Problem 70 - Chemistry: Principles and Reactions

Question

A \(0.143 M\) solution of the weak base \(\mathrm{RNH}_{2}\) has \(\mathrm{pH} 9.11\) at \(25^{\circ} \mathrm{C}\). What is \(\Delta G^{\circ}\) for the dissociation of the weak base in water? $$ \mathrm{RNH}_{2}(a q)+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{RNH}_{3}{\underline{\phantom{xx}}}^{+}(a q)+\mathrm{OH}^{-}(a q) $$

Step-by-Step Solution

Answer

Answer: The standard Gibbs free energy (ΔG°) for the dissociation of the weak base RNH2 in water is approximately 61.0 kJ/mol.

Step 1: Determine the concentration of OH- ions

Since we are given the pH of the solution (9.11), we can determine the concentration of OH- ions by using the relationship between pH and pOH (pH + pOH = 14 at 25°C). pOH = 14 - pH pOH = 14 - 9.11 pOH = 4.89 Now, we can find the concentration of OH- ions using pOH = -log10[OH-]: [OH-] = 10^(-pOH) = 10^(-4.89) ≈ 1.29 × 10^(-5) M

Step 2: Determine K_b (base dissociation constant) for RNH2

We can write the expression for K_b using the dissociation formula provided in the exercise: K_b = [RNH3+][OH-] / [RNH2] We know that the initial concentration of RNH2 ([RNH2]_initial) is 0.143 M and [OH-] is 1.29 × 10^(-5) M at equilibrium. Assuming x M of RNH2 dissociates at equilibrium: [RNH3+]_eq = [OH-]_eq = 1.29 × 10^(-5) + x, as equal amounts of RNH3+ and OH- are formed [RNH2]_eq = 0.143 - x, the initial concentration minus the amount that dissociated K_b = [(1.29 × 10^(-5) + x)(1.29 × 10^(-5) + x)] / (0.143 - x) Since x is very small compared to 1.29 × 10^(-5) and 0.143, we can approximate and simplify the expression as follows: K_b ≈ (1.29 × 10^(-5))^2 / 0.143 K_b ≈ 1.14 × 10^(-11)

Step 3: Calculate ΔG° using the relationship between ΔG° and K_b

The relationship between ΔG° and K is given by: ΔG° = -RT ln(K) Where R is the gas constant (8.314 J/(mol·K)), T is the temperature in Kelvin (298 K in this case), and K is the equilibrium constant (K_b in this case). ΔG° = - (8.314 J/(mol·K)) × (298 K) × ln(1.14 × 10^(-11)) ΔG° ≈ 61.0 kJ/mol The standard Gibbs free energy for the dissociation of the weak base RNH2 in water is approximately 61.0 kJ/mol.

Key Concepts

Weak Base

A weak base is a substance that does not completely ionize in solution. This means that only a small fraction of the molecule reacts with water to produce hydroxide ions (OH^−) and the conjugate acid of the base. By understanding this concept, you can predict the extent of ionization in weak base solutions by observing how weakly or strongly the base dissociates.

When you dissolve a weak base like RNH2 in water, it reaches an equilibrium state described by:

• The base RNH2 partially dissociates to form RNH3^+ and OH^−.
• Not all molecules of the base convert to OH^- ions.
• At equilibrium, the concentrations of the base, its ions, and hydroxide ions remain constant.

This equilibrium characteristic is crucial in distinguishing weak bases from strong bases, which dissociate almost completely in water.

Dissociation Constant

The dissociation constant, specifically referred to as the base dissociation constant (K_b) for weak bases, measures the strength of the base in a solution. It tells us the extent to which the base reacts with water to form ions.

For the base RNH2, its dissociation can be expressed as:

• K_b = \([ \text{RNH}_3^+][ \text{OH}^- ] / [\text{RNH}_2]\)
• This constant is derived from the equilibrium concentrations of RNH3^+, OH^-, and RNH2 in the solution.

The value of K_b is crucial. A smaller K_b value suggests a weaker base that ionizes less in solution. It informs us how much of the base remains un-ionized versus the amount that has dissociated. The relationship and calculation of K_b allow us to further explore the nature and strength of chemical equilibria in weak base solutions.

pH and pOH

Understanding pH and pOH is essential when dealing with acid-base solutions. These terms help you track the concentration of hydrogen ions (H^+) and hydroxide ions (OH^−), respectively.

Here's a simple way to remember:

• **pH** measures the acidity of the solution; lower values indicate more acidic solutions.
• **pOH** measures the basicity; lower values show more alkaline solutions.
• At 25°C, these two values must equate to 14: pH + pOH = 14.

If you know the pH, you can easily find the pOH. For the solution of RNH2, given pH = 9.11, the pOH is calculated as 14 - 9.11 = 4.89. This indicates that the solution is basic, which correlates with the presence of a weak base.

Equilibrium Constant

The equilibrium constant (K) is essential for understanding reactions at equilibrium, determining the direction in which a reaction naturally proceeds. For weak bases, you calculate the base dissociation equilibrium constant K_b.

The formula to find the equilibrium constant derived from the base's interactions is:

• K = \[ \frac{[\text{Products}]}{[\text{Reactants}]} \]
• Signing this to the base dissociation gives: K_b = \([ \text{RNH}_3^+ ][ \text{OH}^-] / [\text{RNH}_2]\)
• A known K_b value helps calculate other properties, like Gibbs Free Energy (ΔG°), using the formula:ΔG° = -RT ln(K)

The equilibrium constant shows whether products or reactants are favored in a reaction at equilibrium. It serves as a fundamental principle in predicting and quantifying the extent of chemical reactions.