Understanding pH within a nanoscopic water pool

The nature of pH within a water pool that is too small to have a constant population of ions formed by water dissociation is a long-standing question. The breakdown of the conventional pH definition is due to the rare and intermittent presence of ion pairs (H^+ and OH^-) in these pools, leading to pH = -log_10 (0). To characterize water ion pair lifetimes and populations in such systems, we have performed a set of stochastic kinetics simulations of the water dissociation reaction in pools ranging from 10^3 to 10^10 waters. We extract a kinetically derived availability coefficient, α_(i>0), as a suitable parameter to quantify the intermittent presence of a number i of ion pairs during an observation period. In this way, pH in confinement is intrinsically connected with the transient ion pair lifetimes and the stochastics associated with their formation. We propose that α_(i>0) is equivalent to an activity coefficient, and use α_(i>0) along with the thermodynamic definition of pH to estimate an effective pH for a nanoscopic H_2 O pool. As the availability coefficient quickly converges to 1 for pool containing well over 5×10^9 waters, the effective pH converges to the pure water bulk pH of 7. The implications of the effective pH concept for confined aqueous environments are discussed.