Quantifying Equilibrium Constants

In the introductory chapter we stated that the formation of chemical compounds with the metal ion in a variety of formal oxidation states is a characteristic of transition metals. We also saw in Chapter 8 how we may quantify the thermodynamic stability of a coordination compound in terms of the stability constant K. It is convenient to be able to assess the relative ease by which a metal is transformed from one oxidation state to another, and you will recall that the standard electrode potential, E , is a convenient measure of this. Remember that the standard free energy change for a reaction, AG , is related both to the equilibrium constant (Eq. 9.1)... 

Yatsimirskii (1970) attempted to quantify HSAB theory and produced hardness indices for adds and bases. These indices were obtained by plotting the logarithms of the equilibrium constants for the reactions of bases with the proton (the hardest add) against similar values for the reactions with CHjHg (one of the softest adds). For adds, the hydroxyl ion (the hardest base) and the chloride ion (a soft base) were chosen. 

Consequences of the Snyder and Soczewinski model are manifold, and their praetieal importance is very signifieant. The most speetaeular conclusions of this model are (1) a possibility to quantify adsorbents ehromatographic activity and (2) a possibility to dehne and quantify chromatographic polarity of solvents (known as the solvents elution strength). These two conclusions could only be drawn on the assumption as to the displacement mechanism of solute retention. An obvious necessity was to quantify the effect of displacement, which resulted in the following relationship for the thermodynamic equilibrium constant of adsorption, K,, in the case of an active chromatographic adsorbent and of the monocomponent eluent ... 

In practice, measurement of the individual rate constants or equilibrium constants for these various chemical steps requires specialized methodologies, such as transient state kinetics (see Johnson, 1992, Copeland, 2000, and Fersht, 1999, for discussion of such methods) and/or a variety of biophysical methods for measuring equilibrium binding (Copeland, 2000). These specialized methods are beyond the scope of the present text. More commonly, the overall rate of reaction progress after ES complex formation is quantified experimentally in terms of a composite rate constant given the symbol km. 

The concept of using the base 10 logarithm to express the magnitude is a widespread practice today. Equilibrium constants of chemical reactions are often noted or compared as pK values where pK = — log 10 (magnitude of equilibrium constant). For example, the extent of dissociation of acetic acid, the acid in vinegar, is quantified by an equilibrium constant of 1.8 x 10-5. Here, then, pK = — log,o (1.8 x 1(T5) = 4.74. 

As with all physicochemical processes, the extent of Equation (6.2) may be quantified by an equilibrium constant K. We call it the autoprotolysis constant, as defined by... 

The resultant (modified) equilibrium constant is called the acidity constant of phenol, and has the new symbol Ka, which has a value is 10-10 for phenol. Ka is also called the acid constant, the acid dissociation constant or just the dissociation constant. The value of Ka for phenol is clearly tiny, and quantifies just how small the extent is to which it dissociates to form a solvated proton. 

Vary the equilibrium constants to quantify their influence. 

The strength of an acid HA is quantified by means of the pATa, which is the negative logarithm of the equilibrium constant for the process described in equation 17 ... 

The surface complexation models quantify adsorption with experimentally determined equilibrium constants. Another, less widely used approach considers the relationship between the equilibrium constant for the adsorption reaction and the associated free energy change (James and Healy, 1972). Attempts have been made to determine the chemical contribution to the overall adsorption free energy by fitting adsorption isotherms to the experimental data values of -50, -33 and —45 kj mol were found for the change in chemical free energy associated with adsorption of Cr, Ni and Zn, respectively, on ferrihydrite (Crawford et al., 1993). Values ranging from -21 to 241 kJ mol were found for Ni on hematite the actual value depended upon the hydrolysis species that were assumed to exist (Fuerstenau and Osseo-Assare, 1987). 

The strength of a Brensted-Lowry acid is quantified in terms of the magnitude of the equilibrium constant for the ionization reaction in which the solvent acts as the base, e.g. for the aqueous system, the general reaction is ... 

Kx is the equilibrium constant quantifying enzyme substrate association... 

All other acids (that we ll mention) are weak acids. The vast majority of acids fall into this category. However, weak acids show a range of acid strength. The relative strength of a weak acid is quantified by the equilibrium constant Ka governing the ionization of the weak acid. 

One of the first questions one might ask about forming a metal complex is how strong is the metal ion to ligand binding In other words, what is the equilibrium constant for complex formation A consideration of thermodynamics allows us to quantify this aspect of complex formation and relate it to the electrode potential at which the complex reduces or oxidizes. This will not be the same as the electrode potential of the simple solvated metal ion and will depend on the relative values of the equilibrium constants for forming the oxidized and reduced forms of the complex. The basic thermodynamic equations which are needed here show the relationships between the standard free energy (AG ) of the reaction and the equilibrium constant (K), the heat of reaction, or standard enthalpy (A// ), the standard entropy (AS ) and the standard electrode potential (E for standard reduction of the complex (equations 5.1-5.3). 

The difference between the extended Debye-Hiickel equation and the Pitzer equations has to do with how much of the nonideahty of electrostatic interactions is incorporated into mass action expressions and how much into the activity coefficient expression. It is important to remember that the expression for activity coefficients is inexorably bound up with equilibrium constants and they must be consistent with each other in a chemical model. Ion-parr interactions can be quantified in two ways, explicitly through stability constants (lA method) or implicitly through empirical fits with activity coefficient parameters (Pitzer method). Both approaches can be successful with enough effort to achieve consistency. At the present, the Pitzer method works much better for brines, and the lA method works better for... 

However, for those that are Brpnsted acids or bases, values increase dramatically as the pH is reduced below the value(s) for the compound (Haderlein and Schwarzenbach, 1993 Broholm et al, 2001). (The pK of a Brpnsted acid/base pair is the negative log of the equilibrium constant for their interconversion through the gain or loss of a proton. As such, the pK, also represents the pH value below which the concentration of the acid exceeds that of the base, and above which the base dominates.) Schellenberg et al. (1984) introduced equations that may be used to quantify the effects of pH variations on Kp for Brpnsted acids and bases. 

Accurate predictions of the transport of As in groundwater requires site specific data to model adsorption/desorption reactions. In complex mixtures of minerals, it may not be possible to quantify the adsorption properties of individual minerals. Therefore, it has been suggested that adsorption properties of composite materials should be characterized as a whole (Davis and Kent, 1990). Previously published data for adsorption by pure mineral phases such as the surface complexation database for adsorption by ferrihydrite (Dzombak and Morel, 1990) can be a useful starting point for modeling adsorption of solutes in groundwater however, these equilibrium constants may not reflect the adsorption properties of composite oxide coatings on aquifer solids. For example, incorporation of Si, and to a lesser extent, A1 into Fe oxyhydroxides has been shown to decrease adsorption reactivity towards anions (Ainsworth et al., 1989 Anderson and Benjamin, 1990 Anderson et al, 1985). Therefore, equilibrium constants will likely need to be modified for site-specific studies. 

---