Corresponding states principle simple

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. 

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... 

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 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. 

These forms of a generalized equation of state only require the critical temperature and the critical pressure as substance-specific parameters. Therefore, these correlations are an example for the so-called tsvo-parameter corresponding-states principle, which means that the compressibility factor and thus the related thermodynamic properties for all substances should be equal at the same values of their reduced properties. As an example, the reduced vapor pressure as a function of the reduced temperature should have the same value for all substances, provided that the regarded equation of state can reproduce the PvT behavior of the substance on the basis of the critical data. In reality, the two-parameter corresponding-states principle is only well-suited to reflect the properties of simple, almost spherical, nonpolar molecules (noble gases as Ar, Kr, Xe). For all other molecules, the correlations based on the two-parameter corresponding-states principle reveal considerable deviations. To overcome these limitations, a third parameter was introduced, which is characteristic for a particular substance. The most popular third parameter is the so-called acentric factor, which was introduced by Pitzer ... 

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. 

As discussed in Section 2.5.4, the simple two-parameter corresponding states principle indicates that a generalized equation of state for all substances can be created using only two specific parameters, for example, T, and P. The success of this approach is restricted to simple, spherical molecules like Ar, Kr, Xe, or CH4, where vapor pressure and compressibility factor can be reasonably described. For other molecules, the simple two-parameter corresponding states principle leads to significant errors. A large improvement has been achieved with the introduction of a third parameter which describes the vapor pressure curve (extended three-parameter principle of corresponding states). The most common parameter of this kind is the so-called acentric factor, which is defined as... 

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. 

Since r remains explicit in the reduced equation of state, a simple corresponding-state principle is not, in general, satisfied. For a pol5nneric liquid, however, r oo, and the equation of state is reduced to... 

Later Soave (1972) improved on the RK-EOS by replacing the term a/T f with a more general temperature-dependent term a(T) and proposed a simple form for a oj) for all pure substances, taking advantage of the concept of the acentric factor of Pitzer. Pitzer (1939), and Pitzer. et al. (1955), Pitzer s acentric factor, oj, was intended as an additional parameter for the improvement of the corresponding-states principle. The acentric factor is a measure of the difference in molecular structure between a given component and a gas with spherically symmetric molecules with u) — 0 (such as argon). 

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... 

The second approach is to extend the simple two-parameter corresponding-states principle at its molecular origin. This is accomplished by making the intermolecular potential parameters functions of the additional characterization parameters /I, and the thermodynamic state, for example, the density p and temperature T. This can be justified theoretically on the basis of results obtained by performing angle averaging on a non-spherical model potential and by apparent three-body effects in the intermolecular pair potential. The net result of this substitution is a corresponding-states model that has the same mathematical form as the simple two-parameter model, but the definitions of the dimensionless volume and temperature are more complex. In particular the... 

In order to extend the simple molecular corresponding-states principle to non-spherical fluids, two approaches are possible. The first simply amounts to introducing models for the non-spherical interactions into the intermolecular potential. For example, the intermolecular potential between two axially symmetric molecules whose electrostatic interactions can be represented as point dipoles and quadrupoles can be modeled as ... 

When the above methods fail, estimation methods become important. Schemes based on the Corresponding-States Principle which are particularly important in this respect are described. In order to demonstrate clearly just when the methods of correlation, the theoretical expressions and estimation techniques are applicable, examples are given of transport-property data representation for systems of different complexity simple monatomic fluids, diatomic fluids, polyatomic fluids (specifically, water and refrigerant R134a), nonreacting mixtures and (dilute) alkali-metal vapors as an example of a reacting mixture. 

If the two-parameter theorem of corresponding states were generally valid, the slope S would be the same for all pure fluids. This is observed not to be true each fluid has its own characteristic value of S, which could in principle serve as a third corresponding-states parameter. However, Pitzer noted that all vapor-pressure data for the simple fluids (Ar, Kr, Xe) lie on tlie same line when plotted as log vs. 1/71 and that the line passes tlirough logi j. = —1.0 at Tr = 0.7. This is illustrated in Fig. 3.13. Data for other fluids define other lines whose locations can be fixed in relation to the line for the simple fluids (SF) by the... 

The SRK and PR equations follow the principle of corresponding states in the three-parameter form only the commonly available critical properties T, p, and are required to apply the equation to a substance. The simple vdW mixing rules work well with these equations. Hence they are widely used for the calculation of vapor-liquid equilibrium in mixtures. 

THE PRINCIPLE OF CORRESPONDING STATES FOR THE VISCOSITY OF SIMPLE LIQUIDS. 

The principle of corresponding states is a two-parameter theory and works well only for simple molecules, which are the noble gases and a few nonpolar or very slightly polar ones. Two main approaches have been used in expanding its range of applicability. One is to introduce a third parameter, the most successful being the acentric factor, and the other is based on manipulating the intermolecular potentials. We will mention only the acentric factor. 

The determination of dense fluid properties from ab initio quantum mechanical calculations still appears to be some time from practical completion. Molecular dynamics and Monte Carlo calculations on rigid body motions with simple interacting forces have qualitatively produced all of the essential features of fluid systems and quantitative agreement for the thermodynamic properties of simple pure fluids and their mixtures. These calculations form the basis upon which perturbation methods can be used to obtain properties for polyatomic and polar fluid systems. All this work has provided insight for the development of the principle of corresponding state methods that describe the properties of larger molecules. 

Corresponding-states theory The basis of the simple principle of corresponding states is that... 

The vdW EOS is not very good at representing experimental PvT data, but it has had a profound influence on thermodynamics. Fairly simple, totally empirical modifications of it by Redlich and Kwong, Soave, and Peng and Robinson are very widely used in vapor-liquid equilibrium calculations, as discussed in Chapter 10 and Appendix F. Furthermore, it led to the principle of corresponding states, discussed below, which is very useful. 

Thus, applying the principle of corresponding states to the van der Waals equation leads to a value of 0.375 for the compressibility factor at the critical point for all species. Experimental values for the compressibility factor at the critical point are around 0.29 for simple species and usually less for complex species. Thus, the value predicted by the van der Waals equation is considerably high—vindicating its limitations in predicting PvT behavior. 

We can extend the principle of corresponding states to account for different classes of molecules, based on the particular nature of the intermolecular interactions involved. One way to accomplish this objective is by introducing a third parameter— the Pitzer acentric factor, w.We then write the compressibiUty factor in terms of z , which accounts for simple molecules, and a correction factor for the nonsphericity ... 

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