Partition free energy, definition

The last equation follows from the definition of the partition function, eq. (16.2). Analogously to eq. (16.10) the free energy difference can be evaluated as an ensemble average. 

K generally varies only by factors of three to five for a given solute (12). K typically correlates well with physico-chemical properties of the sorbate, such as aqueous solubility (S) or the octanol-water partition coefficient (K ), again suggesting that hydrophobic interaction predominates. The correlation of Koc with K has led to the definition of linear free-energy relationships (LFER) of the form... 

The partitioning of free energy contributions in the explanation (and for design, the prediction) of binding constants is a subjective matter. Different workers choose different definitions, e.g. of hydrophobic binding, which may or may not include dispersion interaction, and different approaches to factorization of enthalpic and entropic components. 

We can now utilize some of the statistical mechanics relationships derived in Chapter 8 to find expressions for the free energy and the equilibrium constant in term of the molecular partition functions. From the definition of the free energy (Eq. 9.1) the expression for the enthalpy of an ideal gas (Eq. 8.121), and recalling that Ho = Eq (for an ideal gas), we obtain... 

It is also useful to define the chemical potential in terms of the partition function. By the definition of p,k in Eq. 9.24 and the Helmholtz free-energy expression of Eq. 8.114,... 

There is a parameter used in the planar techniques that has not been defined. It was suggested by Martin in 1949 as a way to relate an analyte s structure and its free energy for chromatographic partitioning. It is called Rm and its definition is... 

For Landau free energy (m = 0, - the order parameter), Zt is a partition function of an extended system with additional variable . The integral denoted in square brackets in Eq. (3-12) is simply the configurational partition function of the system with a fixed value of -Z. The statistical definition of the Gibbs free energy function combined with Eq (12) results in the following expression ... 

To apply the preceding concepts of chemical thermodynamics to chemical reaction systems (and to understand how thermodynamic variables such as free energy vary with concentrations of species), we have to develop a formalism for the dependence of free energies and chemical potential on the number of particles in a system. We develop expressions for the change in Helmholtz and Gibbs free energies in chemical reactions based on the definition of A and G in terms of Q and Z. The quantities Q and Z are called the partition functions for the NVT and NPT systems, respectively. 

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

Ben-Naim s definition has many merits it is not limited to dilute solutions, it avoids some assumptions about the structure of the liquid, it allows to use microscopical molecular partition functions moreover, keeping M fixed in both phases is quite useful in order to implement this approach in a computationally transparent QM procedure. The liberation free energy may be discarded when examining infinite isotropic solutions, but it must be reconsidered when M is placed near a solution boundary. 

Expression (2) contains a term, GMm, not present in (3). The reason is that in dealing with free energies we also have to consider entropic contributions. A formal derivation will not be reported here in short, it consists in a formal definition of the partition function of the whole solution, in its factorization into M and 5 components, in the introduction of the continuous distribution of the solvent, and then in the use of standard formulations of statistical mechanics to get free energy (Gibbs or Helmholtz) contributions for the M portion. The formal treatment can be found e.g. in the Ben-Naim s books [14, 15], and the application to our model in ref. [8]. 

Having calculated the partition function, we can use it to obtain the thermodynamics of the system with the standard expressions we have derived in the context of ensemble theory. We first calculate the free energy F from the usual definition ... 

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