Question

A fictional weak base, ZaH_ , reacts with water to yield a solution with a \(\mathrm{OH}^{-}\) ion concentration of \(2.68 \times 10^{-4}\) \(\mathrm{mol} / \mathrm{L}\) . The chemical equation for the reaction is \(\mathrm{ZaH}_{2}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(1) \rightleftharpoons \mathrm{ZaH}_{3}^{+}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq})\) If \(\left[\mathrm{ZaH}_{2}\right]\) at equilibrium is 0.0997 \(\mathrm{mol} / \mathrm{L}\) , what is the value of \(K_{\mathrm{b}}\) for \(\mathrm{ZaH}_{2} ?\)

Short answer

The base dissociation constant (Kb) for ZaH2 is approximately 7.25x10⁻⁶.

Step-by-step solution

Write the base dissociation expression for ZaH2

The base dissociation for ZaH2 can be written as: ZaH2(aq) + H2O(l) ↔ ZaH3+(aq) + OH-(aq)

Determine the expression for the Kb

The expression for the base dissociation constant (Kb) can be written as: \[ K_b = \frac{[ZaH_3^+][OH^-]}{[ZaH_2]} \]

Determine the change in concentrations

We are given the equilibrium concentration of ZaH2 ([ZaH2] = 0.0997 mol/L) and the concentration of OH- ions ([OH-] = 2.68x10⁻⁴ mol/L). Let's denote the initial concentration of ZaH3+ as [ZaH3+]. Now, the change in concentration of OH- is equal to the change in concentration of ZaH3+, so we can write: [ZaH3+] = [OH-] = 2.68x10⁻⁴ mol/L Also, the equilibrium concentration of ZaH2: [ZaH2] = 0.0997 mol/L

Find the Kb value

Now, substituting the equilibrium concentrations in the Kb expression and solving for Kb: \[ K_b = \frac{[ZaH_3^+][OH^-]}{[ZaH_2]} = \frac{(2.68 \times 10^{-4})(2.68 \times 10^{-4})}{0.0997} \] Calculating Kb: Kb ≈ 7.25x10⁻⁶ So, the base dissociation constant (Kb) for ZaH2 is approximately 7.25x10⁻⁶.

Key concepts

Weak Base

A weak base is a base that does not completely ionize or dissociate in water. Instead of fully splitting into its ions, it establishes an equilibrium between the un-dissociated and dissociated forms. This means that in a solution of a weak base, only a small fraction of the base molecules are actually converted into ions.
For the fictional weak base ZaH₂, it partially reacts with water, meaning that not all ZaH₂ molecules will become ZaH₃⁺ and OH⁻. This incomplete association is characteristic of weak bases.

• Weak bases have a low base dissociation constant ( K_b ), indicating limited ionization.
• They differ from strong bases, which dissociate completely in solution.
• The extent of ionization of weak bases affects their equilibrium position in chemical reactions.

Understanding the behavior of weak bases is crucial in predicting the pH of their solutions and how they interact in various chemical processes.

Equilibrium Concentration

Equilibrium concentration refers to the concentration of reactants and products in a chemical reaction once a state of balance, or equilibrium, is reached. At equilibrium, the rate at which the reactants convert to products equals the rate at which the products revert to reactants.
In the case of ZaH₂, the equilibrium concentration is the amount of ZaH₂, ZaH₃⁺, and OH⁻ present in the solution when the reaction has reached equilibrium. Given the information:

• The equilibrium concentration of the weak base ZaH₂ is 0.0997 mol/L.
• The equilibrium concentration of hydroxide ions ( OH^{-} ) is 2.68×10⁻⁴ mol/L.

Calculating equilibrium concentrations is essential to determine the extent of the reaction and to calculate the base dissociation constant ( K_b ), which further describes the propensity of the weak base to dissociate in water.

Chemical Reaction Equation

The chemical reaction equation represents the transformation that occurs when ZaH₂ reacts with water. For this weak base, it's essential to understand how the equation is written because it depicts the process of dissociation and equilibrium.
The balanced chemical reaction for ZaH₂ in water is depicted as: ZaH₂(aq) + H₂O(l) ↔ ZaH₃⁺(aq) + OH⁻(aq) . This equation illustrates several key points:

• **Reactants**: ZaH₂ and H₂O initially present in the reaction.
• **Products**: ZaH₃⁺ and OH⁻ formed as the base dissociates.
• **Reversibility**: The double arrow (↔) signifies that the reaction is not one-way but reaches equilibrium with both forward and reverse processes.

Writing and balancing chemical equations like this one helps predict the behavior of substances in reactions and is foundational in calculating equilibrium states and constants like K_b .

Hydroxide Ion Concentration

The hydroxide ion concentration ( [OH^-] ) is a crucial parameter in understanding the basicity or alkaline property of a solution. In a solution containing a weak base like ZaH₂, the [OH^-] tells us how many hydroxide ions are available.
For ZaH₂'s reaction with water, we've determined the hydroxide ion concentration is 2.68×10⁻⁴ mol/L at equilibrium. This ion concentration is essential for several reasons:

• It determines the pH level of the solution, indicating its basic nature.
• It helps in calculating the base dissociation constant ( K_b ).
• It's a direct outcome of the dissociation of the weak base ZaH₂ in water.

Understanding the concentration of OH^- ions allows us to relate the solution's properties to its chemical behavior and better predict how the base will act in different scenarios.