Question

For the reactions given below the correct relations is: \(\mathrm{HCl}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{Cl}^{\mathrm{c}}\left(\mathrm{K}_{\mathrm{a}}\right)\) \(\mathrm{Cl}^{-}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{HCl}+\mathrm{OH}^{-}\left(\mathrm{K}_{\mathrm{b}}\right)\) (a) \(\mathrm{K}_{\mathrm{b}}=\mathrm{K}_{\mathrm{a}}\) (b) \(\mathrm{K}_{\mathrm{w}}=\mathrm{K}_{\mathrm{a}} \cdot \mathrm{K}_{\mathrm{b}}\) (c) \(\frac{\mathrm{K}_{\mathrm{a}}}{\mathrm{K}_{\mathrm{b}}}=\mathrm{K}_{\mathrm{w}}\) (d) \(\mathrm{K}_{\mathrm{b}}=\frac{1}{\mathrm{~K}_{\mathrm{a}}}\)

Short answer

Option (b) \(\mathrm{K}_{\mathrm{w}}=\mathrm{K}_{\mathrm{a}} \cdot \mathrm{K}_{\mathrm{b}}\) is correct.

Step-by-step solution

Understand the reactions

The provided reactions involve the ionization of hydrochloric acid (HCl) in water and the reverse reaction using chloride ion (Cl⁻). These reactions exhibit the equilibrium constants \(K_a\) for the acid dissociation and \(K_b\) for the base ionization.

Recognize Relevant Equilibrium Constants

For the first reaction, the equilibrium constant \(K_a\) represents the strength of the acid (HCl) dissociating in water. The second reaction's constant \(K_b\) measures the base (Cl⁻) gaining a proton from water to reform HCl.

Apply the Relation of Kw

Recall that for water, the product of acid dissociation constant \(K_a\) and base dissociation constant \(K_b\) is equal to the ion product of water, \(K_w\). Mathematically, this is expressed as \(K_w = K_a \times K_b\).

Match with Given Options

From step 3, we found that \(K_w = K_a \times K_b\). Comparing this with the given options, option (b), \(\mathrm{K}_{\mathrm{w}}=\mathrm{K}_{\mathrm{a}} \cdot \mathrm{K}_{\mathrm{b}}\), matches this relationship.

Key concepts

Acid-Base Reactions

Acid-base reactions are fundamental to understanding many processes in chemistry and biology. In these reactions, an acid donates a proton ( H^+ ) to a base. For example, when hydrochloric acid ( HCl ), a strong acid, is added to water, it dissociates completely to donate a proton to water, forming hydronium ion ( H_3O^+ ) and chloride ion ( Cl^- ).
In this context:

• The acid, HCl , donates a proton, making water act as a base.
• The resulting hydronium ions increase the acidity of the solution.

On the other hand, the reverse reaction involves a base, in this case, the chloride ion ( Cl^- ), attempting to re-gain the proton to form HCl , showing how bases react by accepting protons. These reactions highlight the dynamic nature of acid-base chemistry and the concept of chemical equilibrium, where forward and reverse reactions occur at the same rate, resulting in stable concentrations of reactants and products.
Understanding acid-base reactions allows you to predict the behavior of solutions, calculate pH, and explore buffer systems in various fields ranging from industrial chemical processes to biological systems.

Equilibrium Constants

Equilibrium constants, such as K_a and K_b, are essential for quantifying the equilibrium position of acid-base reactions. These constants indicate the extent to which a reaction proceeds to form products under a given set of conditions.
For an acid denoted HA dissociating in water, K_a represents the equilibrium constant for this dissociation:
\[K_a = \frac{[H_3O^+][A^-]}{[HA]}\]
Here, [A^-] is the concentration of the conjugate base, and [H_3O^+] the concentration of hydronium ions.
For a base ionizing to form OH^- and its conjugate acid, K_b is expressed as:
\[K_b = \frac{[OH^-][HA]}{[A^-]}\]
These values are crucial for:

• Predicting the direction of the chemical reaction.
• Calculating pH and pOH of solutions.
• Understanding the strength of acids and bases.

By determining K_a and K_b, chemists can infer the relative strengths of conjugate acid-base pairs, an important aspect of many chemical applications. The relationship K_w = K_a \times K_b also links these constants to the ionization of water, providing a comprehensive understanding of solution behavior.

Ionization of Water

The ionization of water is a fundamental concept that relates to acid-base reactions and equilibrium constants. Pure water dissociates slightly to form hydronium (H_3O^+) and hydroxide ions (OH^-):
\[2H_2O \rightleftharpoons H_3O^+ + OH^-\]
The equilibrium constant for this pure water auto-ionization is denoted by K_w and is a relatively small value at room temperature, approximately 1.0 \times 10^{-14}.
The relationship between the equilibrium constants of acid and base (K_a and K_b respectively) for conjugate acid-base pairs is tied to K_w in the equation:
\[K_w = K_a \times K_b\]
Key points include:

• The low value of K_w indicates water is predominantly non-ionized.
• It serves as the basis for calculating the pH and pOH of solutions.
• Understanding this concept helps in grasping the auto-protonation/de-protonation balance in aqueous solutions.

Ionization of water explains the intrinsic behavior of water in acid-base chemistry and is crucial for understanding the mechanisms of solution chemistry in both natural environments and laboratory settings.