Refer to:
http://www4.ncsu.edu/~franzen/public_html/CH201/data/Acid_Base_Table.pdf
Second dissociation constant for H₂CO₃ = 4.7 × 10⁻¹¹
Let V L be the volume of K₂CO₃ added.
[CO₃²⁻]/[HCO₃⁻] = (No. of moles of K₂CO₃)/(No. of moles of KHCO₃) = 0.200V/(0.1×0.1) = 20V
HCO₃⁻(aq) ⇌ CO₃²⁻(aq) + H⁺(aq) …… Ka = 4.7 × 10⁻¹¹
pH = pKa + log([CO₃²⁻]/[HCO₃⁻])
11.0 = -log(4.7 × 10⁻¹¹) + log(20V)
log(20V) = 11.0 + log(4.7 × 10⁻¹¹)
20V = 10^[11.0 + log(4.7 × 10⁻¹¹)]
V = 10^[11.0 + log(4.7 × 10⁻¹¹)] / 20
V = 0.235
Volume of K₂CO₃ added = 0.235 L