Corresponding-states principle

Figure A2.3.6 illustrates the corresponding states principle for the reduced vapour pressure P and the second virial coefficient as fiinctions of the reduced temperature showmg that the law of corresponding states is obeyed approximately by the substances indicated in the figures. The useflilness of the law also lies in its predictive value.

Flory (11) improved the notation and form of Prigogine s expressions, and it is essentially the Flory form of Prigogine s free-volume theory that is of most use for design purposes. The Flory work (11) leads to an equation of state which obeys the corresponding-states principle ... 

V. B. Polyakov, V. B., Horita, J., Cole, D. R. and Chialvo, A. A. Novel corresponding-states principle approach for the equation of state of isotopologues H2180 as an example. J. Phys. Chem. B-lll, 393 (2007). 

In the new version, Chapter 10 focuses exclusively on van der Waals forces and their implications for macroscopic phenomena and properties (e.g., structure of materials and surface tension). It also includes new tables and examples and some additional methods for estimating Hamaker constants from macroscopic properties or concepts such as surface tension, the parameters of the van der Waals equation of state, and the corresponding state principle. 

Simha,R., Utracki,L.A. The viscosity of concentrated polymer solutions corresponding states principles. Rheol. Acta 12,455-462 (1973). 

Comparison with nonionic fluids is possible through the corresponding-states principle [37]. If, as usual, the reduced temperature T is defined as the ratio of the thermal energy kBT to the depth of the potential, one finds for a symmetrical Coulomb system with charges q = z+e = z e... 

Calculated values of the quantities (Hr)°/RTc, (HRy/RTc, (Sr)°/R, an (SRy/R are shown by plots of these quantities vs. Pr for various values of Tr i Figs. 6.6 through 6.13. These plots, together with Eqs. (6.56) and (6.57), alio estimation of the residual enthalpy and entropy on the basis of the three-paramet corresponding-states principle as developed by Pitzer (Sec. 3.6). 

The other method has been described by Simha and Utracki in several articles (1963-1973). The method is called a corresponding states principle. A reduced viscosity f/ is plotted against a reduced concentration c, where... 

The compressibility factor is usually calculated using either an equation of state or using the corresponding states principle. Both of these methods will be discussed in this section. 

The two-parameter corresponding-states principle is sufficiently accurate for approximations of the physical properties of simple fluids, and its simplicity makes it attractive for such calculations. It even can provide reasonably accurate predictions for other fluids. 

Note, these values are not meant to be the actual critical properties of the fluid. They are used simply to estimate the properties of the fluid based on the corresponding states principle. 

This remarkable feature is not widely recognized in the literature and suggests a kind of corresponding state principle. It may be recalled that the notion of corresponding states was introduced by van der Waals. If his equation of state [2.5.1] were universally valid, it should be possible to express the critical pressure, p, temperature, T, and volume, in terms of his constants a and b and then rewrite [2.5.1] in terms of reduced pressures, temperatures and volumes, p/p = p, etc. The resulting reduced van der Wools equation reads... 

At this instance it makes sense to recall the corresponding state principle of sec. 2.9 all surface excess entropies are very similar. Assuming this to also be the case for the corresponding interfacial excesses, we conclude that the term in... 

Primitive models satisfy the corresponding states principle [23, 24], The thermodynamic state is completely specified by a reduced temperature T and a reduced ion density p, ... 

Necessarily chain conformations are reflected in solution properties such as viscosity. In a series of papers Utracki and his colleagues (8-11) have studied Newtonian viscosities of polymer solutions they have also studied polymer melts, and nonpoly-meric liquid phases, organic as well as inorganic ones. The aim of the work was to construct a corresponding states principle for viscosity n in function of temperature T and concentration c. 

Utracki, L.A. Roovers, J.E.L. Viscosity and normal stresses of linear and star branched polystyrene solution. I. Application of corresponding states principle to zero-shear viscosities. Macromolecules 1973, 6 (3), 366-372. 

By applying the corresponding states principle, the deviations of the properties of a substance from those of a simple fluid may be correlated in terms of the acentric factor, as described above for vapor pressures (Equation 1.14). The compressibility factor has also been correlated in terms of the acentric factor in the form of a polynomial... 

At low and moderate pressures, the viscosity of a gas is nearly independent of pressure and can be correlated for engineering purposes as a function of temperatnre only. Eqnations have been proposed based on kinetic theory and on corresponding-states principles these are reviewed in The Properties of Gases and Liquids [15], which also inclndes methods for extending the calculations to higher pressures. Most methods contain molecular parameters that may be fitted to data where available. If data are not available, the parameters can be estimated from better-known quantities such as the critical parameters, acentric factor, and dipole moment. The predictive accuracy for gas viscosities is typically within 5%, at least for the sorts of small- and medinm-sized, mostly organic, molecules used to develop the correlations. 

Alternatively, fluid characteristics other than Zc can be used as the additional parameter in the generalization of the simple corresponding-states principle. In fact, since for many substances the critical density, and hence Zc, is known with limited accuracy, if at all, there is some advantage in avoiding the use of Zc-Pitzer has suggested that for nonspherical molecules the acentric factor co be used as the third correlative parameter, where co is defined to be... 

The last several paragraphs have emphasized the shortcomings of a single corresponding-states principle when dealing wit fluids of different molecular classes. However, it is useful to point out that a corresponding-states correlation can be an accurate rep-... 

The aim of the equation development is the accurate and reasonably simple description of the entire saturation boundary, with a clear representation of the corresponding-states principle. 

The basic Equation 3 becomes explicit as soon as the parameters P(PT) and A(Pr) are specified. This specification is the result of a lengthy trial procedure, based on the following factors (a) consistency with known near-critical power laws, (b) approximate consistency with the law of the rectilinear diameter (c) the tendency of the low-pressure vapor curve to form a straight line in logarithmic coordinates, as predicted by Equation 5 (d) imposition of a definite form of the corresponding-states principle and (e) consistency with a large collection of experimental data. 

In his Ph.D. thesis of 1873, van der Waals proposed the first EoS. The relation is frequently written in terms of reduced variables, indicating expected observance of the corresponding states principle ... 

The above formula is acceptable for condensed phases, as solid or liquids, without phase transition, but not for real gases, where the influence of pressure has to be accounted for. Consequently, the formula (5.21), so much used in hand calculations, is of relatively minor importance in process simulation. Instead, computations of enthalpy based on equations of state or corresponding states principle are used. 

Hence, the enthalpy change between T, and Tj. Pi niay be computed from the variation for an ideal gas plus the variation of the departure function, which accounts for non-ideality. The big advantage of the departure functions is that they can be evaluated with z PVT relationship, including the corresponding states principle. Moreover, the use of departure functions leads to a unified framework of computational methods, both for thermodynamic properties and phase equilibrium. 

Two types of PVT representation are used in simulation equation of state and corresponding states principle. The equations of state are today the most applied. Particularly advantageous are the cubic equations of state, since they offer a consistent computation of both thermodynamic properties and phase equilibria. However, there is no single equation of state that could predict accurately the properties of all components, from hydrogen and methane up to polar species and polymers. That is why there are many models, each being accurate for a particular application. 

Other, more complex forms of U(R) have been proposed [81-mai/rig] which have proved very satisfactory for the representation of specific rare gas interactions, for example. However, for the correlation and prediction of second virial coefficient data, empirical methods are generally used. Those methods which are based on the corresponding states principle usually represent the second virial coefficient by a series of terms in inverse powers of temperature, as suggested by results for the (n-6) potentials, but usually with integer powers. 

The theorem of corresponding states goes back to van der Waals, who formulated the principle of corresponding states based on his equation of state. In addition, van der Waals deduced straightforwardly that in reduced coordinates, the vapor pressure curve and the coexistence curve must be the same for all fluids [3]. An extensive treatment of the corresponding state principle has been published by Xiang [4]. 

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