Thermodynamic Derivation of Equations of the Langmuir Type for Ion Equilibria in Soils1

TLDR: In this article, it was shown that the traditional equation of the Langmuir type is not the appropriate equation to describe ion exchange processes because the ion exchanger generally cannot be treated as an ideal solid solution.

Abstract: Classical thermodynamics was employed to derive equations of the Langmuir type with parameters having chemical meaning for homovalent and heterovalent ion exchange processes and for anion-fixation reactions wherein the anion is able to form an insoluble precipitate with some exchangeable cation. The analysis pointed out that the traditional equation of the Langmuir type is not the appropriate equation to describe ion exchange processes because the ion exchanger generally cannot be treated as an ideal solid solution. Nevertheless, for systems with low concentration of one ion in a supporting electrolyte, the solid-phase activity coefficient ratio should be a constant, and an equation of the Langmuir type may describe successfully this portion of the exchange isotherm. It was shown also that anion-fixation isotherms could be completely described by solving simultaneously three equations: a solubility product, a traditional equation of the Langmuir type for cation exchange, and an equivalence equation. Selected experimental data from the literature were utilized to demonstrate the applicability of the derived equations and to determine parameters for comparison and correlation with other data and for the identification of the ionic species undergoing ion exchange. Additional Index Words: adsorption, anion-fixation, ion exchange, specific adsorption. Elprince, A. M., and G. Sposito. 1981. Thermodynamic derivation of equations of the Langmuir type for ion equilibria in soils. Soil Sci. Soc. Am. J. 45:277-282. T APPLICABILITY of an adsorption isotherm equation of the Langmuir type (Langmuir/ 1918) to soil chemical phenomena continues to be the subject of controversy (Griffin and Au, 1977; Harter and Baker, 1977, 1978; Veith and Sposito, 1977; Holford, 1978; Sposito, 1979). The principal focus of discussion in both recent and earlier critiques of the equation 1 Contribution from the Dep. of Soils & Water, King Faisal Univ., Alhasa, Saudi Arabia; and the Dep. of Soil & Environmental Sciences, Univ. of California, Riverside. CA 92521. Received 5 Nov. 1979. Approved 19 Nov. 1980. 3 Associate Professor of Soil Chemistry and Professor of Soil Science, respectively. has been whether experimental data on the reactions between solid soil constituents and ions in aqueous solution should conform to the linear expression £ = _ L + £ q ab b [1] where c is the concentration of the reacting ion in aqueous solution at equilibrium, q is the quantity of the ion that has reacted with unit mass of the solid soil constituent, b is the maximum value of q, and a is an "affinity parameter." With the parameter a assumed to be independent of q, Eq. [1] often has been criticized on experimental grounds by investigators seeking to employ it purely as a means for correlating data on q as a function of c (Cole et al., 1953; Fried and Shapiro, 1956; Olsen and Watanabe, 1957; Rennie and McKercher, 1959; Weir and Soper, 1962; Gunary, 1970; Syers et al., 1973; Holford et al., 1974; Shuman, 1976; Griffin and Au, 1977; Harter and Baker, 1977). On the other hand, Eq. [1] has been criticized on theoretical grounds because it does not seem to account for nonuniformity with respect to affinity in soil adsorption sites (Holford et al., 1974; Holford, 1978); because it cannot be used to distinguish adsorption from secondary precipitation phenomena (Veith and Sposito, 1977) and because it does not show explicitly the effects of competition for adsorption sites in ion exchange phenomena (Griffin and Au, 1977; Harter and Baker, 1977, 1978; Sposito, 1979). Brunauer et al. (1967) have pointed out that many analyses concerning the validity of Eq. [1] overlook the crucial fact that nowhere in the original, kineticsbased derivation of the Langmuir equation (Langmuir, 1918) is it required that the parameter a be a constant, although in subsequent derivations based on statistical thermodynamics this condition was imposed along with other restrictions (Fowler and Guggenheim, 1949; Veith and Sposito, 1977; Sposito, 1979). Therefore, Eq. [ 1 ] could be employed to describe solute adsorption phenomena in a very general (if often inconvenient) fashion by permitting a to be an 278 SOIL SCI. SOC. AM. J., VOL. 45, 1981 arbitrary function of q. Moreover, as emphasized by Brunauer et al. (1967), factors such as nonuniformity in the adsorption sites, interactions among the adsorbed ions, and changes in the configuration of the adsorbed ions actually may compensate one another in such a way as to produce a constant value of a. This possibility was not considered in the statistical thermodynamic derivations of Eq. [ 1 ] (Fowler and Guggenheim, 1949; Sposito, 1979). The principal conclusions that can be drawn from past research on the Langmuir equation are: (i) the chemical significance of the affinity parameter in an equation of the Langmuir type will depend on the specific characteristics of the ion-solid phase reaction to which the equation is applied, and (ii) derivations of an equation of the Langmuir type prescribe sufficient conditions for the validity of the equation, not necessary conditions. In this paper, thermodynamic arguments will be developed to show how an equation of the Langmuir type, with a constant affinity parameter, may be used to interpret certain exchange adsorption and secondary precipitation phenomena in soils. The primary objective of the paper is to illustrate the broad variety of important soil chemical reactions that can be described by Eq. [ 1 ] without unnecessary restrictions being placed on the mechanisms of the reactions. DERIVATION OF EQUATIONS OF THE LANGMUIR TYPE Homovalent Ion-Exchange Processes The thermodynamic equilibrium constant, K, for homovalent ion-exchange processes may be written in the form (MAfAaB)/(MBfBaA) [2] where M is the mole fraction of a component in the solid exchanger phase, a is the activity of an ion, A or B, in aqueous solution, and / is an exchanger-phase activity coefficient. A body of experimental results (Gaines and Thomas, 1953, 1955; Merriam and Thomas, 1956; Frysinger and Thomas, 1960; Martin and Laudelout, 1963; Loven and Thomas, 1965; Helmy and Peineman, 1971; Gast, 1972; Jensen and Babcock, 1973; Udo, 1978) has shown that /A//B is a function of the exchanger composition and generally cannot be set equal to 1.0, as was suggested by Vanselow (1932), who treated the ion exchanger as an ideal solid solution. The experimental determination of fA and fB became possible after the rigorous thermodynamic formulation initiated by Hogfeldt et al. (1950) and Argersinger et al. (1950) and extended later by Gaines and Thomas (1953) and Elprince and Babcock (1975a). Considering A and B as the only ions in the ionexchange system, one may write MA + MB = 1 and employ Eq. [2] to yield the following competitive equation of the Langmuir type (cf. Sposito, 1979, Eq. [10]): (aA/aB)/MA = (aA/aB) [3] where is not a constant because of the dependence of /B//A on the exchanger composition. Hence, a plot of (aA/aB)/MA vs. (aA/aB) covering the whole exchange isotherm (i.e., 0 ^ MA < 1) is expected to be curvilinear instead of a straight line. However, for a short portion of the isotherm, the ratio /A//B sometimes is approximately a constant, and a straight-line plot can be obtained. Apparently this is often the case for relatively low values of the mole fraction of ion A, a result that cannot be predicted from thermodynamics alone but can be observed experimentally (see, e.g., Jensen and Babcock, 1973) as well as predicted from the application of mixture theories to the exchanger phase. For example, the Wilson (1964) mixture theory, when applied to ion-exchange systems with relatively low values of the mole fraction of ion A, yields = Lira (/B//A) = exp [5]