Site balance

A site balance ties these equations together ... 

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... 

The site balance is identical to those in previous examples. 

The site balance is the same as in Example 12.1. Eliminating the unknown surface concentrations gives... 

The last equation is not independent of the others due to the site balance of Eq. (141) hence, in general, we have n-1 equations for a reaction containing n elementary steps. Note that steady state does not imply that surface concentrations are low. They just do not change with time. Hence, in the steady state approximation we can not describe time-dependent phenomena, but the approximation is sufficient to describe many important catalytic processes. 

It is now straightforward to find an expression for the fraction of free sites from the site balance ... 

Although the precise mechanism of the HDS reaction is still under debate, we deliberately chose this scheme because it illustrates the kinetics of processes involving two kinds of sites. Consequently, two site balances exist ... 

Our procedure is to set up a site balance in terms of lattice molecules, i.e.- AgBr ... 

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 ... 

The reactor model adopted for describing the lab-scale experimental setup is an isothermal homogeneous plug-flow model. It is composed of 2NP + 2 ordinary differential equations of the type of Equation 16.11 with the initial condition of Equation 16.12, NP + 3 algebraic equations of the type of Equation 16.13, and the catalytic sites balance (Equation 16.14) ... 

In cases where the antisite defects are balanced, such as a Ga atom on an As site balanced by an As atom on a Ga site, the composition of the compound is unaltered. In cases where this is not so, the composition of the material will drift away from the stoichiometric formula unless a population of compensating defects is also present. For example, the alloy FeAl contains antisite defects consisting of iron atoms on aluminum sites without a balancing population of aluminum atoms on iron sites. The composition will be iron rich unless compensating defects such as A1 interstitials or Fe vacancies are also present in numbers sufficient to restore the stoichiometry. Experiments show that iron vacancies (VFe) are the compensating defects when the composition is maintained at FeAl. 

With respect to the vacant sites, the site balance can be written as... 

If other species are also adsorbed on the surface, the concentrations of the corresponding occupied sites have to be added to the site balance above. The concentration of the active sites is expressed as mole per unit mass of catalyst and is equal to the number of active sites per unit mass of catalyst divided by the Avogadro s number. 

The experimental procedures are those used previously (1). The following saturations were obtained 255.2 X 10 3 equivalent of ethyl-ammonium (EA), 141 X 10 3 equivalent of diethylammonium (DEA), and 65.5 X 10 3 equivalent of triethylammonium (TEA) per 100 grams of Y zeolite equilibrated at a relative humidity of 32%. Thus there is still an appreciable fraction of the lattice negative sites balanced by Na+ cations (6). The crystalline character of the zeolite lattice is not affected during these experiments. 

Cu ions are incorporated into molecular sieve as divalent (Cu2+(HzO)6)2+ or monovalent (Cu2+X (H20)s)+ complex cations [11]. Thus, after dehydration and subsequent reduction, Cu ions are placed in viscinity of one or two framework aluminium atoms [1], As two cationic sites with two close aluminium atoms are present in (A1)MCM-41 (cf. Chapters 3.1. and 3.2.), four different coordinations of the Cu+ ions in (A1)MCM-41 represent two cationic sites with two close aluminium atoms and two cationic sites balanced by a single aluminium atom. Thus, besides two cationic sites accommodating divalent cations, two other cationic sites being enable to accommodate only monovalent cations are present in the (A1)MCM-41 molecular sieve. Because only low exchange degree can be reached for divalent cations (cf. Table 1), sites with isolated aluminium atoms represent majority of cationic sites in (A1)MCM-41 with Si/Al > 20. 

Equations 10 are equivalent to those of eq 9, in which only two are independent. Using this steady state assumption, generally one or more of these unknowns can be eliminated, but one remains. The second assumption is the site balance. The total concentration of active sites is constant and equal to Nt. ... 

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. 

Conservation of the number of active sites leads to the site balance 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. 

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