Pseudo chemical 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 pseudo-chemical potential can be expressed, similarly to (3.53), as a ratio between two partition functions corresponding to the difference in the Helmholtz free energies in (3.88), i.e., 

It is instructive to observe the differences between (3.53) and (3.89). Since the added particle in (3.89) is devoid of the translational degree of freedom, it will not bear a momentum partition function. Hence, we have instead of as in (3.53). For the same reason, the integration in the 

Once we have set up the statistical mechanical expression (3.89), the following formal steps are nearly the same as in the previous section. Relation (3.89) can be rewritten, using the notation of Section 3.5, as 

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Note that the small approximation indicated in Eq. (5.30) becomes exact in the thermodynamic limit of large ensembles. Re-introducing the pseudo-chemical potential as pi — kTIn -,... 

Insertion of a particle at a fixed position the pseudo-chemical potential... 

We recall that the pseudo-chemical potential was defined as the Gibbs energy change for the process of inserting s at a fixed position. Hence, the temperature derivative gives the entropy change for the same process, i.e.,... 

Consider a solute s 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 pseudo-chemical potential of a molecule having a fixed conformation Ps as the change in the Helmholtz energy for the process of introducing s into the... 

Note that the rotational partition function of the entire molecule, as well as the internal partition functions of s, are included in the pseudo-chemical potential. In classical systems, the momentum partition function is independent of the environment, whether it is a gas or a liquid phase. 

Extracting the liberation Helmholtz energy (7.199) from (7.197), we can identify the pseudo-chemical potential of the pair A and B, i.e.,... 

In deriving equation (7.200) for the pseudo-chemical potential p D, we have selected one specific distance between the two particles A and B. This was rendered possible by using (7.194). There is an analogy between the procedure used in section 7.8, figure 7.8, and the procedure that we took in this case. 

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 b = 2r— a at Po> we can write the pseudo-chemical potential of the added solute in the solvent as... 

Likewise, we define the pseudo-chemical potential of water in an ideal gas phase as... 

The pseudo molar volume is the pressure derivative of the pseudo-chemical potential (Ben-Naim, 2006). 

Note that the chemical potential of each species is equal to pLw. However, the pseudo-chemical potentials are different for each species. Since is the same for all species we can rewrite (2.7.64) as... 

This relation is the same as the relationships between the pseudo-chemical potentials of different isomers in chemical equilibrium. The interpretation of (2.7.67) is quite simple. In view of (2.7.63), we can translate (2.7.67) into... 

Thus, the general expression for the pseudo-chemical potential of the water molecule is... 

Fig. 3.11 The chemical potential and the pseudo-chemical potential, p-l is the change in Gibbs energy when adding one particle to the system. is the change in the Gibbs energy when adding one solute s at a fixed point.

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 analog of the pseudo-chemical potential in this system is... 

It is instructive to calculate the canonical partition function as above, but where a solute s is first placed at a specific site in the system. The pseudo-chemical potential is given by... 

The KB/FST inversion procedure is the process of obtaining expressions for the particle number fluctuations or KBIs in terms of experimentally available (isothermal-isobaric) data. Again, there are multiple approaches to the inversion procedure (Ben-Naim 1977 O Connell 1994 Smith 2008). Arguably, the simplest approach involves the pseudo chemical potential and partial molar volumes (Ben-Naim 2006). First, we note that combining Equations 1.46 and 1.47 provides... 

Here, is the work of introducing the particle to the system. The term /Z is the corresponding pseudo-chemical potential and A y is its momentum partition function. The latter two quantities differ very slightly from the corresponding (unprimed) quantities in (3.92). The essential major... 

We now turn to another useful relation, between the quantity AAcb,y(Rq, hard-sphere solute. 

In Section 3.6, we obtained the pseudo-chemical potential for a one-component system. We repeat the same process here, but instead of adding the (N + l)th particle, we add a hard-sphere particle of diameter cths to a fixed position Rq in a system of particles having an effective hard-core diameter cr . The work associated with this process, keeping T, F, N constant, is given by... 

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