Principle of corresponding states

3 Principle of Corresponding States 4.2.3.1 Concept of Corresponding States 

According to this concept, a reduced property is expressed as a function of two variables, and I/, and of the acentric factor, cd  

Chapter 4. METHODS FOR THE CALCULATION OF HYDROCARBON PHYSICAL PROPERTIES 111 

For non-polar components like hydrocarbons, the results are very satisfactory for calculations of vapor pressure, density, enthalpy, and specific, heat and reasonably close for viscosity and conductivity provided that is greater than 0.10. 

This concept can be extended to mixtures if the pseudo-critical constants of the mixture and a mixture reduction group are defined. This gives the 

In quantitative form, it says there exists a universal function that applies to all species if we scale the potential energy to the energy parameter and the distance between molecules to the size parameter. Therefore, we can write  

Equation (4.11) is not restricted to the Leimard-Jones potential function but rather it says the dimensionless potential energy is some universal function of the dimensionless distance. 

We can extend the principle of corresponding states to macroscopic thermodynamic properties. In this form, we can write a general equation of state that applies to all species if we scale the measured properties P, v, and T appropriately. Van der Waals recognized that, for a given species, it was particularly suitable to scale the values to those at its critical point. The critical point represents a unique state, and that state is determined by the intermolecular interactions characteristic of a given species. Thus, we can construct a reduced coordinate system with the following three dimensionless groupings  

The principle of corresponding states says that there is some universal function that is the same (i.e., the same form and the same constants) for offl substances  

Alternatively, one of the dimensionless groupings might be the compressibility factor, z. Thus, 

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Using the principle of corresponding states for the following characteristics avoids the use of the contributing groups method ... 

Using the principle of corresponding states requires knowledge of pseudo-critical constants of petroleum fractions these should be estimated starting from characteristic properties which are the normal boiling temperature and the standard specific gravity. 

The correction for pressure is often determined by applying the principle of corresponding states. 

Hexane, for example, is a component whose properties are well known and follow the principle of corresponding states very closely. The acentric factor recommended by the DIPPR is 0.3046 and is considered by convention not to vary with temperature. 

The other method is to employ the principle of corresponding states and calculate the Cp/ of the mixture in the liquid phase starting from the mixture in the ideal gas state and applying an appropriate correction ... 

The principle of corresponding states enables the enthalpy of a liquid mixture to be expressed starting from that of an ideal gas mixture and a reduced correction for enthalpy ... 

The specific heat of gases at constant pressure is calculated using the principle of corresponding states. The for a mixture in the gaseous state is equal to the sum of the C g of the ideal gas and a pressure correction term ... 

Calculation of thermophysical properties of gases relies on the principle of corresponding states. Viscosity and conductivity are expressed as the sum of the ideal gas property and a function of the reduced density ... 

Guggenheim E A 1945 The principle of corresponding states J. Chem. Phys. 13 253... 

Cook and Rowlinson J S 1953 Deviations form the principles of corresponding states Proc. R. Soc. A 219 405... 

The physical properties of argon, krypton, and xenon are frequendy selected as standard substances to which the properties of other substances are compared. Examples are the dipole moments, nonspherical shapes, quantum mechanical effects, etc. The principle of corresponding states asserts that the reduced properties of all substances are similar. The reduced properties are dimensionless ratios such as the ratio of a material s temperature to its critical... 

According to equation 184, all fluids having the same value of CO have identical values of Z when compared at the same T and P. This principle of corresponding states is presumed vaHd for all T and P and therefore provides generalized correlations for properties derived from Z, ie, for residual properties and fugacity coefficients, which depend on T and P through Z and its derivatives. 

Generalized Correlations. Generalized correlations are often the only recourse when a property value cannot be determined from empirical correlations or by other means. Several powerful correlating techniques fall under this category, including the principle of corresponding states (3,17), reduced property models (1), and the Polanyi-type characteristic curve for microporous adsorbents (14). 

Generalized Surface Tension Correlations. Use of the principle of corresponding states has provided a practical and accurate method for the estimation of surface tensions. The functional relationship for the surface tension of a pure substance (85) is... 

Since air is a mixture of predominantly nitrogen, oxygen, and a host of lesser impurities, there has been less interest in developing precise thermodynamic properties. The only recent correlation of thermodynamic properties is that published by Vasserman, et al. (Barouch, Israel Program for Scientific Translations, Jerusalem, 1970), and is based on the principle of corresponding states because of the scarcity of experimental data. 

Fig. 10. The mole fraction of carbon dioxide in saturated solutions in air at — 110°C (above the lower critical end point). The full line is the experimental curve of Webster and the dashed curves are 1, an ideal gas mixture 2, an ideal gas mixture with Poynting s correction and 3, the solubility calculated from Eq. 8 and the principle of corresponding states.

While virial coefficients can be calculated from statistical-mechanical formulas, for practical work it is usually more convenient to employ semi-empirical correlations. Most of these correlations are based on the principle of corresponding states and as a result their applicability is limited to normal... 

The principle of corresponding states will be used as the basis for comparing the different equations of state. This principle states that, to a reasonable approximation, all gases show the same (p. Fni. T) behavior when compared in terms of the reduced variables. The extent to which this principle is followed is... 

Figure A3.2 Graph of the compressibility factor r for a number of gases versus their reduced pressure at several reduced temperatures. Reprinted with permission, taken from Goug-Jen Su, Ind. Eng. Chem.. 38,803 (1946), the data illustrate the validity of the principle of corresponding states. The line is Goug-Jen Su s estimate of the average value for r.

Lee and Kesler (1975) extended the Benidict- Webb-Rubin equation to a wider variety of substances, using the principle of corresponding states. The method was modified further by Plocker et al. (1978). 

As we have already seen, the universal chart of gases assumes that all gaseous species exhibit the same sort of deviation from ideal behavior at the same values of 7, and V. This fact, known as the principle of corresponding states, is analytically expressed by the deviation parameter (or compressibility factor ) Z. For n =, ... 

Confirmation of the validity of the principle of corresponding states is shown in figure 9.4A, where we see the analogies in the reduced isotherms of H2O, CO2, and N2 according to reduced pressure. However, figure 9.4B shows that the principle of corresponding states is only an approximation of actual behavior in the vicinity of the critical point in the case of H2O, there are in fact discrepancies between predicted and actual behavior. 

Figure 9,4 (A) Reduced isotherms of N2, CO2, and H2O vapor according to reduced pressure. (B) Comparison between experimental isotherms of H2O and f predicted by principle of corresponding states.

Under the heading "The empijrical principle of corresponding states , Hirschfelder et al (Ref 4, p 235), stated ... 

The empirical principle of corresponding states may then be stated in the form All substances obey the same equation of state in terms of the reduced variables ... 

In general, the state of a system may be described by any two of the three variables p, V, and T. Thus, accdg to the "principle of corresponding states , any dimensionless group is a universal function of any two of the three reduced variables. In particular the compressibility factor, pV/RT of vander Waals equation of state, can be written as a universal function of the reduced volume and te mperature ... 

The principle of corresponding states provides a convenient and rough means for detg the properties of a dense gas or a liquid. The only info required is the value of two of the critical constants for the substance under consideration. As the critical volume s very difficult to measure, even approximately, it is more convenient to use the expression in terms of pf,Tt (Eq 4.1-3) rather than in terms of Vr,Tr (Eq 4.1-2). The values of the critical constants may in turn be estimated from more readily available data, such as bp, mp, etc. For example, if bp of a substance is T, its critical temp Tc, is approximately 3/2 T j and if mp is Tm its Tc is approx... 

This equation contains no explicit reference to either the quantity or the nature of the substance, so that it could be applied in this form to nearly all fluids. It also embodies a very useful concept, the law (or principle) of corresponding states (qv)... 

Slattery, J. C., Diffusion Coefficients and the Principle of Corresponding States. M. S. Thesis, University of Wisconsin, Department of Chemical Engineering, 1955. 

Principle of corresponding states. The principle of corresponding states, originally introduced by van der Waals and applied since to model inter-molecular potentials, transport and equilibrium properties of fluids over a wide range of experimental conditions, was remarkably successful, albeit it is not exact in its original form. An interesting question is whether one could, perhaps, describe the diversity of spectral shapes illustrated above by some reduced profile, in terms of reduced variables. If all known rare-gas spectra are replotted in terms of reduced frequencies and absorption strengths,... 

Principle of corresponding states. Intermolecular potentials, like the induced dipole surfaces, are functionals of the intermolecular interactions. The signatures of electron exchange, dispersion and multipole induction are clearly exhibited in both they have much in common. 

For the studies of intermolecular potentials, the principle of corresponding states has been a useful guide. Reasonably accurate models of the equation of state have been proposed that have just two adjustable parameters (e.g., van der Waals equation). It has been argued that the success of such models suggests that all (the isotropic) intermolecular potentials should be of the same form and functions of just two parameters (such as well depth and position of the minimum). While current research does not exactly bear out this conclusion, not even if the scope is limited to rare gas interactions, it is probably fair to say that the idea of the principle of corresponding states is still being tested and tried in many laboratories around the world, for various purposes and motivations. 

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