= Henderson-Hasselbalch equation
{c}
{title2=$\mathrm{pH}\approx\mathrm pK_a+\log_{10}([\mathrm A^-]/[\mathrm{HA}])$}
{wiki=Henderson–Hasselbalch_equation}
The <Henderson-Hasselbalch equation> estimates <buffer solution> <pH> using a <weak acid>'s <acid dissociation constant> and the <conjugate base> to <weak acid> <concentration> ratio.
Its activity form is an exact rearrangement of the chosen <chemical equilibrium> relation. Write the acid dissociation step as $\mathrm{HA}\rightleftharpoons\mathrm H^++\mathrm A^-$, with hydrated <hydrogen ions> understood in <water> and the <solvent> convention incorporated into the <acid dissociation constant>. For consistent <standard states>,
$$
K_a=\frac{a_{\mathrm H^+}a_{\mathrm A^-}}{a_{\mathrm{HA}}},\qquad
\mathrm pK_a=-\log_{10}K_a,\qquad
\mathrm{pH}=\mathrm pK_a+\log_{10}\!\left(\frac{a_{\mathrm A^-}}{a_{\mathrm{HA}}}\right).
$$
Here each $a_i$ is a <dimensionless> <thermodynamic activity> at <chemical equilibrium>. Exactness refers to this specified <acid dissociation constant> relation and the chosen single-<ion> <thermodynamic activity> convention, not to identifying the ratio with quantities initially mixed. If <water> is instead written explicitly as a <reactant>, its <thermodynamic activity> must be treated consistently. See https://eprints.whiterose.ac.uk/id/eprint/167121/1/BM_5_2020_Unterricht_Hippler_Metcalfe.pdf[Hippler and Metcalfe, Using activities to correct the Henderson-Hasselbalch equation (2020), equations 3–4].
For a toy <weak acid> with stipulated <pKa> $5$, use the ideal dilute <concentration> approximation and put $r=[\mathrm A^-]/[\mathrm{HA}]$. The <Henderson-Hasselbalch equation> becomes $\mathrm{pH}\approx5+\log_{10}r$:
$$
r=0.1\ \Rightarrow\ \mathrm{pH}\approx4,\qquad
r=1\ \Rightarrow\ \mathrm{pH}\approx5,\qquad
r=10\ \Rightarrow\ \mathrm{pH}\approx6.
$$
More <conjugate base> relative to <weak acid> means higher <pH>. Each tenfold increase in the ratio contributes exactly one to the calculated <pH>. The stipulated <pKa> is not a tabulated value for an identified substance.
\Image[/buffer-ph-versus-species-ratio.png]
{title=Toy <pKa> $5$: ratios $0.1$, $1$, and $10$ give calculated <pH> $4$, $5$, and $6$}
{description=Both panels show the same <Henderson-Hasselbalch equation> in the ideal <concentration> approximation, with linear and <logarithmic scales> for the <conjugate base> to <weak acid> ratio. The <logarithmic scale> turns the curve into a straight line. The plotted range is illustrative, rather than a guarantee of useful <buffer capacity>.}
The distinction between <thermodynamic activity> and <concentration> is explicit when $a_i=\gamma_i[i]/c^\circ$, where $\gamma_i$ is an <activity coefficient> and $c^\circ$ is the reference <concentration>:
$$
\mathrm{pH}=\mathrm pK_a+\log_{10}\!\left(\frac{[\mathrm A^-]}{[\mathrm{HA}]}\right)+\log_{10}\!\left(\frac{\gamma_{\mathrm A^-}}{\gamma_{\mathrm{HA}}}\right).
$$
Neglecting the final term gives the usual approximate <Henderson-Hasselbalch equation>. It is justified when the ratio of <activity coefficients> is sufficiently close to one, as in a suitable dilute ideal approximation. <Ionic strength>, including contributions from other dissolved <ions>, affects <activity coefficients>; a nominally dilute <weak acid> alone does not establish that all corrections are small. The ratio must use <chemical equilibrium> <concentrations>. Replacing these by prepared amounts or post-<neutralization> inventories is a further approximation, suitable only when subsequent redistribution is small. See https://eprints.whiterose.ac.uk/id/eprint/167121/1/BM_5_2020_Unterricht_Hippler_Metcalfe.pdf[Hippler and Metcalfe, concentration, activity, and analytical-concentration approximations].
The short derivation is to solve the <acid dissociation constant> relation for $a_{\mathrm H^+}$, take a negative base-ten <logarithm>, and use the definitions of <pH> and <pKa>. Its useful beauty is that a <chemical equilibrium> becomes a simple rule about a ratio: select a <weak acid> with <pKa> near the desired <pH>, then adjust the partner ratio. Scaling both partner <concentrations> equally leaves the ideal <pH> prediction unchanged, while changing <buffer capacity>. In practical <buffer solutions>, <temperature>, <ionic strength>, and the available inventories still matter.
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