Chemical potentials electrostatic contributions

The electrochemical potential of charged species i, jl is defined as the work done when this species is moved from charge-free infinity to the interior of a homogeneous phase a which carries no net charge [6]. The opposite process is known as the work function and is familiar for the case of removal of an electron from a metal. In fact, the work function for single ions in electrolyte solutions can also be measured experimentally, as described in detail in chapter 8. This means that the electrochemical potential is an experimentally determinable quantity. However, separation of the electrochemical potential into chemical and electrostatic contributions is arbitrary, even though it is conceptually very useful. 

Thermodynamic information can also be obtained from simulations. Currently we are measuring the differences in chemical potential of various small molecules in dimethylimidazolium chloride. This involves gradually transforming one molecule into another and is a computationally intensive process. One preliminary result is that the difference in chemical potential of propane and dimethyl ether is about 17.5 kj/mol. These molecules are similar in size, but differ in their polarity. Not surprisingly, the polar ether is stabilized relative to the non-polar propane in the presence of the ionic liquid. One can also investigate the local arrangement of the ions around the solute and the contribution of different parts of the interaction to the energy. Thus, while both molecules have a favorable Lennard-Jones interaction with the cation, the main electrostatic interaction is that between the chloride ion and the ether molecule. 

The ions in solution are subject to two types of forces those of interaction with the solvent (solvation) and those of electrostatic interaction with other ions. The interionic forces decrease as the solution is made more dilute and the mean distance between the ions increases in highly dilute solutions their contribution is small. However, solvation occurs even in highly dilute solutions, since each ion is always surrounded by solvent molecules. This implies that the solvation energy, which to a first approximation is independent of concentration, is included in the standard chemical potential and has no influence on the activity. 

Here G is the Gibbs free energy of the system without external electrostatic potential, and qis refers to the energy contribution coming from the interaction of an apphed constant electrostatic potential s (which will be specified later) with the charge qt of the species. The first term on the right-hand side of (5.1) is the usual chemical potential /r,(T, Ci), which, for an ideal solution, is given by... 

The final part is devoted to a survey of molecular properties of special interest to the medicinal chemist. The Theory of Atoms in Molecules by R. F.W. Bader et al., presented in Chapter 7, enables the quantitative use of chemical concepts, for example those of the functional group in organic chemistry or molecular similarity in medicinal chemistry, for prediction and understanding of chemical processes. This contribution also discusses possible applications of the theory to QSAR. Another important property that can be derived by use of QC calculations is the molecular electrostatic potential. J.S. Murray and P. Politzer describe the use of this property for description of noncovalent interactions between ligand and receptor, and the design of new compounds with specific features (Chapter 8). In Chapter 9, H.D. and M. Holtje describe the use of QC methods to parameterize force-field parameters, and applications to a pharmacophore search of enzyme inhibitors. The authors also show the use of QC methods for investigation of charge-transfer complexes. 

It is however possible to obtain a physically meaningful representation of 0(r) for cations, in the context of density functional theory. The basic expression here is the fundamental stationary principle of DFT, which relates the electronic chemical potential ju, with the electrostatic potential and the functional derivatives of the kinetic and exchange-correlation contributions [20] ... 

Whereas the charging approach could be applied to any geometry but only at constant surface charge or potential, the Langmuir expression could be employed for any surface conditions but only for parallel planar plates. The addition of electrostatic, entropic, and chemical contributions would allow the calculation of the free energy of interaction for systems of any shape and any surface conditions, if one could derive a general expression for the chemical free energy contribution. 

This free energy change (AUe) represents the electrostatic contribution to the chemical potential of the ion, that is, the electrical woik necessary to charge the ideal solution, and it is responsible for deviations of the solution from ideal behavior. The activity coefficients of single ions are not measurable experimentally [35] for an electrolyte EpHq, the medium activity coefficient is... 

The genius of Debye and Hiiekel lay in their formulation of a very simple but powerful model for the time-averaged distribution of ions in very dilute solutions of electrolytes. From this distribution they were able to obtain the electrostatic potential contributed by the surrounding ions to the total electrostatic potential at the reference ion and hence the chemical-potential change arising from ion-ion interactions [Eq.(3.3)]. Attention will now be focused on their approach. 

Figure 3.2 Variation of excess chemical potential of CHjF as a function of distance of the carbon atom from the liquid water-hexane interface at 310K (Pohorille and Wilson, 1996 Pratt and Pohorille, 2002). The hydrophobic contribution, obtained by eliminating electrostatic interactions, is the dot-dash curve and the electrostatic contribution is the dashed curve, lowest on the right. The water equimolar surface is at z = 0. The combination of these two contributions leads to interfacial activity for this simple solute.

Figure 5.2 Assessment of electrostatic contributions to the excess chemical potential of water, following Eq. (5.15) redrawn from Hummer et at. (1995). The temperature is T = 298 K and the density is p = 0.03333 A The SPC model of water was used and the reference system interactions are those interactions with all partial charges given the value zero.

Similarly to the RPM case the pressure, the chemical potentials and free energy contain three different contributions the hard-sphere contributions (HS), the contributions from the mass action law (MAL) and electrostatic contribution (EL). 

Clearly, Eq. (13) concerns the electrostatic interactions only, so that a suitably chosen hard-core contribution, e.g. of Camahan-Starling type [25] must be added to the free energy densities. Differentiation with respect to the densities of the species finally yields the chemical potential and the activity coefficients required for evaluating the mass action law determining the concentrations of free ions and ion pairs. 

In section 3 the contribution to the surface chemical potentials arising from the field of the potential drop Aif was derived from the combination of classical thermodynamics with the electrostatic theory of dielectrics. This contribution may be alternatively calculated as follows. Suppose that in general an adsorbed layer with M lattice sites, area A and thickness 1 consists of N neutral molecules of the i-th species, i 1, 2,. .., N. If... 

Discussion of non-equilibrium processes involving ions in terms of the micropotential is especially helpful because it focuses attention on the fact that major source of non-ideality in these systems is electrical in character. The arbitrary nature of the separation of the electrochemical potential into chemical and electrical contributions has often been pointed out in the literature. In fact, chemical interactions are fundamentally electrical in nature. However, the formal separation discussed here is conceptually important. Its usefulness becomes clear when one tackles problems related to the movement of ions in electrolyte solutions under the influence of concentration and electrostatic potential gradients. These problems are discussed in the following section. 

The increase in the HOMO-LUMO gap due to pure ionic bonding is easily calculated. For NaCl, at the equilibrium distance of 2.36 A for the diatomic molecule, the electrostatic contribution to 77 is 6.1 eV. The observed value of 4.8 eV reflects some of the same factors that decrease the gap for covalent bonding. It is interesting to note that ionic bonding should not change the electronic chemical potential, fi. The HOMO is lowered by 6.1 eV, and the LUMO is raised by 6.1 eV, in the case of NaCl. Therefore the midpoint is unchanged. 

The electrochemical way considers the electrochemical potential of electrons as the sum of their chemical potential in the solid, Pe> and the contribution due to the electrostatic (inner) potential ... 

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