The Sheindorf–Rebhun–Sheintuch (SRS) equation has been developed to describe competitive or multicomponent sorption where it is assumed that the single-component sorption follows the Freundlich equation (Sheindorf et al., 1981). The derivation of SRS equation was based on the assumption of an exponential distribution of adsorption energies for each component.
Specifically, the SRS model was developed to describe competitive equilib- rium sorption for multicomponent systems where the sorption isotherms of single component follow the Freundlich equation. A general form of the SRS equation can be written as
Si=KiCi
∑l j=1αi,jCj
ni−1
(22)
where the subscripts i and j denote the metal components i and j, l is the total number of components, and αi,j is the dimensionless competition coefficient for the adsorption of component i in the presence of component j. The parameters Ki and ni are the Freundlich parameters representing a single- component system i as described in Eqn (1) above. By definition, αi,j equals 1 when i = j. If there is no competition, i.e. αi,j = 0 for all j≠i, Eqn (2) yields a single-species Freundlich equation for component i identical to Eqn (1). The estimated αNi–Cd for Ni adsorption, in the presence of Cd, was larger than 1 for Windsor and Olivier soils, indicating noticeable decrease of Ni in the presence of Cd. In contrast, αNi–Cd for Ni adsorption on Webster soil was <1, which is indicative of small influence of competing Cd ions. These results are in agreement with the competitive sorption reported by Antoniadis and
Tsadilas (2007). Such small αNi–Cd implies that Ni adsorption in Webster soil was least affected in a competitive Ni–Cd system in comparison to the other two soils. Moreover, the estimated αCd–Ni for Cd adsorption was 0.61 for Windsor and 0.82 for Olivier, whereas the competitive coefficient of Cd/
Ni was 4.00 for Webster Soil. Although the SRS equation may be regarded as a multicomponent model and does not imply certain reaction mecha- nisms, differences of competitive sorption between the neutral and the two acidic soils were illustrated based on the SRS models’ competitive selectiv- ity parameters. In fact, Roy et al. (1986) suggested that the SRS parameters could be used to describe the degree of the competition under specific experimental conditions. Calculated results using the estimated αNi–Cd are given in Figs 5.8 and 9 and illustrate the capability of the SRS model in describing experimental data for competitive adsorption of Ni and Cd (Liao and Selim, 2009). An F-test indicated that there was no statistical difference between our experimental results and SRS model calculations (at the 95%
confidence level). Based on these calculations, the SRS model was capable of quantifying competitive adsorption for Ni and Cd. However, for both Ni and Cd, the SRS model deviated considerably from experimental data for high concentrations of the competing ions. This finding is consistent with the application of SRS made earlier by Gutierrez and Fuentes (1993) and illustrates the need for model improvement to better describe competitive adsorption of heavy metals over the entire range of concentrations.
The suitability of the multicomponent SRS equation for describing the competitive adsorption isotherms of trace elements on soil and soil minerals have been investigated by several researchers. A general procedure of applying the SRS equation is first to obtain the Freundlich distribution coefficient KF and reaction exponent b or n by fitting the single-component isotherms to Freundlich equation, followed by estimating the competition coefficients αi,j
through fitting the experimental isotherms of binary and ternary mixtures to the SRS equation (Roy et al., 1986). Although the SRS equation does not imply specific reaction mechanisms, the competition coefficients αi,j in the equation can be used to evaluate the relative selectivity of the sorbent to the heavy metal species. It is demonstrated that SRS equation with com- petition coefficients estimated through nonlinear least-squares optimization successfully described the experimental competitive adsorption isotherms of Ni and Cd on three different soils (Liao and Selim, 2009). Gutierrez and Fuentes (1993) employed the SRS equation to represent the competi- tive adsorption of Sr, Cs and Co in Ca-montmorillonite suspensions. They found that the SRS competition coefficients αi,j obtained from experiment
Windsor
0.0 0.2 0.4 0.6
Ni Sorbed (mmol/kg)
0.0 0.5 1.0 1.5 2.0
0.000 0.047 0.235 0.766
Calculated by SRS
Olivier
0.0 0.1 0.2 0.3
0.0 2.0 4.0 6.0
0.000 0.047 0.235 0.766
Calculated by SRS Initial Cd Con. (mM)
Initial Cd Con. (mM)
Webster
Ni Concentration (mM)
0.00 0.03 0.06 0.09
0.0 2.0 4.0 6.0 8.0
0.000 0.235 0.047 0.766
Calculated by SRS Initial Cd Con. (mM) Cd Sorbed (mmol/kg)Cd Sorbed (mmol/kg)
Figure 5.8 Competitive adsorption isotherms for Ni in the presence of different concentrations of Cd. Solid curves are SRS model calculations.
Olivier
0.00 0.05 0.10 0.15 0.20
0.0 2.0 4.0 6.0 8.0
0.000 0.047 0.235 0.766
Calculated by SRS
Windsor
0.0 0.2 0.4 0.6
Cd Sorbed (mmol/kg)
0.0 1.0 2.0 3.0 4.0
0.000 0.047 0.235 0.766
Calculated by SRS Initial Ni Con. (mM)
Initial Ni Con. (mM)
Webster
Cd Concentration (mM)
0.0 0.1 0.2 0.3
0.0 2.0 4.0 6.0 8.0
0.000 0.047 0.235 0.766
Calculated by SRS Initial Ni Con. (mM)
Cd Sorbed (mmol/kg)Cd Sorbed (mmol/kg)
Figure 5.9 Competitive adsorption isotherms for Cd in the presence of different concentrations of Ni. Solid curves are SRS model calculations.
data of binary mixtures successfully predicted the competitive adsorption of ternary mixture Sr–Cs–Co. Similarly, Bibak (1997) found that values of SRS competitive coefficients obtained from binary sorption experiments successfully predicted sorption data of the ternary solute mixture Cu–Ni–
Zn. The SRS equation was successfully used to describe for competitive sorption of Cd, Ni, and Zn on a clay soil by Antoniadis and Tsadilas (2007).
In addition, SRS equation was also used by Wu et al. (2002) in representing the competitive adsorption of molybdate, sulfate, selenate, and selenite on γ-Al2O3 surface where relative affinity coefficient was used instead of com- petitive coefficients. The relative affinity coefficients were calculated as the ratios of the proton coefficients of competing anions. The simulation result showed that the sorption affinity of anions on γ-Al2O3 surface decreased in the order of MoO42− > SeO32− > SeO42− > SO42−.