Consider the titration of 50.0 mL of a 0.100 M solution of the protonated form of the amino acid alanine (H₂A⁺: K_a1 = 4.6 × 10^–3, K_a2 = 2.0 × 10^–10) with 0.100 M NaOH. Calculate the pH after the addition of each of the following volumes of base. (b) 25.0 mL (c) 50.0 mL

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1. Determine the initial moles of H2A+ in the solution by multiplying the volume (50.0 mL) by the concentration (0.100 M). Convert the volume to liters before multiplying.
2. Calculate the moles of NaOH added by multiplying the volume of NaOH added (25.0 mL for part b and 50.0 mL for part c) by its concentration (0.100 M). Convert the volume to liters before multiplying.
3. For part (b), compare the moles of NaOH added to the initial moles of H2A+. Since 25.0 mL of NaOH is added, it is half the equivalence point, and the solution is a buffer. Use the Henderson-Hasselbalch equation to find the pH, considering the first dissociation (Ka1).
4. For part (c), since 50.0 mL of NaOH is added, it reaches the first equivalence point where all H2A+ is converted to HA. Calculate the pH by considering the hydrolysis of HA using the second dissociation constant (Ka2) and the concentration of HA at this point.
5. Use the appropriate equilibrium expressions and assumptions to solve for the pH in each case, ensuring to check if any simplifications can be made based on the relative magnitudes of Ka1 and Ka2.