Question

Which of the following is correct? 1) \(K_{\mathrm{a}}\) (weak acid) \(\times K_{\mathrm{b}}\) (conjugate weak base) \(=K_{\mathrm{v}}\) 2) \(K_{\mathrm{a}}\) (strong acid) \(\times K_{\mathrm{b}}\) (conjugate strong base) \(=K_{\mathrm{w}}\) 3) \(K_{\mathrm{a}}\) (weak acid) \(\times K_{\mathrm{b}}\) (weak base) \(=K_{\mathrm{T}}\) 4) \(K_{\mathrm{a}}\) (weak acid) \(\times K_{\mathrm{b}}\) (conjugate strong base) \(=K_{\mathrm{w}}\)

Short answer

Option 1: \( K_{\text{a}} \text{(weak acid)} \times K_{\text{b}} \text{(conjugate weak base)} = K_{\text{w}} \)

Step-by-step solution

Understand the Given Equations

To find the correct statement, first understand the given relationships. Here, \( K_{\text{a}} \) represents the acid dissociation constant and \( K_{\text{b}} \) represents the base dissociation constant. The product of these constants for a conjugate acid-base pair is equal to the ion-product constant for water, \( K_{\text{w}} \): \[ K_{\text{a}} \times K_{\text{b}} = K_{\text{w}} \].

Write the Expression

For any weak acid (HA) and its conjugate base (A⁻): \[ K_{\text{a}}(HA) \times K_{\text{b}}(A^-) = K_{\text{w}} \]. This is a well-known relationship in acid-base chemistry. \( K_{\text{w}} \) is a constant value of \( 1.0 \times 10^{-14} \) at 25°C.

Analyze the Provided Options

Next, analyze each option provided in the exercise: 1. \( K_{\text{a}} \text{(weak acid)} \times K_{\text{b}} \text{(conjugate weak base)} = K_{\text{w}} \) 2. \( K_{\text{a}} \text{(strong acid)} \times K_{\text{b}} \text{(conjugate strong base)} = K_{\text{w}} \) 3. \( K_{\text{a}} \text{(weak acid)} \times K_{\text{b}} \text{(weak base)} = K_{\text{T}} \) 4. \( K_{\text{a}} \text{(weak acid)} \times K_{\text{b}} \text{(conjugate strong base)} = K_{\text{w}} \).

Eliminate Incorrect Options

Eliminate the options that do not fit the correct expression: - Option 2 is incorrect because a strong acid's \( K_{\text{a}} \) is so large that it would not balance with \( K_{\text{w}} \). - Option 3 is incorrect because there is no relationship involving \( K_{\text{T}} \). - Option 4 is incorrect because a weak acid's conjugate base should also be weak.

Confirm the Correct Option

The correct option is Option 1 because it correctly represents the relationship \( K_{\text{a}} \text{(weak acid)} \times K_{\text{b}} \text{(conjugate weak base)} = K_{\text{w}} \).

Key concepts

Acid Dissociation Constant (Ka)

The acid dissociation constant, denoted as \( K_{\text{a}} \), is a crucial concept in understanding acid-base equilibrium. It measures the strength of an acid in water.
When an acid, \( HA \), dissolves in water, it dissociates into its ions: \( H^+ \) (hydrogen ion) and \( A^- \) (conjugate base). This process can be summarized with the equation:

\[ HA \rightleftharpoons H^+ + A^- \]

The acid dissociation constant \( K_{\text{a}} \) is given by the formula:

\[ K_{\text{a}} = \frac{[H^+][A^-]}{[HA]} \]

Here, \[ H^+ \] and \[ A^- \] are the concentrations of the dissociated ions and \[ HA \] is the concentration of the undissociated acid. A large \( K_{\text{a}} \) value indicates a strong acid, which dissociates extensively in water. Conversely, a small \( K_{\text{a}} \) value suggests a weak acid, where only a small fraction dissociates.

Base Dissociation Constant (Kb)

The base dissociation constant, represented by \( K_{\text{b}} \), is a measure of the strength of a base in water. When a base, \( B \), dissolves in water, it accepts a proton from water, forming its conjugate acid (\( HB^+ \)) and hydroxide ion (\( OH^- \)). This can be written as:

\[ B + H_2O \rightleftharpoons HB^+ + OH^- \]

The formula for the base dissociation constant \( K_{\text{b}} \) is:

\[ K_{\text{b}} = \frac{[HB^+][OH^-]}{[B]} \]

Here, \[ HB^+ \] and \[ OH^- \] are the concentrations of the products, and \[ B \] is the concentration of the undissociated base. A large \( K_{\text{b}} \) value indicates a strong base that dissociates well in water. A small \( K_{\text{b}} \) value points to a weak base with limited dissociation.

Ion-Product Constant for Water (Kw)

The ion-product constant for water, symbolized as \( K_{\text{w}} \), is a fundamental value in acid-base chemistry. It represents the product of the molar concentrations of the hydrogen ion and the hydroxide ion in water. This is expressed by the equation:

\[ K_{\text{w}} = [H^+][OH^-] \]

At 25°C, \( K_{\text{w}} \) has a constant value of \( 1.0 \times 10^{-14} \). This means that in pure water or in any aqueous solution for that matter, the product of the hydrogen ion concentration and the hydroxide ion concentration always equals \( 1.0 \times 10^{-14} \).

This relationship helps to understand the behavior of acids and bases in water. For example, if the concentration of \( H^+ \) is known, the concentration of \( OH^- \) can be calculated using \( K_{\text{w}} \), and vice versa. Additionally, the relationship between \( K_{\text{a}} \) and \( K_{\text{b}} \) complements \( K_{\text{w}} \), further illustrating the interconnected nature of these constants in maintaining the balance of acid-base systems.