Problem 33 - Chemistry and Chemical Reactivity

Question

Chloroacetic acid \(\left(\mathrm{Cl} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H}\right)\) has \(K_{\mathrm{a}}=1.41 \times\) \(10^{-3} .\) What is the value of \(K_{\mathrm{b}}\) for the chloroacetate \(\operatorname{ion}\left(\mathrm{Cl} \mathrm{CH}_{2} \mathrm{CO}_{2}^{-}\right) ?\)

Step-by-Step Solution

Answer

\( K_b = 7.09 \times 10^{-12} \)

Step 1: Understand the Relationship Between Ka and Kb

In a conjugate acid-base pair, the product of the acid dissociation constant \( K_a \) and the base dissociation constant \( K_b \) is equal to the ion product of water \( K_w \). The relationship is given by the equation:\[K_a \times K_b = K_w\]where \( K_w = 1.0 \times 10^{-14} \) at 25°C.

Step 2: Rearrange the Equation to Solve for Kb

We need to find the value of \( K_b \) for the chloroacetate ion. Using the relationship from Step 1, we rearrange the equation to solve for \( K_b \):\[K_b = \frac{K_w}{K_a}\]

Step 3: Substitute the Given Values

Substitute \( K_a = 1.41 \times 10^{-3} \) and \( K_w = 1.0 \times 10^{-14} \) into the equation from Step 2 to find \( K_b \):\[K_b = \frac{1.0 \times 10^{-14}}{1.41 \times 10^{-3}}\]

Step 4: Perform the Calculation

Calculate \( K_b \) by performing the division:\[K_b = \frac{1.0 \times 10^{-14}}{1.41 \times 10^{-3}} \approx 7.09 \times 10^{-12}\]

Key Concepts

Conjugate Acid-Base Pair

In the world of chemistry, understanding conjugate acid-base pairs is essential. These pairs form when an acid donates a proton and becomes its conjugate base, or when a base accepts a proton, turning into its conjugate acid. This might sound a bit tricky, but let's break it down.

A conjugate acid-base pair consists of two species that differ by a single proton (\( ext{H}^+\)). For example, when chloroacetic acid (\( ext{ClCH}_2 ext{COOH}\)) donates a proton, it transforms into its conjugate base, the chloroacetate ion (\( ext{ClCH}_2 ext{COO}^-\)).

• Conjugate acid – The species formed by the acceptance of a proton.
• Conjugate base – The species remaining after an acid has donated a proton.

Understanding conjugate pairs helps in predicting the direction of acid-base reactions and the strengths of acids and bases in different solutions. The stronger an acid, the weaker its conjugate base, and vice versa.

Dissociation Constant

Dissociation constants are crucial for understanding the strength of acids and bases. For acids, we talk about the acid dissociation constant (\(K_a\)). This constant gives us insight into how completely an acid dissociates into its ions in a solution.

Considering chloroacetic acid, which has a \(K_a\) of \(1.41 \times 10^{-3}\), it tells us that while the acid is relatively strong, it doesn't fully dissociate in water. The value of \(K_a\) informs us of the equilibrium concentration of the ions produced as the acid dissociates.

Let's not forget about the base dissociation constant (\(K_b\)) which works similarly for bases. The smaller the \(K_b\) value, the weaker the base. This is why calculating the \(K_b\) of the chloroacetate ion involves using the acid's \(K_a\), thus showing the inverse relationship between the strengths of conjugate acid-base pairs.

Ion Product of Water

The ion product of water (\(K_w\)) is a fascinating concept. It's the equilibrium constant for water's self-ionization, where water molecules dissociate into hydrogen ions (\( ext{H}^+\)) and hydroxide ions (\( ext{OH}^-\)). In pure water at 25°C, the value of \(K_w\) is \(1.0 \times 10^{-14}\).

Knowing \(K_w\) is beneficial when we deal with calculations involving ion concentrations in solutions. For instance, when working out the \(K_b\) from a given \(K_a\), we rely on the equation:\[K_a \times K_b = K_w\]This equation is pivotal for relationships between acid and base strengths, like finding the \(K_b\) of chloroacetate. You rearrange this equation to \(K_b = \frac{K_w}{K_a}\) to find the dissociation constant for the base. This interrelationship helps us understand the balance and adjustment of pH in a solution, fundamental for various scientific applications.