The Corresponding States Principle

Chemical Engineering Department, Colorado School of Mines, Golden, CO 80401-1887, USA 

Edited by A. R. H. Goodwin, J. V. Sengers and C. J. Peters International Union of Pure and Applied Chemistry 2010 Published by the Royal Society of Chemistry, www.rsc.org 

The original, two-parameter corresponding-states principle leads to an equation of state which expresses the residual compressibility factor (or compression factor) in terms of a universal function of the dimensionless temperature and molar volume (or density)  

Equation 6.1 implies all substances obey the same reduced equation of state and we can make a slight transformation of this result to relate directly the properties of one fluid to another. For two fluids j and 0 which obey the simple corresponding-states principle we can write from eq 6.1 

Experimental evidence has shown that the two-parameter corresponding-states equation is obeyed only by the higher molar mass noble gases (Ar, Kr and Xe) and nearly spherical molecules such as methane, nitrogen and oxygen. In order to extend the corresponding-states theory to a larger spectrum of fluids, additional characterization parameters have been introduced into the 

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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 ... 

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. 

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... 

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. 

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. 

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, ... 

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... 

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 ... 

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. 

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]. 

Xiang, H.W. Thermodynamic properties. In The Corresponding-States Principle and its Practice Thermodynamics, Transport and Surface Properties of Fluids, 1st edn. Chap. 5, pp. 49-148. Elsevier, Amsterdam (2005)... 

Casting the equations of state into reduced variables implies commonality of behavior for all fluids for example, common PVT surface, or the common value of compressibility at the critical point P V /RT = 3/8. This commonality is known as the corresponding-states principle (CSP), discussed later. Since Eq. (6.3) is general, so is the critical point. Thus, originally for the low-molecular-weight gases and liquids, its experimental critical coordinates, Pc,Vc, and Tc, were used as the characteristic reducing parameters. 

The first successful theoretical approach of an equation of state model for polymer solutions was the Prigogine-Flory-Patterson theory. It became popular in the version by Flory, Orwoll and Vrij and is a van-der-Waals-like theory based on the corresponding-states principle. Details of its derivation can be found in numerous papers and books and need not be repeated here. The equation of state is usually expressed in reduced form and reads ... 

Figure 2.25. The reduced surface tension as a function of the reduced temperature based on the corresponding states principle of Dee and Sauer (1995).

Gases and Vapors Through Ptffyethylene. Usefulness of the Corresponding States Principle. AIChEJ., 15,64 (1969). 

A modification of the corresponding-state principle by introducing a parameter related to the vapor pressure curve is reasonable, because experimental vapor pressure data as a function of temperature are easy to retrieve. Furthermore, the vapor-liquid equilibrium is a very sensitive indicator for deviations from the simple corresponding-state principle. The value Tr = 0.7 was chosen because this temperature is not far away from the normal boiling point for most substances. Additionally, the reduced vapor pressure at Tr = 0.7 of the simple fluids has the value = 0.1 (log = —1). As a consequence, the acentric factor of simple fluids is 0 and the three-parameter correlation simplifies to the two-parameter correlation. 

For the estimation of Cp the method of Rowlinson/Bondi can be used [32], which is based on the ideal gas heat capacity and the corresponding states principle ... 

Stem, S.A., J.T. Mullhaupt, and P.J. Gareis, The effect of pressure on the permeation of gases and vapors through polyethylene. Usefulness of the corresponding states principle. Aiche JoumaL 1969.15(1) p. 64—73. 

All theoretical equations of state suggest a corresponding state behavior of PVT properties that requires three scaling parameters P, V, and T ). These define the corresponding state and are used to scale/reduce the PVT toward P =P/P, V = V/V, f = T/T. In his Ph.D. thesis of 1873, van der Waals proposed the first EoS formulated in terms of corresponding state (reduced) variables. The relation can be written in terms of PVT or in terms of reduced variables PVf, indicating expected observance of the corresponding state principle ... 

Assuming the validity of the corresponding states principle, the eos is expressed in reduced variable form, where the reducing parameters, P, T, and V, depend on the two Lennard-Jones potential parameters e and v ... 

With this complexity in mind, the most powerful tool available today (just as 25 years ago) for making highly accurate, yet mathematically simple, predictions of the thermophysical properties of fluids and fluid mixtures is the corresponding states principle. 

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