Gibbs solvation energies

First, electric work wx and w2 is calculated for a single ion species denoted as k. For wx the same procedure as for the quantity w2 in the Born treatment of solvation Gibbs energy (Eq. (1.2.5) will be used giving (e = De0))... 

The difference between the electronic energies of the final and initial states must include the energy of ionization of the ion B(z-1)+ in vacuo (where its ionization potential is complemented by the entropy term TA5/), the interaction energy of the ions Bz+ and B(z-1)+ with the surroundings, i.e. the solvation Gibbs energies, and finally the energy of an electron at the Fermi level in the electrode. These quantities can be expressed most simply... 

The essential simplification of the model is to replace the alkyl substituents, from mediyl to tert-butyl, by Lennard-Jones spheres of increasing diameters. In this model we cannot calculate the exact values of k and 2 (in fact, these cannot be calculated even if we had the true rotational potential the main missing information is the solvation Gibbs energies of the molecules involved). Nevertheless, we can demonstrate with this simplified model the two major experimental findings regarding the proton-proton correlation in these series of molecules, as shown in Table 4.6. 

Recall that includes the solvation Gibbs energy of the proton, which is not known. Actually, is not needed to calculate the correlation function g j,. However, to compute kj and X2 one needs a value for Xq. Therefore, Xq together with D. and Dj. were fitted to obtain the values of Xj and hence of gj j of succinic acid. Once... 

The specific processes discussed above are all special cases of the general process (9.2.1). In all of these cases we have seen the explicit modification of the equilibrium constant of the corresponding process. As indicated in Eq. (9.2.3), the general modification requires knowledge of the solvation Gibbs energies of all the components involved in the process. For macromolecules such as proteins or nucleic acid, none of these is known, however. Nevertheless, some specific solvation effects are examined in Sections 9.4 and 9.5. 

We conclude this section by presenting the general statistical mechanical expression for the solvation Gibbs energy of any solute ot, ... 

In Section 9.4 we shall decompose the solvation Gibbs energy of a macromolecule a into various components or ingredients, which will allow us to examine and, in principle, estimate some specific contributions of the solvation to cooperativity. 

Before we examine some specific solvation effects on cooperativity we must first consider various aspects of the solvation Gibbs energy of a macromolecule a. We present here one possible decomposition of AG which will be useful for our purposes. Consider a globular protein a which, for simplicity, is assumed to be compactly packed so that there are no solvent molecules within some spherical region to which we refer as the hard core of the protein. The interaction energy between a and the fth solvent molecule (the solvent is presmned to be water, w) is written as... 

This expanded form of AG may be derived exactly from the definition (9.2.20) and from the specific form of the pair potential (9.4.1). We shall not derive this expression here. Instead, we present a qualitative description of the various terms on the rhs of Eq. (9.4.2) that must sum to the total solvation Gibbs energy AG. ... 

Next, we solvate all the functional groups (equivalently, we turn on the interactions U(k, X,) that were turned off in the first step]. One way to do this is to solvate all the functional groups simultaneously. The resulting free-energy change would be AG and the total solvation Gibbs energy would have been written... 

This is not the case for groups 2 and 3, for which the order of solvation is important, i.e., we obtain different conditional solvation Gibbs energies when we first solvate 2 and then 3, or first 3 and then 2. The corresponding solvation free energies are... 

In this section we restrict ourselves to solvent effects that are due to the first term in the expansion of AG in Eq. (9.4.2). This is equivalent to the assumption that all the particles involved are hard particles, hence only their sizes affect the solvation Gibbs energies. We shall also assume for simplicity that the solvent molecules are hard spheres with diameter a. All other molecules may have any other geometrical shape. 

In the following model example, we assume that each species involved in the binding process has a spherical shape and that the FGs on its surface are distributed in such a way that each pair of FGs on the surface (i.e., exposed to the solvent) is independently solvated. In other words, the conditional solvation Gibbs energy of the ith FG (given the hard core H) is independent of the presence or absence of any other FGs. Formally, this is equivalent to taking only the first sum over i in the expansion on the rhs of Eq. (9.4.2). 

The quantity AG is the solvation Gibbs energy of the hard part of the interaction and has been dealt with in the previous section. The second expression is the conditional solvation Gibbs energy of the kth FG given that the hard part of the interaction has already been solvated. The conditional probability density is... 

For this particular solvent we can now combine the solvation Gibbs energies for the process (9.6.1) to obtain... 

Figure 7.1 Schematic diagram showing the relation between standard free energies of reaction in two phases (ideal gas and a liquid), and the corresponding solvation Gibbs energies of all the species involved in the reaction.

Gibbs energy in the liquid phase, we have to transfer each of the species from the ideal-gas phase into the liquid. This transfer involves the solvation Gibbs energies of all the species participating in the reaction. 

The second situation, where the solvation Gibbs energy features naturally, is in the distribution of a solute s between two phases a and ft. Let a solute s be distributed between two phases a and / . At equilibrium, we have... 

Thus the distribution of s between the two phases a and [I is determined by the difference in the solvation Gibbs energies of s in the two phases. As we shall see in the next sections, relation (7.14) is actually used to measure the solvation Gibbs energy when one of the phases is chosen to be an ideal-gas phase. Thus, when a, is an ideal-gas phase then... 

---