Absolute entropy data

Enthalpy of Formation and Absolute Entropy Data for Hydrogen-Oxygen Fuel Cell... 

Find AH°, AS°, and AG° at 298.15 K for each of the following reactions, using formation data and absolute entropy data. Compare the value of AH° — TAS° with that of AG° to test the consistency of the data. 

Draw the curve of Cp vs. T and Cp/T vs. T from the following heat capacity data for solid chlorine and determine the absolute entropy of solid chlorine at 70.0 K... 

A cryogenic calorimeter measures Cp,m as a function of temperature. We have seen that with the aid of the Third Law, the Cp,m data (along with AHm for phase changes) can be integrated to give the absolute entropy... 

The standard states of these materials are taken as the pure components at 298 °K and a pressure of 101.3 kPa. The following data on the absolute entropies of the hydrocarbons at 298 °K are available. 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

In practice, then, it is fairly straightforward to convert the potential energy determined from an electronic structure calculation into a wealth of thennodynamic data - all that is required is an optimized structure with its associated vibrational frequencies. Given the many levels of electronic structure theory for which analytic second derivatives are available, it is usually worth the effort required to compute the frequencies and then the thermodynamic variables, especially since experimental data are typically measured in this form. For one such quantity, the absolute entropy 5°, which is computed as the sum of Eqs. (10.13), (10.18), (10.24) (for non-linear molecules), and (10.30), theory and experiment are directly comparable. Hout, Levi, and Hehre (1982) computed absolute entropies at 300 K for a large number of small molecules at the MP2/6-31G(d) level and obtained agreement with experiment within 0.1 e.u. for many cases. Absolute heat capacities at constant volume can also be computed using the thermodynamic definition... 

Unfortunately, there axe at this time no low temperature heat capacity data on polymers of known crystallinity so that absolute entropies at 25° C can be calculated. This gap in our knowledge of polymers represents a scientific vacuum that will rapidly be filled1. 

Using the data in Appendix C, calculate ArxnH° from heats of formation, ArxnS° from absolute entropies and ArxnG° from free energies of formation for the reaction 2CO + 02 —> 2C02 at 298 K. Verify that your results satisfy ArxnG° = AmH°— TkrxnS°. 

Using standard absolute molar entropy data, S° Equation (11.10) applies. 

Necessary data for estimating the standard heat of formation and the absolute entropy by Equations 3 and 4 are the elemental analysis, structural parameters, fa and a, and the normal boiling point. For a practical purpose, it will be more convenient if we could calculate AHf° and S° only from elemental analysis data and normal boiling point. The aromaticity fa for coal liquids may be estimated by the correlation shown in Figure 1. On the other hand, the value of 0 may be taken as 0.3 for its average value based on the reported data (lf3, 20, 21). Substitution of these relations into Equations 3 and 4 gives... 

The standard heat of formation AHf ° and absolute entropy S° of the substances appeared in this process have been estimated by the proposed methods and are tabulated in Table V, where AHf ° and S° for each substance were evaluated for the value per unit mole of carbon in the substance. AHf° and S° for residuum were estimated by formulas for coal, i.e., by Equations 1 and 2. The heat capacity of the coal-derived liquids was estimated by refering the measured data by Lee and Bechtold (4). 

Measurements of heat capacities at very low temperatures provide data for the calculation from Eq. (5.11) of entropy changes down to 0 K. When these calculations are made for different crystalline forms of the same chemical species, the entropy at 0 K appears to be the same for all forms. When the form is noncrystalline, e.g., amorphous or glassy, calculations show that the entropy of the more random form is greater than that of the crystalline form. Such calculations, which are summarized elsewhere,t lead to the postulate that the absolute entropy is zero for all perfect crystalline substances at absolute zero temperature. While the essential ideas were advanced by Nemst and Planck at the beginning of the twentieth century, more recent studies at very low temperatures have increased our confidence in this postulate, which is now accepted as the third law. 

If the entropy is zero at T = 0 K, then Eq. (5.11) lends itself to the calculation of absolute entropies. With T — 0 as the lower limit of integration, the absolute entropy of a gas at temperature T based on calorimetric data follows from Eq. (5.11) integrated to give ... 

Both the classical and statistical equations [Eqs. (5.22) and (5.23)] yield absolute values of entropy. Equation (5.23) is known as the Boltzmann equation and, with Eq. (5.20) and quantum statistics, has been used for calculation of entropies in the ideal-gas state for many chemical species. Good agreement between these calculations and those based on calorimetric data provides some of the most impressive evidence for the validity of statistical mechanics and quantum theory. In some instances results based on Eq. (5.23) are considered more reliable because of uncertainties in heat-capacity data or about the crystallinity of the substance near absolute zero. Absolute entropies provide much of the data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 15. 

Calculate the absolute entropy of liquid n-hexanol at 20°C (68°F) and 1 atm (101.3 kPa) from these heat-capacity data ... 

Estimate absolute entropy of crystalline n-hexanol. Since no experimental data are available below 18.3 K, estimate the entropy change below this temperature using the Debye-Einstein equation. Use the crystal entropy value of 1.695 cal/(g mol)(°K) at 18.3 K to evaluate the coefficient a. Hence a = 1.695/18.33 = 0.2766 x 10 3. The A term in Eq. 1.1 therefore is... 

FIGURE 1.10 Calculation of absolute entropy from heat-capacity data (Example 1.16). 

Values of AG° for matty formation reactions are tabulated in standard references. The reported values of AG are not measured experimentally, but are calculated by Eq. (13.16). The detennination of A5 may be based on the tliird law of thennodynamics, discussed in Sec. 5.10. Combination of valnes from Eq. (5.40) for the absolute entropies of the species taking part in the reaction gives the valne of AS. Entropies (and heat capacities) are also commonly determined from statistical calcnlations based on spectroscopic data. ... 

Equation (16.20) for the molar entropy of an ideal gas allows calculation of absolute entropies for tile ideal-gas state. The data required for evaluation of the last two terms on tlie right are tlie bond distances and bond angles in the molecules, and the vibration frequencies associated witli tlie various bonds, as determined from spectroscopic data. The procedure lias been very successful in the evaluation of ideal-gas entropies for molecules whose atomic stractures are known. 

The third-law method is based on a knowledge of the absolute entropy of the reactants and products. It allows the calculation of a reaction enthalpy from each data point when the change in the Gibbs energy function for the reaction is known. The Gibbs energy function used here is defined as... 

Tables for this defined reference state, including the heat capacity, the heat content relative to 298.15° K., the absolute entropy, and the free energy function at even 100° intervals from 298.15° to 3000° K. have b n assembled for the first 92 elements. These tables are arranged alphabetically beginning on page 36. The choice of 298.15° K. as the reference temperature is made because the low temperature heat capacities of many elements and compounds are not known. Most of the thermodynamic data now reported in the literature refer to 25° C., which, when combined with the recent international agreement on 273.15° K. for the ice point (319) gives a reference temperature of 298.15° K. The figure 298° K. quoted in the tables and text should be understood to be the reference temperature, 298.15° K. For those who prefer to use 0° K. as the reference temperature, we have included, for cases in which it is known, the heat content at 298.15° K. relative to 0° K.

The Gibbs energy of formation of PbSe(cr) was measured in the temperature ranges 490 to 600 K and 650 to 858 K using electrochemical cells. The determined enthalpies and entropies of reaction at the mean temperatures of the temperature intervals are given in Table A-95. The quantities were recalculated to 298.15 K by the review and are tabulated in Table A-96. In addition, the corresponding values of the absolute entropy of PbSe(cr) at 298.15 K were calculated. All calculations employed the selected heat capacity of PbSe(cr), the selected thermodynamic properties of selenium, and the data for Pb(cr, I) in [89COX/WAG]. 

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