Question

How many grams of sodium lactate \(\left[\mathrm{CH}_{3} \mathrm{CH}(\mathrm{OH}) \mathrm{COONa}\right.\) or \(\left.\mathrm{NaC}_{3} \mathrm{H}_{5} \mathrm{O}_{3}\right]\) should be added to \(1.00 \mathrm{~L}\) of \(0.150 \mathrm{M}\) 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 form a buffer solution with \(\mathrm{pH} 4.00\) ? Assume that no volume change occurs when the sodium lactate is added.

Short answer

To create a buffer solution with a pH of 4.00, approximately 23.2 grams of sodium lactate should be added to the 1.00 L of 0.150 M lactic acid solution.

Step-by-step solution

Write the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by: \(pH = pK_a + \log\frac{[\textrm{base}]}{[\textrm{acid}]}\)

Find the pKa value of lactic acid

The acidic dissociation constant, Ka, of lactic acid is given as \(\textrm{Ka} = 1.38 \times 10^{-4}\). To find the pKa, we take the negative logarithm of Ka: \(pK_a = -\log(\textrm{Ka}) = -\log(1.38 \times 10^{-4})\) Calculating this value, we get: \(pK_a = 3.86\)

Calculate the ratio of base/acid concentrations

Using the Henderson-Hasselbalch equation, we can find the ratio of base/acid concentrations: \(pH = pK_a + \log\frac{[\textrm{base}]}{[\textrm{acid}]}\) \(4.00 = 3.86 + \log\frac{[\textrm{base}]}{[\textrm{acid}]}\) \(0.14 = \log\frac{[\textrm{base}]}{[\textrm{acid}]}\) Now, we can calculate the base/acid concentration ratio: \(\frac{[\textrm{base}]}{[\textrm{acid}]} = 10^{0.14} = 1.38\)

Calculate the concentration of the base

We are given that the concentration of the lactic acid solution is 0.150 M. To find the concentration of the base, we can use the calculated ratio: \([\textrm{base}] = [\textrm{acid}] \times \frac{[\textrm{base}]}{[\textrm{acid}]}\) \([\textrm{base}] = 0.150 \mathrm{M} \times 1.38\) \([\textrm{base}] = 0.207 \mathrm{M}\)

Calculate the moles of the base needed

We now know the required concentration of the base, and we are given the volume of the solution (1.00 L). To find the moles of sodium lactate needed, we can use the following formula: \(\textrm{moles} = \mathrm{concentration}\times\mathrm{volume}\) \(\textrm{moles} = 0.207 \mathrm{M}\times 1.00\mathrm{ L}\) \(\textrm{moles} = 0.207\, \textrm{moles}\)

Calculate the grams of sodium lactate needed

Finally, we can calculate the mass of sodium lactate needed. The molar mass of sodium lactate (NaC3H5O3) is approximately 112.06 g/mol. So, the mass of sodium lactate needed can be calculated as follows: \(\textrm{mass} = \textrm{moles}\times\mathrm{molar \ mass}\) \(\textrm{mass} = 0.207 \textrm{ moles} \times 112.06 \frac{\textrm{g}}{\textrm{mol}}\) \(\textrm{mass} \approx 23.2\, \textrm{g}\) To create a buffer solution with a pH of 4.00, approximately 23.2 grams of sodium lactate should be added to the 1.00 L of 0.150 M lactic acid solution.