acid base - Buffer Capacity Calculation

I know buffer capacity is the following: $$β=\frac{Δ(\ce{H+})}{Δ(\mathrm{pH})}$$ specifically the amount of acid/base that needs to be added to change pH by 1 unit.

1. 
   
   

   If I have data about how pH of a protein has changed upon adding specific amounts of acid, how do I calculate buffer capacity?

   
   


2. 
   

   Is there any reason why the change in pH needs to be 1? If I calculate how much acid needs to be added for the pH to change by 1.1, can this be scaled to determine amount of acid needed to change pH by 1?

   
   

Answer

The buffer capacity of a weak acid-conjugate base buffer is defined as the number of moles of strong acid needed to change the $\ce{pH}$ by 1 unit. $$\beta = \frac{\mathrm{d}[A]}{\mathrm{dpH}} $$ and the acid is present as $$[A]= \frac{K_\mathrm{w}}{[\ce{H+}]}-[\ce{H+}] +\frac{C_\mathrm{B}K_\mathrm{a}}{[\ce{H+}]+K_\mathrm{a}}$$ where $K_\mathrm{w}$ is the water ionization equilibrium constant, $10^{-14} $, $K_\mathrm{a}$ is the acid dissociation constant, and $C_\mathrm{B}$ is the total concentration of buffer. You assume that your protein is the weak acid. Change $\ce{pH}$ to $\ce{pH}=-\log_{10}([\ce{H+}])$ to differentiate or use the product rule. You should then find that $$\beta = 2.303\left[ \frac{k_\mathrm{w}}{[\ce{H+}]}+[\ce{H+}] +\frac{C_\mathrm{B}K_\mathrm{a}[\ce{H+}]}{([\ce{H+}]+\ce{K_\mathrm{a}})^2}\right]$$ you can then plot $\beta$ vs $\ce{pH}$ and from this you should be able to find what you want.