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 \(\mathrm{pH}=3.75 ?\)

Short 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-by-step solution

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.

Set up the Henderson-Hasselbalch equation

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

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]: \(\frac{[A-]}{[HA]} = 10^{-0.11}\) \(\frac{[A-]}{[HA]} \approx 0.776\)

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: \(\frac{x}{0.0025-x} \approx 0.776\)

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: \(x \approx 0.00137 \text{ mol}\)

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.