Problem 79 - Chemistry The Central Science

Question

(a) Given that \(K_{a}\) for acetic acid is \(1.8 \times 10^{-5}\) and that for hypochlorous acid is \(3.0 \times 10^{-8}\), which is the stronger acid? (b) Which is the stronger base, the acetate ion or the hypochlorite ion? (c) Calculate \(K_{b}\) values for \(\mathrm{CH}_{3} \mathrm{COO}^{-}\) and \(\mathrm{ClO}^{-}\).

Step-by-Step Solution

Answer

(a) Acetic acid is the stronger acid. (b) Hypochlorite ion (ClO⁻) is the stronger base. (c) The Kb values are: \(K_{b}(\mathrm{CH}_{3}\mathrm{COO}^{-}) \approx 5.56 \times 10^{-10}\) and \(K_{b}(\mathrm{ClO}^{-}) \approx 3.33 \times 10^{-7}\).

Step 1: Identify the stronger acid

Compare the Ka values of acetic and hypochlorous acids. The stronger acid will have a higher Ka value. The given Ka values are: \(K_{a}(\text{acetic\: acid}) = 1.8 \times 10^{-5}\) \(K_{a}(\text{hypochlorous\: acid}) = 3.0 \times 10^{-8}\) Since \(1.8 \times 10^{-5} > 3.0 \times 10^{-8}\), acetic acid is the stronger acid.

Step 2: Identify the stronger base

The stronger base will be the conjugate base of the weaker acid. In this case, since hypochlorous acid is the weaker acid, its conjugate base, hypochlorite ion (ClO⁻), is the stronger base. The conjugate base of acetic acid, acetate ion (CH₃COO⁻), is the weaker base.

Step 3: Calculate the Kb values

To calculate the Kb values for the CH₃COO⁻ and ClO⁻, we can use the relationship between Ka and Kb, and the ion product of water (Kw): \(K_{a} \times K_{b} = K_{w}\) Where \(K_w\) is the ion product of water and is equal to \(1.0 \times 10^{-14}\) at 25°C. (a) For CH₃COO⁻ (acetate ion): \(K_{b} = \frac{K_{w}}{K_{a}(\text{acetic\: acid})}\) \(K_{b} = \frac{1.0 \times 10^{-14}}{1.8 \times 10^{-5}}\) \(K_{b} \approx 5.56 \times 10^{-10}\) (b) For ClO⁻ (hypochlorite ion): \(K_{b} = \frac{K_{w}}{K_{a}(\text{hypochlorous\: acid})}\) \(K_{b} = \frac{1.0 \times 10^{-14}}{3.0 \times 10^{-8}}\) \(K_{b} \approx 3.33 \times 10^{-7}\) In conclusion: (a) Acetic acid is the stronger acid. (b) Hypochlorite ion (ClO⁻) is the stronger base. (c) The Kb values are: \(K_{b}(\mathrm{CH}_{3}\mathrm{COO}^{-}) \approx 5.56 \times 10^{-10}\) and \(K_{b}(\mathrm{ClO}^{-}) \approx 3.33 \times 10^{-7}\).

Key Concepts

Acid Strength Comparison

When comparing the strength of acids, the acid dissociation constant, known as \( K_a \), is crucial. It tells us how well an acid donates protons in a solution. The bigger the \( K_a \), the more protons are donated, making the acid stronger. In simpler terms, a strong acid easily gives up its hydrogen ion.
In our exercise, we're given the \( K_a \) values for acetic acid and hypochlorous acid:

• Acetic acid has a \( K_a \) of \( 1.8 \times 10^{-5} \).
• Hypochlorous acid has a \( K_a \) of \( 3.0 \times 10^{-8} \).

We see that acetic acid has a higher \( K_a \) than hypochlorous acid. Therefore, acetic acid is the stronger of the two. Higher \( K_a \) indicates stronger proton donation, reflecting greater acid strength.
Remember, understanding \( K_a \) requires looking at the number and not just the label of the acid. Think about \( K_a \) as a measure of the acid's willingness to give up protons.

Conjugate Base Strength

The concept of conjugate bases is linked directly to the acid it came from. When an acid donates a proton, it becomes its conjugate base. The strength of a conjugate base is inversely related to its parent acid's strength.
This means if an acid is strong and readily gives up a proton, its conjugate base will be quite weak because it doesn't have a strong inclination to take the proton back. Conversely, a weak acid forms a relatively strong conjugate base.
In our example:

• Acetic acid, being stronger, forms the acetate ion \( \text{CH}_3\text{COO}^- \), which is a weaker base.
• Hypochlorous acid, being weaker, forms the hypochlorite ion \( \text{ClO}^- \), which is a stronger base.

So, the strength relationship is clear: weaker acids make stronger bases. It's all about balance––the pendulum of donating and accepting protons.

Acid and Base Dissociation Constants

The dissociation constants \( K_a \) and \( K_b \) are central to understanding acid and base behavior in water. These constants are linked via the ion product of water, \( K_w \), which equals \( 1.0 \times 10^{-14} \) at 25°C.
The relationship between \( K_a \) and \( K_b \) is:

• \( K_a \times K_b = K_w \)

Using this relationship, we can find out the base dissociation constant, \( K_b \), guiding us to understand how readily a base absorbs protons.
Calculating \( K_b \) for the conjugate bases of our acids:

• For the acetate ion \( \text{CH}_3\text{COO}^- \): \( K_b = \frac{1.0 \times 10^{-14}}{1.8 \times 10^{-5}} \approx 5.56 \times 10^{-10} \)
• For the hypochlorite ion \( \text{ClO}^- \): \( K_b = \frac{1.0 \times 10^{-14}}{3.0 \times 10^{-8}} \approx 3.33 \times 10^{-7} \)

This tells us the propensity of each base to accept protons. Higher \( K_b \) values correlate with stronger base tendencies, confirming our earlier conclusions about base strength.