Problem 248 - A Complete Resource Book in Chemistry for JEE Main

Question

Clothings washed in water that has a manganese concentration exceeding \(0.1 \mathrm{mg} \mathrm{L}^{-1}\left(1.8 \times 10^{-6} \mathrm{M}\right)\) may be stained by the manganese, but the amount of \(\mathrm{Mn}^{2+}\) in water can be reduced by adding base. If a laundry wishes to add a buffer to keep the \(\mathrm{pH}\) high enough to precipitate manganese as the hydroxide, \(\mathrm{Mn}(\mathrm{OH})_{2}\), with \(\mathrm{pH}\) required to keep \(\left[\mathrm{Mn}^{2+}\right]\) equal to \(1.8 \times 10^{-6} \mathrm{M}\) is \(2 \mathrm{x}\). Find \(\mathrm{x}\) (nearest integral value). \(\mathrm{K}_{\mathrm{sp}}\) of \(\mathrm{Mn}(\mathrm{OH})_{2}\) is \(4.5 \times 10^{-14}\).

Step-by-Step Solution

Answer

x is approximately 5.

Step 1: Write the Expression for Ksp

The solubility product constant (\(\mathrm{K}_{\mathrm{sp}}\)) gives the equilibrium between a solid and its ions in solution. We know the equation for manganese hydroxide is \(\mathrm{Mn(OH)}_2 \rightleftharpoons \mathrm{Mn}^{2+} + 2\mathrm{OH}^-\). Therefore, the \(\mathrm{K}_{\mathrm{sp}}\) expression is:\[\mathrm{K}_{\mathrm{sp}} = [\mathrm{Mn}^{2+}][\mathrm{OH}^-]^2\]

Step 2: Substitute Known Values into Ksp Expression

The problem provides us with \(\mathrm{K}_{\mathrm{sp}} = 4.5 \times 10^{-14}\) and \([\mathrm{Mn}^{2+}] = 1.8 \times 10^{-6} \mathrm{M}\). Substitute these values into the \(\mathrm{K}_{\mathrm{sp}}\) expression:\[4.5 \times 10^{-14} = (1.8 \times 10^{-6})[\mathrm{OH}^-]^2\]

Step 3: Solve for [OH-]

Now, solve for \([\mathrm{OH}^-]^2\) by isolating \([\mathrm{OH}^-]\):\[[\mathrm{OH}^-]^2 = \frac{4.5 \times 10^{-14}}{1.8 \times 10^{-6}}\]Compute the division to find:\[[\mathrm{OH}^-]^2 = 2.5 \times 10^{-8}\]Now take the square root of both sides to find \([\mathrm{OH}^-]\):\[[\mathrm{OH}^-] = \sqrt{2.5 \times 10^{-8}} = 5 \times 10^{-5}\]

Step 4: Find pOH and then pH

The \([\mathrm{OH}^-]\) concentration helps us find \(\mathrm{pOH}\):\[\mathrm{pOH} = -\log(5 \times 10^{-5})\]Calculate \(\mathrm{pOH}\) using the logarithm:\[\mathrm{pOH} = 4.3\]Now, use \(\mathrm{pH} + \mathrm{pOH} = 14\) to find \(\mathrm{pH}\):\[\mathrm{pH} = 14 - 4.3 = 9.7\]

Step 5: Find x in the pH Expression

According to the problem, the required \(\mathrm{pH}\) is expressed as \(2x\). Substitute the \(\mathrm{pH}\) value we found into the equation to solve for \(x\):\[2x = 9.7\]\[x = \frac{9.7}{2} = 4.85\]Rounding to the nearest integer, we find \(x = 5\).

Key Concepts

Manganese Hydroxide Solubility

Manganese hydroxide, denoted as \( \mathrm{Mn(OH)}_2 \), is a compound that can dissolve in water, especially depending on the conditions like pH. The solubility of manganese hydroxide is governed by the solubility product constant, \( \mathrm{K}_{\mathrm{sp}} \), which provides insight into how much of the compound can dissolve before it starts precipitating from the solution. In this specific problem, manganese ions precipitate as \( \mathrm{Mn(OH)}_2 \) to reduce concentration levels, preventing stains on laundry. When calculating the solubility, it's essential to maintain a balance between the solid \( \mathrm{Mn(OH)}_2 \) and its ions in solution—manganese \( \mathrm{Mn}^{2+} \) and hydroxide \( \mathrm{OH}^- \). Adjusting the pH can tip this balance, causing the manganese ions to pair with hydroxide ions and form the solid compound, effectively reducing the manganese concentration in the water.

pH and pOH Relationship

The concept of pH and pOH is central to understanding acidic and basic solutions. The pH scale measures how acidic or basic a solution is, with lower values representing more acidic conditions and higher values indicating basic or alkaline conditions. On the other hand, pOH measures the concentration of hydroxide ions in a solution. Both scales are interrelated by the equation:

• \( \mathrm{pH} + \mathrm{pOH} = 14 \)

This relationship is pivotal when determining the conditions necessary to cause precipitation in a solution, like forming \( \mathrm{Mn(OH)}_2 \). In this instance, once we find the concentration of \( \mathrm{OH}^- \), we can easily calculate the pOH. From there, deriving the pH becomes straightforward. A pH level that falls within basic ranges ensures enough hydroxide ions are present to react with \( \mathrm{Mn}^{2+} \) ions, leading to precipitation.

Buffer Solution Chemistry

Buffers are an essential element in solution chemistry, capable of resisting significant pH changes even when small amounts of acids or bases are introduced. Typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid, buffers help stabilize the pH level of a solution. This principle is vital in the application of maintaining an environment where manganese will crystallize out of the water solution. In our exercise, by adding a buffer that maintains a high enough pH level, more \( \mathrm{OH}^- \) ions are present to precipitate \( \mathrm{Mn(OH)}_2 \). This means that even if the pH naturally tries to drop due to interacting with other substances, the buffer ensures a consistent basic environment conducive to precipitating manganese hydroxide.

Solubility Product Constant

The solubility product constant, \( \mathrm{K}_{\mathrm{sp}} \), is a crucial concept in predicting the solubility of ionic compounds. It describes the product of the molar concentrations of the constituent ions, each raised to the power of its coefficient in the balanced chemical equation. For manganese hydroxide, this is represented as:

• \[ \mathrm{K}_{\mathrm{sp}} = [\mathrm{Mn}^{2+}][\mathrm{OH}^-]^2 \]

Knowing \( \mathrm{K}_{\mathrm{sp}} \) is essential for determining how much of a substance will dissolve in water before excess material begins to precipitate. In the problem, the known \( \mathrm{K}_{\mathrm{sp}} \) allows us to calculate the concentration of hydroxide ions required for precipitation. By substituting known values into the \( \mathrm{K}_{\mathrm{sp}} \) formula, we derive the necessary ion concentrations to achieve precipitation, guiding us toward the optimal conditions for reducing manganese levels.