Corresponding states, law

Table 3.2 presents a selection of the most used thermodynamic options for phase equilibrium with suitable enthalpy and entropy methods. The accuracy of both phase equilibrium and enthalpy/entropy computation must be examined when using EOS models. For example, often a cubic EOS underestimates the enthalpy of vaporisation. In this case other methods are more accurate, as those based on three-parameters corresponding states law (Lee-Kesler, Curl-Pitzer, etc.). Mixtures rich in components with particular behaviour, as or CH, need special methods for accurate simulation. When binary interaction parameters for liquid activity models are absent, the UNI FAC predictive method may be employed. It is worth to note that UNIFAC is suitable only for exploratory purposes, but not for final design. When high non-ideal mixtures are involved at higher pressure then the combination of EOS with liquid activity models is recommended (see Chapter 6). 

The computation of the second virial coefficient B can follow different ways. Here we illustrate the use of three-parameter corresponding states law. The following equations can be conveniently represent the behaviour of non-polar molecules ... 

Lee and Kesler (reference cited) found an accurate representation for compressibility of both gases and liquids by combining BWR-EOS with corresponding states law. They generated departure functions for enthalpy, entropy, fugacity coefficient and heat capacity. Tables are given in Reid et al. (1987), whereas illustrative graphs are presented in Perry (1997). The method is similar to that developed for compressibility. As an example, the enthalpy departure function may be calculated with the relation ... 

One can easily notice that the coefficients Vj in the right hand side of the above result comprise all the relevant features of ion-ion and ion-solvent interactions, so they are expected to be functions of the concentration (Varela et al, 2010). For non-correlated ion transport, conductivity maxima are predicted in terms of the probability of jumping between different types of cells, and the behaviour of the conductivity normalized to its maximum value with the scaled ionic concentration is seen to approximately fit to a universal corresponding states law given by... 

Keller J B and Zumino B 1959 Determination of intermolecular potentials from thermodynamic data and the law of corresponding states J. Chem. Phys. 30 1351... 

A theoretical basis for the law of corresponding states can be demonstrated for substances with the same intemiolecular potential energy fimction but with different parameters for each substance. Conversely, the experimental verification of the law implies that the underlying intemiolecular potentials are essentially similar in fomi and can be transfomied from substance to substance by scaling the potential energy parameters. The potentials are then said to be confomial. There are two main assumptions in the derivation ... 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

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.

This is Widom s scaling assumption. It predicts a scaled equation of state, like the law of corresponding states, that has been verified for fluids and magnets [102]. 

Many other equations of state besides van der Waals lead to laws of corresponding states, although naturally these will not be of the form (1). 

G. Meslin (1893) has investigated the conditions under which an equation of state can lead to a law of corresponding states. The most general form of equation possible is ... 

By means of these equations we can eliminate three constants from (2). But, if the equation (2) is now reduced, as is required by the law of corresponding states, it must not contain any constants characteristic of the substance, hence (2) can contain only three independent characteristic constants. In this case (2) can always be written in the form ... 

The necessary condition that an equation of state shall lead to a law of corresponding states is, therefore, that it shall contain only three characteristic constants, and shall exhibit a critical point. 

Madame Ivirstine Meyer (1900) has shown that the discrepancies are not to be explained by errors in the critical data the law of corresponding states can be tested without making use of these constants, and differences between the observed and calculated magnitudes are still apparent. D. Berthelot (Journ. de Phys., 1903) has deduced some new equations. 

Equation (6.30) leads to a final method of obtaining an approximate value for In4> by making use of the law of corresponding states. This law states that all gases obey the same equation of state when expressed in terms of the reduced variables T, — T/Tc, pT - p/pc. and V, — V/Vc, where T., pc. and Vc are the critical temperature, pressure, and volume, respectively. 

At temperatures greater than the critical temperature, the law of corresponding states is a reasonably good approximation for most gases. Goug-Jen Su5 showed the correspondence by graphing the compressibility... 

The law of corresponding states indicates that all gases should show the same behavior in applying equation (6.31). This enables one to construct a chart showing Tt isotherms of z against px, such as that shown in Figure 6.4, from which can be estimated. 

Mostinski, I.L. Brit. Chem. Eng. 8 (1963) 580. Calculation of boiling heat transfer coefficients, based on the law of corresponding states. 

Mostinski IL (1963) Calculation of Boiling Heat Transfer Coefficients, Based on the Law of Corresponding States, Br Chem Eng, 8 580. 

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