Entropy description

Free Volume Versus Configurational Entropy Descriptions of Glass Formation Isothermal Compressibility, Specific Volume, Shear Modulus, and Jamming Influence of Side Group Size on Glass Formation Temperature Dependence of Structural Relaxation Times Influence of Pressure on Glass Formation... 

VI. FREE VOLUME VERSUS CONFIGURATIONAL ENTROPY DESCRIPTIONS OF GLASS FORMATION... 

Figure 7.72 illustrates a large number of glass transition data of polymer solutions with comparisons to the Gibbs-DiMarzio (DM), Fox (F), and Schneider (S) equations described in Fig. 7.69 [30]. The upper left displays two sets of literature data on poly(vinylidene fluoride)-poly(methyl methacrylate) solutions (B,A). The glass transition shows a positive deviation from simple additivity of the properties of the pure components, which can only be represented with the help of the indicated interaction parameters of the Schneider equation. The lower left set of data illustrates poly(oxyethylene)-poly (methyl methacrylate) solutions (, o). They are well described by all three of the equations, indicating rather small specific interactions and great similarity between volume and entropy descriptions.

It is well known that organic plasticizers decrease a polymer s glass-transition temperature, as described by previous researchers via entropy continuity, volume continuity, free volume concepts, and the conformational entropy description of when... 

Fokker-Planck operator, the first term in eqn (13.11). This result is in line with the common sense view that the Hamiltonian description at microscopic level can be replaced by the free energy and entropy description in non-equilibrium statistical dynamics. 

Thus the thermodynamic description of the Langmuir model is that the energy of adsorption Q is constant and that the entropy of adsorption varies with 6 according to Eq. XVII-37. 

Neither the thermodynamic nor the rheological description of surface mobility has been very useful in the case of chemisorbed films. From the experimental point of view, the first is complicated by the many factors that can affect adsorption entropies and the latter by the lack of any methodology. 

Thermodynamics is one of the most well-developed mathematical descriptions of chemistry. It is the held of thermodynamics that dehnes many of the concepts of energy, free energy and entropy. This is covered in physical chemistry text books. 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

Various equations of state have been developed to treat association ia supercritical fluids. Two of the most often used are the statistical association fluid theory (SAET) (60,61) and the lattice fluid hydrogen bonding model (LEHB) (62). These models iaclude parameters that describe the enthalpy and entropy of association. The most detailed description of association ia supercritical water has been obtained usiag molecular dynamics and Monte Carlo computer simulations (63), but this requires much larger amounts of computer time (64—66). 

To conclude this section let us note that already, with this very simple model, we find a variety of behaviors. There is a clear effect of the asymmetry of the ions. We have obtained a simple description of the role of the major constituents of the phenomena—coulombic interaction, ideal entropy, and specific interaction. In the Lie group invariant (78) Coulombic attraction leads to the term -cr /2. Ideal entropy yields a contribution proportional to the kinetic pressure 2 g +g ) and the specific part yields a contribution which retains the bilinear form a g +a g g + a g. At high charge densities the asymptotic behavior is determined by the opposition of the coulombic and specific non-coulombic contributions. At low charge densities the entropic contribution is important and, in the case of a totally symmetric electrolyte, the effect of the specific non-coulombic interaction is cancelled so that the behavior of the system is determined by coulombic and entropic contributions. 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

In Chapter 1, we describe the fundamental thermodynamic variables pressure (p), volume (V), temperature (T), internal energy ((/), entropy (5), and moles (n). From these fundamental variables we then define the derived variables enthalpy (//), Helmholtz free energy (A) and Gibbs free energy (G). Also included in this chapter is a review of the verbal and mathematical language that we will rely upon for discussions and descriptions in subsequent chapters. 

It is important to note that, for important sub-cases of case /), which will be discussed in more detail in Sect. 2.4, there is a low extent of disorder entropy effects, if any, are small and changes of the lattice dimensions are absent or small. These particular disordered forms are not considered as mesomorphic. In such cases, the limiting models which are fully ordered or fully disordered may be designated respectively as ordered or disordered crystalline modifications, if their consideration is useful for the structural description of a polymeric material. Note... 

What Are the Key Ideas Tlic direction of natural change coi responds 10 the increasing disorder of energy and matter. Disorder is measured by the thermodynamic quantity called entropy. A related quantity—the Gibbs free energy—provides a link between thermodynamics and the description of chemical equilibrium. 

The details of the derivation are complicated, but the essence of this equation is that the more possible descriptions the system has, the greater is its entropy. The equation states that entropy increases in proportion to the natural logarithm of W, the proportionality being given by the Boltzmann constant, k — 1.3 806 x lO V/r. Equation also establishes a starting point for entropy. If there is only one way to describe the system, it is fully constrained and W — 1. Because ln(l)=0,S = 0 when W — 1. 

The Boltzmann equation, S = kXxiW, establishes the zero point for entropies. Because ln(l) = 0, this equation predicts that the entropy will be zero for a system with only one possible description. Figure 14-1 la shows a system with W =, nine identical marbles placed in nine separate compartments. The figure also shows two types of conditions where > 0. Figure 14-llZ) shows a condition when the marbles are not identical. If one marble is a different color than the other eight, there are nine different places to place the different marble. Figure I4-IIc shows a condition where there are more compartments than marbles. If there are more places to put the marbles than there are marbles, there are multiple ways to place the marbles in the compartments. 

Phase changes, which convert a substance from one phase to another, have characteristic thermodynamic properties Any change from a more constrained phase to a less constrained phase increases both the enthalpy and the entropy of the substance. Recall from our description of phase changes in Chapter 11 that enthalpy increases because energy must be provided to overcome the intermolecular forces that hold the molecules in the more constrained phase. Entropy increases because the molecules are more dispersed in the less constrained phase. Thus, when a solid melts or sublimes or a liquid vaporizes, both A H and A S are positive. Figure 14-18 summarizes these features. 

The simplicity and accuracy of such models for the hydration of small molecule solutes has been surprising, as well as extensively scrutinized (Pratt, 2002). In the context of biophysical applications, these models can be viewed as providing a basis for considering specific physical mechanisms that contribute to hydrophobicity in more complex systems. For example, a natural explanation of entropy convergence in the temperature dependence of hydrophobic hydration and the heat denaturation of proteins emerges from this model (Garde et al., 1996), as well as a mechanistic description of the pressure dependence of hydrophobic... 

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