To find the theoretical pH after each addition of 1 mL of 0.10 M HCl to your buffer solution of acetic acid (0.05 M) and sodium acetate (0.05 M), we can use the Henderson-Hasselbalch equation:

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

Initial Conditions:

• Volume of acetic acid (HA): 10 mL of 0.05 M → $n_{HA} = 0.05 \times 0.01 = 0.0005 \, \text{mol}$
• Volume of sodium acetate (A⁻): 10 mL of 0.05 M → $n_{A^-} = 0.05 \times 0.01 = 0.0005 \, \text{mol}$
• Total Volume: 20 mL

Initial pH Calculation:

Since the initial conditions are already given, we start with:

$\text{Initial pH} = 4.75$

After Each Addition of 1 mL of 0.10 M HCl:

1. Amount of HCl added:

   • After each addition, $1 \, \text{mL}$ of $0.10 \, \text{M}$ HCl adds $0.001 \, \text{mol}$ of H⁺.

2. Reaction with Acetic Acid and Sodium Acetate:

   • H⁺ reacts with the acetate ion (A⁻) to form acetic acid (HA).

Iterative Calculations:

1. After 1 mL HCl:

   • Moles of A⁻ after addition: $n_{A^-} = 0.0005 - 0.001 = -0.0005$ (excess H⁺).
   • Moles of HA after addition: $n_{HA} = 0.0005 + 0.001 = 0.0015$.
   • Total moles = 0.0015 + 0 (since A⁻ becomes effectively zero).
   • New concentrations in 21 mL: $\text{pH} = 4.75 + \log\left(\frac{0}{0.0015}\right) \text{ (undefined)}$

2. After 2 mL HCl:

   • Total H⁺ added = 0.002 mol → Moles of HA = 0.0025, Moles of A⁻ = -0.0015.
   • New concentrations: $\text{pH} = 4.75 + \log\left(\frac{0}{0.0025}\right) \text{ (undefined)}$

3. After 3 mL HCl:

   • Total H⁺ added = 0.003 mol → Moles of HA = 0.0035, Moles of A⁻ = -0.0025.
   • New concentrations: $\text{pH} = 4.75 + \log\left(\frac{0}{0.0035}\right) \text{ (undefined)}$

4. After 4 mL HCl:

   • Total H⁺ added = 0.004 mol → Moles of HA = 0.0045, Moles of A⁻ = -0.0035.
   • New concentrations: $\text{pH} = 4.75 + \log\left(\frac{0}{0.0045}\right) \text{ (undefined)}$

Conclusion:

After the addition of 1 mL of HCl, the buffer system is overwhelmed, and the pH will drop significantly, approaching the pH of a strong acid solution as all acetate ions are consumed.

Summary of pH Changes:

1. Initial pH: 4.75
2. After 1 mL HCl: pH significantly decreases (undefined due to lack of A⁻).
3. After 2 mL HCl: pH drops further (undefined).
4. After 3 mL HCl: pH continues to drop (undefined).
5. After 4 mL HCl: pH continues to drop (undefined).

Questions for Further Exploration:

1. How would the pH change if a different buffer was used?
2. What would happen if you used a weak base instead of HCl?
3. How does the concentration of the buffer components affect pH stability?
4. Can you calculate the exact pH after adding a specific volume of a strong base?
5. What are the implications of exceeding buffer capacity in biological systems?

Tip:

When working with buffers, always consider the capacity and the limits of the buffering system to maintain pH.