Precision calculated equilibria

Equilibrium potentials can be calculated thermodynamically (for more details, see Chapter 3) when the corresponding electrode reaction is known precisely, even when they cannot be reached experimentally (i.e., when the electrode potential is nonequilibrium despite the fact that the current is practically zero). The open-circuit voltage of any galvanic cell where at least one of the two electrodes has an nonequilibrium open-circuit potential will also be nonequilibrium. Particularly in thermodynamic calculations, the term EMF is often used for measured or calculated equilibrium OCV values. 

The combination of high temperatures and densities at early times leads to the existence of a phase close to thermal equilibrium, when the signihcant particle reaction rates dominated over the expansion rate. This enables precise calculations to be made. 

A precise calculation of AGd as a function of the cationic radius would be very difficult because it would involve a complete conformational analysis of a large and complicated ligand system (82). Nevertheless, the dependency of the cation selectivity on steric interactions is capable of illustration. The term AGd can be estimated very crudely by using Hooke s law. As is shown in Fig. 16, ligands that are differentiated only by the radius of their equilibrium cavities can easily discriminate between cations of different size. This may explain why valinomycin and antamanide, two antibiotics with similar coordination spheres (54, 66), do not prefer the same cation (82). As it is no easy task to predict the exact dimensions of the cavity for a proposed ligand, the tailored synthesis of such ligands is conceivable yet problematic. 

AU,) - the differences between the depths of different minima so going down from a shallower to a deeper minimum of the molecular PES means decreasing the Gibbs free energy and such a process generally has an equilibrium constant larger than unity. However, for precise calculations, the terms coming from translations, rotations and vibrations also must be taken into account. 

Because of difficulties in precisely calculating the total ion activity coefficient (y) of calcium and carbonate ions in seawater, and the effects of temperature and pressure on the activity coefficients, a semi-empirical approach has been generally adopted by chemical oceanographers for calculating saturation states. This approach utilizes the apparent (stoichiometric) solubility constant (K ), which is the equilibrium ion molal (m) product. Values of K are directly determined in seawater (as ionic medium) at various temperatures, pressures and salinities. In this approach ... 

At pressures below 30 kbar FeO is not in equilibrium with respect to nonstoichiometric wustite. However, in many experimental works and thermodynamic calculations the nonstoichiometry of wustite is not taken into account, which leads to confusion and incompatibility of the results. As a result, values may differ by 2 to 3 kcal, as we showed earlier (Mel nik, 1972b). Such large discrepancies preclude precise calculation of the parameters of wustite stability. 

Table I lists Go (Formula 11) and tocXg/4 evaluated from a standard compilation and also evaluated from more recent data for some isotopi-cally substituted hydrogen molecules. It is seen that the Go values are by no means negligible compared with — ovXg/4, the usual anharmonicity correction to the zero-point energy. Even the Go and togXe/4 values calculated from the more recent data do not precisely follow the theoretically expected /x" mass dependence. If one adjusts the more recently obtained togXe/4 for D2 so that the value may follow the dependence. Go for D2 becomes equal to 4.7 cm." In order to obtain the best values of Go and toeXg/4 for D2, these are adjusted to 4.6 cm. and 15.1 cm. respectively (both following the /x" dependence with respect to the H2 values). The HD values obtained from the more recent data already exhibit the expected mass dependence. Obviously, within the present framework, there is no anharmonicity zero-point energy contribution to the theoretically calculated equilibrium constant for H2 + Do = 2HD.

The selectivity coefficient can be determined experimentally by adding a certain amount of resin material to a solution with known concentrations of X- and HCOf. The resulting concentration of the exchanged ions is determined in the mobile and stationary phase, respectively, after equilibrium is achieved. To precisely calculate the selectivity coefficient, the activities a have to be used instead of the concentrations Cj. As a prerequisite, the determination of the activity coefficient/j according to Eq. (35) is required, which is difficult to perform in the matrix of an ion-exchange resin. 

In modem papers ground state constants are frequently reported with cited uncertainties lxl0" cm" (3 kHz) from infrared work and 1 x lO cm" (0.3 MHz) from Raman studies. In band spectra, two sets of rotational constants are obtained, those of the upper and lower states involved in the transition, and a statistical treatment allows the differences between the constants to be determined to precisions approaching or eqnal to microwave uncertainties (1 kHz or less). Thus equilibrium rotational constants of polar molecules can be quite precisely calculated by using microwave-determined Bq constants and infrared-determined a constants. When the values of some of these a constants are missing, they can be substituted by reliable ab initio values. Despite the recent instmmental improvements, the resolution available from both infrared and Raman studies is still much lower than that from microwave spectroscopy, and therefore, studies are limited to fairly small and simple molecules. However, these techniques are not restricted to polar molecules as is the case for microwave spectroscopy, and thus... 

As shown in Example 12.1, the absolute value of the enthalpy of reaction increases with increasing temperature. Therefore, the more precise calculation will lead to a smaller equilibrium constant and thus a smaller conversion than the calculation with a constant standard enthalpy of reaction at 25 C. Thus, according to Eq. (12.23), the following equilibrium constant is obtained for a temperature of450 C,... 

Validity of equation (3.58) is limited to the interval in which the empirical relationship for molar heat are valid, the precision of the correlation of molar heat values being decisive for the precision of calculated equilibrium constant data. For practical purposes this precision is usually satisfactory. For illustration. Fig. 2 shows the course of the temperature dependence of equilibrium constants for some reactions which are of significance in industry. 

The design becomes somewhat more complex when a mixture of compounds is involved. However, the components of the mixture are usually mutually soluble in the liquid phase, and, as a first approximation for related solvents, it can be assumed that the mixture follows Raoult s law, i.e., the partial pressure of each component in the product gas will be equal to the vapor pressure of the pure component at the gas outlet temperature times its mole fraction in the liquid phase. For more precise calculations and more complex liquid mixtures, it is necessary to use vapor-liquid equilibrium (VLB) data for the specific system. The estimation and correlation of VLB data are discussed in various chemical engineering texts, such as Perry s Handbook (Perry et al., 1984), Reid and Sherwood (1966), and Prausnitz (1969). 

Equation (3) allows the calculation of the distance traveled axially by a solute band before the radial standard deviation of the sample is numerically equal to the column radius. Consider a sample injected precisely at the center of a 4 mm diameter LC column. Now, radial equilibrium will be achieved when (o), the radial standard deviation of the band, is numerically equal to the radius, i.e., o = 0.2 cm. 

Interesting is a comparison of the volumes occupied by individual complexes in solution and in the solid state. The partial molal volumes can be obtained from precise measurements of the solution densities of the complexes as a function of concentration [177]. These values may be subsequently compared with the unit cell volumes per complex molecule derived from the crystal structure. For Fe[HB(pz)3]2, the apparent molal volume in tetrahydrofuran solution was determined as 340.9 em mol Taking into account that the complex in solution forms an equilibrium between 86% LS and 14% HS isomers and employing the volume difference between the two spin states AF° = 23.6 cm mol S the volume of the LS isomer was calculated as 337.6 cm mol This value agrees closely with the volume of 337.3 cm mol for the completely LS complex in solid Fe[HB(pz)3]2 [105]. 

Secular equilibrium materials. For materials that have remained a closed system for sufficient time that secular equilibrium has been achieved, the half-lives of nuclides within the decay chain can be calculated from the relationship A,pP = A,dD. If the atom ratio P/D is measured, and one of the decay constants is well known, then the other can be readily calculated. Limitations on this approach are the ability to measure the atom ratios to sufficient precision, and finding samples that have remained closed systems for a sufficient length of time. This approach has been used to derive the present recommended half lives for °Th and (Cheng et al. 2000 Ludwig et al. 1992). 

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