Problem 62 - Chemistry: The Central Science

Question

Determine the \(\mathrm{pH}\) of each of the following solutions \(\left(K_{a}\right.\) and \(K_{b}\) values are given in Appendix D): (a) \(0.095 M\) hypochlorous acid, (b) \(0.0085 \mathrm{M}\) phenol, (c) \(0.095 \mathrm{M}\) hydroxylamine.

Step-by-Step Solution

Answer

(a) Hypochlorous acid: Using the given \(K_{a}\) value \(3.5 \times 10^{-8}\), we find the [H₃O⁺] concentration is approximately \(1.69 \times 10^{-5} M\). Thus, the pH is around 4.77. (b) Phenol: Using the given \(K_{a}\) value \(1.3 \times 10^{-10}\), we find the [H₃O⁺] concentration is approximately \(3.32 \times 10^{-6} M\). Thus, the pH is around 5.48. (c) Hydroxylamine: Using the given \(K_{b}\) value \(1.1 \times 10^{-8}\), we find the [OH⁻] concentration is approximately \(3.03 \times 10^{-5} M\). The pOH is around 4.52, indicating a pH of approximately 9.48.

Step 1: (a) Hypochlorous Acid Solution

1. Write the ionization equilibrium equation for hypochlorous acid (HClO) in water: \[HClO + H_{2}O \leftrightharpoons H_{3}O^{+} + ClO^{-}\] 2. Write the equilibrium expression using the \(K_{a}\) value: \[K_{a}=\frac{[H_{3}O^{+}][ClO^{-}]}{[HClO]}\] 3. Set up an ICE (initial, change, equilibrium) table to find the equilibrium concentrations: \[ \begin{array}{c | c c c} & HClO & H_{3}O^{+} & ClO^{-} \\ \hline \text{Initial} & 0.095 & 0 & 0 \\ \text{Change} & -x & +x & +x \\ \text{Equilibrium} & 0.095-x & x & x \end{array}\] 4. Substitute equilibrium concentrations into the expression and solve for x: \[K_{a}=\frac{x^2}{0.095-x}\] 5. Use Appendix D to find the \(K_{a}\) value for hypochlorous acid and solve for x, the [H₃O⁺] concentration. 6. Calculate the pH: \[pH = -log_{10}[H_{3}O^{+}]\]

Step 2: (b) Phenol Solution

1. Write the ionization equilibrium equation for phenol (C₆H₅OH) in water: \[ C_{6}H_{5}OH + H_{2}O \leftrightharpoons H_{3}O^{+} + C_{6}H_{5}O^{-}\] 2. Write the equilibrium expression using the \(K_{a}\) value: \[K_{a}=\frac{[H_{3}O^{+}][C_{6}H_{5}O^{-}]}{[C_{6}H_{5}OH]}\] 3. Set up an ICE table to find the equilibrium concentrations: \[ \begin{array}{c | c c c} & C_{6}H_{5}OH & H_{3}O^{+} & C_{6}H_{5}O^{-} \\ \hline \text{Initial} & 0.0085 & 0 & 0 \\ \text{Change} & -x & +x & +x \\ \text{Equilibrium} & 0.0085-x & x & x \end{array}\] 4. Substitute equilibrium concentrations into the expression and solve for x: \[K_{a}=\frac{x^2}{0.0085-x}\] 5. Use Appendix D to find the \(K_{a}\) value for phenol and solve for x, the [H₃O⁺] concentration. 6. Calculate the pH: \[pH = -log_{10}[H_{3}O^{+}]\]

Step 3: (c) Hydroxylamine Solution

1. Write the ionization equilibrium equation for hydroxylamine (NH₂OH) in water: \[ NH_{2}OH + H_{2}O \leftrightharpoons NH_{3}OH^{+} + OH^{-}\] 2. Write the equilibrium expression using the \(K_{b}\) value: \[K_{b}=\frac{[NH_{3}OH^{+}][OH^{-}]}{[NH_{2}OH]}\] 3. Set up an ICE table to find the equilibrium concentrations: \[ \begin{array}{c | c c c} & NH_{2}OH & NH_{3}OH^{+} & OH^{-} \\ \hline \text{Initial} & 0.095 & 0 & 0 \\ \text{Change} & -x & +x & +x \\ \text{Equilibrium} & 0.095-x & x & x \end{array}\] 4. Substitute equilibrium concentrations into the expression and solve for x: \[K_{b}=\frac{x^2}{0.095-x}\] 5. Use Appendix D to find the \(K_{b}\) value for hydroxylamine and solve for x, the [OH⁻] concentration. 6. Calculate the pOH using the [OH⁻] concentration: \[pOH = -log_{10}[OH^{-}]\] 7. Calculate the pH using the relation between pH and pOH: \[pH = 14 - pOH\]

Key Concepts

Acid-Base Equilibrium

Acid-base equilibrium revolves around the concept of acids and bases reacting with each other, and how these reactions establish a balance between reactants and products. In aqueous solutions, acids tend to donate protons ( H⁺ ) while bases tend to accept them. In our context, the equilibrium involves weak acids like hypochlorous acid and phenol, or weak bases like hydroxylamine dissociating partially in water. When these substances enter the water, they create a slight ionic environment by producing ions such as H₃O⁺ or OH⁻ .

• Acids generate H₃O⁺ ions increasing the hydronium concentration, contributing to the overall acidity of the solution.
• Bases produce OH⁻ ions, increasing the hydroxide concentration, which increases the basicity of the solution.

An equilibrium is reached when the forward reaction rate of dissociation equals the reverse reaction rate of recombination, meaning concentrations remain stable over time.

Ionization Constant

The ionization constant, represented as K_{a} for acids and K_{b} for bases, is a pivotal parameter in understanding the extent of ionization of an acid or base in solution. It reflects the strength of an acid or base:

• High K_{a} or K_{b} values indicate stronger acids or bases that ionize more completely in water.
• Low values mean weaker ionization.

For a weak acid like hypochlorous acid, its K_{a} is relatively low, showing that only a small portion of the acid ionizes in water. Comparatively, hydroxylamine's K_{b} provides insight into its ability to release OH⁻ ions. Knowing these constants from reference materials (like Appendix D in textbooks) allows us to calculate the equilibrium concentrations required for calculating pH.

ICE Table

An ICE table is a valuable tool used to visually organize and solve equilibrium problems by breaking them down into Initial, Change, and Equilibrium concentrations. This step-by-step approach helps in quantifying the dissociation of acids and bases in solutions by representing initial concentrations, amount of change, and resulting equilibrium concentrations.

• "Initial" row displays the starting concentrations of reactants and products before any reaction has occurred.
• "Change" row shows how concentrations change as the reaction moves toward equilibrium, often denoted by -x for consumption and +x for formation.
• "Equilibrium" row represents the net concentrations after the reaction has reached equilibrium.

By substituting these equilibrium concentrations into the K_{a} or K_{b} expression, one can solve for x to find out the concentrations of H₃O⁺ or OH⁻ , which are essential for further pH calculations.

Acid Dissociation Constant

The acid dissociation constant, K_{a} , is a specific type of ionization constant relevant to acids. It measures how an acid dissociates into its constituent ions in an aqueous solution, indicating its strength. For instance, hypochlorous acid, with its given ionization constant, partially dissociates into H₃O⁺ and ClO⁻ ions. This limited dissociation classifies it as a weak acid.

• Identifying K_{a} assists in predicting how the acid will behave in different solutions or concentrations.
• It also allows us to determine the position of equilibrium between the reactants and products, offering deeper insights into the acidity of the solution.

Understanding K_{a} values not only helps in calculating pH values but also gives a broader picture of the chemical nature and stability of acid in solutions.