To solve this buffer problem, we'll use the Henderson-Hasselbalch equation for part (a) and then adjust for the addition of HCl in part (b).

(a) Calculating the pH of the Buffer

The Henderson-Hasselbalch equation is:

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

Where:

• $[\text{A}^-]$ is the concentration of the conjugate base (sodium acetate, NaC₂H₃O₂).
• $[\text{HA}]$ is the concentration of the weak acid (acetic acid, HC₂H₃O₂).
• $\text{pKa} = -\log(\text{Ka})$.

Given:

• $\text{Ka} = 1.8 \times 10^{-5}$
• $[\text{A}^-] = 0.300 \, \text{mol/L}$
• $[\text{HA}] = 0.300 \, \text{mol/L}$

First, we calculate $\text{pKa}$:

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

Now we plug in the values into the Henderson-Hasselbalch equation:

$\text{pH} = 4.74 + \log\left(\frac{0.300}{0.300}\right) = 4.74 + \log(1) = 4.74 + 0 = 4.74$

So, the pH of the buffer is 4.74.

(b) Calculating the pH After Adding HCl

When HCl is added to the buffer, it will react with the acetate ions ($C_2H_3O_2^-$) to form acetic acid ($HC_2H_3O_2$). The reaction is:

$C_2H_3O_2^- + H_3O^+ \rightleftharpoons HC_2H_3O_2 + H_2O$

1. Initial moles before HCl addition:

   • Moles of $HC_2H_3O_2$ = 0.300 mol
   • Moles of $C_2H_3O_2^-$ = 0.300 mol
   • Moles of $HCl$ added = 0.020 mol

2. Reaction after adding HCl:

   • Moles of $C_2H_3O_2^-$ will decrease by 0.020 mol.
   • Moles of $HC_2H_3O_2$ will increase by 0.020 mol.

So after the reaction, we have:

• Moles of $HC_2H_3O_2$ = $0.300 + 0.020 = 0.320 \, \text{mol}$
• Moles of $C_2H_3O_2^-$ = $0.300 - 0.020 = 0.280 \, \text{mol}$

Now we recalculate the pH using the Henderson-Hasselbalch equation with the new concentrations:

$\text{pH} = 4.74 + \log\left(\frac{0.280}{0.320}\right)$

Calculating the log term:

$\log\left(\frac{0.280}{0.320}\right) = \log(0.875) \approx -0.056$

Now substituting back:

$\text{pH} \approx 4.74 - 0.056 \approx 4.68$

Thus, the pH of the solution after adding 0.020 mol of HCl is approximately 4.68.

Summary of Results

• (a) pH of the buffer: 4.74
• (b) pH after adding HCl: 4.68

Do you have any questions or need further details? Here are some related questions for your consideration:

1. How does the pH change with different concentrations of acid and base?
2. What would happen if a strong base was added instead of HCl?
3. How does the buffer capacity relate to the concentrations of acid and conjugate base?
4. What other buffers could be used for a similar pH range?
5. How can you prepare a buffer with a higher pH?

Tip: When working with buffers, it's crucial to remember that their pH can resist change upon the addition of small amounts of acid or base.