First-order kinetic model, sorption

The First-Order Kinetic Model. Karickhoff (1, 68) has proposed a two-compartment equilibrium-kinetic model for describing the solute uptake or release by a sediment. This model is based on the assumption that two types of sorption sites exist labile sites, S, which are in equilibrium with bulk aqueous solution, and hindered sites, Sjj, which are controlled by a slow first-order rate process. Conceptually, sorption according to this model can be considered either as a two-stage process ... 

As clearly illustrated in Figures 12.9 and 12.11, good descriptions of the effluent results from BC-I and BC-II were achieved when the first-order kinetic model was implemented. Furthermore, increased sorption was realized for the higher input concentration (C0 = 0.005 M) of the BC-I column than for BC-II. This finding is based on the ratio of the parameters (k, k2) that provided the best-fit model (see Table 12.3). For BC-I, the value of (k, k2) was 2.76 compared to a value of 0.98 for BC-II. Such trends are consistent with the upper Bs layer and indicate sorption dependency on the dominant concentration within the soil column as influenced by the S04 input (C0). 

Fig. 6-7. Kinetics of Ni + sorption and associated cation release by Paxton A horizon (0.040 mol L Ni " " added, soil/solution ratio = 1 100). (a) Ratio of Ni " remaining to added as a function of time. The model represents a first-order reversible reaction with kJ = 0.65 and kj 0.95 K = 0.68). (b) Difference between Ni remaining in solution and the first-order kinetic model (Fig. 7a), as a function of The line indicates the area of direct relationship, (c) Cation release during the reaction. The solid line represents (1 — C/Q) or the inverse of the first-order model presented as a solid line in Fig. 6-7a. (d) Difference between cation gain to solution and the first-order kinetic model (Fig. 6-7c), as a function of The line indicates the area of direct relationship. 

Pseudo-first-order kinetic model The sorption of organic molecules from a liquid phase to a solid phase can be considered as a reversible process with equilibrium being established between the solution and the solid phase. Assuming a non-dissociating molecular adsorption of adsorbates onto adsorbent particles, the sorption phenomenon can be described as the diffusion-controlled process (Fogler, 1998). 

Two kinetic models, namely, pseudo-first-order and pseudo-second-order, were used to investigate the adsorption process of methyl orange, methyl blue and safranine T onto synthesised and commercial zeolite. Kinetic parameters along with correlation coefficient for the pseudo-second-order kinetic model are listed in Table A.3. The calculated correlation coefficient is closer to unity for the pseudo-second-order kinetic model than the pseudo-first-order kinetic model. Therefore, the sorption reaction can be approximated more favourably by the pseudo-second-order kinetic model for methyl orange, methyl blue and safranine T onto synthesised and commercial zeolite. MPSD error function values as shown in Table A.3 are also considerably lower for the pseudo-second-order kinetic model, reinforcing the applicability of the pseudo-second-order kinetic model. It may be seen that the initial sorption rate (h) continuously increased with increase in Cq. This is due to the increase in driving force due to the increase in Q. 

A successor to PESTANS has recently been developed which allows the user to vary transformation rate and with depth l.e.. It can describe nonhomogeneous (layered) systems (39,111). This successor actually consists of two models - one for transient water flow and one for solute transport. Consequently, much more Input data and CPU time are required to run this two-dimensional (vertical section), numerical solution. The model assumes Langmuir or Freundllch sorption and first-order kinetics referenced to liquid and/or solid phases, and has been evaluated with data from an aldlcarb-contamlnated site In Long Island. Additional verification Is In progress. Because of Its complexity, It would be more appropriate to use this model In a hl er level, rather than a screening level, of hazard assessment. 

Tillotson et al. (1980) Nitrification and urea hydrolysis by first-order kinetics NH3 volatilization by first-order kinetics from (NH4)2C03 formed from urea hydrolysis NH sorption by linear partition model NH and NOy plant uptake involving diffusion to roots. 

The comparability of the two methods was further tested by attempting to use kinetic-parameter values determined by gas purge to predict breakthrough curves (BTC) obtained from the same soil/solute pairs. A BTC obtained from miscible displacement of PCE through the Eustis soil column is presented in Fig. 11-5. The values for ki and F (fraction of sorbent for which sorption is instantaneous) determined from the Eustis/PCE gas-purge experiment were used to calculate values for /3 (fraction of instantaneous retardation) and w (ratio of hydrodynamic residence time to reaction time). The simulation produced by a first-order bicontinuum model (Brusseau and Rao, 1989a) with these independently determined kinetic parameters is shown in Fig. 11-5. The model simulation compares extremely well with the experimental BTC. Such predictions for organic chemicals, where values for... 

A number of experimental studies have established that both microbial and chemical degradation can be approximately described by first-order kinetics (24). Most pesticide models employ such an approach. As with linear sorption, this relatively naive representation of a fundamentally more complicated process is a simplifying assumption to make mathematical solutions possible and data requirements reasonable. Implicit in the assumption is the belief that the accuracy of simulation of pesticide fate is more dependent upon other factors than a very precise representation of the degradation process. These factors include spatial and temporal variability of the degradation process itself as affected by water, temperature, substrate, and pH, and variability in the transport of pesticide through the soil profile. There is little information to substantiate this assumption, although some field experiments on water and solute movement (discussed below) indicate it to be reasonable at this point in model development. 

A mathematical model is formulated to describe the first-order kinetics of ionic copper released into a marine environment where sorption on suspended solids and complexation with dissolved organic matter occur. Reactions are followed in time until equilibrium, between the three copper states is achieved within about 3 hr (based on laboratory determinations of rate and equilibrium constants). The model is demonstrated by simulation of a hypothetical slug discharge of ionic copper, comparable to an actual accidental release off the California coast that caused an abalone kill. A two-dimensional finite element model, containing the copper submodel, was used to simulate the combined effects of advection, diffusion, and kinetic transformation for 6 hr following discharge of 45 kg of ionic copper. Results are shown graphically. 

Selim et al. (1976) proposed a model where two types of exchange sites were present at the exchange complex, one governed by equilibrium (type 1) sorption, and the other by first-order kinetics (type 2). The total concentration of sorbed solutes S is assumed to be... 

The sorption of Cr(VI) by PPy/Fe O nanocomposite is a typical example [27]. The pseudo-first-order and pseudo-second-order kinetic models... 

Adsorption may also be modeled as a nonequilibrium process using nonequilibrium kinetic equations. In a kinetic model, the solute transport equation is linked to an appropriate equation to describe the rate that the solute is sorbed onto the solid surface and desorbed from the surface (Fetter, 1999). Depending on the nonequilibrium condition, the rate of sorption may he modeled using an irreversible first-order kinetic sorption model, a reversible linear kinetic sorption model, a reversible nonlinear kinetic sorption model, or a bilinear adsorption model (Fetter, 1999). 

Reaction kinetics. The time-development of sorption processes often has been studied in connection with models of adsorption despite the well-known injunction that kinetics data, like thermodynamic data, cannot be used to infer molecular mechanisms (19). Experience with both cationic and anionic adsorptives has shown that sorption reactions typically are rapid initially, operating on time scales of minutes or hours, then diminish in rate gradually, on time scales of days or weeks (16,20-25). This decline in rate usually is not interpreted to be homogeneous The rapid stage of sorption kinetics is described by one rate law (e.g., the Elovich equation), whereas the slow stage is described by another (e.g., an expression of first order in the adsorptive concentration). There is, however, no profound significance to be attached to this observation, since a consensus does not exist as to which rate laws should be used to model either fast or slow sorption processes (16,21,22,24). If a sorption process is initiated from a state of supersaturation with respect to one or more possible solid phases involving an adsorptive, or if the... 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

Kinetic models proposed for sorption/desorption mechanisms including first-order, multiple first-order, Langmuir-type second-order, and various diffusion rate laws are shown in Sects. 3.2 and 3.4. All except the diffusion models conceptualize specific sites to or from which molecules may sorb or desorb in a first-order fashion. The following points should be taken into consideration [ 181,198] ... 

The results of the experiments presented in this paper demonstrate that the migration of nuclides in geologic media can be studied experimentally and treated in at least a semiquantitative fashion using kinetic and partitioning data. The ARDISC model is a useful aid in analyzing nuclide migration data obtained in laboratory experiments. But the ARDISC model is limited to first-order sorption kinetics. 

Calibrations performed using an equilibrium model indicated increasing Kd with time, which is consistent with kinetic effects (i.e., gradual approach to equilibrium). When the kinetic model was calibrated, good model fits were observed for all three columns using a calibrated Kd of 1.4 mL/g and first-order sorption rate constant of 0.15 day 1 (Figure 2). 

Selim et al. (1976b) developed a simplified two-site model to simulate sorption-desorption of reactive solutes applied to soil undergoing steady water flow. The sorption sites were assumed to support either instantaneous (equilibrium sites) or slow (kinetic sites) first-order reactions. As pore-water velocity increased, the residence time of the solute decreased and less time was allowed for kinetic sorption sites to interact (Selim et al., 1976b). The sorption-desorption process was dominated by the equilib-... 

Wu and Gschwend (1986) reviewed and evaluated several kinetic models to investigate sorption kinetics of hydrophobic organic substances on sediments and soils. They evaluated a first-order model (one-box) where the reaction is evaluated with one rate coefficient (k) as well as a two-site model (two-box) whereby there are two classes of sorbing sites, two chemical reactions in series, or a sorbent with easily accessible sites and difficultly accessible sites. Unfortunately, the latter model has three independent fitting parameters kx, the exchange rate from the solution to the first (accessible sites) box k2, the exchange rate from the first box to the... 

In order to assess the feasibility of any nuclear waste disposal concept, mathematical models of radionuclide sorption processes are required. In a later section kinetic descriptions of the three common sorption isotherms (3) are compared with experimental data from the mixing-cell tests. For a radionuclide of concentration C in the groundwater and concentration S on the surface of the granite, the net rate of sorption, by a first-order reversible reaction, is given by... 

Rate-limited sorption can also be modeled assuming a kinetic rate expression coupled with a nonlinear equilibrium expression. If we assume a Freundlich isotherm and a first-order rate expression, we can use the following equation to model sorption kinetics [21] ... 

BIOPLUME III is a public domain transport code that is based on the MOC (and, therefore, is 2-D). The code was developed to simulate the natural attenuation of a hydrocarbon contaminant under both aerobic and anaerobic conditions. Hydrocarbon degradation is assumed due to biologically mediated redox reactions, with the hydrocarbon as the electron donor, and oxygen, nitrate, ferric iron, sulfate, and carbon dioxide, sequentially, as the electron acceptors. Biodegradation kinetics can be modeled as either a first-order, instantaneous, or Monod process. Like the MOC upon which it is based, BIOPLUME III also models advection, dispersion, and linear equilibrium sorption [67]. 

In order to avoid the restrictions to complicated adsorptive reactions in the MOC3D, Selim et al. (1990) developed a simulation system based on the multireaction model (MRM) and multireaction transport model (MRTM). The MRM model includes concurrent and concurrent-consecutive retention processes of the nonlinear kinetic type. It accounts for equilibrium (Freundlich) sorption and irreversible reactions. The processes considered are based on linear (first order) and nonlinear kinetic reactions. The MRM model assumes that the solute in the soil environment is present in the soil solution and in several phases representing retention by various soil... 

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