Thermodynamic functions, of condensed phases

Influence of Temperature and Pressure on the Thermodynamic Functions of Condensed Phases. 

Up to now, the discussion has been about thermodynamic properties of gaseous and liquid Freon-22. The thermodynamic functions of condensed-phase Freon-22, including the solid, are tabulated in Ref. [3.55] at T = 15-232 K (Table 31). This table also includes the values for entropy, enthalpy, and Gibb s function in the vapor phase at Tnbp- Specifically, - //q)g = 9482.8 cal/ mol at Tnbp- On the other hand, data in [3.41, 3.63] show that =... 

Vapor pressures and vapor compositions in equilibrium with a hypostoichiometric plutonium dioxide condensed phase have been calculated for the temperature range 1500 I H 4000 K. Thermodynamic functions for the condensed phase and for each of the gaseous species were combined with an oxygen-potential model, which we extended from the solid into the liquid region to obtain the partial pressures of O2, 0, Pu, PuO and Pu02 as functions of temperature and of condensed phase composition. The calculated oxygen pressures increase rapidly as stoichiometry is approached. At least part of this increase is a consequence of the exclusion of Pu +... 

Thermodynamic Functions of the Condensed Phases. Tabulated thermodynamic functions for the condensed phases of plutonium dioxide and a detailed description of their calculation are given elsewhere (21). The AG (Pu02 c) is represented by the equation given in Table I. The AGf values were calculated using standard thermodynamic relations and the data given below. 

Actually, computational convenience has almost always suggested using pairwise additive potentials for simulations of condensed phases also, though strictly two-body potentials are only acceptable for rarefied gases. The computational convenience of two-body potentials is maintained, however, if non-additive effects are included implicitly, i. e. with the so called two-body effective potentials. All empirical or semi empirical functions whose parameters have been optimized with respect to properties of the system in condensed phase belong to this class. As already observed, this makes these potentials state-dependent, with unpredictable performance under different thermodynamic conditions. 

We now proceed to calculate the thermodynamic functions of a condensed phase. First, by taking the temperature and pressure as independent variables, we may evaluate the molar enthalpy. 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

Values recalculated into SI units from those of Din. Thermodynamic Functions of Gases, vol. 2, Butterworth, London, 1956. Above the solid line the condensed phase is solid below the line it is liquid, t = triple point c = critical point. 

Coopersmith has presented an article largely concerned with the mathematical description of thermodynamic functions near a critical point. Reisman has included classical treatments and modern theories applicable to the study of condensed phase-vapour equilibria. 

Potential energy functions that describe solute molecules and their interactions with the environment have to be developed separately. This usually requires quantum mechanical calculations of the electronic structure and, for con-formationally flexible molecules, energetics of isolated solute molecules [18, 19,26,27]. These calculations are often followed by simulations of condensed phases containing the solute of interest. This could be a simulation of a pure liquid of solute molecules [27] or the solute dissolved in water and a nonpolar liquid (e.g., hexane) resembling the interior of the membrane [16,26]. Only when the calculated and experimental thermodynamic and/or structural properties agree can we consider the potential functions to be reliable. 

Values for the thermodynamic functions as a function of temperature for condensed phases are usually obtained from Third Law measurements. Values for ideal gases are usually calculated from the molecular parameters using the statistical mechanics procedures to be described in Chapter 10. In either... 

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. 

In this paper we describe (1) the gas-phase thermodynamic functions (2) the condensed-phase thermodynamic functions (3) the oxygen potential (and the phase boundaries that are consistent with It) and (4) the resulting vapor pressure and composition as functions of temperature and composition of the condensed phase. 

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. 

A number of other thermodynamic properties of adamantane and diamantane in different phases are reported by Kabo et al. [5]. They include (1) standard molar thermodynamic functions for adamantane in the ideal gas state as calculated by statistical thermodynamics methods and (2) temperature dependence of the heat capacities of adamantane in the condensed state between 340 and 600 K as measured by a scanning calorimeter and reported here in Fig. 8. According to this figure, liquid adamantane converts to a solid plastic with simple cubic crystal structure upon freezing. After further cooling it moves into another solid state, an fee crystalline phase. 

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