Hard spheres solvation

Much attention has been directed since olden times towards ion solvation, which is a key concept for understanding various chemical processes with electrolyte solutions. In 1920, a theoretical equation of ion solvation energy (AG ) was first proposed by Born [1], who considered the ion as a hard sphere of a given radius (r) immersed in a continuous medium of constant permittivity (e), and then defined AG as the electrostatic energy for charging the ion up to ze (z, the charge number of the ion e, the elementary charge) ... 

A particularly simple form of Pr(V ) is obtained for the solvation Helmholtz energy of a in a solvent consisting of N hard-sphere solvent particles of diameter a in a volume V. If the density N/V is very small, so that one can neglect solvent-solvent interactions, the probabihty density Po(R ) is simply and the integral on the rhs of Eq. (9.4.7) reduces to... 

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. 

The attentive reader will realize that we have strayed rather far from the hard spheres of the Einstein theory to find applications for it. It should also be appreciated, however, that the molecules we are discussing are proteins that-through disulfide bridges and hydrogen bonding —have fairly rigid structures. Therefore the application of the theory —amended to allow for solvation and ellipticity —is justified. This would not be the case for synthetic polymers, which are best described as random coils and for which a different formalism is employed. This is the topic of Section 4.9. 

Chen, Y.-G. and Weeks, J. D., Different thermodynamic pathways to the solvation free energy of a spherical cavity in a hard sphere fluid. /. Chem. Phys. 118, 7944-7953... 

A conceptually complementary approach to describe hydrophobic effects has been introduced by Pratt and colleagues (78, 96). Their iifformation theory (IT) model is based on an application of Widom s potential distribution theorem (97) combined with the perception that the solvation free energy of a small hard sphere, which is essentially governed by the probability to find an empty sphere, can be expressed as a limit of the distribution of water molecules in a cavity of the size... 

Hard-sphere diameter Dielectric permittivity Chemical potential of substance A Energy difference, adiabatic interaction energy Dissociation energy counterpoise corrected, adiabatic interaction energy counterpoise corrected Dissociation energy, adiabatic interaction energy Solvation free energies Molar enthalpy of vaporization Dipole moment Tetrafluoroborate 1-Alkyl-3-alkyl imidazolium l-Alkyl-3-methyl imidazolium Dicyanamide... 

Many attempts have been made to improve the Born description of ion solvation. Most of these rely upon continuum descriptions of the solvent in which the permittivity varies from a low value near the ion to the bulk value farther away. This variation is described mathematically either as two or three regions with a constant permittivity in each, or as a solvent with a continuously varying permittivity over a region of a few molecular diameters thick. Macroscopic concepts such as the permittivity are not really valid at molecular dimensions. For this reason these models are not considered further here. Instead, in the next section, a model based on a discrete description of the electrolyte solution as a collection of hard spheres is discussed. 

Acetonitrile is a polar solvent with a relative permittivity of 35.9. It may be represented as a hard sphere with a diameter of 427 pm. Estimate the Gibbs energy of solvation of Na in acetonitrile according to the Born and MSA models. Compare the theoretical estimates with the experimental estimate given that the Gibbs energy of transfer for Na" " from water to acetonitrile is 15.1 kJmoP ... 

When the role of hard spheres, like those depicted in Figure 5.26, is played by the molecules of solvent, the resulting volume exclusion force is called the oscillatory solvation force, or sometimes when the solvent is water, the oscillatory hydration force. The latter should be distinguished from the monotonic hydration force, which has a different physical origin and is considered separately in Section 5.4.5.4 below. 

Basically, the SPT is an approximate procedure to compute the work required to create a cavity at a fixed position in a liquid. The work required to create a cavity in the liquid is fundamental in the study of the solvation of solutes in any solvent. The simplest solute is a hard-sphere (HS) particle, and the simplest solvent also consists of HS particles. The solvation process can always be decomposed into two parts creating a suitable cavity and then turning on the other parts of the solute-solvent interaction. 

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