Question

Determine the volume (in mL) of \(1.00 \mathrm{M} \mathrm{NaOH}\) that must be added to \(250 \mathrm{mL}\) of \(0.50 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{CO}_{2} \mathrm{H}\) to produce a buffer with a pH of 4.50.

Short answer

Add approximately 45.7 mL of 1.00 M NaOH.

Step-by-step solution

Identify the Buffer Equation

In order to solve this problem, we'll use the Henderson-Hasselbalch equation, which is given by \( \text{pH} = \text{pK}_a + \log\left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \). Here, \( [\text{A}^-] \) is the concentration of the conjugate base (acetate ion), and \( [\text{HA}] \) is the concentration of the weak acid (acetic acid).

Calculate \( \text{pK}_a \) of Acetic Acid

For acetic acid, \( \text{pK}_a \approx 4.74 \). Use this value in the Henderson-Hasselbalch equation to find the ratio of base to acid required for the desired pH.

Apply the Henderson-Hasselbalch Equation

Set \( \text{pH} = 4.50 \) and use the equation: \( 4.50 = 4.74 + \log\left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \). Simplifying for the ratio \( \frac{[\text{A}^-]}{[\text{HA}]} \), we have:\[ \log\left( \frac{[\text{A}^-]}{[\text{HA}]} \right) = 4.50 - 4.74 = -0.24 \]This implies \( \frac{[\text{A}^-]}{[\text{HA}]} = 10^{-0.24} \approx 0.575 \).

Define Variables for Acid and Conjugate Base

Let \( n_\text{acid} = 0.25 \, \text{L} \times 0.50 \, \text{M} = 0.125 \, \text{mol} \) be moles of acetic acid initially. Let \( V \) be the volume of NaOH solution we need to add in liters. Since NaOH is a strong base, every mole of NaOH added will convert one mole of acetic acid to acetate ion.

Formulate the Mass Balance Equation

Because each mole of NaOH converts one mole of acetic acid to acetate, the equation becomes:\[ 0.575 = \frac{n_{\text{base}}}{0.125 - n_{\text{base}}} \]where \( n_\text{base} \) is the moles of acetate ion, which is \( n_{\text{base}} = V \times 1.00 \, \text{M} \).

Solve for \( V \)

Insert \( n_{\text{base}} = V \times 1.00 \, \text{M} \) into the mass balance equation:\[ 0.575 = \frac{V \times 1.00}{0.125 - V \times 1.00} \]Solving for \( V \), we rearrange and solve the equation:\[ 0.575(0.125 - V) = V \]\[ 0.071875 = 1.575V \]\[ V = \frac{0.071875}{1.575} \approx 0.0457 \, \text{L} = 45.7 \, \text{mL} \].

Finalize the Calculation

After solving the equation for \( V \), we find that approximately 45.7 mL of the \(1.00 \, \text{M} \, \text{NaOH}\) solution is required. Therefore, adding this volume will result in the desired pH of the buffer.

Key concepts

Buffer Solutions

Buffer solutions are unique in their ability to resist rapid changes in pH when small amounts of acid or base are added. This characteristic makes them essential in various chemical and biological processes where maintaining a stable pH is crucial. Buffer solutions typically consist of a weak acid and its conjugate base or a weak base and its conjugate acid.

• Weak Acid and Conjugate Base: Such as acetic acid ( (CH_3CO_2H) ) and its conjugate base, acetate ion ( (CH_3CO_2^-) ).
• Weak Base and Conjugate Acid: Often used in biological systems to maintain physiological pH levels.

By choosing the right combination of weak acid and conjugate base, we can design a buffer that maintains a solution's pH within a narrow range, even when challenged by the addition of external acidic or basic species.

Acid-Base Titration

In acid-base titration, the main goal is to determine the concentration of an unknown solution (the analyte) by titrating it with a solution of known concentration (the titrant). This process involves a carefully controlled addition of titrant to the analyte until the reaction reaches its equivalence point.

• Equivalence Point: This signifies that stoichiometrically equivalent amounts of acid and base have reacted together.
• Indicator or pH Meter: Used to detect the equivalence point by observing a color change or pH change, respectively.

During the titration, from the start to the equivalence point, and beyond, specific calculations and observations are made. These help estimate how the pH changes, allowing for precise adjustment of buffer systems, like using titration data to calculate how much (NaOH) to add to an acetic acid solution to form a buffer.

pH Calculation

Calculating pH, especially in buffer solutions, is straightforward with the Henderson-Hasselbalch equation. This equation relates pH to the concentrations of the weak acid and its conjugate base. Given by:\[\text{pH} = \text{pK}_a + \log\left( \frac{[\text{A}^-]}{[\text{HA}]} \right)\]- where \([\text{A}^-]\)is the concentration of the conjugate base and \([\text{HA}]\)is the concentration of the weak acid.

• pKa Value: The negative logarithm of the acid dissociation constant, providing a measure of acid strength.
• Importance: Using this equation allows chemists to determine the required amounts of acid and base to create a solution with a specific pH.

For instance, in the problem involving acetic acid, calculating the needed volume of (NaOH) to achieve a particular pH level involves rearranging this equation and solving for the unknown variable.

Acetic Acid

Acetic acid (CH_3CO_2H) is a common weak acid widely used in chemistry labs and industry. It's a clear, colorless liquid that has a distinctive sour taste and mild acidity in solution.

• pKa Value: Around 4.74, making it a weak acid suitable for buffer solutions.
• Uses: Acetic acid appears in various applications, from being a component in vinegar to serving as a chemical reagent in organic synthesis.

In buffer preparation, when mixed with an appropriate amount of a strong base like (NaOH) , acetic acid can maintain a relatively constant pH, ideal for many experimental setups. Understanding the properties and behavior of acetic acid helps in designing experiments, such as preparing buffers that involve precise pH calculations using the Henderson-Hasselbalch equation.