Problem 80 - Chemistry and Chemical Reactivity

Question

Calculate the hydronium ion concentration and the pH of the solution that results when \(20.0 \mathrm{mL}\) of \(0.15 \mathrm{M}\) acetic acid, \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H},\) is mixed with \(5.0 \mathrm{mL}\) of \(0.17 \mathrm{M} \mathrm{NaOH}.\)

Step-by-Step Solution

Answer

pH is 2.92, hydronium ion concentration is \(1.2 \times 10^{-3} \ M\).

Step 1: Calculate Moles of Acetic Acid and NaOH

To start, calculate the moles of acetic acid using the formula: \( ext{moles} = M imes V \). For acetic acid: \( 0.15 ext{ M} imes 0.020 ext{ L} = 0.003 ext{ moles of } ext{CH}_3 ext{CO}_2 ext{H} \). For NaOH: \( 0.17 ext{ M} imes 0.005 ext{ L} = 0.00085 ext{ moles of NaOH} \).

Step 2: Determine Limiting Reactant and Calculate Remaining Moles

NaOH and acetic acid react in a 1:1 molar ratio: \( ext{CH}_3 ext{CO}_2 ext{H} + ext{NaOH} ightarrow ext{CH}_3 ext{CO}_2 ext{Na} + ext{H}_2 ext{O} \). The limiting reactant is NaOH (0.00085 moles). Subtract it from the initial moles of acetic acid: \( 0.003 - 0.00085 = 0.00215 ext{ moles of } ext{CH}_3 ext{CO}_2 ext{H remaining} \).

Step 3: Calculate Concentration of Acetic Acid

Calculate the concentration of the remaining acetic acid. The total volume of the solution is the sum of both solutions: \( 20.0 ext{ mL} + 5.0 ext{ mL} = 25.0 ext{ mL} = 0.025 ext{ L} \). Therefore, the concentration of acetic acid is: \( rac{0.00215 ext{ moles}}{0.025 ext{ L}} = 0.086 ext{ M} \).

Step 4: Calculate the Hydronium Ion Concentration

Acetic acid is a weak acid with a dissociation constant \( K_a = 1.8 imes 10^{-5} \). Use the relationship: \[ K_a = \frac{[H_3O^+][CH_3CO_2^-]}{[CH_3CO_2H]} \]. Assume \([H_3O^+] = [CH_3CO_2^-] = x\) and solve for \(x\): \[ 1.8 imes 10^{-5} = \frac{x^2}{0.086} \], giving \(x = \sqrt{1.8 imes 10^{-5} \times 0.086} = 1.2 imes 10^{-3} ext{ M} \).

Step 5: Calculate the pH of the Solution

The \( pH \) is calculated using the formula: \( pH = -\log [H_3O^+] \). Substitute the \([H_3O^+]\) concentration: \[ pH = -\log(1.2 \times 10^{-3}) \approx 2.92 \].

Key Concepts

Acetic Acid

Acetic acid, commonly found in vinegar, has the chemical formula \( ext{CH}_3 ext{CO}_2 ext{H} \). It is known as a weak acid, which means it doesn’t completely dissociate into ions in water. This characteristic is important in acid-base chemistry because it influences the calculation of pH and hydronium ion concentration. Understanding weak acids like acetic acid is crucial when predicting how they will react with bases such as sodium hydroxide (NaOH). Unlike strong acids, where we assume nearly complete ionization, weak acids partially ionize, creating a dynamic equilibrium between the undissociated acid and the ions.

pH Calculation

pH is a scale used to specify the acidity or basicity of an aqueous solution. It ranges from 0 to 14, where 7 is neutral, low values
indicate acidity, and high values indicate basicity. To calculate pH, we use the formula:

• \( pH = -\log[H_3O^+] \)

In the exercise, hydronium ion (\(H_3O^+\)) concentration was calculated as \(1.2 \times 10^{-3}\, \text{M}\). Immediately, by plugging it into the formula, the calculated pH was approximately 2.92. This value shows the solution is acidic. Remember, every decrease in pH by one unit represents a ten-fold increase in acidity.

Limiting Reactant

In chemical reactions, the limiting reactant is the substance that determines the maximum amount of product that can be formed. In our exercise, acetic acid and sodium hydroxide react in a 1:1 molar ratio:

• \(\text{CH}_3\text{CO}_2\text{H} + \text{NaOH} \rightarrow \text{CH}_3\text{CO}_2\text{Na} + \text{H}_2\text{O} \)

By calculating the moles of each reactant, we identified NaOH as the limiting reactant. It had the fewest moles (0.00085 moles) compared to acetic acid's 0.003 moles. This means NaOH will completely react before the acetic acid, leaving some acetic acid unreacted. Understanding the limiting reactant is key for reacting only what is available and accurately predicting the chemical changes in a solution.

Hydronium Ion Concentration

Hydronium ions (\(H_3O^+\)) are crucial in determining the acidity of a solution. The concentration of hydronium ions directly relates to the pH and gives insight into an acid's strength. In this exercise, to find the hydronium ion concentration, we applied acetic acid's dissociation constant (\(K_a\)) within the formula:

• \[ K_a = \frac{[H_3O^+][\text{CH}_3\text{CO}_2^-]}{[\text{CH}_3\text{CO}_2\text{H}]} \]

Assuming that both \([H_3O^+]\) and \([\text{CH}_3\text{CO}_2^- ]\) are equal (\(x\)), we substituted to solve for \(x\), marking the hydronium ion concentration as \(1.2 \times 10^{-3}\, \text{M}\). This calculation highlights how weak acids' partial dissociation can be quantified and shows the significance of hydronium ions in calculating pH accurately.