To calculate the pH of the buffer solution after the addition of $0.0388$ mol of HCl, we can use the Henderson-Hasselbalch equation and consider the effect of the added HCl on the buffer system.

Initial Information:

• Initial moles of acetic acid ($\text{HC}_2\text{H}_3\text{O}_2$): $0.153$ mol
• Initial moles of acetate ($\text{NaC}_2\text{H}_3\text{O}_2$): $0.194$ mol
• $K_a$ of acetic acid = $1.8 \times 10^{-5}$
• Initial pH calculated: 4.85

Step-by-Step Solution:

1. Determine the effect of adding HCl:

   • $\text{HCl}$ is a strong acid and will dissociate completely, providing $0.0388$ mol of $\text{H}^+$.
   • The $\text{H}^+$ ions from $\text{HCl}$ will react with the acetate ions ($\text{C}_2\text{H}_3\text{O}_2^-$) in the buffer, forming additional acetic acid ($\text{HC}_2\text{H}_3\text{O}_2$).

2. Calculate the new amounts of acetic acid and acetate:

   • After adding $0.0388$ mol of $\text{HCl}$:

     • Acetate ions ($\text{C}_2\text{H}_3\text{O}_2^-$) will decrease by $0.0388$ mol.
     • Acetic acid ($\text{HC}_2\text{H}_3\text{O}_2$) will increase by $0.0388$ mol.

   • New moles of acetic acid = $0.153 + 0.0388 = 0.1918$ mol

   • New moles of acetate = $0.194 - 0.0388 = 0.1552$ mol

3. Use the Henderson-Hasselbalch equation to find the new pH: [ \text{pH} = \text{p}K_a + \log \left( \frac{[\text{A}^-]}{[\text{HA}]} \right) ] where:

   • $\text{p}K_a = -\log(1.8 \times 10^{-5}) \approx 4.74$
   • $[\text{A}^-] = 0.1552$ mol (acetate concentration)
   • $[\text{HA}] = 0.1918$ mol (acetic acid concentration)

4. Plug in the values: $\text{pH} = 4.74 + \log \left( \frac{0.1552}{0.1918} \right)$ $\text{pH} = 4.74 + \log(0.809)$ $\text{pH} = 4.74 - 0.092$ $\text{pH} \approx 4.65$

Final Answer:

The pH of the buffer solution after the addition of $0.0388$ mol of HCl is approximately 4.65.

Would you like further details or explanations on this solution?

Here are some related questions you might find helpful:

1. How does the Henderson-Hasselbalch equation apply to buffer solutions?
2. What is the effect of adding a strong acid to a buffer solution?
3. How would the buffer capacity affect the pH change upon acid addition?
4. What are some real-life applications of buffer solutions in chemistry?
5. How does the pKa value influence the pH of a buffer?

Tip: Remember, buffers work best when the pH is close to the pKa of the acid, within about one pH unit above or below.