Concept Explanation: Equilibrium Constant for Acid-Base Reaction

The given reaction is: NH₃(aq) + HCl(aq) ⇌ NH₄⁺(aq) + Cl^–(aq)

This represents the neutralization reaction between ammonia (a weak base) and hydrochloric acid (a strong acid). To find the equilibrium constant for this reaction, we need to understand how it relates to acid-base dissociation constants.

Step-by-Step Solution:

Step 1: Write the equilibrium expression for the given reaction:

$K=\frac{\left[{\mathrm{NH}}_4^{+}\right]\left[{\mathrm{Cl}}^{–}\right]}{\left[{\mathrm{NH}}_3\right]\left[\mathrm{HCl}\right]}$

Step 2: Recognize that HCl is a strong acid and completely dissociates: HCl → H⁺ + Cl^–

Since HCl dissociation is complete, we can express the reaction as:

NH₃(aq) + H⁺(aq) ⇌ NH₄⁺(aq)

Step 3: Write the equilibrium constant for this proton transfer reaction:

$K=\frac{\left[{\mathrm{NH}}_4^{+}\right]}{\left[{\mathrm{NH}}_3\right]\left[H^{+}\right]}$

Step 4: Recognize that this is the reciprocal of the acid dissociation constant for NH₄⁺:

$K_a\left({\mathrm{NH}}_4^{+}\right)=\frac{\left[{\mathrm{NH}}_3\right]\left[H^{+}\right]}{\left[{\mathrm{NH}}_4^{+}\right]}$

Therefore, $K=\frac{1}{K_a}\left({\mathrm{NH}}_4^{+}\right)$

Step 5: Alternatively, we can relate this to the base dissociation constant of NH₃ and K_w:

$K_b\left({\mathrm{NH}}_3\right)=\frac{\left[{\mathrm{NH}}_4^{+}\right]\left[{\mathrm{OH}}^{–}\right]}{\left[{\mathrm{NH}}_3\right]}$

And since $K_a\left({\mathrm{NH}}_4^{+}\right)=\frac{K_w}{K_b}$, we get:

$K=\frac{1}{K_a}=\frac{K_b}{\left(}$

Final Answer: Both expressions are correct and equivalent. Therefore, the correct option is: Both (1) and (2)

Related Topics:

• Acid-base equilibria
• Weak acid and weak base dissociation
• Neutralization reactions
• Relationship between K_a, K_b, and K_w
• Conjugate acid-base pairs

Important Formulae:

For any conjugate acid-base pair:

$K_a\times K_b=K_w={10}^{-14}\text{at 25°C}$

Equilibrium constant for acid-base neutralization:

$K=\frac{1}{K_a}\text{ or }K=\frac{K_b}{K_w}$