Free energy, Gibbs calculation

We now have all the parameters and analytical forms necessary to perform enthalpy and Gibbs free energy calculations at high P and T conditions ... 

Section 5.1 presents the fundamental method as the heart of the chapter— the statistical thermodynamics approach to hydrate phase equilibria. The basic statistical thermodynamic equations are developed, and relationships to measurable, macroscopic hydrate properties are given. The parameters for the method are determined from both macroscopic (e.g., temperature and pressure) and microscopic (spectroscopic, diffraction) measurements. A Gibbs free energy calculation algorithm is given for multicomponent, multiphase systems for comparison with the methods described in Chapter 4. Finally, Section 5.1 concludes with ab initio modifications to the method, along with an assessment of method accuracy. 

The gas-phase and aqueous basicities of six 1,2,3-triazoles (71-76), have been determined, the former by FTICR and the latter by spectrophotometry and II NMR. The gas-phase experiments are in good agreement with the Gibbs free energies calculated at the B3LYP/6-31G level. In contrast, only semiquantitative ascertainments are possible when basicities in the gas phase and in solution are compared. The equilibria involved are complex due to tautomerism [87],... 

At low pressures, for which the UNIFAC method was developed, the Helmholtz free energy and Gibbs free energy calculated from an EOS are virtually indistinguishable... 

The enthalpy and Gibbs free energy calculated according to their chemical reaction at 2(XX)K ... 

T. Z. M. Denti, W. F. van Gunsteren, and F. Diederich,/. Am. Chem. Soc., 118,6044 (1996). Computer Simulations of the Solvent Dependence of Apolar Association Strength Gibbs Free Energy Calculations on a Cyclophane-Pyrene Complex in Water and Chloroform. 

Figure 1.5. Gibbs Free Energy Calculated by theRISM Theory AGrmRiSM, the excess free energy calculated by Eq. (1.113) Grmid, the ideal gas contribution GrmRiSM= Grmid + AGrmRiSM Grmexpt, experimental results.

Using model chemistry methods, we also reported AGg s calculations with slightly less accuracy, within 1.1 to 1.6 kcal/mol of experimental values. The model chemistries G3 [70], CBS-QB3 [59], CBS-APNO [60], and W1 [71] produced mean absolute deviations of 1.16, 1.43, 1.06, and 0.95 kcal/mol, respectively [10,13]. This result was confirmed in 2005 where G2, G3, and CBS-APNO predicted accurate values of AGg s for formation of ion-water clusters when compared to experimental results [72-76] Contrary to the previous publications, however, CBS-QB3 was less accurate for these clustered ions than the other model chemistries. Although there are many effective methods of gas phase Gibbs free energy calculations, it would be useful to find computationally less demanding methods that produce a similar accuracy [12]. 

Other publications, however, report more accurate values of B3LYP gas phase Gibbs free energy calculations on aliphatic amines, diamines, and aminoamines. In 2007 Bryantsev et al. reported that B3LYP calculations with the basis set 6-31-h-G had a mean absolute error of 0.78 kcal/mol from experimental values of the gas phase basicity (AGg s) of the reverse reaction of equation 1 reported in the NIST database [58]. This accuracy is comparable to that of expensive, high level model chemistries, but because the experimental values have uncertainties of 2 kcal/mol, it is difficult to discern exactly how accurate the calculations are in comparison to values in the other publications [81]. The take-home message remains the same always benchmark DFT calculations for the systems you are interested in computing [52]. 

By studying a number of model systems (generally alcoholic solutions of alkylammonium salts), Barthel has also proved experimentally that the latter consideration gives a better approximation to reality. Important results of these studies are the Gibbs free energies, calculated from the temperature dependence of the association constant with the following equation ... 

FIGURE 8.12 Relationships between the amounts of unfrozen water and changes in the Gibbs free energy calculated from the NMR spectra recorded in (a) air or (b) CDCI3 for wheat seed roots after germinating for 3 or 7 days in different media. 

Appendix 1. Change in Molar Gibbs Free Energy Calculations. .. 391... 

Change in Molar Gibbs Free Energy Calculations... 

Each of the ordered structures represents under the respective conditions the state with the lowest Gibbs free energy. Calculations of the Gibbs free energies and comparisons between the various lattices and the homogeneous phase can therefore provide an understanding of the phase diagram. In addition, they enable us to a determine the structure parameters. 

Table 3. Possible reactions taking place during sintering of UHTC with MoSi or TaSi. When available, the Gibbs free energy calculated in standard conditions from room temperature (RT) to 1900°C is reported. When the free Gibbs energy passes from positive to negative values, the transition temperature is given in brackets. The probability of occurrence of each reaction is assigned according to the micro-structural features.

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