Question

Hydrazine, \(\mathrm{N}_{2} \mathrm{H}_{4}\), is a weak base and is used as fuel in the space shuttle. If the \(\mathrm{pH}\) of a \(0.139 \mathrm{M}\) solution is 11.00 , what is the ionization constant of the base?

Short answer

The ionization constant of the base is approximately \( 7.19 \times 10^{-6} \).

Step-by-step solution

Understand the Relationship between pH and pOH

Since the given solution has a pH of 11.00, we need to determine the pOH. We know that \[ \mathrm{pH} + \mathrm{pOH} = 14 \].Given \( \mathrm{pH} = 11.00 \), we can compute \( \mathrm{pOH} = 14 - 11 = 3.00 \).

Calculate Hydroxide Ion Concentration

From the \( \mathrm{pOH} \), we can determine the \( \mathrm{OH}^{-} \) concentration using the formula \[ \mathrm{pOH} = -\log[\mathrm{OH}^{-}] \].Thus, \[ [\mathrm{OH}^{-}] = 10^{-\mathrm{pOH}} = 10^{-3.00} = 1.0 \times 10^{-3} \text{ M} \].

Setup the Ionization Reaction of Hydrazine

The ionization of hydrazine in water can be written as follows:\[ \mathrm{N}_2\mathrm{H}_4 + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{N}_2\mathrm{H}_5^{+} + \mathrm{OH}^{-} \].Initially, the concentration of \( \mathrm{N}_2\mathrm{H}_4 \) is 0.139 M, and that of \( \mathrm{N}_2\mathrm{H}_5^{+} \) and \( \mathrm{OH}^{-} \) is 0.

Apply the Change in Concentration

Assume that \( x \) is the concentration of \( \mathrm{OH}^{-} \) at equilibrium, corresponding from the ionization of \( \mathrm{N}_2\mathrm{H}_4 \). At equilibrium, we have \[ [\mathrm{OH}^{-}] = x = 1.0 \times 10^{-3} \],\[ [\mathrm{N}_2\mathrm{H}_4] = 0.139 - x \approx 0.139 \],and\[ [\mathrm{N}_2\mathrm{H}_5^{+}] = x = 1.0 \times 10^{-3} \text{ M}. \]

Calculate the Ionization Constant

The ionization constant \( K_b \) is given by \[ K_b = \frac{[\mathrm{N}_2\mathrm{H}_5^{+}][\mathrm{OH}^{-}]}{[\mathrm{N}_2\mathrm{H}_4]} = \frac{(1.0 \times 10^{-3})(1.0 \times 10^{-3})}{0.139} \].Calculate:\[ K_b = \frac{1.0 \times 10^{-6}}{0.139} \approx 7.19 \times 10^{-6}. \]

Key concepts

pH

pH is a measure of the acidity or basicity of a solution, expressed on a logarithmic scale from 0 to 14. A pH of 7 is considered neutral, with lower values being acidic and higher values being basic. Calculating the pH is crucial in chemistry, as it helps define the properties of a solution.
For a basic solution like the one in our exercise, having a high pH (11.00 in this case) indicates a low concentration of hydrogen ions \( [\mathrm{H}^+] \). Instead, it points to higher amounts of hydroxide ions. The formula for understanding the relationship between pH and hydroxide concentrations is given by \( \text{pH} + \text{pOH} = 14 \). This allows us to calculate pOH from the known pH.
Understanding the pH is critical when working with buffer solutions, acids, and bases, making it a cornerstone in many chemical equations and solutions.

Hydroxide ion concentration

The hydroxide ion concentration \( [\mathrm{OH}^-] \) is a vital aspect when examining solutions with basic tendencies. It tells us about the actual amount of hydroxide ions present in a solution. By finding the pOH, one can determine the hydroxide ion concentration using the equation \( \text{pOH} = -\log[\mathrm{OH}^-] \).

• In the exercise, we achieved a pOH of 3.00, allowing for the calculation \[ [\mathrm{OH}^-] = 10^{-3.00} = 1.0 \times 10^{-3} \text{ M}. \]

This reveals the amount of hydroxide ions contributing to the basic nature of the solution. Precisely knowing the \( [\mathrm{OH}^-] \) concentration helps to analyze a solution's overall alkalinity.

Base ionization

Base ionization refers to the process where a base reacts with water to form hydroxide ions. This occurs when molecules of the base, like hydrazine \( \mathrm{N}_2\mathrm{H}_4 \), interact with water molecules, resulting in the dissociation to yield hydroxide ions \( [\mathrm{OH}^-] \), as shown in the chemical equation:

• \( \mathrm{N}_2\mathrm{H}_4 + \mathrm{H}_2\mathrm{O} \rightleftharpoons \mathrm{N}_2\mathrm{H}_5^{+} + \mathrm{OH}^- \)

This process increases the solution's pH, rendering it basic.
Ionization strength is quantified by the ionization constant \( K_b \), which is pivotal in understanding the capacity of a base to dissociate in water. The strength and behavior of hydrazine as a weak base can be analyzed through its ionization constant, aiding in identifying its efficiency and reactivity within chemical systems.

Equilibrium concentration

Equilibrium concentration describes the concentrations of all species in a reaction that remains constant over time at equilibrium. In a base ionization reaction, like that of hydrazine, it's crucial to establish the equilibrium concentrations of the reactants and products to understand the system's stability.
Initially, hydrazine has a concentration of 0.139 M. After ionization, the concentration of various species at equilibrium is derived. The given exercise reveals:

• \([\mathrm{OH}^-] = x = 1.0 \times 10^{-3} \text{ M}\)
• \([\mathrm{N}_2\mathrm{H}_5^{+}] = x = 1.0 \times 10^{-3} \text{ M}\)
• \([\mathrm{N}_2\mathrm{H}_4] \approx 0.139\)

Balancing these values helps calculate the ionization constant \( K_b \). Comprehension of equilibrium concentration is imperative in evaluating how the system adapts over time, especially when subjected to changes.