pH calculator
The pH of a strong acid or base, worked out from the charge balance rather than the formula in the textbook. The two agree until they spectacularly do not. Nothing you type here leaves your browser — it is worked out on your own device, so there is nothing to send and nothing to log.
The textbook formula says an acid is alkaline
Every introduction to pH gives you pH = −log₁₀(c). Put in 10−8 M hydrochloric acid — a real thing you can make, just very dilute — and it returns pH 8.
pH 8 is alkaline. Adding acid to water cannot make it alkaline. There is no quantity of acid, however small, that turns water basic. The formula has produced an answer that is not merely inaccurate but impossible, and it does it without any complaint at all.
What it forgot is that water is not an inert background. Water ionises on its own, and at 10−8 M the water is supplying about ten times more H+ than the acid is. Once you count both:
[H+] = c + Kw/[H+] → [H+]² − c[H+] − Kw = 0 → [H+] = (c + √(c² + 4Kw)) / 2
which gives pH 6.97. Acidic, barely, which is exactly what a very dilute acid should be.
It starts going wrong long before that
The interesting question is not where the shortcut becomes absurd but where it stops being right. Sweeping four thousand concentrations across twelve decades, the two answers first differ by more than 0.01 pH at about 6.5×10−7 M — around pH 6.2, which is an ordinary dilute solution and not an exotic edge case.
| Concentration | Shortcut | Charge balance |
|---|---|---|
| 0.1 M | 1.00 | 1.00 |
| 10−6 M | 6.00 | 5.99 |
| 10−7 M | 7.00 | 6.79 |
| 10−8 M | 8.00 | 6.97 |
| 10−10 M | 10.00 | 7.00 |
Notice the bottom row. As the acid gets infinitely dilute the honest answer approaches neutral water, which is what must happen. The shortcut runs away to infinity.
And neutral is not 7
pH 7 being neutral is a fact about water at 25 °C, and it is not even exactly true there. IAPWS give pKw = 13.99 at 25 °C, so neutral is 6.995. The familiar 1.0×10 −14 is a rounding of that, not the other way round.
Change the temperature and it moves properly:
| Temperature | pKw | Neutral pH |
|---|---|---|
| 0 °C | 14.95 | 7.475 |
| 25 °C | 13.99 | 6.995 |
| 50 °C | 13.26 | 6.630 |
| 100 °C | 12.25 | 6.125 |
Boiling water reading pH 6.13 is not acidic. It is neutral. A pH meter that has not been told the temperature will call it acidic, and will be wrong by most of a pH unit — which is why laboratory meters have a temperature probe attached and cheap ones do not.
Where the numbers come from
The ionization constant of water on this page is from IAPWS R11-24, Revised Release on the Ionization Constant of H2O (2024), Table 3 — © 2024 The International Association for the Properties of Water and Steam, reproduced under their terms, which permit publication in whole or in part provided IAPWS are credited.
Values between the tabulated temperatures are interpolated linearly in pKw. IAPWS publish an equation that would do better; this reproduces their table exactly at the ten temperatures they give, and says so rather than implying it implements their formulation.