Single site model

Figure 10-4. The double- and single-site titration models for His and Asp groups [42]. (A) In the double site model, only one X is used for describing the equilibrium between the protonated and deprotonated forms, while the tautomer interversion process is represented by the variable x. (B) In the single-site model, protonation at different sites is represented by different X variables. HSP refers to the doubly protonated form of histidine. HSD and HSE refer to the singly protonated histidine with a proton on the h and e nitrogens, respectively. ASP1 and ASP2 refer to the protonated carboxylic acid with a proton on either of the carboxlate oxygens...

However, the amount of error in the data is not generally the limiting factor in data interpretation. Rather, the locations at which the data are taken most severely hinder progress toward a mechanistic model. Reference to Fig. 1 indicates that the decision between the dual- and single-site models would be quite difficult, even with very little error of measurement, if data are taken only in the 2- to 10-atm range. However, quite substantial error can be tolerated if the data lie above 15 atm total pressure (assuming data can be taken here). Techniques are presented that will seek out such critical experiments to be run (Section VII). 

At low 7t, the denominator simplifies to unity in each case and both models are linear in n. For sufficiently high n, the parenthesis in the denominator approaches Ketv, the initial rate for the dual-site model then approaches zero, and that of the single-site model approaches a constant value. Thus the plot of the experimental data will indicate that the dual-site model is preferable if a maximum exists in the data, or that the single-site model is preferable if a horizontal high-pressure asymptote exists. Hence, for the data of Franckaerts and Froment (FI) shown in Fig. 2, the dual-site model is preferred over the single-site model. 

The data should lie along a straight line when plotted as (7i/r0)1/2 vs n, if the dual-site model is adequate. If, in addition, the dual-site model is to be preferred over the single-site model, the plot of njr0 vs n should be curved... 

An example of a model rejection based upon temperature coefficients can be obtained from the single-site model considered by Ayen and Peters (A3) for the reduction of nitric oxide over a commercial catalyst ... 

Unweighted linear least-squares estimates of these parameters are shown in Table I. The negative parameter estimates of the single-site model are of no... 

By using only simple hand calculations, the single-site model has been rejected and the dual-site model has been shown to represent adequately both the initial-rate and the high-conversion data. No replicate runs were available to allow a lack-of-fit test. In fact this entire analysis has been conducted using only 18 conversion-space-time points. Additional discussion of the method and parameter estimates for the proposed dual-site model are presented elsewhere (K5). Note that we have obtained the same result as available through the use of nonintrinsic parameters. 

The C2 plots of Fig. 26, however, indicate that a linear C2 dependence upon pressure will not be adequate. The use of the dual-site model will require three additional parameters for C2, in order that the model approximately describe the data. The single-site model, however, can be used if we take... 

For a single-site model (and, more generally, in the case of proportionate couplings, even for interacting central regions), a very general relationship exists... 

The transmittance is a dimensionless function of energy, just like the transmission. By definition, for a single-site model it equals unity, and is independent of energy. Using our simple model, we shall now calculate it for arbitrarily long chains. 

A general kinetic model should accommodate all chemical processes known to affect the dechlorination process. These include (1) reductive dechlorination takes place on the iron surface, rather than in the aqueous phase, so adsorption must occur (2) other components in the system may affect the dechlorination reaction by competing for the reaction sites (3) surface sites for reduction and for sorption may not be the same, as for the system with PCE and TCE where dechlorination takes place on the reactive sites, but most of the adsorption is clearly on the nonreactive sites (Burris et al., 1995). In the following section we will first discuss a single-site model similar to the one used by Johnson et al. (1998), which has accounted for the first two observations, then develop a two-site model that will also take the third observation into consideration. We aim to illustrate how coadsorbates in the iron system will affect adsorption and reduction of chlorinated solvents. TCE will be used as an example since relevant adsorption and reduction data are available, from which the required parameters for simulation could be estimated. 

A single site model assumes only one type of surface sites, i.e., the same type of sites is responsible for trichloroethylene adsorption and reduction. The kinetic model developed herein is based on the following reaction... 

In order to explain the degradation kinetics of TCE and PCE, for which the adsorption onto the nonreactive sites is significant (Burris et al., 1995), a two-site model is developed. The basic assumption for the single-site model, i.e., pre-adsorption equilibrium followed by reductive dechlorination, is still valid here. In addition, the two-site model assumes that there are both reactive and nonreactive sites on the iron surface, and while the adsorption of TCE and coadsorbate can occur on both types of sites, reductive dechlorination of TCE only takes place on the reactive sites. Coadsorbate is not involved in redox reactions. The reaction scheme for this model is ... 

Figure 2. Single-site model simulation for the adsorption and reduction of TCE on zerovalent iron surfaces. Modelling parameters are ...

Apparently, analytical solution for these coupled equations is not feasible and we have to use numerical approaches. Model parameters are obtained similar to the single-site model. Additionally, since the adsorption experiments can only provide information on the total concentration of surface sites, we need to know the relative abundance of reactive versus nonreactive sites. That has not been determined unambiguously through experiments, so we simply assume that the reactive sites account for 10% of the total surface sites. 

Although reductive dechlorination is modelled either based on an empirical approach or a single site model, TCE dechlorination in the presence of cysteine apparently requires the use of a two-site model. The simulation as shown in Figure 3b indicates that quantitatively, the two-site model proposed here is sufficient for the interpretation of the experimental results. To enhance the predictive capability of the model, further research is needed to determine adsorption parameters independently. 

A model file used in Scientist1 software package to obtain the simulation of the single site model in the present of a coadsorbate (Figure 2b). 

TCE degradation on metallic iron single site model with competitive adsorbate // Based on adsorption equilibrium and assume that the reaction rate is directly... 

The model fit variances are given in Table IV and calculated rate parameters for the two reactions in Table V. In nearly all cases an improvement in model fit was observed. This is typified by the comparison between the fit gg a single first-order and double first-order (DFO) model to the 0 Co data, as shown in Figure 5. Although the DFO model shows considerable improvement over single-site models, residual plots indicate a small systematic deviation at higher concentrations. This is discussed further in the following section. 

Model Fitting to Mixing-Cell Data Multiple-site kinetic models have been used to describe pesticide and herbicide movement in soils (15,16,17), cesium migration in columns (18), and strontium migration in a sandy aquifer over a twenty-year time period (1L). The results of the selective extraction procedures in all experiments discussed here suggest that a multi-site model should provide a better fit of the data than a single-site model. This hypothesis is supported by the variances in Table I, with the possible exception of selenium. 

DFO model fit variances for cesium show a considerable improvement over the single-site model variances given in Table I. 

With the possible exception of selenium, a two-site, double first-order (DFO) model shows an improvement over single site-models for describing sorption of the radionuclides studied. The dependence of sorption on alteration history in the majority of cases indicates that experiments with systems representative of well-weathered fracture systems are necessary to obtain data applicable to actual disposal vault conditions. 

In the single site model with two (spin-up and spin-down) levels it is possible to make the direct comparison between our Ansatz and the master equation methods. For the latter, we used the well known master equations for quantum dots [180,181],... 

Fig. 12.12 Top Interaction of 4e with human serum albumin as probed by isothermal titration calorimetry, a single-site model yielded a stoichiometry of 5 1 of 4e HSA with a kn of 2 pM. Middle Inhibition of hemolysis of 4e by HSA. Bottom Identical potency of NO inhibition of 4e solubilized in DMSO or in HSA...

In many reactions it has been demonstrated that more than one site is involved in the catalytic process. This is particularly often the case for dissociation reactions. The same procedure as depicted above for a single site model can be used for the derivation of the rate expression for a dual site model, but the result is somewhat different. This is exemplified for the following dissociation reaction A 2B, which is thought to proceed according to the three step sequence ... 

A further simplification of the rate expressions is obtained if the product concentrations or partial pressures are negligibly small. For the single site model this results in the following expressions ... 

---