Problem 119 - Chemistry The Molecular Science

Question

A \(0.100-M\) acetic acid solution has \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\) \(0.00134 \mathrm{~mol} / \mathrm{L}\) (a) Calculate the percent ionization of the acid. (b) Sufficient sodium acetate is added to the \(0.100-\mathrm{M}\) acetic acid solution so that its acetate ion concentration is \(0.050 \mathrm{~mol} / \mathrm{L}\). Calculate the percent ionization of the acetic acid in this solution.

Step-by-Step Solution

Answer

(a) 1.34%, (b) 0.036%

Step 1: Define Percent Ionization Formula

The percent ionization of an acid is given by the formula:\[\text{Percent Ionization} = \left(\frac{\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]}{\text{Initial concentration of acid}}\right) \times 100\%\]We'll use this formula for both parts of the problem.

Step 2: Calculate Initial Percent Ionization

Use the given concentrations to find the percent ionization for part (a):\[\text{Percent Ionization} = \left(\frac{0.00134}{0.100}\right) \times 100\% = 1.34\%\]

Step 3: Apply Common Ion Effect for Part (b)

For part (b), when sodium acetate is added, there is a common ion effect. This means the ionization of acetic acid decreases due to the added acetate ions.

Step 4: Determine New H3O+ Concentration

Calculate the new equilibrium concentration of \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) using the weak acid equilibrium equation:\[K_a = \frac{\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\left[\mathrm{CH}_3\mathrm{COO}^−\right]}{\left[\mathrm{CH}_3\mathrm{COOH}\right]}\]We have:\(\left[\mathrm{CH}_3\mathrm{COO}^−\right] = 0.050\),\(\left[\mathrm{CH}_3\mathrm{COOH}\right] \approx 0.100\) due to minimal ionization.We assume initial \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) is very small due to common ion effect and hence negligible for initial approximations.Thus:\[K_a = 1.8 \times 10^{-5} = \frac{\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] \times 0.050}{0.100}\]Solving gives \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right] = 3.6 \times 10^{-5}\ \text{mol/L}\).

Step 5: Calculate New Percent Ionization

Use the new \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) value to find the percent ionization:\[\text{Percent Ionization} = \left(\frac{3.6 \times 10^{-5}}{0.100}\right) \times 100\% = 0.036\%\]

Step 6: Conclusion

The initial percent ionization of the \(0.100\, \text{M}\) acetic acid is \(1.34\%\), and after adding sodium acetate, the percent ionization decreases to \(0.036\%\).

Key Concepts

Percent Ionization

Percent ionization is a measure of the extent to which a weak acid dissociates into its ions in a solution. It's a useful way to gauge the strength of an acid in solution. For any weak acid, the percent ionization provides a snapshot of how much the acid dissociates under certain conditions.

To calculate percent ionization, use the formula:

• Percent Ionization = \( \left( \frac{[\mathrm{H}_3\mathrm{O}^+]}{\text{Initial concentration of the acid}} \right) \times 100\% \)

This formula makes it clear how to find the percentage of the acid that has ionized, using the concentration of hydrogen ions (\( [\mathrm{H}_3\mathrm{O}^+] \)), which reflects the amount of acid in its ionized form.

In our example with acetic acid, the initial percent ionization of a 0.100 M acetic acid solution is \( 1.34\% \), showing a relatively low degree of ionization. This low percentage is typical for weak acids, as they only partially dissociate in water.

Common Ion Effect

The common ion effect is a principle that helps us understand how the presence of a shared ion influences the ionization of weak acids or bases. In a solution containing a weak acid, if a salt containing a common ion is added, the ionization of the acid decreases.

This happens because the addition of a common ion shifts the equilibrium according to Le Chatelier's Principle. Consider a weak acid equilibrium:

• HA ⇌ H\(^+\) + A\(^{-}\)

Adding a salt that provides more A\(^{-}\) (such as sodium acetate in the case of acetic acid) increases the concentration of the ionized species, thus pushing the equilibrium towards the left.

In part (b) of our acetic acid example, adding sodium acetate increases the concentration of acetate ions, which decreases the acid's ionization and lowers the percent ionization from \( 1.34\% \) to \( 0.036\% \). This demonstrates the effect a common ion can have in reducing the ionization of a weak acid.

Weak Acid Equilibrium

Weak acids do not completely ionize in solution, establishing an equilibrium between the undissociated acid and the ions formed. The equilibrium can be represented by the equation:

• HA ⇌ H\(^+\) + A\(^{-}\)

The equilibrium is quantified using the acid dissociation constant, \( K_a \), which for acetic acid is \( 1.8 \times 10^{-5} \).

When calculating the concentration of ions at equilibrium, it's vital to use the initial concentrations and assumptions like assuming minimal ionization changes the initial concentration. This approximation simplifies calculations.

In our scenario, we calculated the equilibrium hydronium ion concentration in the presence of a common ion. This involved solving the equation using the known \( K_a \) value and the provided concentrations for acetate ions. The outcome reflected the influence of equilibrium on weak acid behavior, indicating the importance of considering all components when determining the acidity of solutions.