pH is defined operationally as follows. For a solution X, first measure the electromotive force E of the galvanic cell
:reference electrode | concentrated solution of KCl | solution X | H | Pt
and then also measure the electromotive force E of a galvanic cell that differs from the above one only by the replacement of the solution X of unknown pH, pH(X), by a solution S of a known standard pH, pH(S). Then obtain the pH of X as
:pH(X) = pH(S) + (E − E) F / (RT ln 10)
where
:F is the Faraday constant;
:R is the molar gas constant;
:T is the thermodynamic temperature.
Defined this way, pH is a quantity of dimension 1, that is it has no unit. Values pH(S) for a range of standard solutions S are listed in [http://www.iupac.org/publications/pac/1985/pdf/5703x0531.pdf Definitions of pH scales, standard reference values, measurement of pH, and related terminology]. Pure Appl. Chem. (1985), 57, pp 531–542, where further details can be found.
pH has no fundamental meaning; its official definition is a practical one. However, in the restricted range of dilute aqueous solutions having amount-of-substance concentrations less than 0.1 mol/L, and being neither strongly alkaline nor strongly acidic (2 < pH < 12), the definition is such that
:pH = −log[c(H) y / (1 mol/L)] ± 0.02
where c(H) denotes the amount-of-substance concentration of hydrogen ion H and y denotes the activity coefficient of a typical uni-univalent electrolyte in the solution.