A powerful tool in modelling the effect of water quality changes on water distribution infrastructure is the Saturation Index

Ω = IAP/K

Where IAP is the ion-activity product and K is the equilibrium constant. Let us explore these two quantities as they apply to water distribution systems.

Chemical Equilibrium        

Chemical systems show the property that after a reaction has gone to completion, the proportions remaining are constant.  That is if substance A reacts with substance B to make a new substance C,

                        A + B à  C,

the activities after the completion of the reaction are related to each other by

                        a_C / (a_A × a_B)  =  K

 no matter what the original proportions of A and B were, or for that matter if one had started with only C for the reverse reaction

                        C à A + B.

 However, for the reaction written this way, the equilibrium constant expression would be

                        K' = 1/K = (a_A × a_B)/a_C.

This concept of an equilibrium mixture of substances that have fixed relative concentrations is a powerful one in geochemistry, and enables us to predict concentrations at equilibrium for a number of substances of interest under various conditions.  Special attention must be given, however, to the qualification "at equilibrium." Many geochemical systems, particularly at low temperatures, are not in equilibrium, and therefore rates of reactions need to be considered.   Unfortunately, relatively little is known about the kinetics of heterogeneous reactions in complicated systems like natural waters.  On the other hand, reactions occurring within the water itself tend to reach equilibrium rapidly, and so homogeneous reactions in surface and groundwaters can conveniently be modeled by equilibrium expressions.  For certain minerals (e.g. carbonates), groundwater flow is slow enough that their reactions can also be modeled conveniently by assuming equilibrium.

Approach to Equilibrium

Note that distribution systems operate far from equilibrium because (1) flow rates are high; (2) flow is episodic. However, we do see approach to equilibrium in many cases during overnight stagnation. Moreover, understanding the system at equilibrium helps us predict the directions of changes. The classic example of the use of this concept is the Langelier Index, which compares pH at equilibrium for CaCO₃ to measured pH.

We can generalize the concept underlying the Langelier Index by defining the Ion Activity Product as a function having the same form as the equilibrium constant, but with measured values rather than equilibrium values. For the dissolution of CaCO₃,

CaCO3 à Ca++ + CO3=

Ksp = (a_{Ca, equil})(a_{CO3, equil}) and IAP = (a_Ca,meas)(a_{CO3, meas})

When IAP = K, the ratio IAP/K is one and the log (IAP/K) = 0, and we say the system is at equilibrium, or in the case of dissolving solids, the solution is saturated with respect to the compound of interest.

Where the symbol Ω is used as a single parameter quantity for degree of approach to saturation.

Because of uncertainties in the underlying thermodynamic data for minerals and complex ions in solution, it is common to regard log (IAP/K) values between -0.5 and +0.5 as at equilibriium.

Calculating Degree of Saturation

Because it is necessary to calculate activity coefficients and extent of complexing for each substance that is present in significant amounts, the calculation of degree of approach to equilibrium (or to saturation for solids) requires the iterative solving of an extensive set of simultaneous equations. Several computer codes have been developed to do these calculations. See the page Approach to Equilibrium.

  Degree of Saturation in Distribution Systems

Most lead surfaces in distribution systems are coated with Pb oxide, carbonate, or phosphate minerals, often in successive layers. Among the most common are the Pb carbonates cerussite and hydrocerussite. A survey of representative systems - shown below - reveals that most are undersaturated with either of these minerals at the Pb action level (0.015 mg/L). Therefore water reacting with, say, a Pb service line, will, given time, dissolve either of these minerals to yield Pb concentrations above the action level. That is, the waters are corrosive to Pb scales at the action level. Also notice that the degree of undersaturation is greater, on average, for hydrocerussite. Hydrocerussite dominates over cerussite at pHs above about 8.5, whereas the median pH for this dataset is only 7.7.