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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.
If I have data about how pH of a protein has changed upon adding specific amounts of acid, how do I calculate buffer capacity?
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?
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.
You can't just willy nilly add acid or base to measure a pH change of 1.1 and then back calculate how much acid/base would be needed to change the pH by 1.0 pH units.
In order to do the back calculation you'd need to know what pH you started at, what pH you finished at, and the pKa and pKb values for all the species that interact with an acid or base in that pH range.
Think about being given a buffer solution as an unknown. You're adding base and watching the pH increase. Is it a single protonic acid or a double protonic acid? A difference of 1 pH unit is more than enough to have two pKa's be significant.
To calculate the buffering capacity you'd need to know how much of each ionizing group were present and what each of their pK's is. To measure the buffering capacity you simply make incremental small additions of acid and/or base and, for each addition, plot the total amount of base added against the pH observed after that addition. I know it's traditional to plot pH against added base but that's not what we want here. Try to make the base additions small enough that the curve is smooth and shows any inflections. When you have a good looking curve fit a polynomial to it. Now differentiate the polynomial. The result is the buffering capacity consistent with the definition in the question.
There is confusion often because the buffering capacity isn't the quantity of protons that must be withdrawn or added to bring about a change of 1 pH. It is the slope of the line tangent to the titration curve at the pH of interest expressed in units of Eq/pH.
