Problem 81 - 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) The hypochlorite ion is the stronger base. (c) The \(K_b\) values are: \(\mathrm{CH}_{3} \mathrm{COO}^{-}\): \(5.56 \times 10^{-10}\) and \(\mathrm{ClO}^{-}\): \(3.33 \times 10^{-7}\).

Step 1: Compare Ka values

In order to determine which acid is stronger, we must compare the \(K_{a}\) values for acetic acid and hypochlorous acid. A higher \(K_{a}\) value corresponds to a stronger acid, since it indicates a larger extent of dissociation. Acetic acid: \(K_{a1} = 1.8 \times 10^{-5}\) Hypochlorous acid: \(K_{a2} = 3.0 \times 10^{-8}\) Comparing the two values, we can see that \(K_{a1} > K_{a2}\). Therefore, acetic acid is the stronger acid.

Step 2: Determine the stronger base

The stronger an acid is, the weaker its conjugate base. Since we know that acetic acid is the stronger acid, its conjugate base, the acetate ion (\(\mathrm{CH}_{3} \mathrm{COO}^{-}\)), is a weaker base than the hypochlorite ion (\(\mathrm{ClO}^{-}\)), which is the conjugate base of hypochlorous acid. So, the hypochlorite ion is the stronger base.

Step 3: Calculate Kb values

To calculate the \(K_b\) value for each conjugate base, we will use the relationship between \(K_a\), \(K_b\), and \(K_w\). The ion product constant of water, \(K_w\), is equal to \(1.0 \times 10^{-14}\) at 25°C. The relationship between \(K_a\), \(K_b\), and \(K_w\) is given by: \[K_a \times K_b = K_w\] Now, let's calculate the \(K_b\) values for the acetate ion and hypochlorite ion using their corresponding \(K_a\) values: For the acetate ion (\(\mathrm{CH}_{3} \mathrm{COO}^{-}\)): \[K_{b1} = \frac{K_w}{K_{a1}} =\frac{1.0 \times 10^{-14}}{1.8 \times 10^{-5}} = 5.56 \times 10^{-10}\] For the hypochlorite ion (\(\mathrm{ClO}^{-}\)): \[K_{b2} = \frac{K_w}{K_{a2}} =\frac{1.0 \times 10^{-14}}{3.0 \times 10^{-8}} = 3.33 \times 10^{-7}\] To summarize the results: (a) Acetic acid is the stronger acid. (b) The hypochlorite ion is the stronger base. (c) The \(K_b\) values are: \(\mathrm{CH}_{3} \mathrm{COO}^{-}\): \(5.56 \times 10^{-10}\) \(\mathrm{ClO}^{-}\): \(3.33 \times 10^{-7}\)

Key Concepts

Acid Dissociation Constant

The acid dissociation constant, represented as \(K_a\), is a quantitative measure of the strength of an acid in solution. It is the equilibrium constant for a chemical reaction known as the dissociation of acid molecules in water. This process can be represented by the equation for a generic acid HA:
\[HA \rightleftharpoons H^{+} + A^{-}\]
When an acid dissociates, it releases hydrogen ions (\(H^+\)) into the solution, making the solution acidic. The strength of an acid is indicated by its \(K_a\) value: the larger the \(K_a\), the more the acid dissociates, implying a stronger acid. In the given exercise, acetic acid with a \(K_a\) of \(1.8 \times 10^{-5}\) is stronger than hypochlorous acid, which has a \(K_a\) of \(3.0 \times 10^{-8}\). This means that acetic acid donates \(H^+\) ions to the solution more readily than hypochlorous acid.

To improve comprehension in this area, it's essential to recognize that \(K_a\) provides insight into the balance between intact acid molecules and the ions formed upon dissociation. So, this balance can be leveraged to predict the behavior of acids in various chemical reactions, directly affecting the pH-level of the solutions they are in.

Conjugate Bases

In the context of acid-base chemistry, a conjugate base is the species that remains after an acid has donated a proton during a chemical reaction. The general formula for this reaction is:
\[HA \rightleftharpoons H^{+} + A^{-}\]
Here, \(HA\) is the acid, \(H^+\) (the hydrogen ion or proton) is what the acid donates, and \(A^-\) is the conjugate base. The strength of a conjugate base is inversely related to the strength of its parent acid; a strong acid will have a weak conjugate base, and vice versa. For instance, since acetic acid is a stronger acid than hypochlorous acid, acetate ion (\(CH_3COO^−\)), which is the conjugate base of acetic acid, is a weaker base than the hypochlorite ion (\(ClO^−\)).

Understanding conjugate bases is vital in predicting the outcome of acid-base reactions and in processing the behavior of substances in their ionic form. To help students grasp this concept, emphasize that the principle of conjugate pairs helps maintain the acid-base equilibrium in solutions, and this balance plays a critical role in buffering and neutral processes within chemical systems or biological systems such as human blood.

Ion Product Constant of Water

The ion product constant of water, denoted as \(K_w\), indicates the product of the concentrations of hydrogen ions (\(H^+\)) and hydroxide ions (\(OH^-\)) present in pure water. At 25°C (298 K), this equilibrium constant value is
\[K_w = [H^{+}][OH^{-}] = 1.0 \times 10^{-14}\]
In water, hydrogen and hydroxide ions are produced in equal amounts through the self-ionization of water molecules, leading to a neutral pH of 7. The concept of \(K_w\) is especially useful in acid-base equilibrium calculations. For instance, \(K_w\) provides a link between the acid dissociation constant (\(K_a\)) and the base dissociation constant (\(K_b\)) through the relationship:
\[K_a \times K_b = K_w\]
In exercises involving acid-base equilibria, understanding this relationship allows students to calculate the strength of a conjugate base, given the strength of its conjugate acid, and vice versa. This concept informs the intrinsic connectedness of acids and bases in water, with the neutral condition of water providing a frame of reference for differentiating between acidic or basic solutions based on their pH levels.