The Sorensen definition of pH as hydrogen ion concentration

The original concept of pH was developed by Søren Peder Lauritz Sørensen, and involved the concentration rather than the activity of hydrogen ions. This was the result of earlier work by Svante Arrhenius whose 1884 definition of an acid was "something that dissociates in solution to produce hydrogen ions".

Sørensen's work was centered around research on the effect of ion concentrations in the analysis of proteins. At this stage, measurements of acidity and alkalinity were dependent on the colour change of indicator solutions, which was an unsatisfying method, owing to its poor sensitivity. Given the importance of acidity in enzymatic reactions, and its contemporary definition as a function of hydrogen concentration, Sørensen pursued a method of simply and conveniently expressing the accurate hydrogen ion concentration of a solution.

Sørensen used the term potenz (power) to describe the magnitude of hydrogen ion concentration, corresponding to the negative power of 10. This is revealed by his own papers, and supported by some of his biographers. In his own words (translated from the impenetrable German of the original):

"The magnitude of the hydrogen ion concentration will accordingly be represented by means of the normality factor with regard to the hydrogen ion, and this factor will be written in the form of a negative potenz (power) of 10. Since I refer to the above in a later section (see page 159), here I will mention only that I employ the name "hydrogen ion exponent" and the symbol pH for the numerical value of this potenz (power)."

How could it be clearer than that? But to this day the exact origin of the little p in pH is still debated. Chemistry trolls claim that it was arbitrary (using p and q to label the test solution and the reference solution), that it stands for pondus Hydrogenii (the power of Hydrogen), and various others.

The symbol has mutated over the years. The little "p" wasn't even little to begin with- Sørensen himself used "P_H" in his original works:

A variety of symbols followed, such as "P_H+", "Ph", and "P_h,". The "P" was decapitalised in a 1920 publication by W. M. Clark, apparently for typographical convenience; this modern form became convention owing to its adoption by the widely circulated Journal of Biological Chemistry.

Given that the concentration of hydrogen ions spans several scales of magnitude (eg. from 12 mol/L to 10^-14 mol/L), the handling of this property in the literature was awkward, and for convenience Sørensen defined pH as the negative logarithm of hydrogen ion concentration. This was not convenient for everybody. Though wholly supportive of the use of hydrogen ion concentration as a descriptor of acid-base status, and the major contributior to the dissemination of this concept, Clark was apparently quite resistant to the use of pH as a means of representing it:

"...Both convenience and the nature of the physical facts invite us directly to operate with some logarithmic fucntion of [H⁺]. It is unfortunate that a mode of expression so well adapted to the treatment of various relations should conflict with a mental habit. [H⁺] represents the hydrogen ion concentration, the quntity usually thought of in conversation when we speak of increases or deceases in acidity. pH varies inversely as [H⁺]. This is confusing."

In order to precisely measure the hydrogen ion concentration without resorting to colour-change tests, Sørensen devised an experiment in which the concentration gradient of ions could be related to the electric gradient between electrodes in an electrochemical cell.

Sørensen's experiment

There is little information out there about the precise setup of this experiment, and the diagram is extrapolated from articles about Sørensen, where the experiment and its implications are explained in a much more detailed and accurate manner. Sørensen's original article does not have a depiction of his apparatus - only the densely German description of it. Fortunately, Prof. William Jensen from the university of Cincinnati has reproduced in one of his commentaries an advertisement from 1922 which depicts the sort of apparatus Sørensen would have used. The pdf was available to me, and I was delighted to discover that Arthur H Thomas Company's ad for their "Electrometric H-Ion Outfit" was embedded into the pdf document as a high resolution image, revealing the detailed structure of the apparatus.

After a brief encounter with several gentle photoshop filters, the apparatus presents itself thus:

Again, Professor Jensen's article does this a far greater justice than my two cents ever could, as would make sense (given his position as the Oesper Professor of Chemical Education and the History of Chemistry). I therefore direct the reader to his online works for a detailed explanation of the chemistry involved. My contribution here will be limited to a diagram reproducing Sørensen's technique in a stylized and simplified fashion.

Sørensen made use of two existing techniques, namely the saturated calomel electrode and the hydrogen electrode. The calomel electrode (calomel being the other name for mercurous chloride, Hg₂Cl₂) was used as a reference electrode, as its behaviour is predictable, and the activity of the chloride ion is fixed (by the fact that the KCl is saturated).The other electrode is a normal hydrogen electrode.

The Nernst Equation butchered in the image below is used to represent the relationship of concentration gradients and electric gradients across within this two-cell system.

With the hydrogen electrode immersed in 1 mmol/L HCl, with H₂ bubbling at 1 atmosphere of pressure and at 18°C, Sørensen reported a cell potential of 0.338 V. This would be the E₂.

E₁ would be the cell potential with some sort of experimental solution.

Thus, the calculation of pH resembled the following equation:

pH = (E - 0.338) / 0.05916

In this equation, 0.338 V is the potential of the hydrogen electrode (i.e. the baseline value), E is the cell potential generated in the experiment, and the 0.05916 is the "Nernst Slope", the change in potential associated with a tenfold change in concentration (it is 59.16 mV at 25 C°).

In this fashion, the pH values for numerous buffers were calculated from their corresponding E measurements.

Disadvantages of using hydrogen ion concentration as a definition of pH

Apart from the fact that the Nernst equation describes only "ideal" systems (i.e. it is not likely to ever accurately describe a realistic experimental set-up), the idea of using hydrogen ion concentration falls apart as soon as one abandons the Arrhenius definition of acids and bases. Sørensen's response to this was to redefine pH as a negative logarithm of the activity of hydrogen. This has been incorporated into the modern IUPAC definition, with the expectation that this activity is experimentally measured against a calibrated electrode of a known activity within a Harned cell.

The modern definition of pH and its various peculiarities is discussed elsewhere.