Adsorption fraction occupied

Adsorption occurs on the microscopic level, with molecules of adsorbate sticking to the atoms on the surface of the substrate (in this case, the pan). In practice, it is rare for each and every adsorption site to have a dye molecule adsorbed to it. The coverage is only fractional. We give the name isotherm to the fraction of the total adsorption sites occupied by adsorbate. An isotherm is denoted with the Greek symbol 0 (theta). 

In this form the Langmuir equation shows how the fraction of surface adsorption sites occupied by solute increases as the solute activity in solution increases. From now on we drop the subscript 2 and the superscript b. Since Equation (67) is written solely in terms of the solute, these designations are redundant. 

In this expression 0 is the fraction of surface adsorption sites occupied, nx is the concentration of the adsorbed ions in the solution, and A- is a constant. 

The Lundstrom model70 is given in Fig. 13b. He assumes that protein adsorbs with a rate constant ka into State 1. Upon adsorption, some of the adsorbed proteins in State 1 (native) may conformationally change (via rate constant kr) to State 2 (denatured). Letting n, and n2 be the number of molecules per unit area in States 1 and 2, and and a2 be the area fractions occupied in each state, he says (noting that the unoccupied area fraction = (1 — — a2n2) that70) ... 

To model the process, a mass balance is made for the adsorbed species, and the partial pressures of the reactants are considered as constant. The surface coverage of each adsorbed species is defined as the fraction of the total number of available adsorption sites occupied by that species and denoted by 0A or 0B. The differential equations for this mechanism are ... 

Another way of disappearance of nonequilibrium charge carriers is their recombination at the particle surface (radiative with the rate constant ks>r, and nonradiative with the rate constant ks>n). Of basic importance is the question of whether the surface recombination sites are the sites of the quencher adsorption. In other words, is the quencher adsorption able to result in disappearance of the surface recombination sites. With positive answer, the expression for the surface recombination rate should be written as (k + ks,n)-S-(l - 0a)-e-h, where S is the particle surface area, and 0a is the surface fraction occupied by the quencher (electron acceptor). Otherwise, the latter multiplier (1 - 0J should be excluded. Further we will consider the both cases (1 and 2), compare them with experimental data and choose the case providing a better description of the phenomena observed. 

While natural waters will certainly not contain total lead or other heavy metal concentrations so large as the 5.0 X 10 M value used here to illustrate the effect of cation adsorption on adsorption potential, sediments in contact with natural waters will undoubtedly have a large fraction of their adsorption sites occupied with major cations such as Ca " ", K ", or Na . The... 

Derive from kinetic principles the competitive Langmuir equation for gas adsorption, that is, a function that expresses the fraction of adsorption sites occupied by molecule A in the presence of an adsorbing molecule B. 

The total effectiveness will be proportional to the fraction of cellular adsorption sites occupied by complexed (i.e. angiostat bearing) carriers,... 

In Chapter 5, we defined the osmotic pressure of a polymer solution, we indicated how it can be measured, and we described various effects concerning the compressibility and the preferential adsorption. When the polymers are very long and when the volume fraction occupied by the polymers in the solution is small, the complex reality can be represented by a simple model which is the standard continuous model, studied in Chapter 10 in the context of perturbation theory. This model is especially useful because it allows us to perform effective calculations. In particular, it can be used in the limit of long polymers to determine universal quantities because, then, the general properties of long polymers become independent of the chemical microstructure. Calculations are... 

The term 6a in Eq. (P9.4.4) represents the fraction of total adsorption sites occupied by A. If there are two kinds of molecules, A and B, which are competing for the adsorption sites, one modifies Eq. (P9.4.4) to Kapa... 

Intrinsic viscosity scaled to zero-shear-rate viscosity Fraction of surface adsorption sites occupied... 

Impurities and other components may compete for adsorption and occupy a fraction of the sites. Hence a variation of the pressure would probably have less effect than when all the sites are occupied only by reactants and products. This may not significantly change the trends as... 

Being Ca the concentration of the adsorbate in the liquid phase (mg L-i), Icad the kinetic constant of adsorption, ko the kinetic constant of desorption and 6U the fraction of adsorption sites occupied by component A of the total number of sites occupied at... 

The fraction of the adsorption sites occupied at a given time, q, is given by... 

The transformation can be described by considering three surface processes as shown in Equation 2.108 adsorption of the reactant, surface reaction, and desorption of the product. If the reaction is carried out in an open reactor under constant conditions, for example, in a catalytic packed bed reactor, the fractions occupied by and A2 are time invariant obtain ... 

The surface fractions occupied by the reactants Aj and A2 are given by the Langmuir isotherm, supposing competitive adsorption and neglecting the coverage by the product. 

As before the correlation curves 8i(), where is the volume of adsorption space occupied, are calculated from the single solute isotherms and, for given bulk phase compositions the free-energy curves, as functions of (f>, calculated for each component. The fraction of adsorption space occupied by a given component is that in which its free-energy curve is the lowest. In this way. 

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

There is a complication in choosing a catalyst for selective reductions of bifunctional molecules, For a function to be reduced, it must undergo an activated adsorption on a catalytic site, and to be reduced selectively it must occupy preferentially most of the active catalyst sites. The rate at which a function is reduced is a product of the rate constant and the fraction of active sites occupied by the adsorbed function. Regardless of how easily a function can be reduced, no reduction of that function will occur if all of the sites are occupied by something else (a poison, solvent, or other function). 

The fraction of sites occupied by A, designated a, is known as the Langmuir adsorption isotherm. It is given by... 

We can think of a heterogeneous catalyst as a collection of active sites (denoted by ) located at a surface. The total number of sites is constant and equal to N (if there is any chance of confusion with N atoms, we will use the symbol N ). The adsorption of the reactant is formally a reaction with an empty site to give an intermediate I (or more conveniently R if we explicitly want to express that it is the reactant R sitting on an adsorption site). All sites are equivalent and each can be occupied by a single species only. We will use the symbol 6r to indicate the fraction of occupied sites occupied by species R, making N6r the number of occupied sites. Hence, the fraction of unoccupied sites available for reaction will be 1 - 0r The following equations represent the catalytic cycle of Fig. 2.7 ... 

At low values of the bulk concentration Bcy surface coverage is proportional to this concentration, but at high values it tends toward a limit of unity. This equation was derived by Irving Langmuir in 1918 with four basic assumptions (1) the adsorption is reversible (2) the number of adsorption sites is limited, and the value of adsorption cannot exceed A° (3) the surface is homogeneous aU adsorption sites have the same heat of adsorption and hence, the same coefficient B and (4) no interaction forces exist between the adsorbed particles. The rate of adsorption is proportional to the bulk concentration and to the fraction 1-9 of vacant sites on the surface = kjil - 9), while the rate of desorption is proportional to the fraction of sites occupied Vj = kjd. In the steady state these two rates are equal. With the notation kjk = B, we obtain Eq. (10.14). 

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