pH (source code)
= pH
{c}
{title2=$\mathrm{pH}=-\log_{10}a_{\mathrm H^+}$}
{wiki}
The <pH> of a <chemical solution> is the negative base-ten <logarithm> of its <hydrogen ion> <thermodynamic activity>:
$$
\mathrm{pH}=-\log_{10}a_{\mathrm H^+},\qquad a_{\mathrm H^+}=10^{-\mathrm{pH}}.
$$
The <thermodynamic activity> is a <dimensionless quantity>, so the <logarithm> has a <dimensionless> argument. The definition uses <thermodynamic activity>, rather than total <chemical acid> <concentration>. Single-<ion> <thermodynamic activities> require a convention; practical <pH> measurements are tied to calibrated standards. See https://goldbook.iupac.org/terms/view/P04524[IUPAC Gold Book, pH].
In a dilute ideal <chemical solution> in <water>, a useful approximation is
$$
a_{\mathrm H^+}\approx\frac{[\mathrm H^+]}{c^\circ},\qquad c^\circ=1\,\mathrm{mol\,L^{-1}},\qquad
\mathrm{pH}\approx-\log_{10}\!\left(\frac{[\mathrm H^+]}{1\,\mathrm{mol\,L^{-1}}}\right).
$$
Square brackets denote <concentration> in <moles> per litre. In <water>, $[\mathrm H^+]$ is shorthand for the <hydronium> <concentration>, often written $[\mathrm H_3\mathrm O^+]$; it does not describe isolated bare <protons>. Departures from ideal behavior need <activity coefficients>. The <concentration> formula approximates the <thermodynamic activity> definition.
For an ideal <strong acid> supplying one <proton> per initial unit, stipulate a total <concentration> of $0.001\,\mathrm{mol\,L^{-1}}$ and negligible <self-ionization of water>. Then $[\mathrm H^+]\approx10^{-3}\,\mathrm{mol\,L^{-1}}$, so
$$
\mathrm{pH}\approx-\log_{10}(10^{-3})=3.
$$
Reducing that <concentration> tenfold by <dilution> gives $10^{-4}\,\mathrm{mol\,L^{-1}}$ and <pH> $4$, under the same assumptions. More generally, dividing the <hydrogen ion> <thermodynamic activity> by ten raises <pH> by one. This is why the scale compresses an enormous range: equal steps of <pH> represent multiplicative changes, rather than equal differences in <concentration>. These are model calculations, rather than measured samples.
\Image[/ph-logarithmic-scale.png]
{title=One step of <pH> represents a tenfold change in <hydrogen ion> <thermodynamic activity>}
{description=The <logarithm> makes equal spacing on the <pH> axis correspond to factors of ten on the <thermodynamic activity> axis. The marked <strong acid> example uses the dilute ideal <concentration> approximation and negligible <self-ionization of water>. The plotted interval is illustrative; it does not bound possible <pH> values.}
Neutrality depends on <temperature>. In the dilute ideal <water> model, define the <dimensionless> ion product $K_w=([\mathrm H^+]/c^\circ)([\mathrm{OH}^-]/c^\circ)$ and $\mathrm pK_w=-\log_{10}K_w$. Equal <hydronium> and <hydroxide> <concentrations> imply
$$
\mathrm{pH}_{\mathrm{neutral}}=\frac{\mathrm pK_w}{2}.
$$
At $25\,{}^\circ\mathrm C$, the textbook approximation $K_w\approx10^{-14}$ gives neutral <pH> $7$. Since $K_w$ varies with <temperature>, neutrality need not mean <pH> $7$. Nor is <pH> restricted to $0$–$14$: the activity definition permits negative <pH> when $a_{\mathrm H^+}>1$, and <pH> above $14$ when $a_{\mathrm H^+}<10^{-14}$. High <concentrations> can invalidate the dilute ideal approximation. See https://openstax.org/books/chemistry-2e/pages/14-2-ph-and-poh[Flowers et al., Chemistry 2e, pH and pOH].
<S. P. L. Sørensen> introduced <pH> at the Carlsberg Laboratory in 1909, as recorded in the https://carlsbergfondet.dk/en/about-the-foundation/our-history/timeline-of-the-foundation-s-history/[Carlsberg Foundation historical timeline].
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