Gibbs energy from partition function

While a random distribution of atoms is assumed in the regular solution case, a random distribution of pairs of atoms is assumed in the quasi-chemical approximation. It is not possible to obtain analytical equations for the Gibbs energy from the partition function without making approximations. We will not go into detail, but only give and analyze the resulting analytical expressions. 

For thermochemical uses, an expression for the integral Gibbs energy of formation of the compound ABC can be derived by integration of eq. (9.104), but in order to show clearly some of the main implications of the model a more detailed analysis starting from the partition function is preferred [23]. 

It has already been shown in Chapter 4 (section 4.2.1) that from the thermodynamic point of view the process described by Eq. (16.2) can be modeled by the sum of its partial processes (extraction steps), irrespective of whether they really proceed or not. That is because Gibbs free energy is the function of state and its total change does not depend on the reaction path. According to the complex formation-partition model [76], one can distinguish two main steps in extraction of metal ions ... 

When a system at specified T and P contains two species and one of them is in equilibrium with that species in a reservoir through a semipermeable membrane, the transformed Gibbs energy of the system is calculated from the semigrand partition function F(7( P, /q, N2) by use of... 

The thermodynamic parameters in Table II can be used to calculate the thermodynamic quantities necessary to define the partition function. Essentially, the strategy to calculate the folding/unfolding partition function from structural data involves the identification and enumeration of states and the assignment of Gibbs free energies to those states. Once this task is accomplished, the evaluation of the partition function is straightforward, as illustrated in Fig. 4. 

From the partition function (3.5), we can now find the Helmholtz free energy, entropy, and Gibbs free energy of our gas. Using the equation... 

Through the vehicle of partition functions expressions can be derived for macroscopic quantities from molecular parameters. Examples of such qucintities cire the surface pressure, and, for Gibbs monolayers, the adsorbed amount. Fluctuations in extensive quantities, like the number density or the interfacial excess energy, may also be obtained (sec. 1.3.7). 

The Gibbs free energy G can be directly calculated from the partition function ... 

We have defined the solvation process as the process of transfer from a fixed position in an ideal gas phase to a fixed position in a liquid phase. We have seen that if we can neglect the effect of the solvent on the internal partition function of the solvaton s, the Gibbs or the Helmholtz energy of solvation is equal to the coupling work of the solvaton to the solvent (the latter may be a mixture of any number of component, including any concentration of the solute s). In actual calculations, or in some theoretical considerations, it is often convenient to carry out the coupling work in steps. The specific steps chosen to carry out the coupling work depend on the way we choose to write the solute-solvent interaction. 

The Gibbs free energies are calculated using standard thermodynamic tables which are easily usable by machine since they give the data in the form of polynomial coefficients. The data are sometimes limited to 6000 K and it is therefore necessary to make extrapolations or to carry out calculations of partition functions from spectroscopic data In the latter case, which is certainly more reliable, one can determine standard thermodynamic functions with the aid of the classical formulae of statistical thermodynamics. The results may then be fitted to polynomials so that they match the tabulated data . Furthermore the calculation of partition functions is necessary for spectroscopic diagnostics and for the calculations of reaction rate parameters. 

Figure 4. Plot, based on the harmonic oscillator partition function, of the entropy of the oscillator as a function of log(oscillator frequency). On the left-hand ordinate is plotted the entropy in cal/mol deg (EU, entropy units), and on the right-hand ordinate is plotted the entropic contribution to the Gibbs free energy in kcal/mol pentamer at 298°K. This illustrates the increasing contribution of low frequency oscillations to the entropy and free energy of a protein chain segment. Adapted with permission from Urry et al. (1988).

Calculation of Thermodynamic Functions from Molecular Properties The calculation methods for thermodynamic functions (entropy S, heat capacities Cp and Cy, enthalpy H, and therefore Gibbs free energy G) for polyatomic systems from molecular and spectroscopic data with statistical methods through calculation of partition functions and its derivative toward temperature are well established and described in reference books such as Herzberg s Molecular Spectra and Molecular Structure [59] or in the earlier work from Mayer and Mayer [7], who showed, probably for the first time in a comprehensive way, that all basic thermochemical properties can be calculated from the partition function Q and the Avagadro s number N. The calculation details are well described by Irikura [60] and are summarized here. Emphasis will be placed on calculations of internal rotations. 

The last of Eqs. (51) defines a thermodynamic fundamental equation for G = G(N, P, 7) in the Gibbs energy representation. Note that passing from one ensemble to the other amounts to a Legendre transformation in macroscopic thermodynamics [39]. Vq is just an arbitrary volume used to keep the partition function dimensionless. Its choice is not important, as it just adds an arbitrary constant to the free energy. The NPT partition function can also be factorized into the ideal gas and excess contributions. The configurational integral in this case is ... 

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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