Relating pH and buffer capacity

So, here is the equation for a weak acid - strong base pH curve

And here is the expression gotten when the derivative of concentration of hydronium ions with respect to volume of base added is applied: Additionally, here is the buffer capacity formula found: Now, I want to be able to relate these two (buffer capacity versus pH). I've done a buffer capacity versus pH experiment wherein I measured how buffer capacity changes with pH. The graph I got reaches its peak when pH = pK value. My challenge is, how do I relate the two equations for buffer capacity and pH. Additionally, is there any experiment you suggest I do to further the theoretical model of the pH curve (beyond just doing a simple acid-base titration for verification)?

I suggest you plot a logarithmic diagram of your system, i.e. pH = f(log Ci), where Ci is the concentration of each protolyte in the solution. The buffer capacity can then be plotted as

pH = f(log(β/ln10)).

If you have plotted your logarithmic diagram properly, you can directly plot the line for log(β/ln10).

I will give you an example:

Suppose we have a weak acid HA with the pka = 4.8. The line for log(β/ln10) goes from (0; 0) along the line of log [H3O+] until it reaches the crossing with the line for log [A-]. Here it bends upwards 0.3 log C units above the crossing point. It then follows the line for log [A-] close to pH = pka. Here it bends down, passing pH = pka, 0.6 log C units below the line for log Ctot. Then the line for log(β/ln10) will follow the line for log [HA] until it reaches the crossing point with the line for

log [OH-]. Here the line for log(β/ln10) bends up again and passes the crossing point 0.3 units higher. Then the line for log(β/ln10) will follow the line for log [OH-] to pH = 14.

From this example, we can immediately identify a local maximum for the buffer capacity, i.e. at pH = pka. From the diagram, we can also identify two local minima, which will be at about pH 3 and close to pH 9. We can also directly estimate the maximum buffer capacity from the diagram.

Logarithmic diagrams are excellent tools for this kind of problems. If you understand the rules, you can plot the buffer capacity (log(β/ln10) directly in the diagram without any calculations.

I tried to copy in a logarithmic diagram of the example above, but it did not work.

This class of problems is solved quite simply. You start with Henderson - Hasselbalch $r = 10^{pH - pK}$ in which $r$ is the ratio if dissociated to undissociated acetate. It's easy to see that the fraction of the total acetate that is dissociated is $f_1$ = 1/(1+r). This is the charge on acetate ions. The buffering capacity of the acetate is then simply $\partial f_1\partial pH$. Water has a net charge of $Q_w = 10^{-pH} - 10^{pH - pK_w}$ The buffering capacity of water is then $\partial Q_w/\partial pH$ and the buffering of the mixture the sum of those two derivatives.

To see the relatioship bewtween buffering capacity and pH one simply calculates $mf_1 + Q_w$ in which $m$ is the number of moles of acetate in a liter of water and makes a plot vs pH.If you stay away from p < 3 (where you are 'titrating' water - not acetic acid) the curve should look like a stair step with the pK of acetic acid half way up the riser (charge $-m/2$). Differentiating this with respect to $pH$ gives the buffering capacity. Obviously, there will be a major bump at $pK_a$

The fact that the curve is so simple to compute means that if you hace software that will allow the fitting of user defined functions to data you can estimate $m$ and $pK$ from laboratory titration data.