Chemical potential scaled particle theory

We take advantage of this presentation to make mention of the Scaled Particle Theory (SPT) [89,90] that depicts the mechanism of a test particle (or solute particle) insertion. Among the particle of diameter d, there is also a spherical cavity of radius Xd/2 that contains no particle centers. The local density of particles centers in contact with the surface of the cavity is denoted by pG(A, p), where p is the bulk density. G(X, p), whose precise form is not known, is the central function of the SPT. This function is related to the excess chemical potential by... 

Figure B3.3.8. Insertion probability for hard spheres of various diameters (indicated on the right) in the hard sphere fluid, as a function of packing fraction rj, predicted using scaled particle theory. The dashed line is a guide to the lowest acceptable value for chemical potential estimation by the simple Widom method. 

The scaled particle theory SPT) was developed mainly for the study of hard-sphere liquids. It is not an adequate theory for the study of aqueous solutions. Nevertheless, it has been extensively applied for aqueous solutions of simple solutes. The scaled particle theory (SPT) provides a prescription for calculating the work of creating a cavity in liquids. We will not describe the SPT in detail only the essential result relevant to our problem will be quoted. Let aw and as be the effective diameters of the solvent and the solute molecules, respectively. A suitable cavity for accommodating such a solute must have a radius of c ws = ((Tw + cTs) (Fig. 3.20b). The work required to create a cavity of radius a s at a fixed position in the liquid is the same as the pseudo-chemical potential of a hard sphere of radius as. The SPT provides the following approximation for the pseudochemical potential ... 

An expression for the work of insertion W can be obtained from scaled particle theory (SPT) [31]. SPT was developed to derive expressions for the chemical potential and pressure of hard sphere fluids by relating them to the reversible work needed to insert an additional particle in the system. This work W is calculated is by expanding (scaling) the size of the sphere to be inserted from zero to its final size the size of the scaled particle is Act, with X running from 0 to 1. In the limit 2 0, the inserted sphere approaches a point particle. In this limiting case it is very unlikely that the depletion layers overlap. The free volume fraction in this limit can therefore be written as... 

In the treatment of solubility and solvent effects, the chemical potential of introducing a molecule into a fixed position in a solvent may be broken down into a cavity potential term and an interaction potential term. Scaled particle theory can be used to calculate the cavity potential term. The interaction potential between solute and solvent can be calculated separately by some other means. ... 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

The goals of microscopic theories of chemical rate processes are to define the range of validity of the macroscopic law, to provide microscopic expressions for the rate coefficients and to extend the kinetic description to smaller distance and time scales. Traditional approaches model the isomerization reaction by motion of a particle of mass m across a potential barrier in a thermal bath of temperature T. Transition state theory (TST) [2] assumes an equilibrium distribution at the barrier top the rate is given by the equilibrium one-way flux across the barrier. The rate thus obtained has the form ... 

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