Question

What mass of sodium acetate, \(\mathrm{NaCH}_{3} \mathrm{CO}_{2}\), must be added to 1.00 L of \(0.10 \mathrm{M}\) acetic acid to give a solution with a pH of \(4.50 ?\)

Short answer

Add approximately 4.71 g of sodium acetate to the solution.

Step-by-step solution

Understand the Henderson-Hasselbalch Equation

To solve this problem, we need to use the Henderson-Hasselbalch equation, which relates the pH of a buffer solution to the concentration of acid and its conjugate base. The equation is 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 (sodium acetate in this case) and \([\text{HA}]\) is the concentration of the acid (acetic acid).

Find the \(\text{pK}_a\) of Acetic Acid

The \(\text{pK}_a\) is the negative logarithm of the ionization constant (\(K_a\)) of acetic acid. The \(K_a\) of acetic acid is typically \(1.8 \times 10^{-5}\). Thus: \[ \text{pK}_a = -\log(1.8 \times 10^{-5}) \approx 4.74 \]

Calculate the Required Ratio of \([\text{A}^-]\) to \([\text{HA}]\)

Using the Henderson-Hasselbalch equation and the desired pH (4.50), solve for the ratio \(\frac{[\text{A}^-]}{[\text{HA}]}\):\[ 4.50 = 4.74 + \log \left( \frac{[\text{A}^-]}{0.10} \right) \] \[ \log \left( \frac{[\text{A}^-]}{0.10} \right) = 4.50 - 4.74 = -0.24 \] \[ \frac{[\text{A}^-]}{0.10} = 10^{-0.24} \approx 0.575 \] \[ [\text{A}^-] = 0.10 \times 0.575 = 0.0575 \] M.

Calculate the Mass of Sodium Acetate Required

Using the concentration found, calculate the mass of sodium acetate required. The molar mass of sodium acetate \(\text{NaCH}_3\text{CO}_2\) is approximately 82.03 g/mol. Hence, for a 1.00 L solution,\[ \text{mass} = [\text{A}^-] \times \text{volume} \times \text{molar mass} \]\[ \text{mass} = 0.0575 \, \text{mol/L} \times 1.00 \, \text{L} \times 82.03 \, \text{g/mol} \approx 4.71 \, \text{g} \]

Key concepts

Buffer Solution

A buffer solution is a special type of solution that resists changes in pH when small amounts of acid or base are added. This makes buffer solutions essential in many chemical and biological systems where maintaining a stable pH is crucial. In this exercise, the buffer solution consists of acetic acid ( ext{HC}_2 ext{H}_3 ext{O}_2 ext{) and its salt, sodium acetate ( ext{NaC}_2 ext{H}_3 ext{O}_2 ext{). The combination of a weak acid and its conjugate base is what gives the solution its buffering properties.

When an acid (H+) is added to the solution, the acetate ions ( ext{C}_2 ext{H}_3 ext{O}_2^- ext{) capture these extra H+ ions, forming more acetic acid and thus minimizing changes in pH. Conversely, when a base (OH-) is added, the acetic acid donates a proton (H+), neutralizing the OH- and preventing significant pH change. This ability of buffer solutions to maintain a relatively stable pH is key in many natural processes as well as industrial applications.

pH Calculation

The pH is a measure of acidity or alkalinity of a solution. It is given by the formula \(\text{pH} = -\log[\text{H}^+]\), where [H+] is the concentration of hydrogen ions. In the context of buffer solutions, as seen in this exercise, pH calculations are often performed using the Henderson-Hasselbalch equation:

\[ \text{pH} = \text{pK}_a + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) \]

This equation relates the pH, pK_a (negative logarithm of the acid dissociation constant \(K_a\), and the concentrations of the conjugate base [ ext{A}^-] and the acid [ ext{HA}].

In this problem, you are tasked with finding the mass of sodium acetate ( ext{NaCH}_3 ext{CO}_2 ext{) needed to achieve a desired pH. By rearranging the equation, you can easily solve for the needed concentration of sodium acetate to achieve a pH of 4.50. Understanding this calculation is crucial for creating solutions with specific pH levels, often required in numerous scientific and engineering tasks.

Acetic Acid

Acetic acid, commonly known as vinegar when diluted, is a weak organic acid with the formula \(\text{CH}_3\text{COOH}\). In this exercise, acetic acid acts as the weak acid in the buffer solution. It partially dissociates into hydrogen ions ( ext{H}^+ ext{) and acetate ions ( ext{C}_2 ext{H}_3 ext{O}_2^- ext{) in solution.

The \(\text{pK}_a\) value of acetic acid is around 4.74, which derives from its acid dissociation constant \(K_a = 1.8 \times 10^{-5}\). This describes its readiness to dissociate into its components in an aqueous solution.

Acetic acid is involved in many chemical processes and industries, ranging from food preservation to the production of chemicals like polymers. In the context of buffer solutions, its presence, alongside its conjugate base, stabilizes the pH, making it invaluable in contexts where pH regulation is pivotal.

Sodium Acetate

Sodium acetate is the sodium salt of acetic acid, with the chemical formula \(\text{NaC}_2\text{H}_3\text{O}_2\). In buffer solutions, it acts as the conjugate base complementary to acetic acid's role as the weak acid.

In our problem, adding sodium acetate to the acetic acid solution increases the concentration of acetate ions. These ions work to neutralize added acids, increasing the solution's pH resistance. The pH level is then regulated through this buffer system.

Using the Henderson-Hasselbalch equation, one can calculate the exact amount of sodium acetate needed to achieve a specific pH. With a molar mass of approximately 82.03 g/mol, the calculation enables the precise preparation of the buffer solution required by determining how much sodium acetate needs to be dissolved. Sodium acetate is also widely utilized in various fields, like textile and polymer industries, for its buffering capacity.