Temperature dependence, entropy

Experimental data are presented to show that the JG relaxation mimics the structural relaxation in its volume-pressure and entropy-temperature dependences, as well as changes in physical aging. These features indicate that the dependences of molecular mobility on volume-pressure and entropy-temperature have entered into the faster JG relaxation long before structural relaxation, suggesting that the JG relaxation must be considered in any complete theory of the glass transition. 

The Arrhenius relation given above for Are temperature dependence of air elementary reaction rate is used to find Are activation energy, E, aird Are pre-exponential factor. A, from the slope aird intercept, respectively, of a (linear) plot of n(l((T)) against 7 The stairdard enAralpv aird entropy chairges of Are trairsition state (at constairt... 

Figure 1.4. Temperature dependence of the change in Gihhs energy, enthalpy and entropy upon transfer of ethane and butane from the gas phase to water. The data refer to transfer from the vapour phase at 0.101 MPa to a hypothetical solution of unit mole fraction and are taken from ref. 125.

NMR signals of the amino acid ligand that are induced by the ring current of the diamine ligand" ". From the temperature dependence of the stability constants of a number of ternary palladium complexes involving dipeptides and aromatic amines, the arene - arene interaction enthalpies and entropies have been determined" ". It turned out that the interaction is generally enthalpy-driven and counteracted by entropy. Yamauchi et al. hold a charge transfer interaction responsible for this effect. 

Here we have the formation of the activated complex from five molecules of nitric acid, previously free, with a high negative entropy change. The concentration of molecular aggregates needed might increase with a fall in temperature in agreement with the characteristics of the reaction already described. It should be noticed that nitration in nitromethane shows the more common type of temperature-dependence (fig. 3.1). 

The temperature dependence of reaction rates permits evaluation of the enthalpy and entropy components of the free energy of activation. The terms in Eq. (4.4) corresponding to can be expressed as... 

Solubility parameters are generally tabulated, together with the corresponding liquid molar volumes, only at 25°C. Although solubility parameters are themselves temperature-dependent, the combination of quantities in Eq. 70 is not. Differentiating Eq. 70 with respect to temperature gives — the excess entropy, a quantity which has been assumed to be zero in accord with the definition of a regular solution. Thus only data at 25°C are needed. Solubility parameters may be... 

A. Milchev, I. Gutzow. Temperature dependence of the configurational entropy of undercooled melts and the nature of the glass transition. J Macromol Sci B 22 583-615, 1982. 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

The free-energy change AG is made up of two terms, an enthalpy term, AH, and a temperature-dependent entropy term, TAS. Of the two terms, the enthalpy term is often larger and more dominant. 

The partial molar entropy of a component may be measured from the temperature dependence of the activity at constant composition the partial molar enthalpy is then determined as a difference between the partial molar Gibbs free energy and the product of temperature and partial molar entropy. As a consequence, entropy and enthalpy data derived from equilibrium measurements generally have much larger errors than do the data for the free energy. Calorimetric techniques should be used whenever possible to measure the enthalpy of solution. Such techniques are relatively easy for liquid metallic solutions, but decidedly difficult for solid solutions. The most accurate data on solid metallic solutions have been obtained by the indirect method of measuring the heats of dissolution of both the alloy and the mechanical mixture of the components into a liquid metal solvent.05... 

It may reasonably be assumed that the terms in the expression for the entropy which depend on the temperature diminish, like the entropy of a chemically homogeneous condensed phase, to zero when T approaches zero, and the entropy of a condensed solution phase at absolute zero is equal to that part of the expression for the entropy which is independent of temperature, and depends on the composition (Planck, Thennodynamik, 3 Aufi., 279). 

One of the more interesting results of these calculations is the contribution to the heat capacity. Figure 10.10 shows the temperature dependence of this contribution to the heat capacity for CH3-CCU as calculated from Pitzer s tabulation with 7r = 5.25 x 10-47 kg m2 and VQ/R — 1493 K. The heat capacity increases initially, reaches a maximum near the value expected for an anharmonic oscillator, but then decreases asymptotically to the value of / expected for a free rotator as kT increases above Vo. The total entropy calculated for this molecule at 286.53 K is 318.86 J K l-mol l, which compares very favorably with the value of 318.94T 0.6 TK-1-mol 1 calculated from Third Law measurements.7... 

Clearly, the temperature profile is linear. The activation parameters are the sums shown in general, a sum of entropies and enthalpies is the result when constants are multiplied. If values of AS% and Aare known independently, from the temperature dependence of Ka for example, one can then calculate AS and AH by difference. 

The expression derived for the wet glass transition, Tgw, assuming the configurational entropy as the temperature dependent function30) ist ... 

Equilibrium vapor pressures were measured in this study by means of a mass spectrometer/target collection apparatus. Analysis of the temperature dependence of the pressure of each intermetallic yielded heats and entropies of sublimation. Combination of these measured values with corresponding parameters for sublimation of elemental Pu enabled calculation of thermodynamic properties of formation of each condensed phase. Previ ly reported results on the subornation of the PuRu phase and the Pu-Pt and Pu-Ru systems are correlated with current research on the PuOs and Pulr compounds. Thermodynamic properties determined for these Pu-intermetallics are compared to analogous parameters of other actinide compounds in order to establish bonding trends and to test theoretical predictions. 

In addition to chemical reactions, the isokinetic relationship can be applied to various physical processes accompanied by enthalpy change. Correlations of this kind were found between enthalpies and entropies of solution (20, 83-92), vaporization (86, 91), sublimation (93, 94), desorption (95), and diffusion (96, 97) and between the two parameters characterizing the temperature dependence of thermochromic transitions (98). A kind of isokinetic relationship was claimed even for enthalpy and entropy of pure substances when relative values referred to those at 298° K are used (99). Enthalpies and entropies of intermolecular interaction were correlated for solutions, pure liquids, and crystals (6). Quite generally, for any temperature-dependent physical quantity, the activation parameters can be computed in a formal way, and correlations between them have been observed for dielectric absorption (100) and resistance of semiconductors (101-105) or fluidity (40, 106). On the other hand, the isokinetic relationship seems to hold in reactions of widely different kinds, starting from elementary processes in the gas phase (107) and including recombination reactions in the solid phase (108), polymerization reactions (109), and inorganic complex formation (110-112), up to such biochemical reactions as denaturation of proteins (113) and even such biological processes as hemolysis of erythrocytes (114). 

Theoretically, the problem has been attacked by various approaches and on different levels. Simple derivations are connected with the theory of extrathermodynamic relationships and consider a single and simple mechanism of interaction to be a sufficient condition (2, 120). Alternative simple derivations depend on a plurality of mechanisms (4, 121, 122) or a complex mechanism of so called cooperative processes (113), or a particular form of temperature dependence (123). Fundamental studies in the framework of statistical mechanics have been done by Riietschi (96), Ritchie and Sager (124), and Thorn (125). Theories of more limited range of application have been advanced for heterogeneous catalysis (4, 5, 46-48, 122) and for solution enthalpies and entropies (126). However, most theories are concerned with reactions in the condensed phase (6, 127) and assume the controlling factors to be solvent effects (13, 21, 56, 109, 116, 128-130), hydrogen bonding (131), steric (13, 116, 132) and electrostatic (37, 133) effects, and the tunnel effect (4,... 

As to the computation of reaction enthalpies and entropies, AH and AS , the same arguments apply if they have been obtained from the temperature dependence of the equilibrium constant. A different situation arises vdien AH is determined directly from calorimetry, say with a constant relative error 6. The standard entropy AS then has the standard error... 

Many workers have offered the opinion that the isokinetic relationship is confined to reactions in condensed phase (6, 122) or, more specially, may be attributed to solvation effects (13, 21, 37, 43, 56, 112, 116, 124, 126-130) which affect both enthalpy and entropy in the same direction. The most developed theories are based on a model of the half-specific quasi-crystalline solvation (129, 130), or of the nonideal conformal solutions (126). Other explanations have been given in terms of vibrational frequencies involving solute and solvent (13, 124), temperature dependence of solvent fluidity in the quasi-crystalline model (40), or changes of enthalpy and entropy to produce a hole in the solvent (87). 

The third law of thermodynamics establishes a starting point for entropies. At 0 K, any pure perfect crystal is completely constrained and has S = 0 J / K. At any higher temperature, the substance has a positive entropy that depends on the conditions. The molar entropies of many pure substances have been measured at standard thermodynamic conditions, P ° = 1 bar. The same thermodynamic tables that list standard enthalpies of formation usually also list standard molar entropies, designated S °, fbr T — 298 K. Table 14-2 lists representative values of S to give you an idea of the magnitudes of absolute entropies. Appendix D contains a more extensive list. 

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