Macroscopic Equilibrium Constants

Because Kcff applies to a scheme that involves more that one equilibrium (see Eq. (1.7)), it is referred to as a macroscopic equilibrium constant, to distinguish it from the microscopic equilibrium constants KA and E, which describe the individual equilibria. 

From this we see that the relation between the concentration of A and the amount of it that is bound should follow the Hill-Langmuir equation. Ke, the macroscopic dissociation equilibrium constant, is given by ... 

In Appendix 1.6B we obtained an expression for the macroscopic dissociation equilibrium constant, Ke, for the binding of a ligand on the same scheme as in Figure 1.14. Allowing for the difference in terms, KeS and EC50 are seen to be identical. 

In talking about thermodynamics and the properties of chemical equilibrium constants it is very difficult for chemists to avoid attempts to include the influence of forces between molecules on the magnitude of the equilibrium constant and differences between observed equilibrium constants. For the purpose of this chapter, it is convenient to talk first about the equilibrium constant and the macroscopic properties of matter which affect it first. Next, the reader will be introduced to the concept of forces between molecules, their relative magnitude and influence is separations. 

As this system nears equilibrium, the rate of the forward reaction decreases and the rate of the reverse reaction increases. At equilibrium, the macroscopic properties of this system are constant. Changes at the molecular level take place at equal rates. 

The resultant different complexes with the same stoichiometry [SL(1) and SL(2)] generally can t be separated by electrophoresis because of the same size and charge. Only the macroscopic equilibrium constants and K2 can be derived from electrophoresis data. 

The macroscopic equilibrium constants describe not a specific interaction, but the sum of every possible interaction between substrate and ligand at a particular stoichiometry ... 

Table 1 Microscopic and Macroscopic Equilibrium Constants for the Curves in Figs. 2 and 3... 

Titration calorimetry has been successfully employed in the determination of thermodynamic parameters for complexation (Siimer et al., 1987 Tong et al., 1991a). The technique has the advantage of employing direct calorimetric measurements and has been proposed as the most reliable method (Szejtli, 1982). It should be noted that the information derived from multistep series reactions is macroscopic in nature. In contrast to spectrophotometric methods that provide information concerning only the equilibrium constant(s), titration calorimetry also provides information about the reaction enthalpy that is important in explaining the mechanism involved in the inclusion process. 

The same approach can be applied not only to the bulk equilibrium constants (K) but also to the microscopic connection processes (given the symbol k). Recall that the macroscopic equilibrium constant is simply the sum of all the microscopic equilibrium constants. For example, if an acid (H2A) has two non-equivalent ionisable protons there are two distinct but equivalent ways to remove a proton to produce HA- and hence there are two microscopic equilibrium constants kx and k2) for this deprotonation process. Thus the macroscopic acid dissociation constant, K = k1+k2. Don t get confused between microscopic equilibrium constants and rate constants, both of which have the symbol k. So, in terms of... 

For simplicity, let us consider perfectly drained conditions (p = 0) and start from an equilibrium between solid and solute (xjrc — xj/x). The equilibrium is disturbed by application of a constant macroscopic stress X = <5 ( > 0). 

Macroscopic solvent effects can be described by the dielectric constant of a medium, whereas the effects of polarization, induced dipoles, and specific solvation are examples of microscopic solvent effects. Carbenium ions are very strong electrophiles that interact reversibly with several components of the reaction mixture in addition to undergoing initiation, propagation, transfer, and termination. These interactions may be relatively weak as in dispersive interactions, which last less than it takes for a bond vibration (<10 14 sec), and are thus considered to involve "sticky collisions. Stronger interactions lead to long-lived intermediates and/or complex formation, often with a change of hybridization. For example, onium ions are formed with -donors. Even stable trityl ions react very rapidly with amines to form ammonium ions [41], and with water, alcohol, ethers, and esters to form oxonium ions. Onium ion formation is reversible, with the equilibrium constant depending on the nucleophile, cation, solvent, and temperature (cf., Section IV.C.3). 

It is obvious that such a definition of solvent polarity cannot be measured by an individual physical quantity such as the relative permittivity. Indeed, very often it has been found that there is no correlation between the relative permittivity (or its different functions such as l/sr, (sr — l)/(2er + 1), etc.) and the logarithms of rate or equilibrium constants of solvent-dependent chemical reactions. No single macroscopic physical parameter could possibly account for the multitude of solute/solvent interactions on the molecular-microscopic level. Until now the complexity of solute/solvent interactions has also prevented the derivation of generally applicable mathematical expressions that would allow the calculation of reaction rates or equilibrium constants of reactions carried out in solvents of different polarity. 

Now suppose we are dealing with a small but macroscopic system of, say, 1020 particles, much less than a millimole. And suppose the system is not quite at the traditional equilibrium point of Ap = 0 suppose that there is a deviation of 10 10 from the exact equality in Ap/kBT. This means that the exponent determining the equilibrium constant is 1010, so Keq = exp[ 1010]. This tells us something we already knew from the phase rule, that even at such tiny deviations from exact... 

Because the equilibrium constant Keq ranges from zero, when the system is all solid, to infinity, when it is all liquid, (strictly, one should include vaporization, neglected here) it is convenient to introduce another related function, a ratio we call D (for distribution), which contains the same information but ranges from — 1 to +1 D = (Keq — l)/(Keq +1). This allows us to portray graphically the behavior of a system in terms of the amount of each of two phases as a function of temperature. This is done in Fig. 1, for a small system (a), a mid-size system (b), and a large but not truly macroscopic system (c). However, even case (c) in this figure does not... 

Due to practical significance and theoretical interest, much effort has been made to clarify the unique characteristics of metal ion/polyelectrolyte mixture solutions in various disciplines of chemistry. Since a proper equilibrium expression for metal ion binding to polymer molecules is indispensable for the quantification of the physicochemical properties, apparent or macroscopic equilibrium constants have been determined. Unfortunately, however, these overall constants are usually defined arbitrarily, being dependent on the research groups, the experimental techniques, and the systems to be investigated hence they are not comparable with each other nor re-latable to the intrinsic equilibrium constants defined at respective reaction sites. Compared with the situation for the equilibrium analyses of metal complexation with monomer ligands, to which the law of mass action can directly be applied, complete analytical treatment of the metal ion/ polyelectrolyte complexation equilibria has not yet been established even at the present time. There are essential difficulties inherent in the analyses of metal complexation equilibria in polyelectrolyte solutions. 

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