Thermodynamic perturbation theory first-order

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

TPTl first-order thermodynamic perturbation theory... 

Following the original first-order thermodynamic perturbation theory of Wertheim heteronuclear models can be proposed where the segments in a given model molecule are arbitrarily different. The earliest works in this direction... 

TPTl-MSA First-order thermodynamic perturbation theory combined with the mean spherical approximation... 

One of the main assumptions in the development of Wertheim s first-order thermodynamic perturbation theory (TPTl) is that association sites are singly bondable. Indeed, the entire multi-density formalism of Wertheim is constructed to enforce this condition. For the case of hydrogen bonding, the assumption of singly bondable sites is justified however for patchy colloids (see Section I for a background on patchy colloids), it has been shown experimentally [42, 43] that the number of bonds per patch (association site) is dependent on the patch size. It has been 30 years since Wertheim first published his two-density cluster... 

We will later consider the approximation that affects the transition from Eq. (4.4) to Eq. (4.6) in detail. But this result would often be referred to as first-order perturbation theory for the effects of - see Section 5.3, p. 105 - and we will sometimes refer to this result as the van der Waals approximation. The additivity of the two contributions of Eq. (4.1) is consistent with this form, in view of the thermodynamic relation pdpi = dp (constant T). It may be worthwhile to reconsider Exercise 3.5, p. 39. The nominal temperature independence of the last term of Eq. (4.6), is also suggestive. Notice, however, that the last term of Eq. (4.6), as an approximate correction to will depend on temperature in the general case. This temperature dependence arises generally because the averaging ((... ))i. will imply some temperature dependence. Note also that the density of the solution medium is the actual physical density associated with full interactions between all particles with the exception of the sole distinguished molecule. That solution density will typically depend on temperature at fixed pressure and composition. 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

As we described in Section III.G, perturbation theories can be extended in a systematic way using cluster expansion techniques. These techniques have recently been applied to the calculation of the thermodynamic properties and vapor-liquid equilibrium of 12-6 diatomics and seem to offer a clear improvement over the first-order perturbation theories. To illustrate this point. Table I shows values of the critical density and critical temperature predicted by the ISF-ORPA theory and the first-order perturbation theory together with results recently obtained from molecular dynamics... 

In Table VI, the first-order perturbation theory results for the excess thermodynamic properties are compared with experimental results for several mixtures. For these calculations, Grundke et al. (25) used the 6 12 potential with the pure-fluid potential parameters listed in Table VII. Further they assumed Equation 67 and... 

The equilibrium Tl-0 distance and Tl-O-Tl angle were estimated by a calculation based on Schwinger thermodynamic [3] and first- [4] and second-order [2] perturbation theories using harmonic and anharmonic potential constants [2, 3]. 

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