Since Arrhenius started being taught in schools, and since most people started going to school, the concentration of hydrogen ions became a household term, synonymous with pH. Even as Sørensen revised his definition in terms of activity rather than concentration, people were taking up the notion that in every glass of water, some 107 protons and hydroxyl ions were lurking, free and naked, ready to tear molecules to shreds.
Plainly, this is insane.
The idea of being in possession of a macroscopic amount of any subatomic particle is theoretically interesting, having been explored at length by Dan Rutter, and reexplored by a physicist blogger. In this manner, let us address the proton.
A proton, being a bare hydrogen nucleus, can indeed exist in solution.
However it is freakishly rare.
I am not sure where they got this number from, as it is not referenced, but a group of high school chemistry teachers has posted a value estimating the true concentration of free protons in an aqueous solution, which they report as being in the range of 10-130 moles/L.
However, authoritative medical texts on the subject of acid-base balance present us with different figures, like 40 nmol/L of H+ ions at a pH of 7.4.
Nanomoles, they say. Rather different in scale to 10-130.
Are McNamara and Worthley actually referring to naked protons? No of course they are not.
What accounts for this confusing notation is the extreme inconvenience of routinely using a chemically accurate description of the autoionization of water.
Firstly, whenever a proton or hydroxyl ion become free, they only exist for about 20 picoseconds before they reunite again to form H2O. Thus, there is a constant but miniscule pool of dissociated ions in any given volume of water, as statistically speaking at any given moment some ion somewhere has just dissociated, and is about 20 picoseconds away from being reabsorbed by another molecule.
The immediate interaction of a naked proton is typically with a normal H2O molecule, with the formation of a "hydronium" or "hydroxonium" ion ( H3O+).
Even this is an unrealistic and unstable form. The ( H3O+) molecule finds itself rapidly hydrogen bonded to other water molecules. The resulting molecule resembles H3O+(H2O)n where n = some random number determined by local environmental variables, such as temperature, pressure, and the presence of other solutes.
Thus as soon as they form the bare protons find themselves bound ("solvated")to other water molecules, and they form long chains of hydrogen-bound units. These chains are of variable length and can be represented as H5O2+, H13O6+ and so forth. But at no stage are there actual nude protons roaming around.
The really important concept to internalise is this:
When equations discuss [H+] or [H3O+], the concentrations of these species are normally used as a shorthand notation for the total concentrations of all the singly charged small cluster species (eg. H5O2+ and H13O6+). Intelligent people give this "H+" concentration as 40 × 10-9 at 37 °C, for a pH of 7.4
Thus, though it is slightly crazy to discuss the concentration of hydrogen ions, we continue to do so, because it would be cumbersome to represent all the possible H3O+(H2O)n species in any equation. An excellent discussion of these species is carried out in the relevant chapter by M.Chaplin.