Rate site balance

The site balance specihes that the number of empty plus occupied sites is a constant, Sq. Equality of the reaction rates plus the site balance gives four independent equations. Combining them allows a solution for while eliminating the surface concentrations [S], [AS], and [PS]. Substitute the various reaction rates into the site balance to obtain... 

Note that under steady-state conditions the rate of each reaction step equals the overall net rate, 0, 9a, and 6b represent the fractions of the total number of sites that are vacant, or occupied by A and B, respectively. Afr represents the total concentration of active sites. Conservation of the total number of active sites leads to the site balance expression ... 

Expressions for 9 SM and H2 can be derived and related to rate (k) and equilibrium constants (K). The SI and S2 site balances are 9SM +9a + S, = 1 and //2 + S2 = 1 respectively 9sx, S2 are empty sites). Based on Henry s law, the gas-phase hydrogen pressure and the liquid-phase hydrogen concentration may be used interchangeably. The rate expression can be written as follows ... 

Some components in a gas or liquid interact with sites, termed adsorption sites, on a solid surface by virtue of van der Waals forces, electrostatic interactions, or chemical binding forces. The interaction may be selective to specific components in the fluids, depending on the characteristics of both the solid and the components, and thus the specific components are concentrated on the solid surface. It is assumed that adsorbates are reversibly adsorbed at adsorption sites with homogeneous adsorption energy, and that adsorption is under equilibrium at the fluid- adsorbent interface. Let (m" ) be the number of adsorption sites and (m 2) the number of molecules of A adsorbed at equilibrium, both per unit surface area of the adsorbent. Then, the rate of adsorption r (kmol m s ) should be proportional to the concentration of adsorbate A in the fluid phase and the number of unoccupied adsorption sites. Moreover, the rate of desorption should be proportional to the number of occupied sites per unit surface area. Here, we need not consider the effects of mass transfer, as we are discussing equilibrium conditions at the interface. At equilibrium, these two rates should balance. Thus,... 

PK modeling can take the form of relatively simple models that treat the body as one or two compartments. The compartments have no precise physiologic meaning but provide sites into which a chemical can be distributed and from which a chemical can be excreted. Transport rates into (absorption and redistribution) and out of (excretion) these compartments can simulate the buildup of chemical concentration, achievement of a steady state (uptake and elimination rates are balanced), and washout of a chemical from tissues. The one- and two-compartment models typically use first-order linear rate constants for chemical disposition. That means that such processes as absorption, hepatic metabolism, and renal excretion are assumed to be directly related to chemical concentration without the possibility of saturation. Such models constitute the classical approach to PK analysis of therapeutic drugs (Dvorchik and Vesell 1976) and have also been used in selected cases for environmental chemicals (such as hydrazine, dioxins and methyl mercury) (Stem 1997 Lorber and Phillips 2002). As described below, these models can be used to relate biomonitoring results to exposure dose under some circumstances. 

The deactivation of catalysts concerns the decrease in concentration of active sites on the catalyst Nj. This should not be confused with the reversible inhibition of the active sites by competitive adsorption, which is treated above. The deactivation can have various causes, such as sintering, irreversible adsorption and fouling (for example coking or metal depositions in petrochemical conversions). It is generally attempted to express the deactivation in a time-dependent expression in order to be able to predict the catalyst s life time. An important reason for deactivation in industry is coking, which may arise from a side path of the main catalytic reaction or from a precursor that adsorbs strongly on the active sites, but which cannot be related to a measurable gas phase concentration. For example for the reaction A B the site balance contains also the concentration of blocked sites C. A deactivation function is now defined by cq 24, which is used in the rate expression. 

Rate expression (3.9) has been derived for a relatively simple kinetic model by application of the site balance and the steady state hypothesis. More complex models will result in more complex expressions, which are hard to handle. Fortunately, some simplifications can be applied. 

By application of the steady-state hypothesis, a site balance and the assumption that the surface dissociation is rate determining while the other steps are in quasi-equilibrium, the following rate expression is derived ... 

Equations 9 and 10 show how the thiophene mole fraction in the gas and the concentration of unpoisoned sites varies with time (T) and position (m ) in the reactor and both of these equations can be graphed using the calculated values for the rate of thiophene poisoning and active site concentration as well as the experimental conditions. Figures 2 and 3 show the ratio of the instantaneous to initial thiophene mole fraction and active site balance, respectively, as a function of reduced length down the bed at several different reduced times for Catalyst A . Figures 4 and 5 show the same ratios for Catalyst C . 

The total number of adsorption sites on the surface now appears explicitly in the rate expression for this elementary adsorption step. Since the site balance is the same as before (Equation 5.2.5), the equilibrium adsorption isotherm can be calculated in the manner described above ... 

The site balance and the Langmuir adsorption isotherm can be used to derive the forward rate expression ... 

The application of the quasi-steady-state approximation and the site balance (assuming A is the mart) gives the following expression for the reaction rate ... 

Substituting for the inert sites in the site balance, the rate law for sinface reaction control when an adsorbing inert is present is... 

Rate expression (3.9) has been derived for a relatively simple kinetic model by application of the site balance and the steady-state hypothesis. More complex models will result in more complex expressions (see Refs. [1] and [2] for general derivation), which are hard to handle. Fortunately, some simplifications can usually be applied. The rate expression then follows by computing the net rates of intermediate formation rj, which are the sum of the intermediate formation (forward) and intermediate removal (backward) reaction steps ... 

The steady-state approach generally yields complex rate expressions. A simplification is obtained by the introduction of one or several rate-determining step(s) and ( wasi-equilibrium steps, and further by the initial reaction rate approach. For complex reaction schemes, identifying the most abundant reaction intermediates ("mari") and making use of the site balance can simplify the kinetic models and rate expressions. 

Here Kj is the equilibrium constant of the y-th elementary reaction (as the ratio of the forward and reverse rate constants), d yd ik =, 2,...,q)src the surface coverages of the active sites and surface intermediates, respectively, and P, i = 1,2,..., ) represents the partial pressure of each terminal species. The surface coverages are subject to the site balance constraint... 

By substituting quasi-equilibrium expressions into the site balance, the concentration of vacant sites is obtained and the final forms of the rate equations become ... 

II. Let Reaction 4 be the rate-limiting step and assume all other reactions are at equilibrium. Assume that adsorption from the gas phase follows a Langmuir adsorption isotherm. Neglect the reverse of Reaction 4. When performing a site balance, assume the surface is either vacant or covered with adsorbed ethylene and ethyl, i.e., the coverage of atomic hydrogen and adsorbed ethane are negligibly small. 

Rate expressions of the form of Equation 5.153 are known as Hougen Watson or Langmuir-Hinshelwood kinetics [17, This form of kinetic expression is often used to describe the species production rates for heterogeneously catalyzed reactions. We complete the section on the kinetics of elementary surface reactions by returning to the methane synthesis reaction listed in Section 5.2. The development proceeds exactly as outlined in Section 5.2. But now it is necessary to add a site-balance expression (Equation 5,129) in Step 3. 

Considering the site balance equation (1 = 0NH3+fi H2O+fi ) and by replacing equation 17 and 18 into equation 16, the following Rideal rate expression is eventually obtained ... 

Additional simplification is acquired when assumptions are made regarding the surface species to be included in the site balance. Frequently it is neither possible nor necessary to include every intermediate because many concentrations are going to be very small compared to others. By selecting those expected to dominate based upon the behavior of the rate and any available spectroscopic information, the number of important unknown surface coverages can be significantly decreased. At the limit, if a MARI exists, then only one surface species becomes important in the final rate derivation. 

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