Entropy thermodynamic calculations

Now that we have considered the calculation of entropy from thermal data, we can obtain values of the change in the Gibbs function for chemical reactions from thermal data alone as well as from equilibrium data. From this function, we can calculate equilibrium constants, as in Equations (10.22) and (10.90.). We shall also consider the results of statistical thermodynamic calculations, although the theory is beyond the scope of this work. We restrict our discussion to the Gibbs function since most chemical reactions are carried out at constant temperature and pressure. 

Thermodynamic methods, which have been those most widely used in the past, utilize isotherms and heats of adsorption as their foundations. Entropy changes calculated from such data are not easy to transform unambiguously into specific descriptions of the adsorbed phase. 

In order to estimate the true heat of activation, AH° for reaction (2) has to be evaluated by non-thermodynamic calculations [5]. Since AH cannot be directly evaluated, the true pre-exponential factors, and hence true entropies of activation, from eqn. (101) cannot be measured. Weaver... 

Fig. 4.7 Scheme of statistical thermodynamic calculations of ideal-gas entropy for the compounds without internal rotation... 

For thermodynamic calculation of equilibria useful in petroleum science, combustion data of extreme accuracy are required because the heats of formation of water and carbon dioxide are large in comparison with those in the hydrocarbons. Great accuracy is also required of the specific heat data for the calculation of free energy or entropy. Much care must be exercised in selecting values from the literature for these purposes, since many of those available were determined before the development of modem calorimetric techniques. 

Symbol Gibbs free energy is denoted eponymously by G, after Josiah Willard Gibbs, ca. 1873, who single-handedly created much of chemical thermodynamics. In the older literature F was sometimes used. Equation since G = H TS, the free energy of a molecule can be calculated from its enthalpy (above) and entropy at temperature 7 the entropy is calculated by standard statistical mechanics methods [130]. 

As entropy is a state function, we are free to choose a path from the initial and final states. The path along which a process takes place reversibly would be most convenient for thermodynamic calculations, because the heat absorbed or released can be directly related to the entropy change ... 

Thus, the configurational entropy. S, (7. p) expresses the number of local potential energy minima and hence can be evaluated by potential energy landscape and thermodynamic integration methodology [53]. The vibration entropy was calculated within the framework of a harmonic approximation to each basin [165,166], an approximation that is valid at low temperatures [53]. Three notable results were obtained by Sastry [53] ... 

Figure 3.6 shows schematically the molar entropy of a pure substance as a function of temperature. If a structural transformation occurs in the solid state, an additional increase in the molar entropy comes from the heat of the transformations. As shown in the figure, the molar entropy of a pure substance increases with increasing temperature. In chemical handbooks we see the tabulated numerical values of the molar entropy calculated for a number of pure substances in the standard state at temperature 298 K and pressure 101.3 kPa. A few of them will be listed as the standard molar entropy, s , in Table 5.1. Note that the molar entropy thus calculated based on the third law of thermodynamics is occasionally called absolute entropy. 

Based upon experimentally observed spectroscopic data, statistical thermodynamic calculations provide thermodynamic data which would not be obtained readily from direct experimental measurements for the species and temperature of interest to rocket propulsion. If the results of the calculations are summarized in terms of specific heat as a function of temperature, the other required properties for a particular specie, for example, enthalpy, entropy, the Gibb s function, and equilibrium constant may be obtained in relation to an arbitrary reference state, usually a pressure of one atmosphere and a temperature of 298.15°K. Or alternately these quantities may be calculated directly. Significant inaccuracies in the thermochemical data are not associated generaUy with the results of such calculations for a particular species, but arise in establishing a valid basis for comparison of different species. 

If a reaction in a mixture of solids is accompanied by the formation of gas or fluid phases (melts, solutions), solid solutions, or by the generation of defects, then, for a more strict thermodynamic forecast, it is necessary to take into account the changes of entropy and specific heat capacity during phase transitions of the components (melting, vaporization, dissolution), changes of volume and other parameters. If these factors are not taken into account, one can come across the contradictions between experimental data and thermodynamic calculations. 

Thermodynamic calculations, based on integrated intensity measurements, show that the crown form has the lower enthalpy, and also the lower entropy (.dS =8.5 ) or 6 2 eu ). The low entropy of the crown finds an explanation in the high symmetry of this conformation, compared to the low symmetry of the boat-chair. The best boat-chair for tetroxocane appears to be the BC-1,3,5,7 conformation. However, the ether functions at positions 3 and 7 have their dipoles close together in an unfavorable orientation but... 

This section outlines procedures for calculating entropy changes that occur in the system and in the surroundings during several types of processes. The final subsections show how entropy changes calculated for the thermodynamic universe (system plus surroundings) predict whether a particular contemplated process can occur spontaneously when attempted in the laboratory or in a chemical plant. 

Somewhat analogous considerations apply to the entropy of water vapor. The result derived from heat capacity measurements is again lower than the statistical value, and this can be accounted for by random orientation of the water molecules in the solid. The situation is complicated, however, by the distribution of hydrogen bonds in the ice crystal, and by other factors. In this instance, also, the crystal is not perfect, and so the entropy would not be zero at 0 K. The statistical value of the entropy is therefore the correct one to be used in thermodynamic calculations. 

Eor thermodynamic calculations and analyses of phase diagrams of binary systems whose components form a binary compound, partially or totally dissociating at melting, it is necessary to know the enthalpies of fusion of the components of the binary compounds, and of both eutectic mixtures, as input quantities. When these data cannot be found in the literature, it is possible to estimate them using entropy or enthalpy balances. 

Finally, for C-J temperatures above the boiling point of the metal, consideration should be given to the possible existence of distinct species in the gas phase. To include them as components in the assumed set of detonation gases requires knowledge of their molecular geometry and ideal-gas thermodynamic properties (e.g., heat capacity as a function of temperature and heat and entropy of formation). In the absence of such data, it is possible to predict the molecular geometry and thermodynamic properties via quantum-chemical and statistical-thermodynamic calculations, respectively, or to estimate them by analogy with known related molecules. 

Cycloamyloses have been separated by h.p.l.c. on a /u-Bondapak-carbohydrate column using acetonitrile-water mixtures as eluant. The molecular dynamics of the inclusion complexes formed between cyclohexa-amylose and some aromatic amino-acids and dipeptides have been studied by n.m.r. spectroscopy. The forces binding the complexes were found to be weak. The c.d. spectra of cyclohepta-amylose which had been complexed with 2-substituted naphthalenes were measured at various concentrations of cyclohepta-amylase and temperatures between 10-70 C. The complex with 2-naphthoxyacetic acid showed 1 1 stoicheiometry. The molar ellipticity and thermodynamic parameters were determined and enthalpy and entropy ranges calculated. The correlation was explained by a cyclohepta-amylose guest molecule interaction where the guest molecule was highly solvated. The induced c.d. spectra of cyclohepta-amylose complexes with substituted benzenes confirmed that an axial inclusion... 

It is of interest to estimate the thermodynamic functions of siuface layers of different composition. Since a binary system is considered, it is possible to make thermodynamic calculations for mixtures of different concentrations at different temperatures (Fig. 2.27). This was done using the equation that connects the surface pressure with the change in entropy and enthalpy of the surface layers [232] ... 

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