Intrinsic constants

Table 3. Representative affinity constants for the binding of metal to transport sites or whole cells/organisms. Ionic strengths and pH values are given for the conditional constants. In the column Comments , information on the method of determination (Km = Michaelis-Menten constant WC = whole-cell titrations) the type of constant (CC = conditional constant IC = intrinsic constant) and special conditions (Cl = competitive inhibitors NICA = nonideal competitive adsorption) are given...

We return to the complex formation equilibria described in Chapter 2 (Eqs. 2.1 -2.10). The equilibrium constants as given in these equations are essentially intrinsic constants valid for a (hypothetically) uncharged surface. In many cases we can use these constants as apparent constants (in a similar way as non-activity corrected constants are being used) to illustrate some of the principal features of the interdependent variables that affect adsorption. Although it is impossible to separate the chemical and electrical contribution to the total energy of interaction with a surface without making non-thermodynamic assumptions, it is useful to operationally break down the interaction energy into a chemical and a Coulombic part ... 

Linear-free energy relation between the rate constants for water exchange k w [s 1] and the intrinsic adsorption rate constants katjS(int) [M 1 s 1] from the pressure jump experiments of Hachiya et al. The intrinsic constants refer to an uncharged surface. The linear-free energy relations based on the experimental points are extended to some ions with lower H20 exchange rate in order to predict adsorption rates. 

The information in Figure 14 was produced in the following way. The slope (or the Kurbatov coefficients shown in Table V) and position of the fractional absorption gdges in Figure 3 were used as the criteria of model fit. Kcd an< CdOH were used as the fitting parameters and all other parameters were held constant. Consequently, the intrinsic constants shown in Figure 14a represent best fit parameters and, given that all other surface and solution association constants are invariant, constitute a unique solution set for each adsorption density. 

Table IV. Intrinsic Constants and Capacitance Values for Divalent Metal Ion Binding...

Since is presumed to be independent of C, all the intrinsic constants are also independent of C. Note also that the factor 2(0,0, 0) does not appear explicitly in n, although it is contained in the definitions of all the binding constants. 

The individual BI can be obtained either fi om the probabilities read from the GPF, as we have done in Section 2.1, or from Eq. (2.2.8). For instance, the individual BI of site a, expressed in terms of the intrinsic constants, is... 

We say that the sites are identical in a weak sense whenever the three PFs 0(1,0,0), 0(0,1,0), and 0(0,0, 1) have the same value. This is identical to the requirement that the single-site intrinsic constant is the same for any specific site. In this case we can replace these three PFs by three times one representative PF, as is done on the rhs of Eq. (2.2.22). We shall say that the sites are identical in a strict sense whenever the PF of any given occupation number is independent of the specific group of occupied sites. For instance, in an equilateral triangle all PFs with two sites occupied are equal. Hence we can replace the sum on the rhs of Eq. (2.2.23) by three times one representative PF. This cannot be done, in general, for a linear arrangement of the three sites, in which case 0(1,1,0) is different from 0(1,0,1), even when the sites are identical in the weak sense (see Chapter 5). Similarly, for... 

We proceed with the case of three identical sites in the strict sense and define the corresponding intrinsic constants... 

Again, we stress that these conditional constants always refer to a specific configuration before and after the addition of the ligand k, k, and /t,/, may be referred to as the intrinsic constants for the rsr, second, and third ligands. These should be distinguished from the normally used first, second, and third thermodynamic constants, defined in Section 2.3. In the latter, the specification of the sites is not required. 

Although we shall not use nonintrinsic constants in this book, we mention them here since they are sometimes used in the literature. The nonintrinsic constants are obtained from the intrinsic constants by simply removing the requirement of a specific set of sites. For the three-site case, these are defined by... 

This is the BI expressed in terms of the measurable equilibrium constants K. Since we require that the BI as represented by the thermodynamic constants be the same as that represented by the intrinsic constant, we can make the identifications ... 

These equations relate the sequential thermodynamic constants (first, second, and third) to the sequential intrinsic constants. The difference between the two sets arises from the requirement to specify the sites in the latter but not in the former. The generalization to m identical sites (in the strict sense) is quite straightforward. 

The ratio between the thermodynamic constant and the corresponding intrinsic constant is the ratio between the number of specific configurations of the ligands before and after the addition of the yth ligand. 

The more remarkable and well-known fact is that the two systems a and b are also equivalent (in the above sense) to a system c of Af identical and independent molecules, each having two identical but dependent sites, with one intrinsic constant k and pair correlation S. The GPF of this system c is... 

Here, all sites have identical intrinsic constants but the pairs that originally belonged to the same molecules are left uncorrelated. All the pair correlations are here assigned to the newly formed four pairs of type LH. 

The distinction between Kj and k becomes clearer when the two sites are not equivalent (e.g., glycine), in which case 2k in Eq. (4.8.6) is the sum of two different intrinsic constants, say the binding to the acidic group k and to the basic group. 

From the experimental binding isotherm n = n(C), one can determine the two experimental constants and K2. If the two sites are identical, then one can convert from and K2 into fcj and via Eq. (4.8.5). This is not possible when the two sites are not identical in which case one cannot obtain the intrinsic constants, say k and solely from knowledge of and K2 (see Subsection 4.8.3). 

Clearly, from the experimental values of and one caimot obtain the three independent intrinsic constants and k =kjc/k. ... 

We conclude this section by noting that when the subunits become identical, and A = Aj, = A = A and = q = q, the sites are, in general, not identical in either the weak or the strict sense. This is in contrast to the model discussed at the end of Section 5.5, where we found only one intrinsic constant. Here, in general, there are two different intrinsic binding constants, denoted by and for binding on the first (or third) site and on the second site, respectively. The general expression is complicated, but for q = 1 we have the following simple expressions ... 

From now on we use only the GPF for the cyclic system and drop the subscript C. Since our system has m identical units, the sites will always be identical in the weak sense. There is always one intrinsic constant for the first site but, in general, we have more than one pair correlation, triplet correlation, etc. As in Section 7.1 we develop, for simplicity, the case of two states/= 2, but most of the results are quite general. 

For three different sites, denoted by a, b, and c, the relations between the thermodynamic constants and intrinsic constants are... 

Observable constants, usually thermodynamic dissociation constants, for the association of a particular ligand with two or more sites on a larger molecular entity (e.g., a macromolecule). Macroscopic constants are composites of microscopic (i.e., intrinsic) constants. 

Assuming a correlation between surface complexation and aqueous hydrolysis exists, the trend in strengths of surfaces complexes for An in different oxidation states onto a given mineral would be in the order An4+ > AnC>2+ > An3+ > AnOj. Several authors have provided evidence for linear relations between the first hydrolysis constant of metals and the intrinsic constant associated to the formation of surface species of metals as S-OMamorphous silica (Schindler Stumm 1987), hydrous ferric oxides (Dzombak Morel 1990), aluminum (hydr-)oxides and kaolinite (Del Nero et al. 1997, 1999a). 

Parametric Studies. To provide a feel for the relative importance of some of the model parameters, and to understand observed differences in performance between REV and USV catalysts (Figure 1), key parameters such as the cracking and coking intrinsic constants kj and Aj, the heats of reaction AHj, and the order of coke deactivation, n-, were varied. The base case model parameters and the rate... 

K, values are intrinsic constants, whereas IC50 values are extrinsic constants. Theoretically, IC50 values, in contrast to K, values, are dependent on the type of substrate, the concentration of substrate, and incubation conditions (protein concentration or incubation times, etc). 

Because they are intrinsic constants, K values can theoretically be reproduced from one laboratory to another. 

There are four sites available for binding 02, but only one from which bound 02 can dissociate. Thus, from the law of mass action, the (overall) extrinsic equilibrium constant is equal to 4k /k- = 4X. Similarly, the extrinsic constant denoted by X2 is given by 1K2, where K2 is the intrinsic constant, and by the same reasoning K% = and Ke4 = JK4. 

The intrinsic constants are thermod3mamic constants written for reactions occurring at a hypothetical isolated site on the surface. Actual activities on the surface cannot be directly determined but Q or apparent stability quotients can be calculated based on measurable bulk concentrations. The intrinsic constants and apparent stability quotients are related by considering the electrostatic correction for an ion in solution near the surface compared to an isolated ion on the surface. In an idealized planar model, is the mean potential at the plane of surface charge created by the ionization of the surface functional groups and the formation of surface complexes and is the mean potential at the plane of adsorbed counter ions at a distance 3 from the surface (17). The electrostatic interaction energies at the surface and at a distance 3 are expressed as exponentials. Therefore ... 

Summary of Complex-Ionization Reactions and Intrinsic Constants... 

For Ca", Mg", and SO there is a shift in the pH(PZC) which is expected in situations of nonsymmetrical specific adsorption (17). A change in the pH(PZC) due to specific adsorption is accompanied by a shift in the pHjgp in the opposite direction (10). Under these conditions, extrapolation to zero charge to obtain K NT does not necessarily correspond with extrapolation to zero potential. Thus there may be some error involved in the intrinsic constants for Ca" , Mg" " and SOI. ... 

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