Thermodynamic functions condensed phases

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... 

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 +... 

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. 

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. 

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. 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

Use the NIST WEB BOOK (http //webbook.nist.gov) to find the vapor pressure of water as a function of temperature over the range from 300 K to 600 K. When you reach the home page foir the WEBBOOK, cUck on the NIST Chemistry Webbook, cfick on Name under search options, type water in the space for name, cfick on thermodynamic data, cfick on condensed phase, cfick on saturation properties, and insert the temperature... 

With these formulas for Um and Sm, equations can be directly derived for the other thermodynamic functions Hm, Am, and Gm. For condensed phases Hm = Um to a very good approximation. For gases, assumed ideal, H0 = U0 and Hm = Um + RT, giving... 

Figure 4 Equilibrium CVD phase diagram for the Nb-Ge-H-CI system. The diagram was constructed from thermodynamic calculation results and depicts the condensed phases which form as a function of experimental variables. The Nb/(Nb+Ge) values are reactant gas concentrations. After Wan.9...

The evaluation of the free energy is essential to quantitatively treat a chemical process in condensed phase. In this section, we review methods of free-energy calculation within the context of classical statistical mechanics. We start with the standard free-energy perturbation and thermodynamic integration methods. We then introduce the method of distribution functions in solution. The method of energy representation is described in its classical form in this section, and is combined with the QM/MM methodology in the next section. 

This condition will cause Reaction 13 to proceed to the right, and Ca(OH)2 will be stable where the CaO is in contact with, the gas phase. We can represent the possible reaction product as a function of gas chemistry in order to construct a thermochemical diagram that depicts the stability range of various condensed phases as functions of the thermodynamic activities of the two components of reactive gas such as CO and H O in air, etc. Based on the above analysis, we can build the thermochemical diagram for CaO and CaC03 and Ca(OH)2 at 500°C by using literature values as a function of pCO and pH O (see Eigure 12). 

Actually, the problem stems from the very nature of empirical potentials whose parameters have been optimized to reproduce by simulation very few thermodynamic and/or dynamic properties at a specified P and T. This makes the resulting function a state-dependent potential whose behavior under different P and T conditions might be unpredictable. For instance, two-body effective potentials that include non-additive effects in an implicit way might be poorly transferable under conditions when their effect on the well depth becomes significant. On the other hand, potential models with parameters optimized on a wide base of diverse experimental data of gas and condensed phase could perform better from this point of view. 

---