= Buffer capacity and concentration
Stipulate <pKa> $5$ and $K_w=10^{-14}$; these are model inputs, not measurements of an identified substance. At <pH> $5$ the pair contribution is $0.575646\ldots C$. For <analytical concentrations> $C=0.010$ and $0.100\,\mathrm{mol\,L^{-1}}$, respectively, it is
$$
\beta_{\mathrm{pair}}=0.00575646\ldots\quad\text{and}\quad0.0575646\ldots\;\mathrm{mol\,L^{-1}\ per\ pH}.
$$
The free <hydrogen ion> and <hydroxide> contribution is the same at this <pH>, $\beta_{\mathrm{water}}=(\ln10)(10^{-5}+10^{-9})\,\mathrm{mol\,L^{-1}}=0.0000230282\ldots\,\mathrm{mol\,L^{-1}}$ per <pH>. The full <buffer capacities> are therefore approximately $0.00577949$ and $0.0575877\,\mathrm{mol\,L^{-1}}$ per <pH>. Identical <pH> fixes the free <hydrogen ion> <thermodynamic activity>; it does not fix the inventory of <weak acid> and <conjugate base> available for <neutralization>.
\Image[/monoprotic-buffer-capacity-concentration.png]
{title=At fixed <pKa>, the pair contribution to <buffer capacity> scales with <analytical concentration>}
{id=monoprotic-buffer-capacity-concentration}
{description=Ideal <monoprotic acid> model with stipulated <pKa> $5$ and $K_w=10^{-14}$. Left: the pair contribution for two <analytical concentrations>, with its maximum at <pH> equal to <pKa>. Right: pair, free <hydrogen ion> and <hydroxide>, and total contributions for $C=0.010\,\mathrm{mol\,L^{-1}}$, on a <logarithmic scale>. Curves are calculated, not measured; the constant ideal <activity coefficient> assumption limits their interpretation in real samples.}
At one unit of <pH> on either side of <pKa>, $r=10$ or $0.1$, giving $\beta_{\mathrm{pair}}/\beta_{\mathrm{pair,max}}=40/121\approx0.331$. The partners still form a <buffer solution>, but its local resistance is only about one third of the maximum for that <analytical concentration>. At fixed <pH>, the pair contribution scales exactly with $C$; the full <buffer capacity> contains the additional $C$-independent <water> contribution. Consequently, extreme <dilution> does not justify dropping <self-ionization of water>.
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