Question

Lactic acid is a common by-product of cellular respiration and is often said to cause the "burn" associated with strenuous activity. A 25.0 -mL sample of 0.100 \(M\) lactic acid (HC \(_{3} \mathrm{H}_{5} \mathrm{O}_{3}\), \(\mathrm{p} K_{\mathrm{a}}=3.86\) is titrated with \(0.100 \mathrm{M}\) NaOH solution. Calculate the \(\mathrm{pH}\) after the addition of \(0.0 \mathrm{mL}, 4.0 \mathrm{mL}, 8.0 \mathrm{mL}, 12.5 \mathrm{mL}\) \(20.0 \mathrm{mL}, 24.0 \mathrm{mL}, 24.5 \mathrm{mL}, 24.9 \mathrm{mL}, 25.0 \mathrm{mL}, 25.1 \mathrm{mL}\) \(26.0 \mathrm{mL}, 28.0 \mathrm{mL},\) and \(30.0 \mathrm{mL}\) of the NaOH. Plot the results of your calculations as pH versus milliliters of NaOH added.

Short answer

In summary, the pH of the lactic acid solution can be calculated at different volumes of NaOH added using the given steps: determine the moles of lactic acid and NaOH at each step, identify the reaction status (before, at, or after the equivalence point), calculate the pH using appropriate equations, and plot the results as pH versus milliliters of NaOH added.

Step-by-step solution

Calculate moles of lactic acid and NaOH at each step

To find the moles of lactic acid and NaOH at each given volume, we will use the formula: moles = concentration × volume We know the concentration of lactic acid is 0.100 M and the volume is 25.0 mL (0.025 L). For NaOH, the concentration is 0.100 M and the volume varies for each step.

Determine the reaction between lactic acid and NaOH based on the volume of NaOH added at each step

Lactic acid (HC3H5O3) reacts with NaOH as follows: HC3H5O3 + OH- → C3H5O3- + H2O Depending on the volume of NaOH added, there can be three possible situations: 1. Before equivalence point: Excess lactic acid (buffer solution) 2. At equivalence point: Equal moles of lactic acid and NaOH 3. After equivalence point: Excess NaOH

Calculate the pH at each step according to the reaction status

Using the three reaction situations mentioned above, we can now calculate the pH at each volume of NaOH added: 1. Before equivalence point (buffer solution): Use the Henderson-Hasselbalch equation: pH = pKa + log([C3H5O3-]/[HC3H5O3]) 2. At equivalence point: Calculate the concentration of the conjugate base (C3H5O3-) and use the following formula to find pH: pOH = -log([OH-]) pH = 14 - pOH 3. After equivalence point: Calculate the concentration of excess OH- ions and calculate the pH using the following formula: pOH = -log([OH-]) pH = 14 - pOH

Calculate the pH at each requested volume of NaOH added

Using the information from steps 1, 2, and 3, the pH can be calculated for all given volumes of NaOH added. Perform these calculations for each volume of NaOH added (0.0 mL, 4.0 mL, 8.0 mL, 12.5 mL, 20.0 mL, 24.0 mL, 24.5 mL, 24.9 mL, 25.0 mL, 25.1 mL, 26.0 mL, 28.0 mL, and 30.0 mL).

Plot the results

Finally, plot the calculated pH values at each volume of NaOH added on a graph, with the x-axis representing the volume of NaOH added in milliliters and the y-axis representing the pH. The resulting plot is the titration curve for the lactic acid titrated with NaOH.

Key concepts

Buffer solution

A buffer solution is a special type of solution that maintains a stable pH when small amounts of an acid or base are added. In the case of titrating lactic acid with NaOH, a buffer solution is formed before reaching the equivalence point.
This happens because some of the lactic acid reacts with NaOH to form its conjugate base, lactate \(C_3H_5O_3^-\).

• Maintains pH stability: The presence of both lactic acid and lactate \(C_3H_5O_3^-\) in the solution helps neutralize any additional H\(^+\) ions or OH\(^-\) ions, resisting drastic pH changes.
• Effective pH range: Buffer solutions are most effective within 1 pH unit of the pKa of the acid involved, which for lactic acid is 3.86.

Before reaching the equivalence point in titration, the solution acts as a buffer, protecting the pH from rapid changes.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a fundamental formula used to estimate the pH of buffer solutions. When dealing with the titration of lactic acid by NaOH, this equation helps calculate the pH before the equivalence point is reached.
For lactic acid, we use:\[pH = ext{pKa} + ext{log} rac{[C_3H_5O_3^-]}{[HC_3H_5O_3]}\]

• pKa: This is the negative logarithm of the acid dissociation constant (Ka). For lactic acid, pKa = 3.86.
• [C\(_3H_5O_3^−\)]: This represents the concentration of the conjugate base in the solution.
• [HC\(_3H_5O_3\)] : This is the concentration of the remaining lactic acid in the solution.

As the titration proceeds, concentrations of acid and conjugate base change, but the equation helps us track the pH just before the equivalence point.

Equivalence point

The equivalence point in a titration is a critical stage where the amount of titrant added is exactly enough to completely react with the substance being titrated. For the titration of lactic acid with NaOH, it means equal moles of lactic acid and NaOH are present.
At this stage:

• No original acid (HC\(_3H_5O_3\)) remains, only its conjugate base (C\(_3H_5O_3^-\)).
• The pH corresponds to the pH of the solution formed by the conjugate base, calculated using pOH and converting it to pH.

Following the equivalence point:

• Any additional NaOH added results in an excess, increasing the solution's pH significantly past neutrality.
• Knowing the equivalence point is key for determining the endpoint of titration and analyzing the titration curve.