Question

What is the relationship between the acid ionization constant for a weak acid \(\left(K_{\mathrm{a}}\right)\) and the base ionization constant for its conjugate base \(\left(K_{\mathrm{b}}\right) ?\)

Short answer

The relationship between the acid ionization constant \(K_{\mathrm{a}}\) for a weak acid and the base ionization constant \(K_{\mathrm{b}}\) for its conjugate base is given by the equation \(K_{\mathrm{a}} \times K_{\mathrm{b}} = K_{\mathrm{w}}\), where \(K_{\mathrm{w}}\) is the autoionization constant of water.

Step-by-step solution

Understanding the Concepts

Recognize that the acid ionization constant, represented as \(K_{\mathrm{a}}\), is a measure of the strength of an acid in solution. It represents the equilibrium constant for the reaction of the acid donating a proton to water. Similarly, the base ionization constant, \(K_{\mathrm{b}}\), describes the strength of a base to accept a proton in a water solution. The relationship between the two constants involves the product of \(K_{\mathrm{a}}\) for a weak acid and \(K_{\mathrm{b}}\) for its conjugate base being equal to the autoionization constant of water \(K_{\mathrm{w}}\), which at 25°C is \(1.0 \times 10^{-14}\).

Writing the Equilibrium Expressions

Write down the equilibrium expression for the dissociation of a weak acid (HA) in water: \({HA + H_2O \rightleftharpoons A^{-} + H_3O^{+}\) with \(K_{\mathrm{a}} = \frac{[A^{-}][H_3O^{+}]}{[HA]}\). Also write the equilibrium expression for the reaction of the conjugate base (\(A^{-}\)) with water: \({A^{-} + H_2O \rightleftharpoons HA + OH^{-}\) with \(K_{\mathrm{b}} = \frac{[HA][OH^{-}]}{[A^{-}]}\).

Relating \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\) through \(K_{\mathrm{w}}\)

Establish the relationship using the water autoionization equilibrium \({H_2O + H_2O \rightleftharpoons H_3O^{+} + OH^{-}\) which has the constant \(K_{\mathrm{w}} = [H_3O^{+}][OH^{-}] = 1.0 \times 10^{-14}\) at 25°C. Since the product of the concentrations of \(H_3O^{+}\) and \(OH^{-}\) is constant for water, it follows that \(K_{\mathrm{a}} \times K_{\mathrm{b}} = K_{\mathrm{w}}\). This relationship indicates that the stronger the acid (higher \(K_{\mathrm{a}}\)), the weaker its conjugate base (lower \(K_{\mathrm{b}}\)), and vice versa.

Key concepts

Acid Ionization Constant

Understanding the acid ionization constant, denoted as \(K_{\mathrm{a}}\), is essential for grasping the nature of weak acids. This constant quantifies how much an acid can dissociate in a solution to form a proton, or hydrogen ion, \(H^+\), and its conjugate base. The mathematical expression for \(K_{\mathrm{a}}\) is defined as the ratio of the product of the concentrations of the hydrogen ion and the conjugate base to the concentration of the undissociated acid.

For the generic acid dissociation reaction:\[ HA \rightarrow H^+ + A^- \]
\(K_{\mathrm{a}}\) is expressed as:\[ K_{\mathrm{a}} = \frac{[H^+][A^-]}{[HA]} \]
The higher the value of \(K_{\mathrm{a}}\), the stronger the acid, indicating a greater tendency to donate a proton. In essence, \(K_{\mathrm{a}}\) reflects the acid's strength and its propensity to participate in chemical reactions.

Base Ionization Constant

The counterpart to an acid's ionization is the base ionization constant, labeled as \(K_{\mathrm{b}}\). This constant measures the strength of a base in its ability to accept a proton. For a weak base \(B\), which accepts a proton to form its conjugate acid \(HB^+\), the reaction is typically written as:

\[ B + H_2O \rightleftharpoons HB^+ + OH^- \]
Here, the base ionization constant is calculated using the formula:\[ K_{\mathrm{b}} = \frac{[HB^+][OH^-]}{[B]} \]
Similar to the acid constant, a higher \(K_{\mathrm{b}}\) value signifies a stronger base. Understanding both \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\) allows students to predict the behavior of substances in an aqueous environment, essentially equipping them to better comprehend their chemistry curriculum.

Conjugate Acid-Base Pair

The concept of a conjugate acid-base pair is fundamental in acid-base chemistry. It manifests when an acid donates a proton, becoming its conjugate base, or when a base accepts a proton, becoming its conjugate acid. Conjugate pairs are always found on the opposite sides of a chemical equation representing acid-base reactions.

For example, if we take the dissociation of acetic acid (CH3COOH), the equation is:\[ CH3COOH + H2O \rightleftharpoons CH3COO^- + H3O^+ \]
In this case, \(CH3COOH\) is the acid and \(CH3COO^-\) is its conjugate base. The water molecule \(H2O\) acts as a base here, and \(H3O^+\) is its conjugate acid. This relationship is reciprocal; thus, understanding this concept is pivotal for predicting the outcomes of acid-base reactions and for making sense of the equilibrium that exists in such solutions.

Autoionization of Water

Water's remarkable property of self-ionization, also termed autoionization, is represented by the equilibrium constant \(K_{\mathrm{w}}\). At room temperature (25°C), the water molecules undergo a reversible reaction where two water molecules react to produce a hydronium ion (\(H3O^+\)) and a hydroxide ion (\(OH^-\)). This is a crucial concept since it provides the baseline for the ion product of water.

The chemical reaction is written as:\[ 2H_2O \rightleftharpoons H3O^+ + OH^- \]
And the associated equilibrium expression is:\[ K_{\mathrm{w}} = [H3O^+][OH^-] \]
At 25°C, \(K_{\mathrm{w}}\) is always \(1.0 \times 10^{-14}\). This value is vital for understanding the interplay between the concentrations of hydronium and hydroxide ions in any aqueous solution, which in turn is related to pH and pOH values. The concept of autoionization is a cornerstone in comprehending acid-base balances within chemistry.