The Pseudochemical Potential

The chemical potential is the work (here, at T, V constant) associated with the addition of one particle to a macroscopically large system  

The statistical-mechanical expression for the pseudochemical potential can be expressed, similarly to (5.9.10), as a ratio between two partition functions corresponding to the difference in the Helmholtz energies in (5.9.34), i.e.. 

It is instructive to observe the differences between (5.9.10) and (5.9.35). Since the added particle in (5.9.35) is devoid of the translational degree of freedom, it will not bear a momentum partition function. Hence, we have instead of as in (5.9.10). For 

Once we have set up the statistical-mechanical expression (5.9.35), the following formal steps are nearly the same as in the previous section. The result is 

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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 ... 

The general expressions for the chemical potential and the pseudochemical potential in the T, F, N ensemble have been developed in Sections 3.5 and 3.6. The analogous relations in the T, P, N ensemble are derived in Appendix 9-F. Here, we summarize the basic quantities that will be required. The chemical potential and the pseudo-chemical potential of a component A are defined in the T, P, Nj, ensemble by... 

In section 2.2 we introduced the concept of the pseudochemical potential of an ideal gas. A similar concept is defined here for the adsorbed molecules. The pseudochemical... 

The latter quantity for this particular model (no internal degrees of freedom assigned to the ligand) is equal to the pseudochemical potential. 

Since our particles are structureless, in the sense that they do not have any internal partition function, it is sufficient to study the pseudochemical potential of the species of which the solvation thermodynamics is under study. 

We have already obtained the pseudochemical potentials of a species in a mixture of hard rods, in section 4.5.3. We generalize here the expression for mixtures of interacting particles. 

Cavity Formation and the Pseudochemical Potential of a Hard Sphere... 

We derive here a useful relation between the work required to form a cavity and the pseudochemical potential of a hard-sphere solute. We shall still discuss only a system of spherical particles in the T, F, N ensemble. The results are also valid for more general systems and in other ensembles. 

In section 5.9.3, we obtained the pseudochemical potential for a one-component system. We repeat the same process here, but instead of adding the (A + l)th particle, we add a hard-sphere particle of diameter cjhs to a fixed position Ro in a system of particles having an effective hard-core diameter da. The work associated with this process, keeping T, V, N constant, is given by... 

Having the pseudochemical potential of a hard sphere in any liquid we can always add the liberation Helmholtz energy to obtain the chemical potential of the hard sphere in the same system. The same cannot be done to the quantity A cav Therefore it is meaningful to speak about the chemical potential of hard spheres in a liquid, but there is no meaning to the analogous concept of the chemical potential of cavities in a liquid. 

Since the work required to create a cavity of radius r is the same as the work required to insert a hard sphere of diameter 6 = 2r — a at Ro, we can write the pseudochemical-potential of the added solute in the solvent as... 

From now on we shall assume that qs is the same PF of a single molecule s, whether in an ideal gas or in the liquid. Our focus in the study of solvation will be on the coupling work W s l). We have seen in Chapters 5 and 6 that this quantity conveys the effect of molecular interactions between the particles on the chemical potential. In the limit of very low densities, or when interactions are negligible, this term vanishes and the chemical potential (6.13.1) reduces to that of a molecule of species s in an ideal gas. In section 5.9.3 we presented a convenient interpretation of the pseudochemical potential p7 as the work involved in placing 5 at a fixed location in the liquid. Within the classical treatment of our systems the process of fixing the location of a specific particle is meaningful. We shall adopt this interpretation of p7 throughout the book. 

Let 5 be a molecule with internal rotational degrees of freedom. We assume that the vibrational, electronic, and nuclear partition functions are separable and independent of the configuration of the molecules in the system. We define the pseudochemical potential (PCP) of a molecule with a fixed conformation as the change in the Helmholtz energy for the process of introducing s into the system / (at fixed T, K), in such a way that its center of mass is at a fixed position R. If we release the constraint on the fixed position of the center of mass, we can define the chemical potential of the conformer in the gas and liquid phases as follows ... 

If we require that F = , then the densities p = NjV in Eq. (E.2) and p = N/ < F> in Eq. (E.4) will also be the same. Hence, the liberation Helmholtz energy and the liberation Gibbs energy are equal, and therefore the pseudochemical potentials in these two systems are also equal ... 

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