Problem 63 - Chemistry: The Central Science in SI Units, Global Edition

Question

Calculate the percent ionization of hydrazoic acid \(\left(\mathrm{HN}_{3}\right)\) in solutions of each of the following concentrations \(\left(K_{a}\right.\) is (c) \(0.0400 \mathrm{M}\). given in Appendix \(\mathrm{D}):(\mathbf{a}) 0.400 \mathrm{M},(\mathbf{b}) 0.100 \mathrm{M}\)

Step-by-Step Solution

Answer

For a 0.400 M solution of hydrazoic acid (HN3), the percent ionization is calculated as follows: 1. x = \(\sqrt{(1.9 \times 10^{-5})(0.400)} \approx 8.68 \times 10^{-4}\) 2. Percent Ionization = \(\frac{8.68 \times 10^{-4}}{0.400}\) × 100% ≈ 0.217% For a 0.100 M solution of hydrazoic acid (HN3), the percent ionization is calculated as follows: 1. x = \(\sqrt{(1.9 \times 10^{-5})(0.100)} \approx 4.36 \times 10^{-4}\) 2. Percent Ionization = \(\frac{4.36 \times 10^{-4}}{0.100}\) × 100% ≈ 0.436%

Step 1: (General Formula for Percent Ionization)

(The formula for percent ionization is given by: Percent Ionization = \(\frac{[\text{H}^+]}{[\text{HA}]_\text{initial}}\) x 100%)

Step 2: (Acid Ionization Reaction)

(Express the ionization of hydrazoic acid as an equilibrium reaction: \[ \mathrm{HN}_{3} \rightleftharpoons \mathrm{H}^{+} + \mathrm{N}_{3}^{-} \])

Step 3: (Setting up the ICE Table)

(Establish an ICE table to keep track of the initial, change, and equilibrium concentrations of all species: ``` HN3 --> H+ + N3- Initial [C] 0 0 Change -x +x +x Equilib [C]-x x x ```)

Step 4: (Expression for Ka)

(Write the expression for the equilibrium constant, Ka: \[K_a = \frac{[\mathrm{H}^+][\mathrm{N}_{3}^-]}{[\mathrm{HN}_3]}\])

Step 5: (Substitution of Equilibrium Concentrations in Ka expression)

(Substitute the equilibrium concentrations from the ICE table into the Ka expression: \[K_a = \frac{x^2}{([C] - x})\]) Now, we will proceed to solve the given problems using the above methods.

Step 6: (Percent Ionization for 0.400 M HN3)

(Given concentration, [HA] = 0.400 M, and Ka = 1.9 x 10^-5) 1. First, we will assume that x is very small compared to [C], hence we can approximate the denominator by [C]: \[K_a \approx \frac{x^2}{[C] }\] 2. Solve for x (the concentration of H+ ions at equilibrium): \[x^2 = K_a \cdot [C] \Rightarrow x = \sqrt{K_a \cdot [C]}\] 3. Calculate the percent ionization: Percent Ionization = \(\frac{x}{[HA]}\) × 100%

Step 7: (Percent Ionization for 0.100 M HN3)

(Given concentration, [HA] = 0.100 M, and Ka = 1.9 x 10^-5) 1. First, we will assume that x is very small compared to [C], hence we can approximate the denominator by [C]: \[K_a \approx \frac{x^2}{[C] }\] 2. Solve for x (the concentration of H+ ions at equilibrium): \[x^2 = K_a \cdot [C] \Rightarrow x = \sqrt{K_a \cdot [C]}\] 3. Calculate the percent ionization: Percent Ionization = \(\frac{x}{[HA]}\) × 100%

Key Concepts

Hydrazoic Acid

Hydrazoic acid, with the chemical formula \( \mathrm{HN}_3 \), is a weak acid. This means it does not completely dissociate in water. Instead, only a small fraction of \( \mathrm{HN}_3 \) molecules ionize to produce \( \mathrm{H}^+ \) and \( \mathrm{N}_3^- \) ions.
Understanding the behavior of weak acids like hydrazoic acid is crucial in acid-base chemistry, as it helps predict how such compounds will behave in different chemical environments.
Weak acids typically have a defined equilibrium position that can be quantified using an equilibrium constant (\( K_a \)). When hydrazoic acid is dissolved in water, the system reaches an equilibrium described by the equation: \[ \mathrm{HN}_3 \rightleftharpoons \mathrm{H}^+ + \mathrm{N}_3^-\]
Knowing how to calculate the concentration of \( \mathrm{H}^+ \) ions in solution through the equilibrium constant allows us to determine important properties like percent ionization.

Equilibrium Constant

The equilibrium constant, \( K_a \), is a crucial concept in understanding acid dissociation in aqueous solutions. It quantifies the extent to which an acid dissociates into its component ions at equilibrium.
For a weak acid like hydrazoic acid \( (\mathrm{HN}_3) \), the expression for \( K_a \) is given by:\[ K_a = \frac{[\mathrm{H}^+][\mathrm{N}_3^-]}{[\mathrm{HN}_3]} \]
This expression helps in calculating the relative concentrations of ions in a solution when equilibrium is established.
When dealing with weak acids, it is often necessary to calculate these equilibriums to understand how much of the acid actually ionizes. The size of \( K_a \) reflects the strength of the acid: a smaller \( K_a \) indicates a weaker acid that doesn't dissociate much, while a larger \( K_a \) suggests a stronger acid. For hydrazoic acid, the \( K_a \) is typically low, indicating minimal ionization.

Percent Ionization

Percent ionization is a measure that indicates the extent to which a weak acid ionizes in solution. It is expressed as a percentage of the initial acid concentration that has dissociated into ions.
To calculate the percent ionization for hydrazoic acid, you can use the formula:\[ \text{Percent Ionization} = \frac{[\mathrm{H}^+]}{[\mathrm{HA}]_{\text{initial}}} \times 100\%\]
where \([\mathrm{H}^+]\) is the concentration of hydrogen ions at equilibrium, and \([\mathrm{HA}]_{\text{initial}}\) is the initial concentration of the acid.
The percent ionization is important because it provides information on how effective an acid is at releasing hydrogen ions in solution. This metric is particularly useful in determining whether assumptions made in calculations (like neglecting \( x \)) are justified, especially if \( x \) is much smaller than \([C]\), the initial concentration.

ICE Table

An ICE (Initial, Change, Equilibrium) table is a systematic way to track the concentrations of species in an equilibrium reaction.
For the ionization of hydrazoic acid, the table helps understand the changes each component undergoes as the reaction progresses towards equilibrium.
Here's how it is set up:

• Initial: Start with the initial concentrations of \( \mathrm{HN}_3 \), \( \mathrm{H}^+ \), and \( \mathrm{N}_3^- \). Initially, \([\mathrm{H}^+]\) and \([\mathrm{N}_3^-]\) are zero because no reaction has occurred yet.
• Change: As the reaction progresses, \( \mathrm{HN}_3 \) decreases by \( x \), while \( \mathrm{H}^+ \) and \( \mathrm{N}_3^- \) increase by \( x \).
• Equilibrium: By the time equilibrium is established, the concentrations are \([\mathrm{HN}_3]+x\), \([\mathrm{H}^+]-x\), and \([\mathrm{N}_3^-]-x\).

Using the ICE table, you can substitute these values into the equilibrium expression for \( K_a \), and solve for \( x \). This approach simplifies the computation of percent ionization and helps verify assumptions in weak acid equilibrium calculations.