Problem 173 - A Complete Resource Book in Chemistry for JEE Main

Question

Solid \(\mathrm{AgNO}_{3}\) is added slowly to a buffer solution of \(\mathrm{pH}\) \(=10\) to precipitate \(\mathrm{AgOH}\). The \(\left[\mathrm{Ag}^{+}\right]\)concentration in the solution is \(\left[\mathrm{K}_{\mathrm{sp}}(\mathrm{AgOH})=10^{-10}\right]\) (a) \(10^{-4} \mathrm{M}\) (b) \(10^{-5} \mathrm{M}\) (c) \(10^{-6} \mathrm{M}\) (d) \(10^{-7} \mathrm{M}\)

Step-by-Step Solution

Answer

The concentration of \([\mathrm{Ag^+}]\) is \(10^{-6}\,\mathrm{M}\) thus the answer is (c).

Step 1: Understand the Equation

Silver hydroxide, \(\mathrm{AgOH}\), dissolves according to the equation: \(\mathrm{AgOH}_{(s)} \rightleftharpoons \mathrm{Ag^+}_{(aq)} + \mathrm{OH^-}_{(aq)}\). From this equation, the solubility product is given by \(K_{sp} = [\mathrm{Ag^+}][\mathrm{OH^-}]\).

Step 2: Calculate the Hydroxide Ion Concentration

At \(\mathrm{pH} = 10\), \(\mathrm{pOH} = 14 - \mathrm{pH} = 4\). The hydroxide ion concentration, \([\mathrm{OH^-}]\), can be calculated as \([\mathrm{OH^-}] = 10^{-\mathrm{pOH}} = 10^{-4}\,\mathrm{M}\).

Step 3: Apply the Solubility Product Formula

Using the solubility product expression, \(K_{sp} = [\mathrm{Ag^+}][\mathrm{OH^-}]\), and substituting the known values: \(10^{-10} = [\mathrm{Ag^+}](10^{-4}\,\mathrm{M})\).

Step 4: Solve for Silver Ion Concentration

Rearrange the solubility product equation to find \([\mathrm{Ag^+}]\): \([\mathrm{Ag^+}] = \frac{10^{-10}}{10^{-4}} = 10^{-6}\,\mathrm{M}\).

Step 5: Match the Calculation to Answer Choices

The calculated concentration \([\mathrm{Ag^+}] = 10^{-6}\,\mathrm{M}\) matches option (c).

Key Concepts

Buffer Solution

A buffer solution is a special kind of solution that resists changes in pH when acids or bases are added to it. It is typically composed of a weak acid and its conjugate base or a weak base and its conjugate acid.
This property is important because it allows the solution to maintain a relatively constant pH in various chemical processes or reactions.

• For instance, a buffer solution with a pH of 10 implies it is alkaline. Such a buffer might be made by mixing a weak base with its salt.
• In the exercise, the buffer solution helps maintain a pH of 10 as silver ions (\(\mathrm{Ag^+}\)) are added, which reduces significant fluctuations in the pH.

By stabilizing the pH level, the buffer creates a controlled environment where precise chemical reactions, such as precipitation in this case, can occur reliably. Understanding buffers is crucial in both experimental chemistry and various biological systems.

Hydroxide Ion Concentration

The concentration of hydroxide ions (\(\mathrm{OH^-}\)) in a solution is a key factor that determines the basicity, or alkalinity, of that solution.
To calculate the hydroxide ion concentration, we can use the pH of the solution.

• For a solution with pH 10, first calculate the pOH: \[\mathrm{pOH} = 14 - \mathrm{pH} = 4\]
• Then, the hydroxide ion concentration is given by: \[\mathrm{[OH^-]} = 10^{-\mathrm{pOH}} = 10^{-4} \ \mathrm{M}\]

This calculation is critical for various applications, including determining the solubility of compounds in a solution.
In the case of the exercise, the hydroxide ion concentration directly influences the solubility product of \(\mathrm{AgOH}\), as the product of \([\mathrm{Ag^+}]\) and \([\mathrm{OH^-}]\) must equal the solubility product constant, \(K_{sp}\). Understanding how to derive the \(\mathrm{OH^-}\) concentration from pH is an essential skill in chemistry.

Silver Hydroxide

Silver hydroxide (\(\mathrm{AgOH}\)) is a compound that, when dissolved in water, disassociates into silver ions (\(\mathrm{Ag^+}\)) and hydroxide ions (\(\mathrm{OH^-}\)).
Though \(\mathrm{AgOH}\) is not highly soluble in water, its solubility is crucial in many chemical processes.

• Solubility product (\(K_{sp}\)) is a constant that at equilibrium represents the maximum amount of a solid that can dissolve in a solvent. For \(\mathrm{AgOH}\), the \(K_{sp}\) given is \(10^{-10}\).
• In the exercise, we used the solubility product expression: \[K_{sp} = [\mathrm{Ag^+}][\mathrm{OH^-}]\]
• By knowing the \([\mathrm{OH^-}]\), we could solve for \([\mathrm{Ag^+}]\), illustrating the fine balance of ions in solutions.

When \(\mathrm{AgNO_3}\) is added to our buffer solution, \(\mathrm{Ag^+}\) ions interact with \(\mathrm{OH^-}\) ions.
This calculation demonstrates how a complex chemical equilibrium is achieved, impacting the precipitation of \(\mathrm{AgOH}\). It also highlights the broader principles of solubility and equilibria in chemistry.