Entropy and heat capacity

As has been the approach for most of the author s other reviews on organic thermochemistry, the current chapter will be primarily devoted to the relatively restricted scope of enthalpy of formation (more commonly and colloquially called heat of formation) and write this quantity as A//f, instead of the increasingly more commonly used and also proper A//f° and AfHm No discussion will be made in this chapter on other thermochemical properties such as Gibbs energy, entropy, heat capacity and excess enthalpy. Additionally (following thermochemical convention), the temperature and pressure are tacitly assumed to be 25 °C ( 298 K ) and 1 atmosphere (taken as either 101,325 or 100,000 Pa) respectively3 and the energy units are chosen to be kJmol-1 instead of kcalmol-1 (where 4.184 kJ = 1 kcal, 1 kJ = 0.2390 kcal). 

W. Weltner, Jr., andK. S. Pitzer, Methyl alcohol The entropy, heat capacity and polymerization equilibria in the vapor, and potential barrier to internal rotation. J. Am. Chem. Soc. 73, 2606 2610 (1951). 

Alpha-beta (a-jS) transitions of the condensed forms of Si02 quartz, try-dimite, and cristobalite may all be regarded as lambda transformations. Their kinetics are higher than those of quartz-trydimite, quartz-cristobahte, and quartz-coesite, which are first-order transformations. Figure 2.7 plots in detail the evolution of enthalpy, entropy, heat capacity, and volume at the transition zone... 

Figure 2J Enthalpy, entropy, heat capacity, and volume modification in a-quartz/ /3-quartz transition region. Reprinted from H. C. Helgeson, J. Delany, and D. K. Bird, American Journal of Science, 278A, 1-229, with permission.

The heat of formation (A//f) for 1,2,4-thiadiazole has been reported <90JPR885> but values for other thermodynamic functions (entropy, heat capacity, and free energy) are not available. 

The entropies, heat capacities, and thermodynamic functions of gaseous cyclooctaselenium have been calculated from spectroscopic and structural data for temperatures of up to 3000 K (64). Both the heat capacities and entropies of sulfur rings S (n = 6, 7, 8, 12) at a given temperature depend linearly on the ring size n (65). Therefore, it has been assumed that analogous relationships exist for the cyclic Se molecules, and the following equations have been derived from the data of Se2 and Seg at 298 K (64) ... 

Thermochemical data on the separate phases in equUibrium are needed to constmct accurate phase diagrams. The Gibbs energy of formation for a pure substance as a function of temperature must be calculated from experimentally determined temperature-dependent thermodynamic properties such as enthalpy, entropy, heat capacity, and equihbrium constants. By a pure substance, one generally means a stoichiometric compound in which the atomic constituents ate present in an exact, simple reproducible ratio. 

ERD2] Erdos, E., Thermodynamic properties of sulphites 11. Absolute entropies, heat capacities and dissoeiation pressures. Collect. Czech. Chem. Commun., 27, (1962), 2273-2283. Cited on page 186. 

The same methods used to describe activity coefficients here can also be applied to other thermodynamic properties such as excess volumes, enthalpies, entropies, heat capacities, and so on by manipulating the defining equation (17.38) appropriately (see Pitzer, 1987). Experimental data useful in deriving the ion-interaction parameters of... 

XLl.1.3.1 Standard entropy, heat capacity and enthalpy of formation... 

Entropy, Heat Capacity and Hindered Rotation Contribution to Thermodynamic Parameters of the Target Species... 

Thermodynamic properties of solutions are not only useful for estimating the feasibility of reactions in solution, but they also offer one of the better methods of investigating the theoretical aspects of solution structure. This is particularly true for the standard partial molal entropy, heat capacity, and volume of the solutes, values of which are sensitive to the arrangement of solvent molecules around a solute molecule. They have been examined extensively in aqueous solution for the purpose of structure interpretation and more recently in non-aqueous solutions. Enthalpies and free energies of solvation and transfer between... 

The most fundamental starting point for any theoretical approach is the quantum mechanical partition function PF), and the fundamental connection between the partition function and the corresponding thermodynamic potential. Once we have a PF, either exact or approximate, we can derive all the thermodynamic quantities by using standard relationships. Statistical mechanics is a general and very powerful tool to connect between microscopic properties of atoms and molecules, such as mass, dipole moment, polarizability, and intermolecular interaction energy, on the one hand, and macroscopic properties of the bulk matter, such as the energy, entropy, heat capacity, and compressibility, on the other. 

There was, however, one important follow-up paper, by Buff and Brout (1955). The reader may have noticed that the Kirkwood-Buff paper concerns exclusively those properties of solutions that can be obtained from the grand potential by differentiation with respect to pressure or particle number. Those such as partial molar energies, entropies, heat capacities, and so forth, are completely ignored. The original KB theory is an isothermal theory. The Buff-Brout paper completes the story by extending the theory to those properties derivable by differentiation with respect to the temperature. Because these functions can involve molecular distribution functions of higher order than the second, they are not as useful as the original KB theory. Yet they do provide a coherent framework for a complete theory of solution thermodynamics and not just the isothermal part. 

For practical applications it is necessary to have values of thermodynamic functions available, and in particular it is necessary to have values of enthalpy, entropy, heat capacity, and Gibbs energy for a wide range of temperatures and pressures. The critical evaluation of data is a difficult and important art, often made more difficult by the failure of authors of original papers to supply necessary information. Some comments on this subject can be found in a recent article. ... 

Gibbs energies, enthalpies, entropies, heat capacities, and volumes, as well as intensive properties, such as permitlivities or viscosities. The excess functions of extensive properties over those for ideal mixtures of the components, symbolized by y (or the respective increments for intensive quantities, symbolized by AT), are usually defined in terms of the mole fraction composition with respect to the pure components ... 

Equation 17.54 is a useful conclusion. The (translational) partition function, originally defined as an infinite sum of negative exponentials of the energy levels, is equal to an expression in terms of the mass of the gas particles, the absolute temperature, the system volume, and some fundamental universal constants. This expression lets us calculate explicit values for q, which can then be used to determine values for energy, entropy, heat capacity, and so on. These calculated values—determined from a statistical rather than a phenomenological perspective—can then be compared to experimental values. We will thus get the first chance to see how well a statistical approach to thermodynamics compares with experiment. 

The most thorough treatment of uranium and plutonium aquo-ion equilibria over extended temperatures is that of Lemire and Tremaine [71]. This paper uses the systematic relationships developed by Criss and Cobble [72], which relate aquo-ion entropies, heat capacities, and their high-temperature behavior. Although the experimental determination of aquo-ion heat capacities has been dramatically advanced by the development of flow microcalorimeters [73,74], the only measurements of f-block aquo-ion heat capacities were made before this innovation [75,76]. Therefore, Lemire and Tremaine had to rely on estimated heat capacities for almost all of their calculations, and most of their equilibrium constants are uncertain by two or more orders of magnitude. Lemire [77] has also written a report on neptunium aquo-ion equilibria over extended temperatures. 

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