Problem 70 - Chemistry: Principles and Reactions

Question

Find the value of \(K_{\mathrm{a}}\) for the conjugate acid of the following bases. (a) pyridine, a pesticide; \(K_{\mathrm{b}}=1.5 \times 10^{-9}\) (b) aniline, an important dye intermediate; \(K_{\mathrm{b}}=3.8 \times 10^{-10}\)

Step-by-Step Solution

Answer

The value of \(K_{\mathrm{a}}\) for pyridine's conjugate acid is \(6.67 \times 10^{-6}\), and the value of \(K_{\mathrm{a}}\) for aniline's conjugate acid is \(2.63 \times 10^{-5}\).

Step 1: Identify the given values

For each base, we are given their respective values of \(K_{\mathrm{b}}\). Write down the values: (a) Pyridine: \(K_{\mathrm{b}} = 1.5 \times 10^{-9}\) (b) Aniline: \(K_{\mathrm{b}} = 3.8 \times 10^{-10}\) ##Step 2: Recall the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\)##

Step 2: Recall the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\)

The relationship between the \(K_{\mathrm{a}}\) for a conjugate acid and the \(K_{\mathrm{b}}\) for a weak base is given by: $$ K_{\mathrm{a}} \cdot K_{\mathrm{b}} = K_{\mathrm{w}} $$ ##Step 3: Calculate \(K_{\mathrm{a}}\) for pyridine's conjugate acid##

Step 3: Calculate \(K_{\mathrm{a}}\) for pyridine's conjugate acid

Using the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\), we can calculate \(K_{\mathrm{a}}\) for the conjugate acid of pyridine. Since \(K_{\mathrm{b}}=1.5 \times 10^{-9}\) and \(K_{\mathrm{w}} = 1 \times 10^{-14}\), we have: $$ K_{\mathrm{a}} = \frac{K_{\mathrm{w}}}{K_{\mathrm{b}}} = \frac{1 \times 10^{-14}}{1.5 \times 10^{-9}} $$ $$ K_{\mathrm{a}} = 6.67 \times 10^{-6} $$ The value of \(K_{\mathrm{a}}\) for pyridine's conjugate acid is \(6.67 \times 10^{-6}\). ##Step 4: Calculate \(K_{\mathrm{a}}\) for aniline's conjugate acid##

Step 4: Calculate \(K_{\mathrm{a}}\) for aniline's conjugate acid

Using the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\), we can calculate \(K_{\mathrm{a}}\) for the conjugate acid of aniline. Since \(K_{\mathrm{b}}=3.8 \times 10^{-10}\) and \(K_{\mathrm{w}} = 1 \times 10^{-14}\), we have: $$ K_{\mathrm{a}} = \frac{K_{\mathrm{w}}}{K_{\mathrm{b}}} = \frac{1 \times 10^{-14}}{3.8 \times 10^{-10}} $$ $$ K_{\mathrm{a}} = 2.63 \times 10^{-5} $$ The value of \(K_{\mathrm{a}}\) for aniline's conjugate acid is \(2.63 \times 10^{-5}\). The conjugate acid of pyridine has a \(K_{\mathrm{a}}\) value of \(6.67 \times 10^{-6}\), and the conjugate acid of aniline has a \(K_{\mathrm{a}}\) value of \(2.63 \times 10^{-5}\).

Key Concepts

Weak Acids

In chemistry, weak acids are substances that do not completely dissociate in water. Instead of releasing all their hydrogen ions (\( \text{H}^+ \)) into the solution, they only partially ionize. This is in contrast to strong acids, which fully dissociate in solution. The degree of ionization for weak acids can have a significant impact on their behavior in chemical reactions.

Understanding weak acids is crucial because:

• They play a significant role in chemical equilibrium.
• Their behavior affects pH levels in solutions.
• They participate actively in buffer systems, which help maintain stable pH levels in many biological and chemical contexts.

In an equilibrium scenario, the dissociation of a weak acid in water can be represented as:\[\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\]where \( \text{HA} \) is the weak acid and \( \text{A}^- \) is the conjugate base.

Conjugate Acids

A conjugate acid is formed when a base gains a proton. In the framework of the Brønsted-Lowry acid-base theory, acids and bases can transform into each other through proton transfer. This theory helps explain the concept of conjugate acid-base pairs.

Key points about conjugate acids include:

• Every base has a corresponding conjugate acid.
• The strength of the conjugate acid is inversely related to the strength of the base from which it formed.
• When a weak base gains a hydrogen ion, it becomes a conjugate acid. For example, pyridine gains \( \text{H}^+ \) and becomes pyridinium, its conjugate acid.

In the example from the original exercise, pyridine and aniline transform into their respective conjugate acids by gaining hydrogen ions. The reaction can be demonstrated as:\[\text{Weak Base} + \text{H}_2\text{O} \rightleftharpoons \text{Conjugate Acid} + \text{OH}^-\]

Equilibrium Constant

The equilibrium constant, symbolized as \( K \), describes the ratio of the concentrations of products to reactants at equilibrium. Different types of equilibrium constants correspond to different types of reactions. Specifically, \( K_a \) and \( K_b \) are the equilibrium constants for acids and bases respectively.

For acid-base reactions, these constants are defined as:

• \( K_a \) for acids: \( K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} \)
• \( K_b \) for bases: \( K_b = \frac{[\text{BH}^+][\text{OH}^-]}{[\text{B}]} \)

The relationship between the equilibrium constants of conjugate acids and bases allows conversion from base strength to acid strength through:\[K_a \cdot K_b = K_w\]where \( K_w \) is the ionic product of water, \( 1 \times 10^{-14} \) at room temperature. This formula was used in the original solution to calculate the \( K_a \) values for pyridine's and aniline's conjugate acids, illustrating the interplay between the strengths of conjugate acid-base pairs.