Momentum partition functions

In (1.62) we have introduced the momentum partition function, defined by... 

It is instructive to recognize the three different sources that contribute to the liberation free energy. First, the particle at a fixed position is devoid of momentum partition function (though it still has all other internal partition functions such as rotational and vibrational). Upon liberation, the particle... 

Note that in writing (7.16) and (7.17), we have assumed that A ( is the same in the two phases. Again, classically speaking, the momentum partition function depends only on the temperature and is not affected by the interaction of the solvaton with the rest of the system. Any other degrees of freedom might or might not be affected by the interactions. In most sections of this chapter, we assume that qs is unaffected by the interaction of the solvaton with the rest of the system. Hence, in this case, (7.18) reduces to... 

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. 

The mass is a molecular parameter that enters into the momentum partition function, but this parameter does not enter in the calculation of the work required to create a cavity. 

Here qrot,s and qvib,s are the microscopic partition functions for rotation and vibration of M in the solution while Hm.s and Am, indicate the related numeral densities and the momentum partition functions, respectively. The functional (G ) has now the meaning of the free energy of the entire solute-solvent system, at the temperature T, with respect to a reference state given by non-interaction nuclei and electron, supplemented by the imperturbed pure liquid S, at the same temperature. Then, the fundamental energetic quantity connected with the insertion of the solute in the solvent, i.e. the free energy of solvation can be obtained as... 

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... 

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... 

Here, q is the internal partition function of a single molecule, V the volume of the system, N the number of particles, and the momentum partition function. 

If we analyze carefully the various factors involved in the expression of the chemical potential of the d and / molecules, we find that the momentum partition function as well as any other internal partition function of each molecule, as well as the volume accessible to each molecule, are unchanged during the entire process. The only change that does take place is in the assimilation terms of d and / hence the entropy change in (2.3.12) is due to the deassimilation of 2N identical molecules into two subgroups of distinguishable molecules, N of one kind and of a second kind. This process, along with an alternative route for its realization, is described in Fig. 2.4. 

Here is the internal partition function of S (=1 for a strnctureless particle, such as a noble gas atom), is its number density, and Ag is its momentum partition function. The qnantity p in either phase pertains to the particle at a fixed position where it is devoid of the translational degrees of freedom, and kj In pJl expresses its liberation from this constraint. The Gibbs energy of solvation, is expressed as the change in the chemical potential of S at equilibrium, where is then ... 

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