Gibbs energy hard spheres

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. 

FIGURE 11.10 Gibbs free energy, G, versus osmotic pressure, II, for a suspension of hard spheres showing the intersection of the disordered and the ordered curves corresponding to the disorder-order transition with 4a-a%/3 = 0.74. Adapted from Cast et al. [63]. Reprinted with permission Academic Press. 

Accounting for size differences can also be realized in terms of distribution functions, assuming certain interaction energies. Simply because of size differences between molecules preferential adsorption will take place, i.e. fractionation occurs near a phase boundary. In theories where molecular geometries are not constrained by a lattice, this distribution function is virtually determined by the repulsive part of the interaction. An example of this kind has been provided by Chan et al. who considered binary mixtures of adhesive hard spheres in the Percus-Yevick approximation. The theory incorporates a definition of the Gibbs dividing plane in terms of distribution functions. A more formal thermodynamic description for multicomponent mixtures has been given by Schlby and Ruckenstein ). 

For an ideal monoatomic gas, the Gibbs energy at constant temperature and pressure varies with the logarithm of the molecular density. The additional terms in in equation (2.9.25) give the contribution to G due to the non-ideality of the hard-sphere system with respect to an ideal gas. 

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 ... 

The first-order result is of particular interest because it provides an exact upper bound to the free energy of the perturbed system if the properties of the unperturbed (reference) state are known, as in the case of a hard-sphere fluid. This is obtained from the Gibbs-Bogliubov relationship,177 which has the form (for a one-component system)... 

A much-discussed electrostatic treatment of hydration based on the Born equation relates the Gibbs energy of solvation AG to the ionic radius and the solvent dielectric constant [128], For the hydration process [Eq. (1)] we take hydration as a specific case of solvation and set AGh = AGs- The change in Gibbs energy may be obtained readily if the ions are treated as hard spheres in a continuous dielectric ... 

In Appendix G we derive the solvation Gibbs energy of a point hard sphere. This is the same as the work required to create a cavity of radius r < aj2 in a liquid consisting of hard spheres with diameter a and density p. A cavity of radius cr/2 can be formed by a hard solute of diameter zero (Fig. 3.20). We shall refer to such a particle as a hard point. The solvation Gibbs... 

It should be noted that the SPT is not a pure molecular theory in the following sense. A molecular theory is supposed to provide, say, the Gibbs free energy as a function of T, P, N as well as of the molecular parameters of the system. Once this function is available, the density of the system can be computed from the relation p = (9/x/9 )t (with pi = G/N). The SPT utilizes the effective diameter of the solvent molecules as the only molecular parameter (which is the case for a hard-sphere fluid) and, in addition to the specification of T and P, the solvent density Pw is also used as input in the theory. The latter being a measurable quantity carries with it implicitly any other molecular properties of the system. The first application of the SPT to calculate the thermodynamics of solvation in liquids was carried out by Pierotti (1963, 1965). 

In the following, we elaborate on two generalizations of (4.4.40), one for two real solutes and the second for any number of either real or hard-sphere solutes. Before doing that, we note that relation (4.4.40) means that the HrpO interaction at zero separation is equivalent to the reversal of the solvation process. This is one example of the exact relation between the H[Pg.463]

Appendix G The Statistical Mechanical Expression for the Solvation Gibbs Energy of Hard Spheres and the Work of Cavity Formation... 

The simplest nontrivial solvation phenomenon is the case of a hard-sphere (HS) particle in a fluid of HS particles. This case was discussed in connection with the scaled-particle theory in section 5.11. Here we note that the solvation Gibbs energy of an HS solvaton in an HS solvent is always positive, and it increases monotonically as a function of the size of the HS solvaton or, equivalently, the radius of the corresponding cavity (see section 5.11). 

The simplest solvatons are two hard spheres in water. If we are in 2lT, P,N system, the corresponding work is the Gibbs energy change... 

Thus, the work required to bring two such hard spheres to a final distance R is equal to the difference in the solvation Gibbs energy of the pair ss at R (viewed as a single molecular entity) and the solvation Gibbs energy of two single hard spheres. 

Figure 7. Reduced Gibbs free energies and entropies for the quantum hard-sphere fluid along...

Fig. 8. Gibbs free energy for the formation of a cluster of n hard-spheres ApPa = 16, after fitting the results for the free energy in the different windows to one polynomial. The inset shows the sequence (unbiased + biased) of measured Gibbs free energies AGi(n)/kgT + bi...

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