pH and pOH calculator

Two things go wrong on pH pages, and both of them look like answers. The first is treating a weak acid as though almost none of it dissociated, which is a fine assumption at 0.1 mol/L and a bad one at 0.0001 mol/L, where the shortcut overstates the pH. The second is the very dilute solution: take −lg(c) at 10⁻⁸ mol/L and the page tells you an acid made the water alkaline, which is not a rounding error but a wrong sign. Both come from the same omission — the ions water supplies on its own — so this page does not omit them. It solves the charge balance, hydrogen ions against hydroxide plus dissociated acid, and an acid can therefore never come out above neutral here however dilute you make it. What it does not carry is a table of Ka values: type the one your own source gives.

How it is calculated

[H+] − Kw/[H+] − Ka·c/(Ka + [H+]) = 0, pH = −lg[H+], pH + pOH = pKw = 13.99 at 25 °C

The equation solved is the charge balance: every hydrogen ion in the solution came either from the acid or from water, and the amount from the acid depends on how much of it has dissociated at that very concentration. It has exactly one positive root, found here by bisection in whole double precision, and a strong acid is the same equation with Ka taken to infinity, which collapses to [H+] = (c + √(c² + 4Kw))/2. IUPAC defines pH through the activity of the hydrogen ion rather than its concentration, and this page — like every school exercise — uses concentration in its place, which is one reason a pH quoted to four decimal places is fiction. The link between pH and pOH is the ionization constant of water, 13.99 at 25 °C rather than a round 14, so pure water is neutral at pH 6.995. Everything here assumes 25 °C, water as the solvent, and one proton per molecule.

Source: IUPAC, Quantities, Units and Symbols in Physical Chemistry (the Green Book), 3rd edition, 2nd printing 2012, Sec. 2.13.1 (viii), definition of pH, p. 75 — "The quantity pH is defined in terms of the activity of hydrogen(1+) ions (hydrogen ions) in solution: pH = paH+ = − lg(aH+)", with the p operator defined as px = − lg x, which is what makes pOH = − lg(aOH−); the same section calls it a notional definition, "since pH is defined in terms of a quantity that cannot be measured independently", and quotes ±0.003 in pH as the uncertainty of a primary standard

Questions people ask

Why is there no list of acids to pick from?

Because a Ka is a measurement, and published values for the same acid differ in the second decimal place between compilations. Silently picking one and labelling it "acetic acid" would make the page look more authoritative than the number underneath it, which is exactly the failure this site tries not to have. Type the pKa your own textbook or datasheet gives and the arithmetic on top of it is exact.

What approximations are left in here?

Two, and neither is the one that usually bites. The first is using concentration where IUPAC defines pH through activity, which is exact only in an infinitely dilute solution and drifts as the ionic strength climbs — expect a tenth of a pH unit or so in a concentrated solution, and no calculator can fix it without knowing your other solutes. The second is temperature: everything here is 25 °C. The approximation that is NOT here is the textbook one, dropping water's own ions to leave a quadratic; that is what fails in dilute solution, and this page solves the charge balance instead. It tells you on the result when water is supplying more than a hundredth of the ions, which is where the two methods start to part company.

Why is neutral 6.995 and not 7?

Because pKw is not exactly 14. IAPWS, which maintains the reference formulation for water, gives 13.99 at 25 °C and atmospheric pressure, so neutral water — where [H+] equals [OH−] — sits at half of that. The familiar 7 comes from rounding Kw to 1.0×10⁻¹⁴, which is a perfectly good round number and not a measurement. It also moves with temperature: the same table gives pKw = 12.25 at 100 °C, where neutral water is pH 6.1 and still neutral.

Can I use it for sulfuric or phosphoric acid?

Not properly. Everything here is monoprotic — one proton per molecule, one dissociation constant. A diprotic acid has a second constant, usually far smaller, and its contribution near the first equivalence point is not something a single dissociation can carry. For a rough first proton you can enter the first pKa and read the answer as an upper bound on the pH, but do not quote it.

How many decimal places should I trust?

Three at the very most, and usually two. IUPAC notes that the uncertainty on a primary standard buffer is around ±0.003 in pH, and that is a value established with a Harned cell under laboratory conditions. Anything a calculator adds past the third decimal is arithmetic, not chemistry. This page rounds pH and pOH to three places for that reason and shows the raw ion concentrations separately, where the extra digits are at least meaningful.

Why does the answer change if I switch acid and base at the same pK?

It should. A pKa describes how readily a species gives a proton away and a pKb how readily it takes one, and they are not the same quantity — for a conjugate pair at 25 °C they add up to pKw, so a base with pKb 4.75 is the partner of an acid with pKa 9.24, not of one with pKa 4.75. If you have a pKa for a base, subtract it from 13.99 before entering it here.

Sources

The documents this page reads its numbers out of, linked so you can check them yourself.

1. IUPAC, Quantities, Units and Symbols in Physical Chemistry (the Green Book), 3rd edition, 2nd printing 2012, Sec. 2.13.1 (viii), definition of pH, p. 75 — "The quantity pH is defined in terms of the activity of hydrogen(1+) ions (hydrogen ions) in solution: pH = paH+ = − lg(aH+)", with the p operator defined as px = − lg x, which is what makes pOH = − lg(aOH−); the same section calls it a notional definition, "since pH is defined in terms of a quantity that cannot be measured independently", and quotes ±0.003 in pH as the uncertainty of a primary standard
2. IAPWS R11-24, Revised Release on the Ionization Constant of H2O — Table 3, calculated values of pKw at 0 °C to 1000 °C, gives pKw = 13.99 at 25 °C with the density of the liquid taken at 0.1 MPa; Kw is the ionization constant of water at the standard molality of 1 mol/kg

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