Problem 165 - A Complete Resource Book in Chemistry for JEE Main

Question

What is the minimum \(\mathrm{pH}\) required to prevent the precipitation of \(\mathrm{ZnS}\) in a solution that is \(0.01 \mathrm{M} \mathrm{ZnCl}_{2}\) and saturated with \(0.10 \mathrm{M} \mathrm{H}_{2} \mathrm{~S} ?\) [Given: \(\mathrm{K}_{\mathrm{sp}}\) of \(\mathrm{ZnS}=10^{-21}, \mathrm{~K}_{2_{1}} \times \mathrm{K}_{\mathrm{a}_{2}}\) of \(\left.\mathrm{H}_{2} \mathrm{~S}=10^{-20}\right]\) (a) 4 (b) 3 (c) 2 (d) 1

Step-by-Step Solution

Answer

The minimum pH to prevent ZnS precipitation is greater than 12.

Step 1: Write the equilibrium expressions

The dissolution of zinc sulfide, \( \mathrm{ZnS} \), can be represented by the equation: \( \mathrm{ZnS} \rightleftharpoons \mathrm{Zn}^{2+} + \mathrm{S}^{2-} \). The solubility product \( \mathrm{K}_{\mathrm{sp}} \) is given by \( \mathrm{K}_{\mathrm{sp}} = [\mathrm{Zn}^{2+}][\mathrm{S}^{2-}] \).The dissociation of \( \mathrm{H}_2\mathrm{S} \) is: \( \mathrm{H}_2\mathrm{S} \rightleftharpoons \mathrm{H}^+ + \mathrm{HS}^- \) and \( \mathrm{HS}^- \rightleftharpoons \mathrm{H}^+ + \mathrm{S}^{2-} \). The products of equilibrium constants for these reactions are given as \( \mathrm{K}_{a_1} \times \mathrm{K}_{a_2} = 10^{-20} \).

Step 2: Express [S2-] in terms of pH

For \( \mathrm{S}^{2-} \) derived from \( \mathrm{H}_2\mathrm{S} \) dissociation, \( [\mathrm{S}^{2-}] = \frac{[\mathrm{HS}^-][\mathrm{H}^+]}{\mathrm{K}_{a_2}} \approx \frac{[\mathrm{H}^+]^2}{10^{-20}} \) because the intermediate \( [\mathrm{HS}^-] = \frac{[\mathrm{H}_2\mathrm{S}][\mathrm{H}^+]}{10^{-7}} \) can be approximated as being small for a saturated solution.

Step 3: Set up the equation for precipitation prevention

To prevent precipitation, \( [\mathrm{Zn}^{2+}][\mathrm{S}^{2-}] < \mathrm{K}_{\mathrm{sp}} \). Substituting \( [\mathrm{Zn}^{2+}] = 0.01 \) into this gives: \( 0.01 \times \frac{[\mathrm{H}^+]^2}{10^{-20}} < 10^{-21} \).

Step 4: Solve for [H+]

Solving the inequality \( 0.01 \times \frac{[\mathrm{H}^+]^2}{10^{-20}} < 10^{-21} \), we find: \[ [\mathrm{H}^+]^2 < 10^{-3} \cdot 10^{-21} \]\[ [\mathrm{H}^+]^2 < 10^{-24} \]Thus, \( [\mathrm{H}^+] < 10^{-12} \).

Step 5: Calculate the pH

Given \( [\mathrm{H}^+] < 10^{-12} \), the corresponding \( \mathrm{pH} > 12 \). Thus, a pH greater than 12 is required to prevent precipitation.

Key Concepts

Ksp (Solubility Product Constant)

The Solubility Product Constant, often abbreviated as \( K_{sp} \), is a crucial concept when it comes to understanding solubility and precipitation of compounds in a solution. \( K_{sp} \) is a type of equilibrium constant, specifically for the dissolution of slightly soluble salts. It represents the peak concentrations of ions that can be present in a saturated solution before precipitation occurs. In essence, \( K_{sp} \) helps predict whether a salt will dissolve or precipitate in a particular solute.

For example, in the dissolution of zinc sulfide \( (\mathrm{ZnS}) \), the equilibrium expression can be written as:

• \( \mathrm{ZnS} \rightleftharpoons \mathrm{Zn}^{2+} + \mathrm{S}^{2-} \)
• The \( K_{sp} = [\mathrm{Zn}^{2+}][\mathrm{S}^{2-}] \)

This formula helps us calculate the extent to which Zinc and Sulfide ions can coexist in solution without forming a solid precipitate. The lower the \( K_{sp} \), the less soluble the compound is. In our exercise, we use the \( K_{sp} \) value of \( 10^{-21} \) for ZnS to determine the concentration limits that prevent the deposition of ZnS from the solution.

Equilibrium Expressions

Equilibrium expressions are fundamental in chemistry as they relate the concentrations of reactants and products at equilibrium. They're used to calculate the extent of reactions and to gauge whether the system has reached an equilibrium state or if reactions will proceed toward forming products or reactants.

For our problem with zinc sulfide and hydrogen sulfide, equilibrium expressions are used to show how different substances dissociate and how their ions relate:

• \( \mathrm{ZnS} \rightleftharpoons \mathrm{Zn}^{2+} + \mathrm{S}^{2-} \)
• \( \mathrm{H}_2\mathrm{S} \rightleftharpoons \mathrm{H}^+ + \mathrm{HS}^- \)
• \( \mathrm{HS}^- \rightleftharpoons \mathrm{H}^+ + \mathrm{S}^{2-} \)

The equilibrium expressions help us determine the concentrations of sulfur ions \( \mathrm{S}^{2-} \) impacting dissociation and precipitation. By knowing the concentration of hydrogen ions \( (\mathrm{H}^+) \), we can set up inequalities to avoid precipitation, essentially controlling the reaction's direction and outcome.

Chemical Precipitation

Chemical precipitation is the process where dissolved ions in a solution form insoluble compounds, resulting in solid particles separating from the liquid. Precipitation occurs when the product of the concentrations of the ions in solution exceeds the \( K_{sp} \) of the compound.

In our scenario with \( \mathrm{ZnS} \), we aim to avoid precipitation by controlling conditions to keep the ion product below the solubility product constant. Here's how that principle guides the solution:

• Ensure \( [\mathrm{Zn}^{2+}][\mathrm{S}^{2-}] < K_{sp} \)
• Leads to \( 0.01 \times \frac{[\mathrm{H}^+]^2}{10^{-20}} < 10^{-21} \)

In this expression, maintaining \( [\mathrm{H}^+] \) at a particular level prevents the sulfide ion concentration from reaching the threshold that causes \( \mathrm{ZnS} \) to precipitate. Knowing how to manipulate conditions to govern precipitation is a significant aspect of controlling reactions in industrial and laboratory settings.

Dissociation Constant (Ka)

The dissociation constant, denoted as \( K_a \), measures the propensity of a compound to dissociate into ions in a solution. For weak acids, such as hydrogen sulfide \( \mathrm{H}_2\mathrm{S} \), it provides insight into the degree of dissociation and stability of the substances in solution.

The hydrogen sulfide has two dissociation constants because it can lose two hydrogen ions, each step having its equilibrium:

• \( \mathrm{H}_2\mathrm{S} \rightleftharpoons \mathrm{H}^+ + \mathrm{HS}^- \)
• \( \mathrm{HS}^- \rightleftharpoons \mathrm{H}^+ + \mathrm{S}^{2-} \)

Given as \( \mathrm{K}_{a1} \times \mathrm{K}_{a2} = 10^{-20} \), the dissociation constants are pivotal for working out the sulfur ion concentrations and directly influence the calculations for \( pH \).

Understanding \( K_a \) allows us to comprehend how much \( \mathrm{H}_2\mathrm{S} \) will dissociate into its ions in a solution, ultimately affecting the solution's acidity and the conditions under which precipitation will occur. By controlling the \( pH \), we manage these dissociations, effectively preventing unwanted precipitation of \( \mathrm{ZnS} \).