The modern concept of pH can be defined as a number expressing the acidity or alkalinity of a solution as the logarithm of the reciprocal of hydrogen ion concentration or hydrogen ion activity.

This is not the definition. One might search far and wide for a simple lay definition, and one would meet no interesting or informative obstacles for many miles. To uncover a proper definition one must descend into the atmosphere-of-Venus-like environment of Pure and Applied Chemistry. There, one may finally discover the 2002 IUPAC definition, as well as the IUPAC-specified international standard for the procedures of pH measurement.

This official definition can be reproduced in a boring grey box, for the purposes of revision.

pH = -log10 (aH+)         ... or...    = -log10 ( mHγ H / m°)

Where

  • aH+ is the relative activity of the hydrogen ion in solution
  • γH is the molal activity coefficient of the hydrogen ion at the molality mH
  • m° is the standard molality

Or, in terms humans might understand,

pH is the negative logarithm (base 10) of the hydrogen ion activity in a solution

Though that may seem straightforward, it is not. The "hydrogen ion activity" as defined by the mHγ H / m° fraction is actually not a quantity that is measurable by any sort of thermodynamically valid method.

Therefore, we can only ever have a notional definition of pH. That is to say, a totally imaginary definition, not based in scientific measurement, and existing only in a magical world of make-belief.

The reasons for this is that pH ends up being a single ion quantity, i.e. it is not determinable in terms of a base unit of any system of measurement. It is a convention which became widely accepted because of its convenience rather than because of its scientific validity.

Such validity came later, as a "primary method of measurement" was defined by IUPAC.

In order to hold water, a convention must be measured by a method which is described by a well-defined equation, in which all variables can be determined experimentally by SI units. Thus, the definition of pH borrows its validity from the Nernst equation, which describes two half-cells and relates the concentration gradient between them to the electric gradient that balances it. Both the concentration gradient and the electric gradient are satisfactory variables from the point of view of the above definition. The concentration of ions in at least one half-cell is controlled by the experimenter, and the electric gradient can be measured, which means the concentration gradients can be calculated. From these relationships it is possible to arrive at a concentration of hydrogen ions.

The gold standard of pH measurement is the Harned cell, which is discussed in greater detail in the next chapter.

References

For the history of acids and bases, apart from the lucid and comprehensive chapter from Kerry Brandis, one can turn to acid-base.com by Alan W. Grogono; the linked chapter draws heavily on the famous (in certain circles) book by Astrup and Severinghaus: The history of blood gases, acids, and bases ( Munksgaard, 1986). This opus is not available for free online, which is very sad (I'm looking at you, Google books).

Astrup and Severinghaus also wrote a series of review articles for the Journal of clinical monitoring, which are also locked up behind paywalls.

Severinghaus, John W., and Paul B. Astrup. "History of blood gas analysis. I. The development of electrochemistry." Journal of clinical monitoring 1.3 (1985): 180-192.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. II. pH and acid-base balance measurements." Journal of clinical monitoring 1.4 (1985): 259-277.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. III. Carbon dioxide tension." Journal of clinical monitoring 2.1 (1986): 60-73.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. IV. Leland Clark's oxygen electrode." Journal of clinical monitoring 2.2 (1986): 125-139.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. V. Oxygen measurement." Journal of clinical monitoring 2.3 (1986): 174-189.

Severinghaus, John W., and Poul B. Astrup. "History of blood gas analysis. VI. Oximetry." Journal of clinical monitoring 2.4 (1986): 270-288.

Severinghaus, John W., and Yoshiyuki Honda. "History of blood gas analysis. VII. Pulse oximetry." Journal of clinical monitoring 3.2 (1987): 135-138.

Buck, R. P., et al. "Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002). " Pure and applied chemistry 74.11 (2002): 2169-2200.

Arrhenius, Svante. Über die Dissociation der in Wasser gelösten Stoffe. Verlag von Wilhelm Engelmann, 1887. -The link points to a translation of the original paper.

Sørensen, Søren Peter Lauritz. "Über die Messung und die Bedeutung der Wasserstoffionenkonzentration bei enzymatischen Prozessen." (1909).-The link points to a scanned reproduction of the original paper