Thermodynamic functions estimations

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. 

The process we have followed Is Identical with the one we used previously for the uranium/oxygen (U/0) system (1-2) and Is summarized by the procedure that Is shown In Figure 1. Thermodynamic functions for the gas-phase molecules were obtained previously (3) from experimental spectroscopic data and estimates of molecular parameters. The functions for the condensed phase have been calculated from an assessment of the available data, Including the heat capacity as a function of temperature (4). The oxygen potential Is found from extension Into the liquid phase of a model that was derived for the solid phase. Thus, we have all the Information needed to apply the procedure outlined In Figure 1. 

The uncertainties in the condensed-phase thermodynamic functions arise from (1) the possible existence of a solid-solid phase transition in the temperature range 2160 to 2370 K and (2) the uncertainty in the estimated value of the liquid heat capacity which is on the order of 40%. While these uncertainties affect the partial pressures of plutonium oxides by a factor of 10 at 4000 K, they are not limiting because, at that temperature, the total pressure is due essentially entirely to O2 and 0. 

The partition function provides a direct method to estimate thermodynamic functions statistically. 

In the second approach, the chemical equilibrium between the reactant(s) and the transition state is expressed in terms of conventional thermodynamic functions, i.e., enthalpy and entropy changes. This method is easier to implement and provides useful insights for estimating both the preexponential factors and the activation energies. Consequently, we shall utilize the thermodynamic formulation of the TST in this paper. 

The relevant parameters of the model for the various end-members adopted by Ottonello et al. (1996) are listed in table 5.25. The to limit is the most critical parameter for the estimation of macroscopic thermodynamic functions. Values of to at Ai = 0 (center of the Brillouin zone) based on infrared spectra may be reduced to ai K =... 

The normal vibrations and structural parameters of Sg S, S, and Sjj have been used to calculate several thermodynamic functions of these molecules in the gaseous state. Both the entropy (S°) and the heat capacity (C°) are linear functions of the number of atoms in the ring in this way the corresponding values for Sj, Sg, Sjo and can be estimated by inter- and extrapolation For a recent review of the thermodynamic properties of elemental sulfur see Ref. 

According to the Benson estimations [46], this equilibrium in the case of acetone has the following thermodynamic functions Af/= — 51.9 J mol-1, AS=—161.1 J mol-1 and AG = —3.8 kJ mol-1 (350 K). The calculated ratios of the equilibrium concentrations of two kinds of peroxyl radicals at =350 and different acetone concentrations are presented below. 

With the discussion of the free-energy function G in this chapter, all of the thermodynamic functions needed for chemical equilibrium and kinetic calculations have been introduced. Chapter 8 discussed methods for estimating the internal energy E, entropy S, heat capacity Cv, and enthalpy H. These techniques are very useful when the needed information is not available from experiment. 

Thermodynamics. By correlating thermochemical and electrochemical data for liquid ammonia, it has been possible to estimate thermodynamic functions for a wide variety of ionic species in liquid ammonia at 25 ° C. (22,24). The heats and free energies of formation and entropies (relative to H+) for a few species are given in Table I. These values have been calculated from the data of references 6,19, 20, and 24. 

The enthalpy and the entropy at 5 K were estimated from the extrapolated heat capacity at this temperature according to the Debye T3 law. The thermodynamic functions at higher temperatures were calculated from the obtained Cp(7) dependence. 

Reference data on total energies of forms 19-23 optimized by means of different theoretical methods in the gas phase are given in Table 2. Various energetic characteristics of tetrazoles can be successfully estimated. The vertical adiabatic ionization potentials of both neutral tautomers 20 and 21 were calculated for a- and Tt-radical cations <2000CPL(330)212>. The standard molar thermodynamic functions (enthalpies, heat capacities, and entropies) of... 

Probably the most satisfactory approach which has been used to estimate absolute enthalpies of hydration of single ions is that of Halliwell and Nyburg (39). The method makes use of a simple model of aqueous solutions, and is based on the significance of differences between thermodynamic functions for ions of opposite charge rather than attempting to split the enthalpy of hydration of pairs of ions. Values of the differences between conventional enthalpies of hydration are plotted against (R -fa) - , where R is the effective radius of the ion and a is the effective radius of the water molecule. This plot yields the absolute enthalpy of hydration of the proton. 

The statistical polymer method presented above still is convenient to the equilibrium model only. However, since that allows estimation of all additive parameters of branched polymers, we can evaluate thermodynamic functions which characterize not only equilibrium but also nonequilibrium situation. 

Shaub WM (1982a), Thermochimica Acta 58 11-44.. .Estimated thermodynamic functions for some chlorinated benzenes, phenols and dioxins"... 

The expansions in even powers of normal frequencies are of special interest, because they provide means for obtaining explicit relations between the equations of motion and the thermodynamic quantities, through the use of the method of moments The sum of over all the normal vibrations can be expressed as the trace, or the sum of all the diagonal elements, of a matrix H" obtained by multiplying the Hamiltonian matrix H of the system by itself (n — 1) times. Such expansions thus enable us to estimate the thermodynamic functions and their isotope effects from known force fields and structures without solving the secular equations, or alternatively, to estimate the force fields from experimental data on the thermodynamic quantities and their isotope effects. The expansions explicitly correlate the motions of particles with the thermodynamic quantities. They can also be used to evaluate analytically a characteristic temperature associated with the system, such as the cross-over temperature of an isotope exchange equilibrium. Such possible applications, however, are useful only if the expansion yields a sufficiently close approximation. The precision of results obtainable with orthogonal polynomial expansions will be explored later. 

Statistical mechanics of assemblies of axially symmetric molecules.— To illustrate the way in which the contributions of the directional forces to the thermodynamic functions can be estimated by statistical mechanics, we shall limit ourselves to pure substances in this section. The extension to mixtures will be discussed in the last section. The Helmholtz free energy of an assembly of N identical molecules occupying a volume V at temperature T is given by... 

Bondi and Simkin presented a similar calculation for liquids, using the heat of vaporization (AHv) (241, 242). The purpose was to obtain an estimate of but in the process a value of the H bond increment, 5(OH) is derived. This increment is .. . a measure of (but not identical with) the heat of formation of the H bond,. . Values for 5(OH) are very similar to enthalpy values, though they are not intended as thermodynamic functions. Bondi gives an extensive discussion of their use with simple and polyhydric alcohols and a few other compound classes. Table 7-V compares 6(OH) and AH values. 

Electronic levels (T ) and vibrational-rotational constants of the observed states are from the optical study of Barrow et al. (J ) and the microwave work of Tiemann et al. (2). Other low-lying electronic states and their vibrational-rotational constants are estimated in isoconfIgurational groups by analogy with BaO (8) and from trends observed in the known states of the other alkaline-earth oxides and sulfides. Thermodynamic functions are calculated using first-order anharmonlc corrections to and in the partition function Q = exp(-c ej /T). Uncertainty in the energy and molecular constants for the... 

The Mo-Br bond length of 2.42 A and the ground state vibrational frequency of 300 cm" are estimates from Brewer (1 ). The electronic contributions are assumed to be the same as those of the free gaseous ion, Mo, as suggested by Brewer ( ). All states up to 17,344 cm" as listed by Moore (3 ) are Included in the calculation. Our thermodynamic functions differ slightly from those... 

The thermodynamic functions of this table are essentially the same as those calculated by (11). Uncertainty in the entropy is estimated to vary from 0.12 to 0.5 cal/(K mol) over the range from 300 to 1000 K. Uncertainty in Vg and neglect of anharraonicity are the major sources of error. Discussions of the magnitude of anharmonic effects appear on the tables for... 

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