1.4.1 Nonmechanistic Techniques
Nonmechanistic techniques are based on the determination of the mass distribution of a substance between the solid and solution phases at equilibrium (Essington, 2004). This is most commonly achieved by relating the equilibrium surface excess (S) (amount of a compound adsorbed) to the equilibrium solution concentration (c) of the compound. The variation of S with respect to c may then be mathematically described to elucidate the adsorption behaviour of a particular substance (Essington, 2004).
At fixed temperature, pressure and solution chemistry these relationships have been termed sorption isotherms. However, as Barrow (1978) highlights, factors other than temperature and concentration may affect S/c relationship in a soil system and the term isotherm implies a degree of control which does not exist. Essentially, the use of isotherm equations involves a curve fitting procedure implying isotherm parameters are valid only for the chemical conditions under which the experiment was conducted (Goldberg, 2005). It is also important to note that conformity to a particular empirical equation does not necessarily mean that the model from which it was generated was appropriate (Barrow, 1978). Because sorption isotherms are only descriptions of macroscopic data and do not definitively prove a reaction mechanism (Sparks, 1995), independent experimental evidence of an adsorption process is required before any chemical meaning can be assigned to isotherm equation
parameters (Goldberg, 2005). The four common equilibrium-based adsorption models reviewed here include the Langmuir, Freundlich, Temkin and the linear or distribution coefficient equations.
1.4.1.1 Linear Equation
The simplest and most widely used adsorption isotherm is that given by a linear relationship between S and c:
S = Kd.c + a
where S is the amount of adsorbate (e.g. phosphate) adsorbed per unit of adsorbent (e.g. WTR), c is the adsorbate concentration in solution (or residual P concentration) and a is the y-intercept. In this case, S is related to c through the distribution coefficient Kd which provides a measure of adsorbate retention (Travis and Etnier, 1981). Conformity of a set of sorption data to this model suggests that S is independent of sorption site saturation. Because of the linear assumption, this equation usually describes ion adsorption data across a restricted solution ion concentration range (Goldberg, 2005).
1.4.1.2 Langmuir Equation
The Langmuir adsorption isotherm was initially developed to describe the adsorption of gases onto clean surfaces and may be written as:
S = K.c.xm/(1 + K.c)
where S and c are as previously described, xm is the maximum adsorption per unit mass and K is an affinity parameter related to the bonding energy of the adsorbing surface. Determination of the parameters K and xm is commonly achieved by plotting c/S against c. The model is based on three assumptions (Atkins and de Paula, 2002):
1. Adsorption cannot proceed beyond monolayer coverage.
2. All sites are equivalent and the surface is uniform.
3. The ability of a molecule to adsorb at a given site is independent of the occupation of neighbouring sites. This assumption supposes that there are no interactions between adsorbed molecules.
A key assumption of the Langmuir monolayer theory is that the free energy of adsorption (described by the parameter K) remains constant and is independent of the number of occupied adsorption sites (Travis and Etnier, 1981). Because these assumptions do not hold true for heterogeneous materials such as soils and WTRs, the Langmuir equation is seldom applicable from a mechanistic perspective (Dayton, 1995). The model has, nevertheless, been used extensively to describe sorption data and is particularly useful when quantifying upper limits of adsorption.
1.4.1.3 Freundlich Equation
The Freundlich equation, which is the oldest of the nonlinear sorption isotherms, was first used to describe gas phase and solute adsorption (Sparks, 1995). Unlike the Langmuir isotherm, the enthalpy of adsorption is assumed to be dependent on the occupation of adsorption sites. In this case, the heat of adsorption (distribution coefficient) is a logarithmic function of surface coverage.
The isotherm is described as:
S = K.c a
where S and c are as previously described, K is the distribution coefficient and a is a correction factor relating to the heterogeneity of adsorption sites, where the smaller a is, the greater the expected heterogeneity (Goldberg, 2005). Although this particular isotherm does not invoke any physical model, adherence to the Freundlich isotherm corresponds to a model of adsorption in which the affinity term (K) decreases exponentially as the amount of adsorption increases (Barrow, 1978).
One limitation of the Freundlich isotherm is that, like the linear isotherm model, it does not imply a maximum quantity of adsorption. The equation may be linearized by plotting log S against log c.
1.4.1.4 Temkin Equation
The model from which the Temkin equation is derived is one in which the affinity term (K1) decreases linearly as the amount of adsorption increases (Barrow, 1978). In contrast to the
Freundlich equation, the Temkin equation assumes that the heat of adsorption is a linear function of surface coverage (Travis and Etnier, 1981). In its simplest form, the equation may be written as:
S = K1.log (K2.c)
where S and c are as previously described K1 is the affinity parameter and K2 is a coefficient. In this case, a decreasing affinity term suggests that the energetically most favourable sites are occupied first (Atkins and de Paula, 2002). The parameters K1 and K2 may be determined by plotting S against log c.
1.4.2 Mechanistic techniques
Mechanistic techniques are based on relating the equilibrium surface excess to a solution property other than substance concentration. This may, for example, involve an evaluation of adsorption as a function of solution pH (Essington, 2004). As the term implies, mechanistic techniques give quantitative insight into potential adsorption mechanisms and often involve the use of chemical surface complexation models. Such models may, for example, consider surface charge arising from protonation-dissociation and ion surface complexation reactions. Many of these models are descriptions of adsorption processes whose molecular features can be given thermodynamic significance (Goldberg, 2005).
Despite their limitations, nonmechanistic methods have been utilised more extensively than mechanistic techniques. This has been attributed in part to their simplicity as well as the ease of estimation of their adjustable parameters (Goldberg, 2005). The description of adsorption data with empirical isotherm equations has the added benefit of allowing the properties of an adsorbent to be summarized using a few numbers rather than having to refer to a curve (Barrow, 1978).