Thermodynamic Functions for Solids

Thermodynamic Functions for Solids.—In the preceding section we have seen how to express the equation of state and specific heat of a solid as functions of pressure, or volume, and temperature. Now we shall investigate the other thermodynamic functions, the internal energy, entropy, Helmholtz free energy, and Gibbs free energy. For the internal 

The internal energy of metallic sodium is shown as a function of volume in Fig. XIII-2, as an illustration. On account of the large compression that can be attained with sodium, more terms of the power series must be retained than are given in Eq. (2.1), but it is easier to show the properties of this metal than of a less compressible one. Let us consider the behavior of the internal energy as a function of volume at fixed temperature. If the thermal expansion is independent of temperature, so that Pq 

The entropy is most easily determined as a function of volume and temperature from the equation (dS/dT)v = Cv/T. At the absolute zero of temperature, the entropy of a solid is zero independent of its volume or pressure. The reason goes back to our fundamental definition of entropy 

In Fig. XIII-4 we show A as a function of volume for a number of temperatures. At the absolute zero, as we have mentioned above, the Helmholtz free energy equals the internal energy, as given in Fig. XIII-2. 

We have already shown, in Figs. XI-4, XI-5, XI-6, and XI-7, the equation of state, entropy, and Gibbs free energy of a substance in all of its three phases. Examination of the parts of those figures dealing with solids will show the similarity of those curves to the ones found in the present section in a more explicit and detailed way. 

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P10.5 The thermodynamic functions for solid, liquid, and gaseous carbonyl chloride (COCL) obtained from Third Law and statistical calculations... 

Thermodynamic functions [1.35]. Thermodynamic functions for solid tungsten are listed in Table 1.13. For more details and data for liquid tungsten, see elsewhere [1.10]. 

SEA/MES] Searcy, A. W., Meschi, D. J., Calculation of integral and partial thermodynamic functions for solids from dissociation-pressure data, Thermodyn. Nucl. Mater., Proc. Symp., pp. 131-142, IAEA, Division of Publications, Vienna, Austria, (1962). Cited on page 131. 

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. 

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. 

The dissolution of an ionic solid in a polar solvent, or the dissociation of an acid, are examples of processes for which AH° and AS0 are both relatively small and are often of equal importance in determining the sign and magnitude of AG°. This was seen to be the case for HF as discussed above, and the thermodynamic functions for the several steps into which its dissociation can be analysed are much greater in magnitude. As discussed further in Section 3.3, the same is true in the analysis of the solubilities of ionic solids. 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

Compared to the critical evaluation by Myers and Graves (1977b) the variation along the LnF3 series is much more smooth, which we feel is mainly due to the improved thermodynamic functions for the solid and gaseous phases derived in the present study. 

Wc have so far studied only perfect gases and have not taken up imperfect gases, liquids, and solids. Before we treat them, it is really necessary to understand what happens when two or more phases are in equilibrium with each other, and the familiar phenomena of melting, boiling, and the critical point and the continuity of the liquid and gaseous states. We shall now proceed to find the thermodynamic condition for the coexistence of two phases and shall apply it to a general discussion of the forms of the various thermodynamic functions for matter in all three states. 

The thermodynamic functions for items 1 and 2 are calculated using the standard equations for bulk gases and solids, respectively, so that the focus for adsorption thermodynamics is on item 3. It follows from Equations (5) and (7) that the grand potential (free energy of immersion) for each pure component is... 

The thermodynamic functions of solids I and II are recorded in Table 6.5 [752]. Those of solid III have not been determined, but it is considered that the heat capacity of solid III is similar to the values for solids I and II. The earlier values of Cp [751] for solid II are valid below 118.3 K, and are valid for solid I above 118.3 K. 

In the same report, the authors reported low-temperature calorimetry heat capacity measurements (9 to 70 K) for anhydrous NiS04(cr). Limited sets of drop-calorimetry measurements were carried out for the same solid (403 to 1001.5 K), and these were used to generate equations for thermodynamic functions for NiS04 (cr) from 298 to 1200 K. 

The atoms in a Debye solid are treated as a system of weakly coupled harmonic oscillators. Normal modes with wavelengths that are large compared to the atomic spacing do not depend on the discrete nature of the crystal lattice, and consequently these normal modes can be obtained by treating the crystal as an isotropic elastic continuum. In the Debye treatment of a solid all of the normal modes are treated as elastic waves. The partition function for a Debye solid cannot be obtained In closed form, but the thermodynamic functions for a Debsre solid have been tabulated as a function of 9p/T- For the pair of Isotopic metals Li(s)... 

Correlations are drawn between the magnimde of the thermodynamic functions for the tetrahedral-octahedral equilibria in chloroform solution and the configuration (tetrahedral or polymeric octahedral) of the bisamine complex in the solid state. While the correlation is good for many of the systems studied certain anomalies suggest that other factors such as the bridging power of X and packing conditions in the crystal can play an important role in deciding solid state stereochemistry. 

Summary. In the calculation of thermodynamic functions for an equilibrium MD system, for gas, fluid, or solid phases, the present knowledge of PBC effects can be summarized as follows. 

Phase equilibria, particularly the solubility of sparingly soluble ionic solids can be calculated if the standard thermodynamic functions for aqueous species, the constants of... 

The National Bureau of Standards Report 6928 lists the properties of compounds of lithium, beryllium, magnesium, and aluminium with hydrogen, oxygen, fluorine, chlorine, nitrogen, and carbon. Thermodynamic functions for the ideal gases, solids, and liquids are tabulated (e.g. values of free energy functions and enthalpy functions , entropy, and heat capacity). 

The thermophysical properties of carbon dioxide presented by Vukalovich and Atunin include phase equilibria, enthalpy, heat capacities, equations of state, and the thermodynamic functions for the ideal gas. The content of Sarkin s work is well represented by the title Gas Dynamics and Thermodynamics of Solid-propellant Rockets . 

A first SIM investigation of sorption-thermodynamic functions for binary [18-20] and ternary [21] mixtures of gases on microporous solids was presented by the Billow group, in the nineteen eighties and 1994, respectively Billow also introduced this technique to the Rees group at the ICSTM London [22]. Since 1989, the latter group published a series of papers, particularly on sorption equilibria for binary mixtures [23-26]. Thermodynamic analyses of the isosteric... 

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