Problem 77 - Chemistry The Central Science In Si Units

Question

(a) Given that \(K_{a}\) for cyanic acid is \(3.5 \times 10^{-4}\) and that for hydrofluoric acid is \(6.8 \times 10^{-4}\), which is the stronger acid? (b) Which is the stronger base, the cyanate ion or the fluoride ion? (c) Calculate \(K_{b}\) values for \(\mathrm{NCO}^{-}\) and \(\mathrm{F}^{-}\).

Step-by-Step Solution

Answer

(a) Hydrofluoric acid is stronger; (b) Cyanate ion is stronger; (c) \(K_b(NCO^{-}) \approx 2.86 \times 10^{-11}\), \(K_b(F^{-}) \approx 1.47 \times 10^{-11}\).

Step 1: Understanding Acid Strength

To determine which acid is stronger, we look at their dissociation constants \(K_{a}\). The larger the \(K_{a}\) value, the stronger the acid. Given that the \(K_{a}\) for cyanic acid is \(3.5 \times 10^{-4}\) and for hydrofluoric acid is \(6.8 \times 10^{-4}\), hydrofluoric acid is the stronger acid as it has a larger \(K_{a}\).

Step 2: Evaluate Base Strength Using Conjugate Acids

The strength of a base is inversely related to the strength of its conjugate acid. The stronger the conjugate acid, the weaker the base. Since hydrofluoric acid (\(HF\)) is stronger than cyanic acid (\(HCNO\)), the fluoride ion \(F^-\) is the weaker base compared to the cyanate ion \(NCO^-\).

Step 3: Calculate \(K_b\) for Cyanate Ion

The relationship between \(K_{a}\) and \(K_{b}\) for a conjugate pair is \(K_{a} \times K_{b} = K_{w}\), where \(K_{w} = 1.0 \times 10^{-14}\). For cyanic acid \(K_{a}(HCNO) = 3.5 \times 10^{-4}\):\[K_{b}(NCO^-) = \frac{K_{w}}{K_{a}} = \frac{1.0 \times 10^{-14}}{3.5 \times 10^{-4}}\approx 2.86 \times 10^{-11}.\]

Step 4: Calculate \(K_b\) for Fluoride Ion

For hydrofluoric acid \(K_{a}(HF) = 6.8 \times 10^{-4}\):\[K_{b}(F^-) = \frac{K_{w}}{K_{a}} = \frac{1.0 \times 10^{-14}}{6.8 \times 10^{-4}} \approx 1.47 \times 10^{-11}.\]

Key Concepts

Acid Strength

Acid strength is a measure of an acid's ability to donate protons (hydrogen ions, H")). The dissociation constant, represented as \( K_a \), indicates how completely an acid ionizes in water:
\[ HA + H_2O \rightleftharpoons H_3O^+ + A^- \]
The larger the \( K_a \), the more an acid dissociates in solution, indicating a stronger acid. For example, hydrofluoric acid (HF) has a \( K_a \) of \( 6.8 \times 10^{-4} \), while cyanic acid (HCNO) has a \( K_a \) of \( 3.5 \times 10^{-4} \).

• This tells us HF is a stronger acid than HCNO because it dissociates more in solution, releasing more hydrogen ions.

Understanding acid strength is important because it helps predict how substances will behave in chemical reactions, especially in biological and environmental contexts.

Base Strength

Base strength is determined by a base's ability to accept protons. It is indirectly related to the acid strength of its conjugate acid. Strong bases have weak conjugate acids and vice versa.
To evaluate base strength using conjugate acids, consider the dissociation of the conjugate acid:

• Fluoride ion \( F^- \) is the conjugate base of the strong acid HF.
• Cyanate ion \( NCO^- \) is the conjugate base of the weaker acid HCNO.

Since HF is stronger, \( F^- \) is a weaker base compared to \( NCO^- \).
When comparing base strengths, remember:

• Stronger acids have weaker conjugate bases.
• Weaker acids have stronger conjugate bases.

This helps predict the direction of equilibrium in acid-base reactions.

Dissociation Constant

The dissociation constant \( K \) quantifies the extent to which an acid or base dissociates in water. For acids, it is denoted as \( K_a \), and for bases, \( K_b \). These constants are crucial for calculating pH and understanding reaction equilibria.
To compute \( K_b \) for a base from its acid dissociation constant, use the formula:
\[ K_a \times K_b = K_w \]
where \( K_w = 1.0 \times 10^{-14} \). This relationship highlights the inverse nature between \( K_a \) and \( K_b \).
As seen in the exercise:

• For cyanate ion \( NCO^- \), \( K_a = 3.5 \times 10^{-4} \), thus \( K_b = \frac{1.0 \times 10^{-14}}{3.5 \times 10^{-4}} \approx 2.86 \times 10^{-11} \).
• For fluoride ion \( F^- \), \( K_b \approx 1.47 \times 10^{-11} \) with \( K_a = 6.8 \times 10^{-4} \).

Dissociation constants help predict the strength and behavior of acids and bases in various chemical environments.

Conjugate Acids and Bases

In acid-base chemistry, conjugate acid-base pairs play a crucial role in reactions and equilibria. When an acid donates a proton, it becomes its conjugate base; when a base accepts a proton, it becomes its conjugate acid.

• For example, when HF (an acid) loses a proton, it forms the fluoride ion \( F^- \), its conjugate base.
• Likewise, when \( NCO^- \) accepts a proton, it forms HCNO, its conjugate acid.

The strength of a conjugate base or acid is inversely related to its parent acid or base. This means understanding conjugate pairs helps in predicting and balancing chemical equations.
The dynamics of conjugate acids and bases assist in determining reactions' directionality and key characteristics of acids and bases, facilitating deeper insights into their chemical behavior.