Order, entropy

Continuing with the mini-theme of computational materials chemistry is Chapter 3 by Professor Thomas M. Truskett and coworkers. As in the previous chapters, the authors quickly frame the problem in terms of mapping atomic (chemical) to macroscopic (physical) properties. The authors then focus our attention on condensed media phenomena, specifically those in glasses and liquids. In this chapter, three properties receive attention—structural order, free volume, and entropy. Order, whether it is in a man-made material or found in nature, may be considered by many as something that is easy to spot, but difficult to quantify yet quantifying order is indeed what Professor Truskett and his coauthors describe. Different types of order are presented, as are various metrics used for their quantification, all the while maintaining theoretical rigor but not at the expense of readability. The authors follow this section of their... 

The enthalpies and entropies of formation of mono-mandelato-complexes have been determined and, in comparison with other hydroxycarboxylic acid complexes, the enthalpy order of stabilization is lactate > a-hydroxyiso-butyrate mandelate > glycolate, whereas the entropy order of stabilization is glycolate > a-hydroxyisobutyrate > mandelate > lactate. The stability constants and enthalpy of formation of mono- and di-malonate complexes have also been measured.The mono-1,1-cyclopentanedicarboxylato-complexes are less stable than the corresponding malonate species. 

We can make this somewhat more quantitative. According to Landau and Lifshitz, the probability of a fluctuation whose change in entropy (order) is As is proportional to exp (As/k) here As is not the molar entropy of the transition, the AS of Eq. [10], but the entropy of ordering of a small region of the material. If this small region contains n molecules (which order to a degree normally displayed by a nematic just at T ), then As can be related to ASy and the probability of such a fluctuation becomes proportional to... 

The surfaces in which the paths satisfying the condition = 0 must lie are, thus, surfaces of constant entropy they do not intersect and can be arranged in an order of increasing or decreasmg numerical value of the constant. S. One half of the second law of thennodynamics, namely that for reversible changes, is now established. 

The principle of tire unattainability of absolute zero in no way limits one s ingenuity in trying to obtain lower and lower thennodynamic temperatures. The third law, in its statistical interpretation, essentially asserts that the ground quantum level of a system is ultimately non-degenerate, that some energy difference As must exist between states, so that at equilibrium at 0 K the system is certainly in that non-degenerate ground state with zero entropy. However, the As may be very small and temperatures of the order of As/Zr (where k is the Boltzmaim constant, the gas constant per molecule) may be obtainable. 

Phase transitions at which the entropy and enthalpy are discontinuous are called first-order transitions because it is the first derivatives of the free energy that are disconthuious. (The molar volume V= (d(i/d p) j is also discontinuous.) Phase transitions at which these derivatives are continuous but second derivatives of G... 

In order to separate the enthalpy and the entropy of activation, the rate is measured as a fiinction of temperature. These data should give a straight line on an Eyrmg plot of log(rate/7) against (1/7) (figure... 

This can be inserted in equation (02.2.3) to give tlie orientational distribution function, and tlius into equation (02.2.6) to deteniiine the orientational order parameters. These are deteniiined self-consistently by variation of tlie interaction strength iin equation (c2.2.7). As pointed out by de Gemies and Frost [20] it is possible to obtain tlie Maier-Saupe potential from a simple variational, maximum entropy metliod based on tlie lowest-order anisotropic distribution function consistent witli a nematic phase. 

If one would ask a chemist not burdened with any knowledge about the peculiar thermodynamics that characterise hydrophobic hydration, what would happen upon transfer of a nonpolar molecule from the gas phase to water, he or she would probably predict that this process is entropy driven and enthalpically highly unfavourable. This opinion, he or she wo ild support with the suggestion that in order to create room for the nonpolar solute in the aqueous solution, hydrogen bonds between water molecules would have to be sacrificed. 

Solubility in Water A familiar physical property of alkanes is contained m the adage oil and water don t mix Alkanes—indeed all hydrocarbons—are virtually insoluble m water In order for a hydrocarbon to dissolve m water the framework of hydrogen bonds between water molecules would become more ordered m the region around each mole cule of the dissolved hydrocarbon This increase m order which corresponds to a decrease m entropy signals a process that can be favorable only if it is reasonably... 

The separation of Hquid crystals as the concentration of ceUulose increases above a critical value (30%) is mosdy because of the higher combinatorial entropy of mixing of the conformationaHy extended ceUulosic chains in the ordered phase. The critical concentration depends on solvent and temperature, and has been estimated from the polymer chain conformation using lattice and virial theories of nematic ordering (102—107). The side-chain substituents govern solubiHty, and if sufficiently bulky and flexible can yield a thermotropic mesophase in an accessible temperature range. AcetoxypropylceUulose [96420-45-8], prepared by acetylating HPC, was the first reported thermotropic ceUulosic (108), and numerous other heavily substituted esters and ethers of hydroxyalkyl ceUuloses also form equUibrium chiral nematic phases, even at ambient temperatures. 

Similarly, the second-order derivative can be shown to be zero (see, Problems, Section 2.20). Evaluating the third term gives an expression for entropy gain along the Hugoniot... 

Thus, to third order in strain, the entropy along the Hugoniot is constant, and weak shock waves are nearly isentropic. For small strains, the Hugoniot can be replaced with the isentrope to a high degree of accuracy. At the initial state, the Hugoniot and isentrope have the same slope and curvature in the P-V plane. 

Driving forces for solid-state phase transformations are about one-third of those for solidification. This is just what we would expect the difference in order between two crystalline phases will be less than the difference in order between a liquid and a crystal the entropy change in the solid-state transformation will be less than in solidification and AH/T will be less than AH/T . 

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