Calculation of the pH Values of Aqueous Systems Containing Carbonic Acid and Significance for Natural Waters, Following (Near-)Exact and Approximated Solutions: The Importance of the Boundary Conditions

Calculating the pH values of carbonic acid solutions is an important task in studies of chemical equilibria in freshwater systems, with applications to environmental chemistry, geology, and hydrology. These pH values are also highly relevant in the context of climate change, since increasing atmospheric CO₂ affects the concentration of dissolved carbon dioxide and carbonic acid, collectively denoted as [H₂CO₃*] = [H₂CO_3(aq)] + [CO_2(aq)]. Solving equilibrium systems to obtain analytical functions is particularly useful when such functions are required, for example, in data fitting. We show here that, although exact or near-exact solutions typically result in third- to fourth-order equations that must be solved numerically, reasonable approximations can be derived that lead to analytical second-order equations. In this framework, the chosen approximations need to meet the boundary conditions of the systems, particularly for c_T → 0 and for high c_T values (where c_T = [H₂CO₃*] + [HCO₃^-] + [CO₃^2-]). Finally, we provide exact solutions for a closed system containing both H₂CO₃* and alkalinity, which enables the description of virtually any aquatic environment without assuming equilibrium with atmospheric CO₂. Implications for pH calculations in natural waters are also briefly discussed.