To solve this problem, we need to use the Henderson-Hasselbalch equation for buffer solutions:

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

where:

• pH is the desired pH of the buffer solution, which is given as 4.91.
• pKa of acetic acid is calculated using the provided $K_a = 1.8 \times 10^{-5}$.
• [A$^-$] represents the concentration of the acetate ion (which will come from sodium hydroxide reacting with acetic acid).
• [HA] represents the concentration of acetic acid (CH$_3$COOH).

Step 1: Calculate pKa of Acetic Acid

Given $K_a = 1.8 \times 10^{-5}$, we calculate $\text{pKa}$:

$\text{pKa} = -\log(K_a) = -\log(1.8 \times 10^{-5}) \approx 4.74$

Step 2: Use the Henderson-Hasselbalch Equation

Substitute the values into the Henderson-Hasselbalch equation:

[ 4.91 = 4.74 + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) ]

Solving for the ratio $\frac{[\text{A}^-]}{[\text{HA}]}$:

[ \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) = 4.91 - 4.74 = 0.17 ]

[ \frac{[\text{A}^-]}{[\text{HA}]} = 10^{0.17} \approx 1.48 ]

Step 3: Calculate the Moles of Acetic Acid

The concentration of acetic acid is 0.462 M, and the volume is 80.0 mL (or 0.080 L). The moles of acetic acid (HA) are:

$\text{moles of HA} = 0.462 \, \text{M} \times 0.080 \, \text{L} = 0.03696 \, \text{moles}$

Step 4: Calculate the Required Moles of Sodium Hydroxide

Since $\frac{[\text{A}^-]}{[\text{HA}]} = 1.48$, the moles of $\text{A}^-$ (acetate ion) needed are:

$\text{moles of } \text{A}^- = 1.48 \times \text{moles of HA} = 1.48 \times 0.03696 \approx 0.0547 \, \text{moles}$

Step 5: Calculate the Mass of Sodium Hydroxide

Each mole of sodium hydroxide (NaOH) provides one mole of acetate ion (A$^-$) by reacting with acetic acid. Thus, we need 0.0547 moles of NaOH. The molar mass of NaOH is approximately 40 g/mol, so the required mass is:

$\text{mass of NaOH} = 0.0547 \, \text{moles} \times 40 \, \text{g/mol} \approx 2.19 \, \text{g}$

Conclusion

The closest answer choice to 2.19 g is not listed, which may indicate an issue with the options provided. However, based on the calculations, 2.19 g of sodium hydroxide is the correct answer to achieve the desired pH of 4.91.

Would you like further clarification on any of these steps?

Additional Questions

1. How does the Henderson-Hasselbalch equation work for buffer solutions?
2. What would happen to the pH if more NaOH is added than calculated?
3. Why is acetic acid a suitable choice for a buffer in this pH range?
4. How does the molar mass of a compound impact buffer preparation?
5. What are practical applications of buffer solutions in daily life?

Tip

Always double-check your units when calculating concentration and volume to avoid errors in pH calculations.