To solve this problem, we will use the Henderson-Hasselbalch equation:pH = pKa + log [A-]/[HA] where pH is the solution's pH, pKa is the acid dissociation constant, [A-] is the concentration of the conjugate base acetate ion , and [HA] is the concentration of the weak acid acetic acid .First, we need to find the pKa of acetic acid. The pKa of acetic acid is 4.76.Given the initial pH of 4 and the pKa of acetic acid, we can calculate the initial ratio of [A-]/[HA]:4 = 4.76 + log [A-]/[HA] -0.76 = log [A-]/[HA] 10^-0.76 = [A-]/[HA]Since the initial concentrations of acetic acid and sodium acetate are both 0.1 M, we can assume that [A-] = [HA] = 0.1 M. Now, let's add the buffer solution containing 0.2 M acetic acid and 0.2 M sodium acetate.To find the new concentrations of acetic acid and acetate ion, we need to consider the volumes of the two solutions. Let V1 be the volume of the initial solution and V2 be the volume of the buffer solution. Then, the new concentrations of acetic acid and acetate ion can be calculated as follows:[HA]_new = [HA]_initial * V1 + [HA]_buffer * V2 / V1 + V2 [A-]_new = [A-]_initial * V1 + [A-]_buffer * V2 / V1 + V2 Since we don't have the volumes of the solutions, we can't calculate the exact concentrations. However, we can still find the new pH of the solution using the Henderson-Hasselbalch equation:pH_new = pKa + log [A-]_new/[HA]_new Since the buffer solution contains equal concentrations of acetic acid and sodium acetate 0.2 M each , the ratio [A-]_new/[HA]_new will be close to 1, and the new pH will be close to the pKa of acetic acid, which is 4.76.In conclusion, after adding the buffer solution, the new pH of the solution will be close to 4.76. The exact concentrations of acetic acid, acetate ion, and hydrogen ion cannot be determined without knowing the volumes of the solutions.