The law of corresponding states

Remember from Chapter 8 that the pressure of monoatomic NVT systems is given by 

Equation 10.42 implies that for fluids for which the pairwise potential has the same functional form (for example, all Lennard-Jones fluids), the pressure is a universal function of the reduced temperature T and number density p. In other words, there is a single, universal p v diagram that describes the behavior of fluids with the same pair potential function. This is the law of corresponding states, which holds for classes of structurally similar chemical substances. 

A dimensionless critical point, which is the same for all fluids, can be defined with 

Then the macroscopic law of corresponding states for all fluids can be written as 

A compressibility factor can be defined that can capture corresponding states equations  

Francis Weston Sears, An Introduction to Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics. Reading, Mass. Addison-Wesley, 1950. 

Using the values ofa,b, and R given by Eqs. (3.18), w e can write the van der Waals equation in the equivalent form 

Equation (3.20) involves only the ratios p/p, T/7, and F/P. This suggests that these ratios, rather than / , T, and V, are the significant variables for the characterization of the gas. These ratios are called the reduced variables of state, n, t, and (j)  

Written in terms of these variables, the van der Waals equation becomes 

The important thing about Eq. (3.21) is that it does not contain any constants that are peculiar to the individual gas therefore it should be capable of describing all gases. 

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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]. 

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 ... 

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. 

The equation does not contain the constants a and 6, characteristic of the substance, and therefore applies to all substances. Hence it is known as the law of corresponding states. 

Each of the two-parameter semi-empirical equations above leads to the law of corresponding states, first formulated by Van der Waals in 1881 [9], which says that all... 

Figure 5.6. Fit of experimental data for ten gases to the law of corresponding states. By permission from G. J. Su, Ind. Eng. Chem. 38, 803 (1946).

The assumption known as the law of corresponding states asserts that the compressibility factor Z should be a function only of the reduced temperature Ti and the reduced pressure Pi, which is approximately correct for many real gases. It is seen that, for a van der Waal gas, the minimum value of Vr = 1/3, which can be achieved only at infinite pressure. From the equation of state written for the reduced temperature and pressure, we can derive the equivalent formula of compressibility as... 

For an ideal gas, Z = 1. In general, the law of corresponding states provides that the compressibility factor depend on the reduced temperature and pressure,... 

Van der Waals emphasizes his conviction that the constant b actually carries significant V dependence, and his astonishment ( to my great joy ) to learn that such dependence does not detract from essential consistency with the law of corresponding states. 

Notice that the shapes of the isotherms of compressibility factors for the three gases given in Figures 3-2, 3-3, and 3-4 are very similar. The realization that this is true for nearly all real gases led to the development of the Law of Corresponding States and the definition of the terms reduced temperature and reduced pressure. Reduced temperature and reduced pressure are defined as... 

The Law of Corresponding States says that all pure gases have the same z-factor at the same values of reduced pressure and reduced temperature. Figure 3-5 gives a test of this theory for compressibility data of methane, propane, n-pentane, and n-hexane.4 Some of the deviation between lines at constant reduced temperatures may be due to experimental error and some due to inexactness of the theory. 

The Law of Corresponding States is more accurate if the gases have similar molecular characteristics. Fortunately most of the gases the petroleum engineer deals with are composed primarily of molecules of the same class of organic compounds known as paraffin hydrocarbons. 

The Law of Corresponding States has been extended to cover mixtures of gases which are closely related. As was brought out in Chapter 2, obtaining the critical point for multicomponent mixtures is somewhat difficult therefore, pseudocritical temperature and pseudocritical pressure have been invented. 

The law of corresponding states can be used to put Equation 6-8 into reduced form. Equation 3-43 can be used to replace the pressure term. 

The similarity of this graph to the graph showing the density of a pure substance indicates that the law of corresponding states should hold for viscosity as well as for volumetric behavior. 

The Compressibility Equation of State —The Law of Corresponding States— The Compressibility Equation of State for Gas Mixtures. 

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