Total number of sites

Since we are explicitly interested in the difference in the sizes of solvent and solute molecules, it is more appropriate to express the values of AU on a per unit volume basis rather than on a molar basis. Accordingly, in Eq. (8.41) we replace the total number of sites N by the total volume of the mixture V and write... 

Dissociation constant, the ratio of the rate of offset of ligand away from a receptor divided by the rate of onset of the ligand approaching the receptor. It has the units of concentration and specifically is the concentration of ligand that occupies 50% of the total number of sites available for ligand binding at equilibrium (see Affinity). 

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 ... 

Note that the rates of product formation and reactant conversion indeed have the dimensions of mol per unit of time, and that these rates are proportional to the number of sites, or, in fact, the amount of catalyst present in the reactor. Also, in the case of a second order reaction, e.g. betv een adsorbed species A and B, we write the rate in the form r = Nk0j 0 by applying the mean-field approximation. Here the rate is proportional to both the total number of sites on the surface and the probability of finding a species A adjacent to a species B on the surface, the latter being proportional to the coverages of A and B. In the mean-field approximation A and B are distributed randomly over the N available sites this only tends to be valid when the adsorbents repel each other. Thus the rate is not r= k(N0/ )(N02,) since the reactants need to be on adjacent sites. Another important consideration is that we want the rate to be linearly proportional to the amount of catalyst in the reactor, in accordance with r = Nk0A0B for a second order surface reaction. 

For each step there is a corresponding rate (for convenience we drop the total number of sites from the expressions, i.e. r becomes a rate per site, or a turnover frequency) ... 

These five defects are based upon an excess in the number of sites available. This excess we call "6". Note that we are not speaking of the ratio of cations to anions, i.e.- stoichiometry, but of the total number of sites. To see how this is possible, consider the Vacancy-Structure type. 

The coefficient y has a high value, of about 12. This implies that an increase in 9q by 9.96 (i.e., an increase in the number of sites taken up by oxygen, which amounts to 6% of the total number of sites) produces a 59% drop in reaction rate. 

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 ... 

Here K is a constant, Te and Tf are the number of empty and filled sites per unit area on the metal surface, , is the adsorption potential, and is the electrostatic potential of the empty site , depends on surface charge. The sum T0 = Te + T/ total number of sites per unit area, depends on the metal, as does fa. 

Keep in mind that in this case the total number of sites is N. The relationship between the values of two successive K values must reflect the difference in probabilities of formation of the corresponding complexes. Therefore, we can write... 

A binary ionic solution must contain at least three kinds of species. One example is a solution of AC and BC. Here we have two cation species A+ and B+ and one common anion species C . The sum of the charge of the cations and the anions must be equal to satisfy electro-neutrality. Hence NA+ + NB+ = N(. = N where NA+, AB+ and Nc are the total number of each of the ions and N is the total number of sites in each sub-lattice. The total number of distinguishable arrangements of A+ and B+ cations on the cation sub-lattice is M/N A, JVg+ . The expression for the molar Gibbs energy of mixing of the ideal ionic solution AC-BC is thus analogous to that derived in Section 9.1 and can be expressed as... 

Reaction between carbon monoxide and dihydrogen. The catalysts used were the Pd/Si02 samples described earlier in this paper. The steady-state reaction was first studied at atmospheric pressure in a flow system (Table II). Under the conditions of this work, selectivity was 100% to methane with all catalysts. The site time yield for methanation, STY, is defined as the number of CH molecules produced per second per site where the total number of sites is measured by dihydrogen chemisorption at RT before use, assuming H/Pd = 1. The values of STY increased almost three times as the particle size decreased. The data obtained by Vannice et al. (11,12) are included in Table II and we can see that the methanation reaction on palladium is structure-sensitive. It must also be noted that no increase of STY occurred by adding methanol to the feed stream which indicates that methane did not come from methanol. 

We turn now to the microscopic description of an imperfect crystal. The various defects in any imperfect crystal can be imagined to be formed from a corresponding perfect crystal by one or more of the following processes (a) remove an atom of species Os from the crystal leaving a vacant lattice site, (b) remove an atom of species Os from the crystal and replace it by an atom of a different species (either Oi or at), (c) add to the crystal an atom of any species to a site on a sublattice unoccupied in the perfect crystal. We refer to the latter as atoms in interstitial positions. Let B be a set of numbers such that Br is the number of sites on sublattice number r in the perfect crystal, and let be the number of sublattices in the crystal (including interstitial sublattices not occupied in the perfect crystal). The total number of sites of all kinds in the perfect crystal is then... 

Endrin has been identified in at least 102 of the 1,430 current and former hazardous waste sites that have been proposed for inclusion in the NPL (HazDat 1996), although the total number of sites evaluated for endrin is not known. The frequency of these sites can be seen in Figure 5-1. Of these sites, 102 are located in the United States. Endrin ketone has been identified in at least 37 of the 1,430 current and former hazardous waste sites that have been proposed for inclusion in the NPL (HazDat 1996). However, the number of sites evaluated for endrin ketone is not known. The frequency of these sites can be seen in Figure 5-2. Of these sites, 37 are located in the United States. 

Ns The total number of sites per unit area on the oxide surface ... 

EPA has identified 1,177 NPL sites. 1,2-Dibromoethane has been found at 9 of the total number of sites evaluated for that compound. We do not know how many of the 1,177 sites have been evaluated for 1, 2-dibromoethane. As more sites are evaluated by EPA, this number may change (View 1989). The frequency of these sites within the United States can be seen in Figure 5-1. 

The general Adair equation for the binding of a ligand X to a multisite protein where K represents the thermodynamic macroscopic association constant for the fth site and where n is the total number of sites is... 

Let S0 equal the number of sites on the surface that are occupied. If the total number of sites is S, then (S S0) is the number of sites on the surface... 

The volume adsorbed at monolayer coverage is simply proportional to the total number of sites irrespective of the depth to which they are covered ... 

Each surface species occupies one or more surface sites. A site is considered to be a location or position on the surface at which a species can reside. A site does not necessarily have to be a particular atom or have a composition itself. The total number of sites per unit area is considered a property of the material surface, and is often assumed to remain constant. 

Let us assume N linear fc-mers adsorbed on M sites. The coverage is given by 0 = kN/M. We can make a mapping L—>L from the original lattice L into an effective lattice L where each empty site of L turns into an empty one of L, while each set of k sites occupied by a k-mer in L is represented by an occupied site in V. Thus, the total number of sites in L is... 

The property being measured (AA) reaches a maximum value AAmax at saturation and when all of compound P has been converted to PX. The ratio of [PX] to the total concentration of all forms of P present [Pit is known as the saturation fraction and is often given the symbol Y. If P has more than one binding site for X, Y is defined as the fraction of the total binding sites occupied. If n is the number of sites per molecule, the total number of sites is n[P], The value of Y is often taken as AA / AAmax, an equality that holds for multisite macromolecules only if the change in A is the same for each successive molecule of X added. This is not always true, but when it is Eq. 7-3 is followed. 

The site fraction is the number of species of a particular component that occupy a particular site divided by the total number of sites of that type. For example, in sodium chloride (NaCl) there is a distinction between cation and anion sites. Impurity species and vacancies may also be present. If there is a total of s distinct types of sites (s = 2 in NaCl) and there is a total number, ATjot, of sites of type j on which are distributed N- atoms (molecules) of component i, the fraction of sites of type j occupied by component i is... 

In simulations [9] Sierpinski gaskets on the 12th stage, containing 177147 or 265722 sites, were used respectively. The number No of randomly distributed A or B particles was 10 percent of the total number of sites. The random mutual annihilation of dissimilar particles was simulated through a minimal process method [10] from all AB pairs at each reaction step one pair was selected randomly, according to its reaction rate (3.1.2) the time... 

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