Entropy of transition

Many substances exist in two or more solid allotropic fomis. At 0 K, the themiodynamically stable fomi is of course the one of lowest energy, but in many cases it is possible to make themiodynamic measurements on another (metastable) fomi down to very low temperatures. Using the measured entropy of transition at equilibrium, the measured heat capacities of both fomis and equation (A2.1.73) to extrapolate to 0 K, one can obtain the entropy of transition at 0 K. Within experimental... 

Transient, or time-resolved, techniques measure tire response of a substance after a rapid perturbation. A swift kick can be provided by any means tliat suddenly moves tire system away from equilibrium—a change in reactant concentration, for instance, or tire photodissociation of a chemical bond. Kinetic properties such as rate constants and amplitudes of chemical reactions or transfonnations of physical state taking place in a material are tlien detennined by measuring tire time course of relaxation to some, possibly new, equilibrium state. Detennining how tire kinetic rate constants vary witli temperature can further yield infonnation about tire tliennodynamic properties (activation entlialpies and entropies) of transition states, tire exceedingly ephemeral species tliat he between reactants, intennediates and products in a chemical reaction. 

The procedures described so far imply that So = 0, but do not rigorously prove that this is so. The final proof comes from a comparison of Sr for the ideal gas, obtained from the integration of Cp data assuming the Third Law is valid combined with the entropies of transition, with values obtained from a calculation of St for the ideal gas by statistical methods. The procedure, to be described in detail in Chapter 10, starts with the Boltzmann equation... 

If a phase transition takes place between T = 0 and the temperature of interest, then we also have to include the corresponding entropy of transition by using Eq. 5 and possibly Eq. 4 (Fig. 7.12). For instance, if we want to know the entropy of... 

The discovery of a transition which we identify with this has been reported by Simon, Mendelssohn, and Ruhemann,16 who measured the heat capacity of hydrogen with nA = 1/2 down to 3°K. They found that the heat capacity, after following the Debye curve down to about 11°K, rose at lower temperatures, having the value 0.4 cal/deg., 25 times that of the Debye function, at 3°K. The observed entropy of transition down to 3°K, at which the transition is not completed, was found to be about 0.5 E.U. That predicted by Eq. (15) for the transition is 2.47 E.U. 

For higher diamondoids very limited data are available in the literature. For triamantane, for example, the enthalpy and entropy of transition from crystalline phase II to crystalline phase I are reported [31] as follows ... 

Figure 1.6 Heat capacity of rhombic and monoclinic sulfur [4,5] and the derived entropy of transition between the two polymorphs.

Thermodynamic representation of transitions often represents a challenge. First-order phase transitions are more easily handled numerically than second-order transitions. The enthalpy and entropy of first-order phase transitions can be calculated at any temperature using the heat capacity of the two phases and the enthalpy and entropy of transition at the equilibrium transition temperature. Small pre-tran-sitional contributions to the heat capacity, often observed experimentally, are most often not included in the polynomial representations since the contribution to the... 

Gibbs energy is small. This contribution is instead incorporated empirically in the enthalpy and entropy of transition. 

The two coefficients KL and Ks are derived empirically. They are related through the entropy of transition and constrained to reproduce the total enthalpy and entropy increments accompanying the phase transition. Since, the Inden model demands a series expansion in order to calculate the entropy, a simpler related equation by Hillert and Jarl [21] is used in many computer programs. 

Note 3 Numerical values of the molar entropy of transition should be given as the dimensionless quantity AxyS/R where R is the gas constant. 

The enthalpy of combustion of rhombic sulfur is -70.96 kcal/mole. The enthalpy of combustion of monoclinic sulfur is -70.88 kcal/mole. Calculate the standard enthalpy and entropy of transition from rhombic to monoclinic sulfur. 

The enthalpy of combustion of diamond is -94 50 kcal/mole The enthalpy of combustion of graphite is -94 05 kcal/mole What is the standard enthalpy and entropy of transition from diamond to graphite0... 

It is for this reason (that AG = 0) that we were able to calculate standard entropies of transition by simply dividing the enthalpies of transition by the Kelvin temperature of transition as we did on p 214. Entropies of reaction cannot be calculated this way the more general expression must be used. 

If we want to calculate the entropy of a liquid, a gas, or a solid phase other than the most stable phase at T =0, we have to add in the entropy of all phase transitions between T = 0 and the temperature of interest (Fig. 7.11). Those entropies of transition are calculated from Eq. 5 or 6. For instance, if we wanted the entropy of water at 25°C, we would measure the heat capacity of ice from T = 0 (or as close to it as we can get), up to T = 273.15 K, determine the entropy of fusion at that temperature from the enthalpy of fusion, then measure the heat capacity of liquid water from T = 273.15 K up to T = 298.15 K. Table 7.3 gives selected values of the standard molar entropy, 5m°, the molar entropy of the pure substance at 1 bar. Note that all the values in the table refer to 298 K. They are all positive, which is consistent with all substances being more disordered at 298 K than at T = 0. 

The transitions between the bottom five phases of Fig. 2 may occur close to equilibrium and can be described as thermodynamic first order transitions (Ehrenfest definition 17)). The transitions to and from the glassy states are limited to the corresponding pairs of mobile and solid phases. In a given time frame, they approach a second order transition (no heat or entropy of transition, but a jump in heat capacity, see Fig. 1). 

Table 2. Examples of Heats and Entropies of Transition of Nematic, Smectic, and Discotic Mesophases...

Table 6 contains data on the transition parameters for a large series of main-chain mesogen macromolecules. Figure 15 shows scanning calorimetry data for poly(oxy-2,2 -dimethylazoxybenzene-4,4 -diyloxydodecanedioyl), entry 8 in Table 6. These data should be compared to data on low molecular mass p-butyl-p -methoxyazoxy-benzene which are shown in Fig. 12. Similar comparisons are available for two benzalazines (entries 14 and 16 of Table 6). Although the entropies of transition from the liquid crystal to the isotropic melt are small for all polymers listed as expected, they are larger than those of the corresponding nematic small molecules [1.6, 3.35, and 3.26 J/(K mol), respectively for the small molecules corresponding to entries 8, 14, and 16]. The main-chain nematics seem to have a somewhat larger entropy of transition than the side-chain nematics for larger flexible spacers.

No general rules about the entropy of transitions, as were found for liquid and plastic crystal transitions, can be set up for condis crystals. Two typical examples may illustrate this point. Polytetrafluoroethylene has a relatively small room-temperature transition-entropy on its change to the condis state and a larger transition entropy for final melting. Polyethylene has, in contrast, a higher condis crystal transition entropy than melting entropy (see Sect. 5.3.2). 

Poly (diethyl siloxane) was suggested by Beatty et al. 1651 based on DSC, dielectric, NMR, and X-ray measurements to possess liquid crystalline type order between about 270 and 300 K. The macromolecule shows two large lower temperature first order transitions, one at about 200 K, the other at about 270 K166 ll,7). The transition of the possible mesophase to the isotropic liquid at 300 K is quite small and irre-producible, so that variable, partial crystallinity was proposed 165) [measured heat of transition about 150 J/mole1S8)], Very little can be said about this state which may even consist of residual crystals. It is of interest, however, to further analyze the high temperature crystal phase between 200 and 270 K. It is produced from the, most likely, fully ordered crystal with an estimated heat and entropy of transition of 5.62kJ/mol and 28J/(Kmol), respectively [calculated from calorimetric data 1S6)... 

Table II. Enthalpy and Entropy of Transitions from Contact Ion Pair to Solvent-Separated Ion Pair, and of Dissociation of Solvent-Separated Ion Pair (K Dis ) of Polystyrylsodium...

The entropy of transition, AS can also be obtained directly from an integration of (ACp/T)dT. As shown in Figure 16.8, the calorimetric method also provides a direct measure of the change in the heat capacity, AtransQ. of the two macrostates at Tm. That is... 

H thermal average distance at the temperature of Atrs 5° entropy of transition... 

Different assumptions for s were proposed in the literature. Barton and Lee (1968) suggested s to be equal to the weight- or mole fraction of the relevant group in relation to the structural unit. Weyland et al. (1970) put s equal to Z , the number of backbone atoms of the contributing group. Becker (1978) and Kreibich and Batzer (1979) identified s, with the number of freely and independently oscillating elements in the backbone of the structural unit. In general s is held as a kind of "entropy of transition". 

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