Problem 96 - Chemistry The Central Science In Si Units

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

You need around 1930 microliters of 1.000 M NaOH solution.

Step 1: Understanding the Problem

We need to calculate the amount of \( \text{NaOH} \) solution required to produce a buffer with a specified pH. A buffer is created when \( \text{NaOH} \) neutralizes some of the \( \text{lactic acid} \), forming its conjugate base. The Henderson-Hasselbalch equation will be used: \[ \text{pH} = \text{pK}_a + \log \left( \frac{[A^-]}{[HA]} \right) \]. Here, \([A^-]\) is the concentration of the conjugate base (lactate ion), and \([HA]\) is the concentration of lactic acid.

Step 2: Finding pKa of Lactic Acid

The \( \text{pK}_a \) of lactic acid \([(\mathrm{HC}_3\mathrm{H}_5\mathrm{O}_3)]\) is given as 3.86. This value is crucial as it will allow us to use the Henderson-Hasselbalch equation.

Step 3: Formulating the Henderson-Hasselbalch Equation

With the desired \( \text{pH} = 3.75 \) and \( \text{pK}_a = 3.86 \), the buffer equation becomes:\[ 3.75 = 3.86 + \log \left( \frac{[A^-]}{[HA]} \right) \].Solving, we find:\[ \log \left( \frac{[A^-]}{[HA]} \right) = 3.75 - 3.86 = -0.11 \].Convert this logarithmic ratio to its linear form:\[ \frac{[A^-]}{[HA]} = 10^{-0.11} \approx 0.776 \].

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is an invaluable tool in chemistry, especially when discussing buffer solutions. Buffers resist changes in pH when small amounts of acid or base are added.

The equation is expressed as: \[\text{pH} = \text{pK}_a + \log \left( \frac{[A^-]}{[HA]} \right)\] Here, \([A^-]\) stands for the concentration of the conjugate base and \([HA]\) for the acid. This formula helps chemists determine the pH of a buffer solution by comparing the concentrations of the acid and its conjugate base.

This equation assumes that the concentrations of the acid and conjugate base in the solution are similar and is most accurate when applied to weak acids and bases. It highlights the direct relationship between the pH of a buffer and the ratio of conjugate base to acid. When this ratio is equal to 1, the pH is equal to the pKₐ, indicating that there are equal concentrations of acid and conjugate base.

Lactic Acid

Lactic acid is a simple carboxylic acid represented by the chemical formula \(\mathrm{CH}_3\mathrm{CH}(\mathrm{OH})\mathrm{COOH}\). It is known for being a part of the fermentation process and can be found in sour milk products.

In the context of buffer solutions, lactic acid serves as the weak acid component. When dissociated in solution, it forms its conjugate base, the lactate ion \(\mathrm{C}_3\mathrm{H}_5\mathrm{O}_3^-\). This dissociation is a key component of how buffers work, as it allows for equilibrium adjustments when strong acids or bases are introduced to the system.

Lactic acid has a \(\text{pK}_a\) of 3.86, which is important when determining the pH range over which it can effectively act as part of a buffer system. Its presence contributes to the stability of the pH, making it an essential part of biochemical processes and industries where pH maintenance is crucial.

pH Calculation

pH calculation is a fundamental concept in chemistry that involves determining the acidity or basicity of an aqueous solution. The pH scale ranges from 0 to 14, where lower values indicate more acidic solutions and higher values signify more basic ones.

A pH of 7 is considered neutral, typically represented by pure water. In a buffered solution, such as one using lactic acid, knowing how to calculate the pH is essential. This is accomplished through the use of the Henderson-Hasselbalch equation by inputting known values of pKₐ and concentrations of the acid and its conjugate base.

For the buffer to be effective, it generally functions best when the pH is close to the pKₐ value of the acid being used. This proximity ensures that the components of the buffer can neutralize added acids or bases without significant fluctuations in pH, thus providing a stable environment in chemical processes.