How To Calculate pH

pH is a compact way of writing a hydrogen ion concentration that would otherwise be a small number with a large negative exponent. It is defined as the negative base ten logarithm of the concentration in moles per litre, so a concentration of 0.001 gives a pH of 3. The logarithm is what makes the scale manageable, and it is also what makes the scale counter-intuitive: a lower pH means a higher concentration of hydrogen ions.

The sign convention exists so that ordinary solutions land on convenient positive numbers. Pure water at 25 degrees Celsius has a hydrogen ion concentration of 0.0000001, which is one times ten to the minus seven, and its pH is therefore 7. Acidic solutions have more hydrogen ions and a pH below 7; alkaline solutions have fewer and a pH above 7.

Because the scale is logarithmic, a change of one pH unit means a tenfold change in concentration. Going from pH 4 to pH 3 is not a small adjustment, it is ten times the hydrogen ion concentration. A change of two units is a hundredfold, and three units is a thousandfold. This is the single most useful fact about pH and the one most often mishandled.

The pOH is defined the same way for hydroxide ions, and at 25 degrees Celsius the two are linked by pH plus pOH equals 14. That constant comes from the self-ionisation of water, whose ion product is one times ten to the minus fourteen. Because the sum is fixed, any statement about pH is simultaneously a statement about pOH, and one can always be read off from the other.

Strong acids dissociate completely, so the hydrogen ion concentration equals the acid concentration and the pH follows directly from the logarithm. A 0.01 mole per litre solution of a strong monoprotic acid therefore has a pH of exactly 2. Weak acids dissociate only partly, and the pH then depends on the acid dissociation constant as well as the concentration.

For a weak acid the calculation uses the half-logarithm shortcut: the pH is approximately one half of the pKa minus the logarithm of the concentration. The pKa is itself minus the logarithm of the acid dissociation constant. This approximation assumes the acid is barely dissociated, which holds for the dilute solutions where the formula is normally used.

Dilution changes the concentration in proportion to the volumes involved. Mixing a small sample volume with a larger volume of water multiplies the concentration by the ratio of the sample volume to the total volume. Ten millilitres made up to one hundred millilitres is a tenfold dilution, and since the concentration falls by a factor of ten the pH rises by exactly one unit.

There is a limit to how far dilution can push the pH, and it is the reason a litre of pure water does not have a pH of 8. Once the acid has been diluted so far that its contribution is smaller than water's own self-ionisation, the water dominates and the pH settles back towards 7. The simple logarithmic formula ignores this and will happily report a pH of 9 for an extremely dilute acid, which is physically impossible.

The measurement is also temperature dependent. The ion product of water grows with temperature, so the neutral point moves below 7 as a solution warms. A pH of 7 is therefore only neutral at 25 degrees Celsius; the definition of pH itself, the logarithm, does not change, but the point where hydrogen and hydroxide concentrations balance does.

In practice pH is measured with a glass electrode that reads a voltage proportional to the logarithm of the activity rather than the raw concentration. Activity differs from concentration in solutions with a high concentration of dissolved salts, which is why a carefully calculated pH can differ slightly from a meter reading. For teaching and for most laboratory work the concentration-based formula is the one to use, and it is the one this page computes.