Problem 87 - Chemistry The Central Science In Si Units

Question

Equal quantities of \(0.010 \mathrm{M}\) solutions of an acid HA and a base B are mixed. The pH of the resulting solution is 9.2 . (a) Write the chemical equation and equilibrium-constant expression for the reaction between HA and 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 \(K_{h}\) for B?

Step-by-Step Solution

Answer

(a) HA + B → A^- + BH^+. (b) Equilibrium constant ≈ 1.56 x 10^-6. (c) Kh for B = 1.25 x 10^-10.

Step 1: Write the Chemical Equation

When equal quantities of an acid (HA) and a base (B) are mixed, they will neutralize each other to form water and a salt. The equation for this neutralization reaction is \( HA + B \rightleftharpoons A^- + BH^+ \).

Step 2: Equilibrium Constant Expression

For the reaction \( HA + B \rightleftharpoons A^- + BH^+ \), the equilibrium constant expression is given by \( K_c = \frac{[A^-][BH^+]}{[HA][B]} \).

Step 3: Determine Reaction Solution from pH

From the pH of the solution (9.2), we can determine the hydroxide ion concentration: \([OH^-] = 10^{-(14 - 9.2)} = 10^{-4.8} = 1.58 \times 10^{-5} \) M. This indicates that the solution is basic and \( BH^+ \) formation occurs.

Step 4: Relate pH and Equilibria

Because the solution is basic, and knowing \( [OH^-] \), the equilibrium for \( B + H_2O \rightleftharpoons BH^+ + OH^- \) leads to a hydroxide concentration equal to 1.58 \times 10^{-5} \, M. This helps us calculate the equilibrium values required.

Step 5: Determine Equilibrium Constant from pKa

For the reverse reaction \( A^- + BH^+ \rightleftharpoons HA + B \), the equilibrium constant \( K_{eq} \) is connected to the dissociation constants by \( K_{eq} = \frac{K_b}{K_a} \). Thus, we calculate \( K_b = \frac{K_w}{K_a} \). For \( K_a = 8.0 \times 10^{-5} \), we find \( K_b = \frac{1.0 \times 10^{-14}}{8.0 \times 10^{-5}} \).

Step 6: Calculate Equilibrium Constant \( K_{eq}\)

Given \( K_a = 8.0 \times 10^{-5} \) for HA, \( K_w = 1.0 \times 10^{-14} \), and the above relation \( K_b \), we calculate \( K_b = 1.25 \times 10^{-10} \). Thus, the equilibrium constant \( K_{eq} \) for the overall neutralization reaction is \( K_{eq} = \frac{1.25 \times 10^{-10}}{8.0 \times 10^{-5}} = 1.56 \times 10^{-6} \).

Step 7: Calculate \( K_h \) for B

Using \( K_b \) for B, we already have: \( K_h (B) = \frac{1.0 \times 10^{-14}}{K_b} = 1.25 \times 10^{-10} \). Therefore, \( K_h \) reflects B's basicity.

Key Concepts

Acid-Base Reaction

Acid-base reactions are a fundamental type of chemical reaction. They involve the transfer of protons (H+) between reactants. Generally, an acid donates a proton to a base. In this exercise, we have an acid (HA) and a base (B). When they are mixed, HA donates a proton to B, forming A- and BH+. This specific reaction can be represented by the equation:

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

This equation illustrates a dynamic equilibrium, where the forward and reverse reactions occur at the same rate. This creates a balance between the reactants and products in a closed system. Understanding these reactions is key to grasping concepts such as pH and acidity, which measure the concentration of hydrogen ions in a solution.

Neutralization Reaction

A neutralization reaction happens when an acid and a base react to form water and a salt. However, in this specific scenario, the acid HA and the base B form ions A- and BH+ instead. The reaction can be written as:

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

Neutralization typically results in a change in the pH of the solution. When equal amounts of acid and base react, they neutralize each other's properties, either increasing or decreasing the pH based on their initial concentrations. For instance, in this situation, the reaction leans slightly towards the formation of BH+ ions, indicating that the resulting solution is basic. This is confirmed by the pH value of 9.2, which is above 7, indicating basicity.

pH Calculation

The pH of a solution is a measure of its acidity or basicity, defined as the negative logarithm of the hydrogen ion concentration.

• If the pH is below 7, the solution is acidic.
• If it's above 7, the solution is basic.
• A pH of exactly 7 indicates a neutral solution.

In this exercise, the pH of 9.2 suggests that the solution is basic, due to the higher concentration of hydroxide ions (\[ [OH^-] \]). You can determine the hydroxide concentration using the formula:

• \[ [OH^-] = 10^{-(14 - ext{pH})} \]

In this particular instance:

• \[ [OH^-] = 10^{-4.8} = 1.58 \times 10^{-5} \, \text{M} \]

Knowing the pH is crucial for understanding the extent of the neutralization reaction and the equilibrium that exists between the products and reactants.

Equilibrium Expression

The equilibrium expression relates the concentrations of reactants and products in a chemical equilibrium. For the reaction between HA and B,

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

The equilibrium constant expression (\( K_c \)) is given by:

• \[ K_c = \left(\frac{[A^-][BH^+]}{[HA][B]}\right) \]

The equilibrium constant indicates the extent to which a reaction proceeds to form products at equilibrium. A higher \( K_c \) value suggests that products are favored, while a lower value indicates reactants are more stable. This helps in predicting the direction of the reaction shift when conditions are altered. In this problem, using the given values and constants such as \( K_a \) and \( K_w \), we can determine the equilibrium constant for both the forward and reverse reactions, providing insight into the reaction dynamics.