Problem 79 - Chemistry: The Central Science

Question

Equal quantities of \(0.010 \mathrm{M}\) solutions of an acid \(\mathrm{HA}\) and a base \(B\) are mixed. The \(p H\) of the resulting solution is 9.2. (a) Write the equilibrium equation and equilibriumconstant expression for the reaction between \(\mathrm{HA}\) and \(\mathrm{B}\). (b) If \(K_{a}\) for HA is \(8.0 \times 10^{-5}\), what is the value of the equilibrium constant for the reaction between HA and B? (c) What is the value of \(K_{b}\) for \(B\) ?

Step-by-Step Solution

Answer

The equilibrium equation for the reaction between HA and B is: \(HA + B \rightleftharpoons A^- + HB\). The value of the equilibrium constant for the reaction between HA and B is 12.67. The value of \(K_b\) for B is \(1.25 \times 10^{-10}\).

Step 1: Write the equilibrium equation and equilibrium constant expression for the reaction between HA and B.

The reaction between the weak acid HA and the weak base B can be represented as: \[HA + B \rightleftharpoons A^- + HB\] Since the weak acid HA ionizes as: \[HA \rightleftharpoons H^+ + A^-\] It has the acid dissociation constant \(K_a\), given by: \[K_a = \frac{[H^+][A^-]}{[HA]}\] Similarly, the weak base B ionizes as: \[B + H_2O \rightleftharpoons HB^+ + OH^-\] It has the base dissociation constant \(K_b\), given by: \[K_b = \frac{[HB^+][OH^-]}{[B]}\]

Step 2: Calculate the concentration of HA and A- after the reaction occurs.

The initial concentrations of HA and B are both 0.010 M. Since equal quantities of HA and B are mixed, the reaction will proceed to completion. This means that the final concentration of A- and HB will be equal to the initial concentration of HA and B: \[[A^-] = [HB] = 0.010\text{ M}\] Since the pH of the resulting solution is 9.2, we can find the concentration of H+: \[[H^+] = 10^{-pH} = 10^{-9.2} = 6.31 \times 10^{-10}\text{ M}\]

Step 3: Calculate the value of the equilibrium constant for the reaction between HA and B.

Since we have the value of \(K_a\) and the concentrations of the species involved, we can solve for the equilibrium constant expression. We can rearrange the Ka expression to find the concentration of HA: \[[HA] = \frac{[H^+][A^-]}{K_a}\] Plugging in the values, we get: \[[HA] = \frac{(6.31 \times 10^{-10})(0.010)}{8.0 \times 10^{-5}} = 7.89 \times 10^{-6}\text{ M}\] Now, we can find the equilibrium constant for the reaction between HA and B, which can be written as: \[K_{eq} = \frac{[A^-][HB]}{[HA][B]}\] Substituting the values, we get: \[K_{eq} = \frac{(0.010)(0.010)}{(7.89\times 10^{-6})(0.010)} = 12.67\]

Step 4: Find the value of Kb for B.

Since \(K_a \times K_b = K_w\) (where \(K_w\) is the ion product constant of water), we can find the value of Kb for B: \[K_b = \frac{K_w}{K_a}\] Since \(K_w\) is \(1.0 \times 10^{-14}\), we have: \[K_b = \frac{1.0 \times 10^{-14}}{8.0 \times 10^{-5}} = 1.25 \times 10^{-10}\] So, the value of Kb for B is \(1.25 \times 10^{-10}\).

Key Concepts

Acid-Base Reactions

Acid-base reactions are a fundamental concept in chemistry, where an acid and a base react in a neutralization process. When an acid, like HA (a weak acid), meets a base, such as B (a weak base), they interact to form their conjugate base and acid, respectively. This particular reaction can be represented as follows:

• \[HA + B \rightleftharpoons A^- + HB\]

In this equilibrium, HA donates a proton to B, resulting in the formation of the products, A^- (the conjugate base of the acid) and HB (the conjugate acid of the base). Given that both HA and B are weak, this type of reaction typically reaches an equilibrium state rather than proceeding to completion. The balance between the reactants and the products defines how acidic or basic the resultant solution will be.
Understanding the behavior of weak acid-weak base reactions is crucial for predicting pH changes in solutions, as the reaction does not completely neutralize pH but rather establishes an equilibrium governed by their respective dissociation constants, \(K_a\) for the acid and \(K_b\) for the base.

Equilibrium Constant

The equilibrium constant is a vital concept that indicates the ratio of concentrations of products to reactants at equilibrium. For acid-base reactions, such as between HA and B, the equilibrium constant \(K_{eq}\) can be expressed as:

• \[K_{eq} = \frac{[A^-][HB]}{[HA][B]}\]

The magnitude of \(K_{eq}\) provides insight into the position of equilibrium. If \(K_{eq}\) is large, it suggests a greater concentration of products relative to reactants, implying the reaction tends to favor the formation of products (equilibrium lies to the right). Conversely, a small \(K_{eq}\) suggests that equilibrium favors the reactants.
For weak acids and bases, the values of \(K_a\) and \(K_b\) are relatively small, indicating limited dissociation. The relationship between \(K_a\) and \(K_b\) is particularly important when calculating ionization in water (\(K_w\)), following the equation:

• \(K_a \times K_b = K_w \)

With \(K_w\) as the ion product constant of water, typically \(1.0 \times 10^{-14}\) at 25°C, this equation helps find one constant if the other is known, aiding in understanding the acid or base strength more thoroughly.

pH Calculation

The pH measurement is an essential part of evaluating the acidity or basicity of a solution. Simply put, pH is calculated as the negative logarithm of the hydrogen ion concentration:

• \[pH = -\log [H^+]\]

In the context of our acid-base reaction, this information is vital for determining the resultant pH after HA and B reach equilibrium. Given an initial pH of 9.2, the concentration of hydrogen ions is calculated as \([H^+] = 10^{-9.2} = 6.31 \times 10^{-10} \text{ M}\). This calculation shows that the solution is basic as pH values exceeding 7 indicate basic conditions, which aligns with the fact that B, being a base, would increase the solution's pH.
Understanding pH further aids in calculating other essential chemical properties and predicting behaviors of different chemical species in solution. In equilibrium and titration experiments, these calculations help students and scientists easily estimate reaction tendencies and outcomes. It is critical to remember the logarithmic nature of the pH scale as it means a small change in pH represents a significant shift in hydrogen ion concentration.