In the T, P, N, ensemble

In Section 9.2 we have defined the Gibbs energy of solvation AGo in the T, P, N ensemble. In the T, V, N (canonical) ensemble the appropriate quantity is A4a, the Helmholtz energy of solvation. It can be shown that the two are equal for macroscopic systems, provided the volume V in the T, V, N ensemble is equal to the average volume of a system in the T, P, N ensemble. 

Similar relationships hold for the enthalpy in the T, P, N ensemble. Thus, using (1.22), we obtain... 

In the T, P, N ensemble there exists fluctuations in the volume of the system, defined by... 

This should be compared with equation (1.49). Thus, the relative fluctuations in the volume in the T, P, N ensemble have the same values as the relative fluctuations in the number of particles in the T, V, /i ensemble, provided that (V) in the former is equal to V in the latter. 

In this section, we used the T, V, N ensemble to obtain relation (2.127). A similar relation can be obtained for any other ensemble. Of particular importance is the analog of (2.127) in the T, P, N ensemble. It has the same form but the events occur in a T, P, N system and instead of the Helmholtz energy change, we need to use the Gibbs energy change. 

Again we note that Vin (3.150) is the average volume in the T, P, N ensemble. Since we already have an expression for (V), we need to express only its derivative with respect to temperature. Using the definition of see... 

In section 3.6, we encountered the following example of a functional. The average volume of a system in the T, P, N ensemble is written as... 

A similar average can be defined also in the T, P, N ensemble (see Sec. 1.7.7). Note also that [ijr) is the average number of HBs that a specific molecule, say 1, forms. Since all molecules are equivalent, it does not matter which one we choose. 

Before we apply relation (4.4.6) to real molecules, we briefly discuss a similar cycle in the T,P,N ensemble. Consider the process of bringing two solute particles from fixed positions at infinite separation to some final separation R, the process being carried out at constant T, P, N. The total work is the change in the Gibbs energy ... 

In this section, we restrict ourselves to processes at T, V constant. The analogous treatment in the T, P, N ensemble will be discussed in Appendix 9-F. Also, for simplicity, we assume that the particles are spherical hence, only the location of the added particle is specified in (3.88). For more complex particles, one can also specify the orientation of the added molecule. 

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... 

The second relation (5.46) is obvious and follows from the fact that the volumes of the VP of all particles add up to give the total volume of the system. [In the T, F, N ensemble, (5.46) gives the exact volume V of the system. On the other hand, in the T, P, N ensemble, V stands for the average volume of the system.]... 

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