in a liter of solution, the moles of \(\mathrm{Zn^{2+}}\) will be equal to the moles of \(\mathrm{Zn}(\mathrm{OH})_{4}^{2-}\), so: \[0.015 = [\mathrm{Zn}(\mathrm{OH})_{4}^{2-}] = K_f \frac{K_s}{[\mathrm{OH}^-]^2}[\mathrm{OH}^-]^4\]

Step 5: Solve for the concentration of \(\mathrm{OH}^-\)

Now, we will solve for the concentration of \(\mathrm{OH}^-\): \[[\mathrm{OH}^-]^6 = \frac{K_s K_f}{0.015}\] \[[\mathrm{OH}^-] = \sqrt[6]{\frac{K_s K_f}{0.015}}\] \[[\mathrm{OH}^-] = \sqrt[6]{\frac{(3.0 \times 10^{-16})(4.6 \times 10^{17})}{0.015}}\] \[[\mathrm{OH}^-] \approx 4.81 \times 10^{-3} \ \text{M}\] Therefore, the concentration of \(\mathrm{OH}^-\) required to dissolve 0.015 mol of \(\mathrm{Zn}(\mathrm{OH})_{\text{2}}\) in a liter of solution is approximately \(4.81 \times 10^{-3} \ \text{M}\).

Key Concepts

Solubility Product

The solubility product, often abbreviated as \( K_s \), is a critical concept in understanding how sparingly soluble compounds dissolve in water. It quantifies the extent to which a compound dissolves and separates into its constituent ions.
In the case of \( \mathrm{Zn}(\mathrm{OH})_2 \), a slightly soluble salt, the solubility product expression is written as:

• \[ K_s = [\mathrm{Zn^{2+}}][\mathrm{OH}^-]^2 \]

For \( \mathrm{Zn}(\mathrm{OH})_2 \), the value is given as \( 3.0 \times 10^{-16} \), indicating very low solubility in water.
This small value hints at the limited extent to which the zinc hydroxide dissociates into zinc and hydroxide ions in solution, which is a characteristic feature of compounds with low solubility products. Understanding how to work with \( K_s \) allows us to predict and manipulate solubility in various chemical scenarios.

Formation Constant

The formation constant, represented as \( K_f \), describes the equilibrium between metal ions and ligands forming a complex ion. It plays a crucial role in the context of complex ion equilibria.
For the hydroxo complex \( \mathrm{Zn}(\mathrm{OH})_{4}^{2-} \), the formation constant expression is:

• \[ K_f = \frac{[\mathrm{Zn}(\mathrm{OH})_4^{2-}]}{[\mathrm{Zn^{2+}}][\mathrm{OH}^-]^4} \]

Given as \( 4.6 \times 10^{17} \), this large number signifies a strong tendency for \( \mathrm{Zn^{2+}} \) and hydroxide ions to combine and form \( \mathrm{Zn}(\mathrm{OH})_{4}^{2-} \).
When \( K_f \) is large, it indicates the formation of the complex ion is favored, which affects the overall solubility of compounds with such complex ions in solution.

Complex Ion Equilibria

Complex ion equilibria involve the formation and dissociation of complex ions. These ions consist of a central metal ion bonded to surrounding ligands, such as hydroxide ions in this context.
Complex ions can often increase solubility by shifting the equilibria in favor of dissolving more solid into the solution.
For \( \mathrm{Zn}(\mathrm{OH})_{4}^{2-} \), the complex ion favored by the large \( K_f \) value shifts the equilibrium.
This means more \( \mathrm{Zn}(\mathrm{OH})_2 \) dissolves as \( \mathrm{Zn}(\mathrm{OH})_{4}^{2-} \) forms, illustrating how the presence of complex ions dramatically alters solubility behavior.
It's particularly important when predicting or calculating solubility in systems where complex formation plays a major role.

Hydroxide Ion Concentration

The hydroxide ion concentration is a crucial factor affecting both solubility and complex ion formation in solutions containing metal hydroxides.
In this exercise, understanding the concentration of \( \mathrm{OH}^- \) ions is key to dissolving \( \mathrm{Zn}(\mathrm{OH})_2 \).
Using the expressions for \( K_s \) and \( K_f \), we can derive the needed \( \mathrm{OH}^- \) concentration. This is done by manipulating both expressions together:

• \[ [\mathrm{OH}^-]^6 = \frac{K_s K_f}{0.015} \]
• \[ [\mathrm{OH}^-] = \sqrt[6]{\frac{3.0 \times 10^{-16} \times 4.6 \times 10^{17}}{0.015}} \]

The resulting concentration of \( \mathrm{OH}^- \) required is approximately \( 4.81 \times 10^{-3} \ \mathrm{M} \).
This precision in calculation ensures that the hydroxide ion concentration is adequate to drive the dissolution of \( \mathrm{Zn}(\mathrm{OH})_2 \) through complex ion formation.