Various Thermodynamic Cycles

Various thermodynamic cycles are used in pKa calculations. Although previously a source of confusion in the field, it is now clear that as long as the most accurate experimental values are used, and no explicit water molecules are added, the choice of cycle should just be a matter of convenience. The most common is based on Eq. (1) and is diagramed in Figure 1, where a molecule is simply deproto-nated, yielding a corresponding base and the proton in solution [1,2,4]. This cycle depends on the accuracy of the continuum model used to determine the anion (reaction 1) or cation (reaction 8) solvation energies, calculations that vary in... 

Other thermodynamic cycles used in various calculations contain explicit water molecules. The effectiveness of this implicit-explicit method in terms of calculating the free energy of solvation is discussed in the preceding section. Since the free energy of solvation is the largest source of error in determining pKa values, the accuracy of the calculation often determines the validity of various thermodynamic cycles. 

The electron-transfer aspect of the reaction was studied by using Marcus theory and various thermodynamic cycles. Thus, the thermochemistries of a number of electron-transfer schemes were considered. Through a series of logical arguments, the scheme settled upon was not electron transfer per se but rather electrophilic attack of PAH by nitrosated nitrogen tetroxide ... 

Figure 4.2 presents various thermodynamic cycles proposed in chronological order in pressure-volume diagram. Considering the ideal case, all the processes are reversible. In Stirling cycle, processes 1-2 and 3-4 are isothermal, while the other two processes are isochoric. As such the efficiency of this cycle is lesser than the... 

According to the analyses and design studies of the gas turbine units applied within various thermodynamic cycles and schemes, the plant efficiency may vary between 31 and 52% depending on the selected scheme and the inlet air temperature, even if the air is heated up to the same temperature of 700°C. 

The general thermodynamic conclusions given above are confirmed by more detailed parametric studies which have been made by several authors of various wet cycles. 

Whilst this Chapter is primarily concerned with the methods of determining the free energies of tautomeric or ionisation equilibria via computer simulation of free energy differences, many of the issues raised relate also to the determination of other molecular properties upon which behaviour of the molecule within the body may depend, such as the redox potential or the partition coefficient.6 In the next section, we shall give a brief explanation of the methods used to calculate these free energy differences -namely the use of a thermodynamic cycle in conjunction with ab initio and free energy perturbation (FEP) methods. This enables an explicit representation of the solvent environment to be used. In depth descriptions of the various simulation protocols, or the accuracy limiting factors of the simulations and methods of validation, have not been included. These are... 

The values for log K and E° in this table are published data from various standard reference sources, hence thermodynamic cycles of the type shown in Fig. 7 may not sum exactly to zero when applied to these data due to differences in experimental conditions. 

The readers should find the papers listed in the Bibliography informative and useful for analysis and design of various finite-time thermodynamic cycles. It is hoped that these papers will provide interest and encouragement for further study in the area of finite-time thermodynamics. 

Figure 11.4 Gas-phase (upper) and solution (lower) reaction coordinates, and the thermodynamic cycles that connect them via free energies of solvation of the various stationary points (vertical lines). Note the significant left to right movement of all stationary points, and particularly the TS structure, on going from the gas phase to solution...

Estimation methods applicable for liquid heat capacities fall into four general categories theoretical, group contribution, corresponding states, and Watson s thermodynamic cycle. An assumption is made that various groups in a molecule contribute with a value to the total molar heat capacity, which is independent of other groups present. 

However, this simple chemical equation conceals a more complicated sequence of events in which the reactants undergo various transformations before the product is formed. These may be summarized in a Born-Haber thermodynamic cycle (Figure 3.1). The first stage of the reaction process is the conversion of M and E into gaseous state atoms, requiring an enthalpy of atomization of A// j(M), and, if E is a solid or liquid, the enthalpy of vaporization, of E or... 

The oxidative addition reaction of I2 with (> -Cp )2Ln (Ln = Sm °" , Eu ° , Ybio.ii) j,ggj3 utilized as a thermodynamic anchor to determine absolute Ln-X bond disruption energies, D(Ln-X), in solution. Suitable thermodynamic cycles can be set up in which the ability to estimate an accurate value for >(Ln-I) is crucial to the accurate calculation of D(Ln-X) for various carbon- and silicon-donor ligands . Values for D(Ln-I) are obtained from reactions such as the one for Sm, where n (probably) equals 2 or 3 ° ... 

Methods other than thermodynamic cycles are often used to calculate acid dissociation constants. Previous publications implement the theoretical relationship between pKa and structural property [6], bond valence methods and bond lengths [33], pKa correlations with highest occupied molecular orbital (HOMO) energies and frontier molecular orbitals [34], and artificial neural networks [35] to predict pKa values. In addition much work has been done using physical properties as quantitative structure-activity relationship (QSAR) descriptors, and regression equations with such descriptors to yield accurate pKa values for specific classes of molecules [36-47]. The correlation of pKas to various molecular properties, however, is often restricted to specific classes of compounds, and it is... 

Due to the numerous potential cycles using explicit molecules, levels of theory, basis sets, and types of molecules, it is impossible to determine one specific method that produces the most accurate pKa values. Rather, this review serves to summarize the current literature and illustrate various schemes that have been successful. Accurate attention to detail and the use of benchmark calculations or experimental values to assist in determination of the correct method to use for a particular system is highly recommended. Further research on thermodynamic cycles using explicit cycles, clustered water structures, conformational effects, and advances in continuum solvation calculations will continue to advance this field. 

Fig. 3.9. Thermodynamic cycle of the interaction of host W with guest Z at two different pH values, the experimentally accessible one pHots and the desired one pHjes- The cycle connects the various desired state functions (AXdes) to the directly observed ones in the supramolecular association (AXobs) atid the changes occuring with the individual species (AXaph). These changes have to be determined separately.

We can conclude that the knowledge of these thermodynamic processes allows us to understand the quantitative parameters of interaction of various biogeochemical cycles in terrestrial and aquatic ecosystems. These parameters can be also used in different biogeochemical models. 

Anyone who has worked with triarylmethyl or pyronin systems has probably been struck by the distinctions between the solid states of various derivatives. Malachite Green chloride, for example, is a highly colored crystalline ionic material, and the carbinol is a colorless, reasonably low nelting [mp 163 °C (14)] solid. This distinction between ionic and covalent iolids can be considered in another thermodynamic cycle ... 

The heats of formation of various ionic compounds show tremendous variations. In a general way, we know that many factors contribute to the over-all heat of formation, namely, the ionization potentials, electron affinities, heats of vaporization and dissociation of the elements, and the lattice energy of the compound. The Born-Haber cycle is a thermodynamic cycle that shows the interrelation of these quantities and enables us to see how variations in heats of formation can be attributed to the variations in these individual quantities. In order to construct the Born-Haber cycle we consider the following thermochemical equations, using NaCl as an example... 

The thermodynamic cycle (Figure 4) showed that changes in redox potential in various solvents were a consequence of the different interaction of the solvent with the dissolved redox couple. One model that has been used to quantify these changes is the electrostatic model, which is based on treating the ion as an ideal sphere in a continuous dielectric the model ignores the effect of any transfer of charge that may occur. The Born expressions for the free energy of solvation of... 

As an example of using the Born-Haber cycle we will calculate the lattice energy of MgO. The values of the various thermodynamic parameters can be found in Kubaschewski et al. (1993), Johnson (1982), and in the NIST-JANAF tables (Chase, 1998). 

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