Problem 96 - Chemistry: The Central Science

Question

How many microliters of \(1.000 \mathrm{M} \mathrm{NaOH}\) solution must be added to \(25.00 \mathrm{~mL}\) of a \(0.1000 \mathrm{M}\) solution of lactic acid \(\left[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COOH}\right.\) or \(\left.\mathrm{HC}_{3} \mathrm{H}_{5} \mathrm{O}_{3}\right]\) to produce a buffer with \(\mathrm{pH}=3.75 ?\)

Step-by-Step Solution

Answer

Approximately 1370 µL of 1.000 M NaOH solution must be added to 25.00 mL of 0.1000 M lactic acid solution to produce a buffer with pH = 3.75.

Step 1: Determine pKa of lactic acid

Lactic acid (CH3CH(OH)COOH or HC3H5O3) has a pKa value of 3.86. This value will be used in the Henderson-Hasselbalch equation to determine the concentration ratio of the weak acid and its conjugate base in the buffer solution.

Step 2: Set up the Henderson-Hasselbalch equation

Now, we will set up the Henderson-Hasselbalch equation with the given pH and pKa values: \(3.75 = 3.86 + \log_{10}\frac{[A-]}{[HA]}\)

Step 3: Solve for [A-]/[HA]

Subtract 3.86 from both sides of the equation and find the ratio of [A-]/[HA]: \(-0.11 = \log_{10}\frac{[A-]}{[HA]}\) Next, find the antilog to the base 10 of both sides to determine the ratio of [A-]/[HA]: \(\frac{[A-]}{[HA]} = 10^{-0.11}\) \(\frac{[A-]}{[HA]} \approx 0.776\)

Step 4: Set up and plug in stoichiometry

Now, we will set up the stoichiometry of the reaction to determine the concentration of lactic acid and its conjugate base after adding a certain volume of NaOH. Let x be the moles of NaOH added. Then x moles of lactic acid will react to produce x moles of its conjugate base. Initial moles of lactic acid = 0.1 M × 0.025 L = 0.0025 mol Moles of lactic acid after adding NaOH = 0.0025 mol - x Moles of the conjugate base after adding NaOH = x Now, plug these values into the ratio from Step 3: \(\frac{x}{0.0025-x} \approx 0.776\)

Step 5: Solve for the moles of NaOH added

Multiply both sides of the equation with the denominator to get rid of the fraction: \(x \approx (0.0025 - x) * 0.776\) Solve for x to find the moles of NaOH added: \(x \approx 0.00137 \text{ mol}\)

Step 6: Convert moles of NaOH to microliters of 1.000 M NaOH solution

To find out the required volume of 1.000 M NaOH solution to add in microliters, use the formula: moles = molarity × volume (in liters) Volume (in liters) = moles / molarity Volume (in liters) = 0.00137 mol / 1.000 M Volume (in liters) ≈ 0.00137 L Convert to microliters: Volume (in microliters) = 0.00137 L × 10^6 µL / L Volume (in microliters) ≈ 1370 µL Therefore, approximately 1370 µL of 1.000 M NaOH solution must be added to 25.00 mL of 0.1000 M lactic acid solution to produce a buffer with pH = 3.75.