Relationship to Equilibrium Constant

In the simplest case, the protein-ligand interaction can be represented as, or modeled as, a reversible bimolecular reaction such as depicted by P-i-l.ol L. The change in Gibbs free energy (AG) for the interaction in the direction indicated is related to the standard free energy change (AG ) by the following equation  

The ratio of complex concentration to the reactant concentrations can be represented by the equilibrium constant Kgq, the reciprocal of fhe equilibrium constant (e.g.. Km, Kj, or KJ, or by some altemative designation in ofher types of studies. Eor fhe example of Kj, substitution into Eq. 3.12 yields 

for the conditions under which most protein-hgand interactions are studied, Eq. 3.13 describes fhe relationship between fhe fhermodynamic parameter AG° and a reaction characteristic (fhe equilibrium constant) fhat can be measured experimentally. Because fhe change in Gibbs free energy is related to the change in enfhalpy and entropy by AG°=AH° - TAS°, Eq. 3.13 can be rearranged to 

Not all such plots turn out to be linear, indicating fhat in those cases the heat capacity change (ACp) is not independent of temperature for the interaction under study. It has also been suggested that AH° values determined using fhe van t Hoff plot mefhod can differ from fhe same values determined using direct calorimetric measurement [30]. However, it has subsequently been reported that discrepancies are relatively minor [31]. 

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Because it applies mostly to electrolytes, it is discussed in Chapter 15. Briefly, Helgeson models the behavior of solutes by developing equations for the standard state partial molar volume (Helgeson and Kirkham 1976) and standard state partial molar heat capacity (Helgeson et al. 1981) as a function of P and T, with adjustable constants such that they can be applied to a wide variety of solutes. If you know these quantities (V°, C°p), you can calculate the variation of the standard state Gibbs energy, and that leads through fundamental relationships to equilibrium constants, enthalpies, and entropies. 

Relationship of change in free energy to equilibrium constants and electrode potentials... 

A slightly different aspect of the same problem makes it exceptionally difficult to estimate certain types of equilibrium constant at high temperatures from data at low temperatures. The well known relationships between equilibrium constant, K, AG, AH and AS may conveniently be written... 

As we have seen, the Hammett equation can, in principle, be applied to both equilibria and rate data. This implies that in certain cases, it is feasible to relate rate constants to equilibrium constants when both reflect the effects of a given structural moiety. In a general form a rate-equilibrium relationship can be written in terms of the corresponding changes in free energies of activation and of equilibration ... 

Equations 2.11 and 2.12 express a linear relationship of free energies known as the Hammett a-p rejationshjp. or simply as the Hammett equation. It can be applied to reaction rales of substituted aromatic compounds as well as to equilibrium constants, and we shall find that it is a very useful tool for obtaining information about reaction mechanisms. (See, for example, Problem 2.1.)... 

One is generally interested in the enthalpies of scrambling reactions since these may be measured thermochemically and thus brought into relationship with equilibrium constants. Real scrambling equilibria, however, deviate more or less from the ideal random case. And, in order to be able to calculate enthalpies, it is assumed that for large values of the enthalpy, (JS)real (JS)rand. Accordingly, we may now estimate the enthalpy of a real scrambling reaction. 

Sverjensky (2003) proposed new standard states, leading to equilibrium constants independent of the surface area, site density, and the amount of the solid sorbent. These new standard states are dependent only on surface site occupancy and can be used with any surface complexation model. Different standard states are defined for the activities of the sorbent sites and the sorbate species. The theoretical relationships that apply for all adsorption reactions are developed below using Eq. (6.6) as an example reaction. 

Homologous changes in small molecules are generally used to probe electron demand at a reaction centre. Classic Bronsted relationships are particularly simple, since they correlate rates and equilibria for a single reaction (aqueous Kn values being necessarily proportional to equilibrium constants for proton... 

The fundamental relationship between equilibrium constant and composition is via the product of the activities of species raised to the corresponding stoichiometric coefficients ... 

The maximum solubility of the solvent is limited by the maximum rate of the reaction at equilibrium. According to the Van t-Hoff relationships, the equilibrium constant K i depends on pressure and temperature. Generally, chemisorption processes are more effective at lower temperature and higher pressure. 

AH is the actual heat of reaction, and can be measured calorimetri-cally. AS is a less tangible quantity, and is evaluated indirectly. Therefore other relationships are useful in evaluating AF. In chemical reactions AF may be related to equilibrium constants by the equation... 

A calculation of this kind requires large amounts of input data, particularly temperature relationships of equilibrium constants and heat capacities. Care must be taken in selecting the H and S values, since a case might occur in which no temperature would exist at which the enthalpy of the equilibrium system would be equal to H or, the entropy to S. It is likewise possible that though such a temperature exists, it lies outside the range of validity of the temperature relationship of equilibrium constants and heat capacities, which fact would then be the source of a considerable error. 

In this chapter, we aim to deepen our understanding of chemical equilibrium. We will begin with a brief discussion of the nature of the equilibrium state and then focus on some key relationships involving equilibrium constants. Then we will make qualitative predictions about the condition of equilibrium finally, we will perform various equilibrium calculations. As we will discover throughout the remainder of the text, the equilibrium condition plays a role in numerous natural phenomena and affects the methods used to produce many important industrial chemicals. 

One can write acid-base equilibrium constants for the species in the inner compact layer and ion pair association constants for the outer compact layer. In these constants, the concentration or activity of an ion is related to that in the bulk by a term e p(-erp/kT), where yp is the potential appropriate to the layer [25]. The charge density in both layers is given by the algebraic sum of the ions present per unit area, which is related to the number of ions removed from solution by, for example, a pH titration. If the capacity of the layers can be estimated, one has a relationship between the charge density and potential and thence to the experimentally measurable zeta potential [26]. 

The equilibrium constant at constant temperature is directly related to the maximum energy, called the free energy AG. which is obtainable from a reaction, the relationship being... 

This corresponds to a distribution of 66% anti and 34% gauche. Table 3.2 gives the relationship between free-energy difference, equilibrium constant, and percent composition of a two-component mixture. 

Classical thermodynamics gives an expression that relates the equilibrium constant (the distribution coefficient (K)) to the change in free energy of a solute when transferring from one phase to the other. The derivation of this relationship is fairly straightforward, but will not be given here, as it is well explained in virtually all books on classical physical chemistry [1,2]. 

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