Question

Acids and bases can be thought of as chemical opposites (acids are proton donors, and bases are proton acceptors). Therefore, one might think that \(K_{\mathrm{a}}=1 / K_{\mathrm{b}}\) . Why isn't this the case? What is the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\) ? Prove it with a derivation.

Short answer

The relationship between the acid dissociation constant, \(K_a\), and the base dissociation constant, \(K_b\), is not simply inverses of each other because it involves the ion product constant of water, Kw. By analyzing the reactions of the acid and its conjugate base and using their equilibrium expressions, we can derive the relationship: \(K_aK_b = \frac{K_w[A^{-}]^{2}}{[HA]^2[H_{2}O]^2}\). This shows that the product of \(K_a\) and \(K_b\) depends on the squares of the concentrations of the acid, the conjugate base, and water, as well as the ion product constant for water, \(K_w\).

Step-by-step solution

Write down the reaction for the acid and its conjugate base

First, let's denote an acid as HA and its conjugate base as A^-. An acid dissociates into its conjugate base and a proton (H^+) in the following reaction: HA + H2O ↔ A^- + H3O^+ The base dissociation reaction involves the conjugate base (A^-) and water: A^- + H2O ↔ HA + OH^-

Write down the expressions for \(K_a\) and \(K_b\)

The expressions can be written based on the concentrations of the products and reactants at equilibrium: Ka = \(\frac{[A^{-}][H_{3}O^{+}]}{[HA][H_{2}O]}\) Kb = \(\frac{[HA][OH^{-}]}{[A^{-}][H_{2}O]}\)

Derive the relationship between \(K_a\) and \(K_b\) using the ion product constant of water

We know that the ion product constant of water, Kw, can be expressed as: Kw = [H3O^+][OH^-] We can solve the previous expressions of Ka and Kb for [H3O^+] and [OH^-] respectively: [H3O^+] = \(\frac{K_{a}[HA][H_{2}O]}{[A^{-}]}\) [OH^-] = \(\frac{K_{b}[A^{-}][H_{2}O]}{[HA]}\) Now, multiply the equations for [H3O^+] and [OH^-]: ([H3O^+])([OH^-]) = \(\frac{K_{a}K_{b}[HA]^{2}[H_{2}O]^{2}}{[A^{-}]^{2}}\) Since the ion product constant of water, Kw, equals [H3O^+][OH^-], we can rewrite the equation as: Kw = \(\frac{K_{a}K_{b}[HA]^{2}[H_{2}O]^{2}}{[A^{-}]^{2}}\) This equation shows the relationship between Ka and Kb, involving the ion product constant of water, Kw. However, it doesn't say that Ka equals the inverse of Kb.

Further simplify the relationship between \(K_a\) and \(K_b\)

To have a clearer understanding of their relationship, we will make the equation solve for the product of Ka and Kb: \(K_aK_b = \frac{K_w[A^{-}]^{2}}{[HA]^2[H_{2}O]^2}\) This equation shows that the product of \(K_a\) and \(K_b\) is not directly equal to 1, rather it depends on the squares of the concentrations of the acid, the conjugate base, and water, along with the ion product constant for water, \(K_w\). Thus, the relationship between \(K_a\) and \(K_b\) is more complex than \(K_{a} \neq 1 / K_{b}\).

Key concepts

Proton Donors

In the context of acid-base chemistry, proton donors are referred to as acids. An acid donates a proton, or an H⁺ ion, to another substance during a chemical reaction. This process is critical in understanding how acids work in various chemical environments. For example, when hydrochloric acid (HCl) is dissolved in water, it donates a proton to a water molecule, forming hydronium ions (H₃O⁺).

Understanding proton donation is essential because it is the backbone of many reactions, not only in chemistry but also in biology and environmental science. It's important to note that the ability to donate a proton and form its conjugate base is what characterizes a substance as an acid.

• Characteristics of Proton Donors: They release H⁺ ions into the solution.
• They have a conjugate base formed after donating the proton.
• The strength of an acid can be determined by how easily it donates its proton.

In essence, a proton donor plays a significant role in determining the acidic nature and behavior of a substance.

Proton Acceptors

In a chemical reaction, proton acceptors are substances that receive protons from donors. These are commonly referred to as bases. The ability of a base to accept a proton is crucial to its function and understanding the reaction dynamics.

For instance, when ammonia (NH₃) is added to water, it accepts a proton from water, resulting in the formation of ammonium ions (NH₄⁺) and hydroxide ions (OH^-). This showcases how bases work to change the chemical makeup of the solution.

• Characteristics of Proton Acceptors: They capture H⁺ ions during a reaction.
• The product of accepting a proton is a conjugate acid.
• The ease of proton acceptance determines the strength and functionality of the base.

Understanding proton acceptors allows one to see the broader picture of acid-base reactions and how equilibrium is maintained in solutions.

Equilibrium Constants

Equilibrium constants (\(K_\mathrm{a}\) and \(K_\mathrm{b}\)) are vital in understanding acid-base reactions. They represent the balance of concentrations of reactants and products in a solution at equilibrium.

For acids, the equilibrium constant, \(K_\mathrm{a}\), reflects its strength and ability to donate protons. It is calculated using the formula:\[K_\mathrm{a} = \frac{[A^-][H_3O^+]}{[HA]} \]
Similarly, the base equilibrium constant, \(K_\mathrm{b}\), provides insight into a base's strength to accept protons:\[K_\mathrm{b} = \frac{[B^+][OH^-]}{[B]}\]
In these formulas, \([A^-]\), \([H_3O^+]\), \([HA]\), \([B^+]\), and \([OH^-]\) represent the concentrations of the ions or molecules at equilibrium.

• Key Points of Equilibrium Constants: Higher values of \(K_\mathrm{a}\) or \(K_\mathrm{b}\) indicate stronger acids or bases respectively.
• The relationship between \(K_\mathrm{a}\) and \(K_\mathrm{b}\) is intricate, showing that their product is equal to the ion product of water, \(K_\mathrm{w}\).
• The constants provide critical insight into how reactions shift under different conditions.

Equilibrium constants are the keys to predicting and understanding reaction behaviors in chemical settings.