= Buffer capacity
{title2=$\beta$}
<Buffer capacity> measures resistance to a change in <pH>: how much additional <chemical acid> or <chemical base> a <chemical solution> accommodates per small change in <pH>. Its differential form is a local <derivative>, rather than the total inventory available before a <buffer solution> is exhausted. Two <buffer solutions> can have identical <pH> and very different <buffer capacities>.
Choose a constant <volume (thermodynamics)> $V$ and fixed <temperature>. Let $n_B$ count added <moles> of fully dissociated, monovalent <chemical base> supplying <hydroxide>, and $n_A$ added <moles> of fully dissociated, monovalent <strong acid> supplying <hydrogen ions>. Neither addition supplies the <weak acid> component itself. The signed addition is $b=(n_B-n_A)/V$, with units $\mathrm{mol\,L^{-1}}$, and
$$
\beta=\frac{\mathrm db}{\mathrm d\mathrm{pH}}=\frac1V\frac{\mathrm dn_B}{\mathrm d\mathrm{pH}}=-\frac1V\frac{\mathrm dn_A}{\mathrm d\mathrm{pH}}.
$$
The last two forms describe separate base-only and acid-only additions. The <buffer capacity> is positive for the stable <chemical equilibrium> model below and is reported in $\mathrm{mol\,L^{-1}}$ per unit of <pH>. Since <pH> is <dimensionless>, the dimensional unit is just $\mathrm{mol\,L^{-1}}$; stating “per <pH>” makes the measurement convention clear. Here constant <volume (thermodynamics)> is an idealization, realized approximately by small additions of concentrated reagent.
For a sufficiently small addition, $\Delta\mathrm{pH}\approx\Delta b/\beta$. A large <buffer capacity> means a small local change of <pH> for a given addition. Over a finite interval, use the changing <buffer capacity> instead:
$$
\Delta b=\int_{\mathrm{pH}_0}^{\mathrm{pH}_1}\beta(p)\,\mathrm dp.
$$
This <integral> gives a finite dose in $\mathrm{mol\,L^{-1}}$; it need not equal the initial <buffer capacity> times the eventual change in <pH>.
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