Prigogine-Defay equation

Laar equation (Eq. 1), as mentioned above. The liquidus line between the two euteetie points ean be caleulated by the Prigogine-Defay equation (Eq. 2), whieh relates the solid eomposition to the melting temperature ... 

From the melting data determined by DSC, chiral purity may be estimated either by measuring the area of the eutectic peak or by applying the Schroder-Van Laar or the Prigogine Defay equation [13]. When the eutectic peak is too small to be detected, the peak asymmetry factor for melting of the major component may be used to estimate the purity [14]. 

The results stated above suppose that simbateness of the values of Oy and as a function of any parameter (temperature, crosslinking density and so on) can be an individual case only. Equation 6.13 demonstrates that the indicated simbateness realisation condition is the criterion = const, or, as follows from Equation 6.14, (p = const. In other words, invariance in polymer structure is the condition of the realisation of the change in the simbateness of Oy and . Let us be reminded that amorphous polymers are thermodynamically non-equilibrium solids, for the description of which two parameters of order, as a minimum, are required according to the principle of Prigogine-Defay. It is obvious that Equation 6.13 satisfies this principle, whereas the linear correlation Oy( ) does not [36]. 

Expressions (10) and (12) will always be true for any material which undergoes an apparent second-order transition. Furthermore, if the volume and entropy surfaces for the transition are the same, as they must be thermodynamically, (10) and (12) can be equated and we arrive at the Prigogine-Defay ratio, R... 

We first consider the case of a two-component solution (biopolymer + solvent) over a moderately low range of biopolymer concentrations, i.e., C < 20 % wt/wt. The quantities pm x in the equations for the chemical potentials of solvent and biopolymer may be expressed as a power series in the biopolymer concentration, with some restriction on the required number of terms, depending on the steepness of the series convergence and the desired accuracy of the calculations (Prigogine and Defay, 1954). This approach is based on simplified equations for the chemical potentials of both components as a virial series in biopolymer concentration, as developed by Ogston (1962) at the level of approximation of just pairwise molecular interactions ... 

The set of equations (3.7-3.9) shows that the sign and magnitude of the second virial coefficient provides information on how the behaviour of the macromolecular solution deviates from that of the thermodynamically ideal state, thus reflecting the nature and intensity of the inter-molecular pair interactions (both biopolymer-biopolymer and biopolymer-solvent) (Prigogine and Defay, 1954 Tanford, 1961 Ogston 1962 ... 

From the experimental temperature dependence of A2 (and the corresponding inferred temperature dependence of juE the other basic excess thermodynamic functions can be determined using general thermodynamic relationships. This then provides a complete thermodynamic characterization of the system as a whole. Thus, for the determination of the excess molar enthalpy of the system at constant pressure, the following equation can be used (Prigogine and Defay, 1954) ... 

There is, however, another statement of the necessary and sufficient condition of thermodynamic stability of the multicomponent system in relation to mutual diffusion and phase separation that is less stringent than equation (3.20) because it may be fulfilled not for every component of the multicomponent system. For example, in the case of the ternary system biopolymeri + biopolymer2 + solvent, it appears enough to fulfil only two of the inequalities (Prigogine and Defay, 1954)... 

Moreover, the conditions in equation (3.20) are always fulfilled for the thermodynamically ideal system so the thermodynamically ideal system is always stable with respect to fluctuations in the system composition. (For a full understanding of the development of the preceding equations, the interested reader should refer to the seminal work of Prigogine and Defay (1954).)... 

The binodal (or coexistence) curve, on which the compositions of the immiscible solutions (phases) lie at equilibrium, can be described by a set of equations involving equilibrium between the chemical potentials of the components in the coexisting phases (Prigogine and Defay, 1954) ... 

It is well known (Defay and Prigogine, 1951) that a spherical interface of radius of curvature r and surface tension y can maintain mechanical equilibrium between two fluids at different pressures p" and p. The phase on the concave side of the interface experiences a pressure p" which is greater than that on the convex side. The mechanical equilibrium condition is given by the Laplace equation ... 

We turn now to the question of validity of the Kelvin equation. Although the thermodynamic basis of the Kelvin equation is well established (Defay and Prigogine 1966), its reliability for pore size analysis is questionable. In this context, there are three related questions (1) What is the exact relation between the meniscus curvature and the pore size and shape (2) Is the Kelvin equation applicable in the range of narrow mesopores (say >vp < 5 nm) (3) Does the surface tension vary with pore width The answers to these questions are still elusive, but recent theoretical work has improved our understanding of mesopore filling and the nature of the condensate. 

The equation of Schroder-Van Laar, which permits the calculation of the liquidus curve, may be applied to the point of the liquidus between the pure enantiomers and the corresponding eutectics ( a r and T aEs- For the part E RE, the equation of Prigogine and Defay applies ... 

Van der Waeds work was well received, but later fell into oblivion. For Instance, it is not acknowledged in Defay and Prigogine s well-known book ) and only in passing in the well-acclaimed publication of Cahn and Hilliard J viz. by referring to van der Waals equation [2.5.291 for the temperature dependence of the surface tension. Only a few authors referred to it Much credit goes to Rowlinson and Widom ) for its reappraisal and modernization. 

Equation (5.40) is a result first derived by Defay, Prigogine( ) and Guggenheim ). When 6 = 0, Eq.(5.40) reduces to the broken-bond equation analog of the ideal solution Eq.(5.13). m is equal to the number of bonds broken per atom upon formation of the surface. Enrichment occurs in the component with the lowest bond strength. When cn equals C22, enrichment or depletion of a component occurs symmetrically around depending on the sign of 6. S and the heat of formation AF = iHx(l — x) are related by ... 

Equation (2.37) shows that for a curved interface, the chemical potentials of each component on both sides of the interface should be equal to achieve equilibrium. The chemical potentials, however, are evaluated at different pressures the pressures on different sides of the interface are related by the second expression in Eq. (2.31). In Eq. (2.20), the work required to increase the surface area of the bubble was expressed by ad A. Defay and Prigogine (1966) derive this work term in a straightforward manner. Consider the system shown in Fig. 2.10, where a spherical liquid droplet of volume V and surface area A is surrounded by its vapor. The total volume of the system, V, is equal to V + V , where V" is the volume of the vapor surrounding the liquid droplet. Suppose part of the droplet vaporizes and the piston moves upward to expand the system to volume V + dV. The work done by the system is, therefore,... 

Equation (2.53), known as the Kelvin equation, reveals that the vapor pressure, P , decreases with increasing interface curvature. So far, we have assumed that the substrate is liquid-wet or that the new phase forms as a bubble. For a droplet or when gas is the wetting phase, the effect of curvature on saturation pressure is formulated shortly. When the radius of a bubble or droplet becomes very small (say r < 10 cm, for a pure substance), the interfacial tension may become a function of the radius (Defay and Prigogine, 1966). However, the derivation of Eq. (2.53) was not based on the assumption of the interfacial tension being independent of r. 

By independent it is meant that no one of the stoichiometric equations can be derived from the others by a linear combination. For further discussion see Prigogine and Defay [1954], Denbigh [1955], and Aris [1965]. 

The reader will recall our discussion of the conditions for the binary spinode (equation 10) and the significance of the inflection points, where (d25T/dX2 )T,P = 0, in bracketing regions of compositional stability (or metastability) and instability with respect to unmixing). It is well-known (Prigogine and Defay, 195, ... 

Ganguly and Kennedy (l97lf). Margules equations for a "simple" ternary mixture (without specific ternary terms see Prigogine and Defay, 195 ) are applied to the quasi-temary system pyrope - grossiilar - (almandine + spessartine) and approximate values for the mixing coefficients (W terms) are derived from existing natural and experimental data. 

We shall snmmarize in this chapter the basic equations oi dassical thermod3mamics of mixtures required in the subsequent chapters of this book. We shall go into a minimum of detail because full accounts of the dassical thermodynamics of mixtures may be foimd in textbooks on chemical thermod5mamics, cl Guggenheim [1949], Prigogine and Defay [1950], English translation by Everett [1954], Haase [1956]. 

The application of the Gibbs-Duhem equation (cf. e.g. Prigogin and Defay [1954]... 

It can be diown that such states are unstable. This means that thermal fluctuations will destroy spontaneously this phase (cf. e.g. Prigogine and Defay [1950], Ch. XV and XVI). The portion V N corresponds to a supersaturated vapour which is metastahle and disappears spontaneously if condensation nuclei are introduced into the system. Similarly M L corresponds to an overexpanded liquid, which again is a metastable state. If we take the loci of the points M and N of a series of isotherms we obtain the curves aC and bC (cf. Fig. 12.2.1) characterized by the equation... 

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