Question

For each strong acid solution, determine \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right],\left[\mathrm{OH}^{-}\right],\) and \(\mathrm{pH} .\) a. \(0.25 \mathrm{MHCl}\) b. \(0.015 \mathrm{MHNO}_{3}\) c. a solution that is \(0.052 \mathrm{M}\) in \(\mathrm{HBr}\) and \(0.020 \mathrm{M}\) in \(\mathrm{HNO}_{3}\) d. a solution that is \(0.655 \% \mathrm{HNO}_{3}\) by mass (assume a density of \(1.01 \mathrm{~g} / \mathrm{mL}\) for the solution \()\)

Short answer

\( [\mathrm{H}_{3}\mathrm{O}^{+}] \) for a. 0.25 M, for b. 0.015 M, for c. 0.072 M. For d, calculate molarity first, then \( [\mathrm{H}_{3}\mathrm{O}^{+}] \). Use these concentrations to calculate \( [\mathrm{OH}^{-}] \) and pH respectively.

Step-by-step solution

Calculate concentration of \(\mathrm{H}_{3}\mathrm{O}^{+}\) for 0.25 M HCl

Since HCl is a strong acid, it will dissociate completely. Therefore, the concentration of \(\mathrm{H}_{3}\mathrm{O}^{+}\) will be equal to the concentration of HCl, which is 0.25 M.

Calculate \(\mathrm{OH}^{-}\) concentration and pH for 0.25 M HCl solution

Use the ion-product of water \(K_{w} = 1.0 \times 10^{-14} \(\mathrm{mol}^{2}\mathrm{L}^{-2}\))\) to calculate \(\mathrm{OH}^{-}\) concentration: \(\left[\mathrm{OH}^{-}\right] = K_{w} / \left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\). Then calculate pH: \(\text{pH} = -\log(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right])\).

Calculate concentrations and pH for 0.015 M \(\mathrm{HNO}_{3}\) solution

For \(\mathrm{HNO}_{3}\), a strong acid, \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) equals to its initial concentration, 0.015 M. Calculate \(\left[\mathrm{OH}^{-}\right]\) and pH using the same methods as in Step 2.

Calculate concentrations and pH for the mixture of 0.052 M HBr and 0.020 M \(\mathrm{HNO}_{3}\)

Add the concentrations of \(\mathrm{H}_{3}\mathrm{O}^{+}\) from both acids since they're strong and dissociate completely: \(0.052 + 0.020 = 0.072 M\). Then calculate \(\left[\mathrm{OH}^{-}\right]\) and pH with the combined \(\mathrm{H}_{3}\mathrm{O}^{+}\) concentration.

Calculate the \(\mathrm{H}_{3}\mathrm{O}^{+}\) concentration for 0.655% \(\mathrm{HNO}_{3}\) by mass

First convert mass percent to molarity: \((0.00655 \times \text{density}) / \text{molar mass of } \mathrm{HNO}_{3}\). Assume 1000 mL of solution for simplicity, so mass of \(\mathrm{HNO}_{3}\) is \((0.00655 \times 1010 \, \text{g})\). Molarity is then \(([\text{mass of } \mathrm{HNO}_{3}] / [\text{molar mass of } \mathrm{HNO}_{3}]) / 1 L\).

Find \(\mathrm{OH}^{-}\) concentration and pH for the 0.655% \(\mathrm{HNO}_{3}\) solution

Calculate the molarity from Step 5, then use it to find \(\left[\mathrm{OH}^{-}\right]\) with \(K_{w}\) and the \(\text{pH}\) by taking the negative logarithm of \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\).

Key concepts

Acid Dissociation in Water

When strong acids like HCl, HNO₃, and HBr dissolve in water, they dissociate completely into their constituent ions. This process is fundamental to understand the behavior of acids in aqueous solutions. For example, HCl dissociates into H₃O⁺ (hydronium ion) and Cl^-. The concentration of dissociated H₃O⁺ directly relates to the original molarity of the acid, as each molecule of acid contributes one hydronium ion to the solution.

To devise a calculation aiding students' comprehension, one should consider the complete dissociation nature of strong acids. Breaking down the steps further, it can be helpful to explain how one molecule of HCl yields one H₃O⁺ ion in solution, emphasizing the one-to-one ratio in strong acid dissociation.

pH and pOH Relationship

The pH of a solution is a measure of its acidity or alkalinity, defined as the negative logarithm of the H₃O⁺ ion concentration. pOH, on the other hand, references the hydroxide ion (OH^-) concentration. Their relationship is inversely connected due to the auto-ionization of water, meaning as the concentration of H₃O⁺ increases, that of OH^- decreases, and vice versa.

The pH and pOH values always add up to 14 at 25°C, due to the ion-product constant of water (K_w). If students struggle with conceptualizing pH as a logarithmic scale, it is useful to present it through visual aids such as a pH scale diagram, reinforcing that a low pH corresponds with high acidity (high H₃O⁺ concentration) and high pH corresponds with high alkalinity (high OH^- concentration).

Molarity of Strong Acids

Molarity, defined as moles of solute per liter of solution, is a critical concept in understanding the strength of an acid in solution. For strong acids, which dissociate fully, the molarity of the acid is equivalent to the concentration of H₃O⁺ ions produced. For instance, a 1M solution of HNO₃ will result in a 1M concentration of H₃O⁺ ions in the solution.

When describing molarity to students, it can be beneficial to use real-world examples, such as likening the molarity of an acid to the intensity level of a sports drink—more solute (or flavor) means a stronger solution (or taste). Additionally, explaining the importance of molarity in calculating pH can help students to understand the practical applications of the concept.

Ion-Product Constant of Water

The ion-product constant of water (K_w) is a constant value at a given temperature that represents the product of the concentrations of H₃O⁺ and OH^- ions in pure water. At 25°C, K_w is 1.0 x 10^-14. This means that the concentrations of H₃O⁺ and OH^- ions in pure water are both 1.0 x 10^-7 M, due to water's property of self-ionization.

In a solution of a strong acid, however, the concentration of H₃O⁺ is higher, making the OH^- concentration lower in accordance with K_w. To enhance understanding, educators might demonstrate this principle through interactive simulations showing the dynamic equilibrium of water's ionization, or by calculation exercises that reinforce the relationship between H₃O⁺ and OH^- ion concentrations and their effect on pH and pOH.