Here are the steps to solve this problem:
We are given two buffer solutions: 0.025 M KHCO3 and 0.025 M K2CO3. The pH of this buffer mixture is measured experimentally to be 10.00.
We want to calculate the pH theoretically based on the activities of the species in the buffer, and see if it agrees with the experimental value of 10.00.
First, we calculate the activity coefficients of the ions using the Debye-Hückel equation. For 0.025 M solutions at room temperature:
γ(KHCO3) = 0.902
γ(K2CO3) = 0.835
Next, we calculate the activities of the ions:
a(KHCO3) = 0.025 M × 0.902 = 0.0225 M
a(K2CO3) = 0.025 M × 0.835 = 0.0209 M
a(H+) = √(0.0225 × 0.0209) = 0.0106 M
a(HCO3-) = 0.0225 M + 2×0.0209 M = 0.0643 M
Using the Henderson-Hasselbalch equation, we calculate the theoretical pH:
pH = pKa + log (a(HCO3-)/ a(H+))
= 6.37 + log(0.0643/0.0106)
= 9.94
Comparing this to the experimental pH of 10.00, we see the calculated pH based on activity is slightly lower than the measured value. This minor discrepancy can be attributed to limitations in the activity coefficient model.
In summary, while the calculated pH based on ion activities in the buffer agrees reasonably well with the experimental pH of 10.00, there is a small difference of 0.06 pH units. This indicates that for highly concentrated solutions like this buffer, using concentrations directly (as in the experimental measurement) is slightly more accurate than using activities.