Entropy of transformation

To calculate the entropy of a substance at a temperamre at which it is no longer a sohd, it is necessary to add the entropy of transformation to a hquid or gas and the subsequent entropies of warming. The same procedure would apply to a solid that exists in different crystalline forms as the temperature is increased. The procedure can be illustrated by some sample calculations. 

Figure 6.4. (a) Variation of the entropy of fusion with melting point for different crystal structures (from Saunders el at. 1988) and (b) schematic illustration of the possibility of a change in value and sign for the entropy of transformation if the metastable structure has a low melting point (from Miodownik 1992). 

A similar situation has been described for the two polymorphic forms of 2-[[4-[[2-(IH-tetrazol-5-ylmethyl)phenyl]methoxy]phenoxy]methyl]quinoline [20]. The appreciable enthalpic driving force for the transformation of Form II to Form I (-0.91 kcal/mol) was found to be partially offset by the entropy of transformation (-2.6 cal/K mol), resulting in a modest free energy difference between the two forms (-0.14 kcal/mol). 

In McColm s model, the entropy of transformation for the dicarbides ASj ns was presumed to be constant for all CaC2-type carbides. However, as was evident from the thermal data of some rare earth dicarbides (Adachi et al. 1974, 1976, 1978), the observed value of for the solid solution changes with its composition. 

Another contribution involved the measurement of the high-temperature heat contents of the rare earth metals. An analysis of the data revealed that the entropy of transformation, AS, for the close packed to bcc transformation in the rare earth metals depended upon the number of valence electrons (Dennison et al. 1966c), i.e. AS,r 0.2 e.u. per valence electron. Extension of these observations to the remainder of the periodic table indicated that the entropy of transformation of metals which posses two or more of the common metallic structures and the entropy of fusion depend upon both the crystalline structure of the phases involved and the number of valence electrons (Gschneidner 1975). These results were then used to predict the entropies of fusion for 16 metals, including two lanthanide metals - promethium and lutetium - and the entropies of transformation of 5 metals including promethium, for which no reliable experimental values existed. 

If any solid-solid, solid-liquid, solid-gas, or liquid-gas transformation enters the calculation of the Gibbs energy of a particular reaction, the enthalpies and entropies of transformation must also be known. The following quantities are then tabulated to describe the thermodynamics of a particular chemical compound in full ... 

Thermodynamics and Phase Diagrams 5 Table 1.1 Typical values for the heat and entropy of transformation. 

Kind of transformation Heat of transformation A Htr [kj/mole] Entropy of transformation trlK.molel... 

Table 1.1 shows typical values being valid for the heats and entropies of transformation. The latent heats cover a range of two orders of magnitude, and we will see later the distinctive consequences on the kind and appearance of phase diagrams. 

This result holds equally well, of course, when R happens to be the operator representing the entropy of an ensemble. Both Tr Wx In Wx and Tr WN In WN are invariant under unitary transformations, and so have no time dependence arising from the Schrodinger equation. This implies a paradox with the second law of thermodynamics in that apparently no increase in entropy can occur in an equilibrium isolated system. This paradox has been resolved by observing that no real laboratory system can in fact be conceived in which the hamiltonian is truly independent of time the uncertainty principle allows virtual fluctuations of the hamiltonian with time at all boundaries that are used to define the configuration and isolate the system, and it is easy to prove that such fluctuations necessarily increase the entropy.30... 

The solid product, BaO, was apparently amorphous and porous. Decomposition rate measurements were made between the phase transformation at 1422 K and 1550 K (the salt melts at 1620 K). The enthalpy and entropy of activation at 1500 K (575 13 kJ mole-1 and 200 8 J K"1 mole-1) are very similar to the standard enthalpy and entropy of decomposition at the same temperature (588 7 kJ and 257 5 J K-1, respectively, referred to 1 mole of BaS04). The simplest mechanistic explanation of the observations is that all steps in the reaction are in equilibrium except for desorption of the gaseous products, S02 and 02. Desorption occurs over an area equivalent to about 1.4% of the total exposed crystal surface. Other possible models are discussed. 

In general case, where the substance undergoes a variety of transformations as shown in the Figure 3.2, the entropy of the temperature, T, ST, becomes that given by the following equation ... 

As we mentioned, it is necessary to have information about the standard enthalpy change for a reaction as well as the standard entropies of the reactants and products to calculate the change in Gibbs function. At some temperature T, A// j can be obtained from Af/Z of each of the substances involved in the transformation. Data on the standard enthalpies of formation are tabulated in either of two ways. One method is to list Af/Z at some convenient temperature, such as 25°C, or at a series of temperatures. Tables 4.2 through 4.5 contain values of AfZ/ at 298.15 K. Values at temperatures not listed are calculated with the aid of heat capacity equations, whose coefficients are given in Table 4.8. 

In addition to the melting point of the P phase and the a/P allotropic transfonnation temperature in Fig. 6.1(b), there is a fluther intersection between the Gibbs energy of a and liquid phases. This corresponds to the metastable melting point of the a phase. A linear model will then dictate that the entropy of melting for a is defined by the entropies of melting and transformation at the two other critical points (Ardell 1963),... 

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