Integral equations thermodynamic consistency

The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

Three different uses of the Gibbs-Duhem equation associated with the integral method are discussed in this section (A) the calculation of the excess chemical potential of one component when that of the other component is known (B) the determination of the minimum number of intensive variables that must be measured in a study of isothermal vapor-liquid equilibria and the calculation of the values of other variables and (C) the study of the thermodynamic consistency of the data when the data are redundant. 

Nevertheless, the reader has to notice that the use of the latter two consistency conditions is not always sufficient in obtaining an accurate description of structural functions (e.g., the bridge function) and, consequently, thermodynamics quantities. The problem of identifying and fulfiling at least a second thermodynamic condition is under the scope in this review article. We will turn back to the cmcial point of thermodynamic consistency in Sections IE and IV when discussing the solution of integral equation theories and their application to simple liquids. 

These expressions are formally exact and the first equality in Eq. (123) comes from Euler s theorem stating that the AT potential u3(rn, r23) is a homogeneous function of order -9 of the variables r12, r13, and r23. Note that Eq. (123) is very convenient to realize the thermodynamic consistency of the integral equation, which is based on the equality between both expressions of the isothermal compressibility stemmed, respectively, from the virial pressure, It = 2 (dp/dE).,., and from the long-wavelength limit S 0) of the structure factor, %T = p[.S (0)/p]. The integral in Eq. (123) explicitly contains the tripledipole interaction and the triplet correlation function g (r12, r13, r23) that is unknown and, according to Kirkwood [86], has to be approximated by the superposition approximation, with the result... 

The consistency test is carried out by plotting In (Y1/Y2) versus Xj and graphically evaluating the integral in Equation 1.31. The curve consists of a positive part and a negative part above and below the line IntYi /Y2) = 0- The data points are considered thermodynamically consistent, and the assumption of negligible effect of variable pressure is deemed valid if the areas above and below the line are equal. 

Now since the total pressure variation in the experiments was small, and V is usually very small for liquid mixtures, the integral on the right side of this equation can be neglected. Thus to test the thermodynamic consistency of the Weissman-Wood activity... 

A quite different approach to thermodynamics of ionic solutions consists of solving the integral equations that relate the correlation functions and the pair potentials. 

Applications of the Integrated Equations. The usefulness of the Gibbs-Duhem equation for establishing the thermodynamic consistency of, and for smoothing, data has been pointed out. The various integrated forms are probably most useful for extending limited data, sometimes from even single measurements, and it is these applications that are most important for present purposes. 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

The use of equation (1) integrated across the whole composition range to obtain (thermodynamic consistency of measurements on binary systems formed from components 1,2, and 3 is becoming an accepted procedure. Analysis of data to obtain (of —o) as a function of X2 is less common. 

The left-hand side is obtained by integration of the single solute data up to a concentration c, and the right-hand side from bisolute isotherms at constant (C1 + C2). The first of these criteria, using the data in the form of the empirical equation for bisolute adsorption previously proposed by Fritz and Schliinder," indicated that the data reported in this paper are thermodynamically consistent within the accuracy of experiment. No mention is made of the application of the second criterion. 

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