Entropy from heat capacity

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

Allen s book discusses equilibrium constants, entropy in terms of disorder, and formulae for the derivation of values of entropy from heat capacity measurements. An important part of the book discusses energy changes in relation to the Periodic classification and to chemical bonding. Gibbs energies of formation are plotted for some series of compounds (e.g. oxides and chlorides) against atomic number. There is a thermodynamic study of the problem of selection of a catalyst for the Deacon process. 

The following tables of properties of carbon dioxide are available enthalpy, entropy, and heat capacity at 0 and 5 MPa (0 and 50 atm, respectively) from 273 to 1273 K pressure—volume product (PV), enthalpy, and isobaric heat capacity (C from 373 to 1273 K at pressures from 5 to 140 MPa (50-1,400 atm) (14). 

The working equations for osmotic and activity coefficients, derived from equation (3) are given as equations (4) and (5), respectively. The various secondary relationships are defined in several additional equations stated and briefly described thereafter. Additional details and derivations of equations for the entropy, the heat capacity, and other related functions can be found in various published papers (11, 20, 23-29, 32-34). 

References (20, 22, 23, 24, 29, and 74) comprise the series of Technical Notes 270 from the Chemical Thermodynamics Data Center at the National Bureau of Standards. These give selected values of enthalpies and Gibbs energies of formation and of entropies and heat capacities of pure compounds and of aqueous species in their standard states at 25 °C. They include all inorganic compounds of one and two carbon atoms per molecule. 

Table III gives values of the changes in Gibbs energy, enthalpy, entropy, and heat capacity of the solution process as calculated from the equations of Table I. Figure 1 shows the recommended noble gas mole fraction solubilities at unit gas partial pressure (atm) as a function of temperature. The temperature of minimum solubility is marked.

RADICALC Bozzelli, J. W. and Ritter, E. R. Chemical and Physical Processes in Combustion, p. 453. The Combustion Institute, Pittsburgh, PA, 1993. A computer code to calculate entropy and heat capacity contributions to transition states and radical species from changes in vibrational frequencies, barriers, moments of inertia, and internal rotations. 

OK can be computed from heat capacity measurements [8] for each crystalline form from near 0 K to the transition temperature (368.6 K) and the heat of transition. The result is zero within experimental error. Hence, both rhombic and monoclinic sulfur ate assigned zero entropy at 0 K. 

Calculation of Entropy and Heat Capacity from Statistical Mechanics... 

Calculation of entropy and Gibbs energy change from heat capacities From Eq. (3.7) and for the limiting case of an infinitely small step in temperature... 

From a computational point of view, the heat of formation, which is derived from the electronic energy of the molecule molecule> is the most difficult thermochemical quantity to predict accurately. Entropies and heat capacities are derived from vibration and rotational constants, all of which can be predicted with considerable accuracy using relatively low levels of theory. Thus, the development of ab initio methods appropriate for a new class of compounds focuses primarily on identifying a level of theory and the basis set(s) needed to achieve sufficient accuracy in the electronic energy [67,68]. 

With the discussion of the free-energy function G in this chapter, all of the thermodynamic functions needed for chemical equilibrium and kinetic calculations have been introduced. Chapter 8 discussed methods for estimating the internal energy E, entropy S, heat capacity Cv, and enthalpy H. These techniques are very useful when the needed information is not available from experiment. 

The result is a discrepancy in the entropy calculated from heat capacity measurements using the Third Law for substances such as CO, N2O, and H2O. 

The thermodynamic properties of a substance in the state of ideal gas are calculated as the sums of contributions from translation and rotation of a molecule as a whole, vibrations and internal rotation in the molecule, and electronic excitation. For example, for entropy and heat capacity the following equations hold ... 

All the normal modes are present in the results of a semiempirical frequency calculation, as is the case for an ab initio or DFT calculation, and animation of these will usually give, approximately, the frequencies of these modes. A very extensive compilation of experimental, MNDO and AMI frequencies has been given by Healy and Holder, who conclude that the AMI error of 10% can be reduced to 6% by an empirical correction, and that entropies and heat capacities are accurately calculated from the frequencies [104], In this regard, Coolidge et al. conclude -surprisingly, in view of our results for the four molecules in Figs. 6.5-6.8 - from a study of 61 molecules that (apart from problems with ring- and heavy atom-stretch for AMI and S-H, P-H and O-H stretch for PM3) both AMI and PM3 should provide results that are close to experimental gas phase spectra [105]. 

An explicit expression relating kinetic fragility to thermodynamic behavior of supercooled liquids was accomplished for the first time by Mohanty and coworkers [55,56] and independently by Speedy [54], These authors derived an expression for the steepness parameter, a measure of kinetic fragility, from the temperature variation of the relation time or viscosity, with the ratio of excess entropy and heat capacity changes at the glass transition temperature [54-56]. A detailed description of this work will be provided later in the review chapter. 

Related Calculations. This general procedure may be used to calculate entropy values from heat-capacity data. However, in many situations involving practical computations, the entropy changes rather than absolute values are required. In such situations, the A term in Eq. 1.1 may not be needed. Entropy changes associated with phase changes, such as melting and vaporization, can be evaluated from the AH/T term (see Example 1.15). 

To obtain ACp they subtracted their measured values from heat capacities of the bulk. The surface enthalpy at 298 K / 298 was taken from solubility experiments. For MgO and using /i 298 = I040 mN/m they obtained a surface tension of 957 mN/m and a surface entropy of 0.28 mNJm-lK l. 

Each of the concentration-dependent G terms in the equations presented earlier can have a temperature dependency given by Eq. 11.38. Similar equations can be written for other thermodynamic properties, from which the Gibbs energy can be computed, such as the enthalpy of formation, entropy, and heat capacity. Equation 11.38 is a much more efficient way of incorporating information into a software database than tables contaming discrete values, which is important for minimizing computer resource requirements. 

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