How do you know when something has greater/smaller capacity? do you add them and the smallest one is for greater, and the biggest number is for smallest capacity?

Question

Which of the following solutions has the smallest buffering capacity?
a. 0.821M HF and 0.217M NaF
b. 0.821M HF and 0.909M NaF
c. 0.100M HF and 0.217M NaF
d. 0.121M HF and 0.667M NaF

Answer 1

The only way I know to do this is to calculate the buffer capacity of each, then compare them.
The buffer capacity is the number of moles of a strong acid or a strong base that causes 1.00 L of the buffer to undergo a 1.00 unit change in pH. So you calculate the pH of the buffer in each case, add 1 to the pH if adding base or subtract 1 from the pH if adding acid, then see moles of the base(or acid) added that makes the change. The largest number wins.
For example, I calculate the pH of A to be 2.56 based on Ka of HF = 7.2E-4 but that may not be the Ka value in your text or notes. Then if we change that by 1.00 by adding acid, the pH would become 1.56 and the amount of acid needed to make that change is 0.190 moles. For B, it has the same initial pH, and a 1.00 change in pH needs 0.865 moles to make the change; therefore, B is a higher buffering capacity than A. The others are done the same way.