Partition function isothermal-isobaric ensemble

Isothermal-isobaric ensemble IIE (each system has constant P, T, and N the walls between systems are flexible and diathermal each system keeps its number of particles, pressure, and temperature, but can trade both volume and energy with neighboring systems). The relevant partition function is the isothermal-isobaric partition function A (P,T,N) ... 

It is mandatory to consider the underlying statistical mechanical formalism in order to systematize the preceding ideas. For this purpose we will examine the partition function for an N-molecule water cluster. The isothermal-isobaric ensemble is appropriate (5) since constant normal stress (denoted below by "pressure" p) seems relevant to most cases of interest. The partition function for this ensemble, Ayv(j3,p),... 

Derive an expression for the partition function A(M, P,T)oi the isothermal-isobaric ensemble. 

Prove that the Gibbs free energy, G N, P, T), is given in terms of the partition function for the isothermal-isobaric ensemble by... 

G[T P] = G = —RTIn A, where A is the isothermal-isobaric partition function and U[T,n ] = —RTInE, where S is the grand canonical ensemble partition function. When a system involves several species, but only one can pass through a membrane to a reservoir, L/(7 jux] = — PTlnT, where T is the semigrand ensemble partition function. The last chapter of the book is on semigrand partition functions. 

There are four main ensembles in statistical thermodynamics for which the independent variables are NVE (microcanonical), NVT (canonical), NpT (Gibbs or isothermal isobaric), and VT (grand canonical). The characteristic fnnetions provided in Equations 1.2 and 1.3 can be expressed in terms of a series of partition functions such that (Hill 1956)... 

We will ignore in this section the contributions to the partition function due to hydrogen bonding. In the isothermal-isobaric statistical ensemble, the physical term, Qp, of the partition function can then be written as... 

In PI work on homogeneous and isotropic fluids one utilizes the three basic statistical ensembles canonical, isothermal-isobaric, and grand canonical. In the canonical ensemble the partition function is given by the trace (Tr) of the density matrix operator... 

In view of the formal identity of the expressions for the Gibbs entropy of quantum and classical isobaric-isothermal ensembles, all the previously made formal connections between the thermodynamics and the statistical mechanics of closed, isothermal classical systems with variable volttme and fixed pressure also apply to closed, isothermal qrrantum systems with variable vol-mne and fixed pressttre. [See Eqs. (86)-(91b).] One need only replace the classical ensemble averages by qrrantrrm ensemble averages and reinterpret the classical isobaric-isothermal partition function as a qrrantum isobaric-isothermal partition function. 

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