I. Activity (effective concentration)
The equilibrium relations should include not the analytical concentration (moles/L or moles/kg reported in a chemical analysis) but some type of effective concentration. For charged solutes (ions) in water, the effective concentration tends to decrease with increases in the total amount of ions in solution. This is due to a "masking" effect, similar to that which occurs at a crowded party. A quantitative evaluation of activity requires first a measure of the total charge in solution. This is quantified by the ionic strength, I, defiined by equation 17.5 in S&Z. Note that this equation should actually use the molal concentration, not the molar concentration. The "charge", z, is also referred to as the "valence". Note that divalent ions have 4 times the effect of monovalent ions on the ionic strength.
Figure 17.1 illustrates the variation in the activity coefficient, which is the factor by which you multiply the analytical concentration to get the activity, with ionic strength. Several formulas are used to estimate the activity coefficients. At very low ionic strengths, the simple Debye-Huckel formula (equation 17.4 in S&Z) does a reasonable job of predicting activity coefficients. For most groundwater problems, we need to use the extended Debye-Huckel formula (equation 17.6, with coefficients A and B that vary with temperature, see Table 17.1). At high ionic strength the Pitzer model is required (e.g. for seawater or brines). The Davies equation is slightly better than the extended Debye-Huckel, but not as good as the Pitzer for high ionic strengths. In this class we will only use the extended Debye-Huckel formula.
An illustration of how this is used is provided in example 17.1 of the text.
II. Acid-Base dissociations
With this method of determining the effective concentrations, we can now examine equilibria for solute-solute reactions. These are quite fast, so an equilibrium approach is appropriate. The simplest type of solute-solute reaction is dissociation of an acid (to yield H+) or of a base (to yield OH-). Water is the simplest acid (as well as the simplest base). This reaction can be represented by the following mass action expressions:
H+ and OH- are ions for which we know how to determine activity coefficients to convert analytical concentrations to activities, but we also need an activity for water. The activity of water in an aqueous (water-based) solution is taken by definition as "unity" (exactly one). Thus we can leave water activity out of the equilibrium expression so that
Dissociation of pure water will yield equal analytical concentrations of H+ and OH-. Also, in dilute solute the activity coefficients for H+ and OH- will be very close to one, and essentially identical. Thus, for pure water
The equilibrium constant K is a function of temperature. At 250C, the equilibrium constant for water dissociation has a value of 1.008 x 10-14 = 10-14.00, so that [H+] in pure water at this temperature will have a value of 10-14.00/2 = 10-7.00 and [OH-] will also have the same value. Water with equal activities of H+ and OH- is considered "neutral". The pH is defined as the negative log of [H+], so this activity of H+ corresponds to a pH of 7.00. Water at this temperature with a pH greater than 7.00 will have a higher activity of OH- than of H+, and is considered basic (or alkaline) while water with a lower pH is acidic. The pH range corresponding to acidic or basic conditions (and the "neutral" pH) varies with temperature since the equilibrium constant varies with temperature.
Strong acids are those for which the equilibrium "dissociation" constant is large, which results in only a small amount of the acid remaining in its "associated" form in solution. HCl is a strong acid. For example, if one adds one mole of HCl to a liter of distilled water, the K of approximately 1000 will cause more than half of the HCl to dissociate to for H+ and Cl-, leaving far less than one mole as dissolved HCl. Similarly, strong bases have large dissociation constants.
Weaker acids (or bases) are those with smaller dissociation constants. A common weak acid in groundwater is carbonic acid, H2CO3*. The presence of this solute in groundwater is due to dissolution of carbon dioxide gas. Carbonic acid is a "polyprotic" acid because it can undergo two dissociation steps to form first bicarbonate (HCO3-) and then carbonate CO32-). The equilibrium constant for the net dissociation reaction that takes carbonic acid to carbonate plus two H+ is equal to the product of the two equilibrium constants for the two dissociation steps. Knowing the pH allows one to calculate the relative activities of carbonic acid, bicarbonate and carbonate in solution (see Figure 18.1 in S&Z).
II. Gas dissolution/exsolution
A reaction for carbon dioxide dissolution in water is listed in Table 18.1 of S&Z. Note that the activity of water in the equilibrium constant equation for this reaction should be one, so it drops out of the equation. Also note that the activity of the gas phase is taken to be its partial pressure equal to its volumetric (and molar) fraction of the gas times the total pressure of the gas. This is the general rule for gas phase activities. The partial pressure of oxygen in the atmosphere is about 0.2 atm since the atmosphere is about 20% oxygen. The concetration of CO2 in the atmosphere in 1960 was about 316 ppm by volume, for a partial pressure of 10-3.50 atm. By the end of 2003, the in the concentration had risen to about 376 ppm, for a partial pressure of 10-3.42 atm. Gas-water reactions are somewhat slower than solute-solute reactions. This explains the fact that rivers and streams can become depleted in oxygen despite the fact that they are in contact with the abundant atmospheric reservoir of oxygen. However, they are still relatively fast compared to groundwater flow rates, so water in the unsaturated zone and at the water table typically comes to equilibrium with respect to CO2 in the surrounding gas phase.
Gas-water reactions are sometimes written as "exsolution" reactions. In the case of carbon dioxide, this would be
and the equilibrium constant (called a Henry's constant according to one convention found in the literature) would be 1/KCO2 as listed in Table 18.1 (i.e. 10+1.46 rather than 10-1.46 ).
III. Ion Associations (Complexes)
Acid-base dissociation reactions are part of the larger class of solute-solute reactions. Most other solute-solute reactions are conventionally treated as "association" reactions.
Ions that form bonds in solution are known as ion-pairs (usually including water associated with the ions) or complexes. The cation is often a metal. The anion in the complex is often known as a ligand. The anion can be inorganic or an organic ion.
Equilibrium constants for net complexation reactions (written as associations) are often referred to as stability constants and represented by the greek letter b. See, for example, the reactions listed in Example 18.2 of S&Z. In this example, the mole balance equation is correct as written in terms of analytical concentrations in moles/kg. However, as in previous examples of reactions in the text, the "mass-law expressions" should be written with activities rather than with analytical concentrations.