Problem 61 - Chemistry

Question

What are the major species present in \(0.250 \mathrm{M}\) solutions of each of the following acids? Calculate the \(\mathrm{pH}\) of each of these solutions. a. \(\mathrm{HNO}_{2}\) b. \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\left(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\right)\)

Step-by-Step Solution

Answer

The major species in a \(0.250\, \mathrm{M}\) solution of \(\mathrm{HNO}_2\) are \(\mathrm{HNO}_2\), \(\mathrm{H}^+\), \(\mathrm{NO}_2^-\), and \(\mathrm{H}_2\mathrm{O}\). The pH of this solution can be calculated as \(pH_1 = -\log x\), where \(x\) is the concentration of \(\mathrm{H}^+\) found using the equilibrium expression with known \(K_a\). For a \(0.250\, \mathrm{M}\) solution of \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\), the major species are \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\), \(\mathrm{H}^+\), \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{O}_{2}^-\), and \(\mathrm{H}_2\mathrm{O}\). The pH of this solution can be calculated as \(pH_2 = -\log y\), where \(y\) is the concentration of \(\mathrm{H}^+\) found using the equilibrium expression with known \(K_a\).

Step 1: Write down the dissociation reactions for the acids

Write the reactions for the dissociation of the two acids in water. We can represent the concentration of water, which does not change, as a constant to simplify the equation. a. For \(\mathrm{HNO}_2\): \[\mathrm{HNO}_{2} (aq) \rightleftharpoons \mathrm{H}^{+} (aq) + \mathrm{NO}_{2}^{-} (aq)\] b. For \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\left(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\right)\): \[\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2} (aq) \rightleftharpoons \mathrm{H}^{+} (aq) + \mathrm{C}_{2}\mathrm{H}_{3}\mathrm{O}_{2}^{-} (aq)\]

Step 2: Set up the equilibrium expressions for the dissociation reactions

Set up the equilibrium expressions for the dissociation reactions of the acids. We will use the acid dissociation constant (\(K_a\)) for each acid, which can be found on a table or in the literature. a. For \(\mathrm{HNO}_2\): \[K_{a1} = \frac{[\mathrm{H}^{+}] [\mathrm{NO}_{2}^-]}{[\mathrm{HNO}_{2}]}\] b. For \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\): \[K_{a2} = \frac{[\mathrm{H}^{+}] [\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{O}_{2}^-]}{[\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}]}\]

Step 3: Calculate the concentration of hydrogen ions

Using the \(K_a\) values and the initial concentrations of the acids, solve for the concentration of hydrogen ions (\([\mathrm{H}^+]\)) for both acids. If the \(K_a\) values for the acids are not given, you can look them up in a table or in the literature. a. For \(\mathrm{HNO}_2\): Given, the initial concentration of \(\mathrm{HNO}_2\) is \(0.250 \, \mathrm{M}\). If we let \(x\) be the concentration of hydrogen ions formed from the dissociation of \(\mathrm{HNO}_2\), then \(x\) will also be the concentration of \(\mathrm{NO}_2^-\). \[K_{a1} = \frac{x^2}{0.250 - x}\] Solve for \(x\) to find the concentration of \(\mathrm{H}^+\). b. For \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\): Similarly, given the initial concentration of \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\) is \(0.250 \,M\). If we let \(y\) be the concentration of hydrogen ions formed from the dissociation of \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\), then \(y\) will also be the concentration of \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{O}_{2}^-\). \[K_{a2} = \frac{y^2}{0.250 - y}\] Solve for \(y\) to find the concentration of \(\mathrm{H}^+\).

Step 4: Calculate the pH of the solutions

Calculate the pH of the solutions using the formula: pH \(= -\log [\mathrm{H}^{+}]\). Use the calculated concentrations of hydrogen ions (\([\mathrm{H}^+]\)) in Step 3. a. For \(\mathrm{HNO}_2\): \[pH_1 = -\log [\mathrm{H}^{+}] = -\log x\] b. For \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\): \[pH_2 = -\log [\mathrm{H}^{+}] = -\log y\]

Step 5: Identify the major species in the solutions

List the major species present in both acidic solutions based on the dissociation reactions and the concentration of the ions. a. In the \(0.250 \, \mathrm{M}\) solution of \(\mathrm{HNO}_2\), the major species are: \(\mathrm{HNO}_2\), \(\mathrm{H}^+\), \(\mathrm{NO}_2^-\), and \(\mathrm{H}_2\mathrm{O}\). b. In the \(0.250 \, \mathrm{M}\) solution of \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\), the major species are: \(\mathrm{HC}_{2}\mathrm{H}_{3}\mathrm{O}_{2}\), \(\mathrm{H}^+\), \(\mathrm{C}_{2}\mathrm{H}_{3}\mathrm{O}_{2}^-\), and \(\mathrm{H}_2\mathrm{O}\). With the pH values found in Step 4 and the major species identified, the exercise is complete.