Problem 137 - A Complete Resource Book in Chemistry for JEE Main

Question

The \(\mathrm{pH}\) of \(0.05 \mathrm{M}\) aqueous solution of diethylamine is 12\. Its \(\mathrm{K}_{b}\) is (a) \(2 \times 10^{-3}\) (b) \(2.5 \times 10^{-3}\) (c) \(3 \times 10^{-3}\) (d) \(4.5 \times 10^{-3}\)

Step-by-Step Solution

Answer

The \(\mathrm{K}_{b}\) of the solution is \(2.5 \times 10^{-3}\), which corresponds to option (b).

Step 1: Determine the concentration of hydroxide ions

We know that \(\text{pH} = 12\), so \(\text{pOH} = 14 - 12 = 2\). This means that the concentration of hydroxide ions (\([OH^-]\)) is \(10^{-2}\, M\) because \([OH^-] = 10^{-\text{pOH}}\).

Step 2: Express the equilibrium expression for diethylamine

Diethylamine (\(C_4H_{11}N\)) is a weak base. The equilibrium reaction is: \[ C_4H_{11}N + H_2O \rightleftharpoons C_4H_{11}NH^+ + OH^- \]Based on the ICE table method, let the initial concentration of diethylamine be \(C = 0.05\, M\). At equilibrium, the change in concentration can be expressed as \([OH^-] = 10^{-2}\, M\).

Step 3: Calculate equilibrium concentrations

In the ICE table:- Initial concentrations: \([C_4H_{11}N] = 0.05\, M\), \([OH^-] = 0\)- Change: \([OH^-] = + 10^{-2}\, M\)- Equilibrium: \([C_4H_{11}N] = 0.05 - 10^{-2}\, M = 0.04\, M\) (approximately), \([OH^-] = 10^{-2}\, M\)The equilibrium concentration of \(C_4H_{11}NH^+\) is also \(10^{-2}\, M\).

Step 4: Apply the base ionization constant expression

The expression for the base ionization constant \(K_b\) is:\[ K_b = \frac{[C_4H_{11}NH^+][OH^-]}{[C_4H_{11}N]} \]Substitute the equilibrium concentrations into this expression:\[ K_b = \frac{10^{-2} \times 10^{-2}}{0.04} = \frac{10^{-4}}{0.04} = \frac{10^{-4}}{4 \times 10^{-2}} = 2.5 \times 10^{-3} \]

Step 5: Identify the correct option

The calculated \(K_b = 2.5 \times 10^{-3}\) matches with option (b) in the provided choices.

Key Concepts

Weak Base Equilibrium

The concept of equilibrium is crucial when dealing with chemical reactions, especially for weak bases. A weak base, like diethylamine, does not completely ionize in water. Instead, it establishes an equilibrium between the unreacted base and the resulting ions in solution.
This can be represented as:

• Base + Water ↔ Conjugate Acid + Hydroxide Ion
• For diethylamine: \[C_4H_{11}N + H_2O \rightleftharpoons C_4H_{11}NH^+ + OH^-\]

In this equilibrium, we see that diethylamine (\(C_4H_{11}N\)) gains a proton from water to form its conjugate acid (\(C_4H_{11}NH^+\)), while water donates a hydroxide ion (\(OH^-\)).
Understanding this balance is key to calculating various chemical properties, such as the pH of the solution.

Hydroxide Ion Concentration

The concentration of hydroxide ions in a solution is a direct indicator of its basicity. A fundamental relationship in chemistry is that between \ \( ext{pH}\) and \ \( ext{pOH}\):

• pH + pOH = 14 (at 25°C)

Given that the \ \(pH\) of our solution is 12, the \ \(pOH\) becomes:

• pOH = 14 - 12 = 2

The hydroxide ion concentration, denoted \ \([OH^-]\), can be found using the formula:

• [OH^-] = 10^{- ext{pOH}} = 10^{-2} ext{M}

This quantifies the solution's basic nature by illustrating the concentration of hydroxide ions present in the solution.

Base Ionization Constant

The base ionization constant, \(K_b\), helps in understanding the strength of a weak base in solution. It is defined as the equilibrium constant for the ionization of a base in water:

• \(K_b = \frac{[Conjugate\ Acid][OH^-]}{[Base]}\)

Using the \(ICE\) (Initial, Change, Equilibrium) table method, you can calculate equilibrium concentrations to determine \(K_b\). Here, the equation for diethylamine at equilibrium is:

• \(K_b = \frac{[C_4H_{11}NH^+][OH^-]}{[C_4H_{11}N]} = \frac{10^{-2} \times 10^{-2}}{0.04}\)
• This calculation yields a \ \(K_b\) of \ \(2.5 \times 10^{-3}\), characterizing diethylamine as a moderately strong weak base.

Diethylamine Solution

Diethylamine is a secondary amine with the molecular formula \ \(C_4H_{11}N\), commonly used in chemical solvents and synthetic processes. In an aqueous solution, it acts as a weak base that partially ionizes.
This partial ionization is reflected in its \ \(pH\) value and influences how it behaves in equilibrium with water. By calculating the \ \(pH\) and \ \(K_b\), one can gauge the solution's basicity and the efficiency of the diethylamine's ionization process.Additionally:

• A 0.05 M solution indicates a relatively low concentration, demonstrating the characteristic of a weak base not being fully dissociated in solution.
• The pH value of 12 reveals its sufficiently basic environment, a result of the equilibrium between diethylamine and its ions.

Understanding these interactions and properties helps in predicting the behavior of diethylamine in various chemical responses and environments.