Three parameter, corresponding-states

This method utilizes essentially the concept developed by Fitzer in 1955. According to the principle of three-parameter corresponding states, the compressibility factor z, for a fluid of acentric factor w, is obtained by interpolating between the compressibilities Zj and Z2 for the two fluids having acentric factors w, and (p -... 

The volumetric properties of fluids are represented not only by equations of state but also by generalized correlations. The most popular generalized correlations are based on a three-parameter theorem of corresponding states which asserts that the compressibiHty factor is a universal function of reduced temperature, reduced pressure, and a parameter CO, called the acentric factor ... 

Three Parameter Models. Most fluids deviate from the predicted corresponding states values. Thus the acentric factor, CO, was introduced to account for asymmetry in molecular stmcture (79). The acentric factor is defined as the deviation of reduced vapor pressure from 0.1, measured at a reduced temperature of 0.7. In equation form this becomes ... 

Pitzer s Corresponding-States Correlation A three-parameter corresponding-states correlation of the type developed by Pitzer, K.S. Thennodynamic.s, 3ded., App. 3, McGraw-HiU, New York, 1995) is described in Sec. 2. It has as its basis an equation for the compressibility factor ... 

To recapitulate, the Flory version of the Prigogine free-volume or corresponding-states polymer solution theory requires three pure-component parameters (p, v, T ) for each component of the solution and one binary parameter (p ) for each pair of components. 

Lee, B. I. and Kesler, M. G. (1975) AIChemEJL 21, 510. A generalized thermodynamic correlation based on three-parameter corresponding states. 

Figure 13.1a shows reduced vapor pressures and Fig. 13.1b reduced liquid molar densities for the parent isotopomers of the reference compounds. Such data can be fit to acceptable precision with an extended four parameter CS model, for example using a modified Van der Waals equation. In each case the parameters are defined in terms of the three critical properties plus one system specific parameter (e.g. Pitzer acentric factor). Were simple corresponding states theory adequate, the data for all... 

It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

A medium with bound electrons can be regarded as an ensemble of oscillators each of which is characterized by three parameters its natural frequency oscillator strength/, and the damping constant y(, which equals the inversed lifetime of the corresponding state ("y, = 1 /r,). The well-known expression for the permittivity of the medium91,95 can be presented in the form89... 

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

The definition of w makes its value zero for argon, krypton, and xenon, and experimental data yield compressibility factors for all three fluids that are correlated by the same curves when Z is represented as a function of Tr and Pr. Thus the basic premise of the three-parameter theorem of corresponding states is that all fluids having the same value of w have the same value of Z when compared at the same Tr and Pr. 

Figures 3.12 and 3.13 for Z°, based on data for the simple fluids, provide a complete two- parameter corresponding-states correlation for Z Since the second term of Eq. (3.45) is a relatively small correction to this two-parameter correlation, its omission does not introduce large errors. Thus Figs. 3.12 and 3.13 may be used alone for quick but less precise estimates of Z than are obtained from the complete three-parameter correlation.

If we take Z = Z° = 0.865, in accord with the two-parameter corresponding states correlation, then V = 1,467.1 cm,mor1, which is less than 1 percent lower than the value given by the three-parameter correlation. 

Calculated values of the quantities (Hr)°/RTc, (HRy/RTc, (Sr)°/R, an (SRy/R are shown by plots of these quantities vs. Pr for various values of Tr i Figs. 6.6 through 6.13. These plots, together with Eqs. (6.56) and (6.57), alio estimation of the residual enthalpy and entropy on the basis of the three-paramet corresponding-states principle as developed by Pitzer (Sec. 3.6). 

In Fig. 4.6 we have plotted a typical heat generation expression (curve a) along with the heat removal line, b. In this case the two curves intersect at three locations corresponding to three different reactor conditions that are possible for the same operating parameters and feed conditions. The low-temperature steady state is uneconomical since the feeds are virtually unconverted. The highest-temperature steady state has nearly complete conversion but may be too hot. Under those conditions side reactions may set in or the reactor pressure becomes too high. The middle steady state strikes a good compromise and is where... 

Figure 49. The energy variance (in units of [So(wi + >>2) /4]) versus the time variable N (in units of T). The system parameters are chosen as //p = 0.01, So 1-77 x 10 , lOn/coi = 12kI(A2 = T = 150. The three curves correspond to three initial states, i.e., below, above, or within the separatrix associated the Arnold diffusion. [From V. Ya Demikhovskii, F. M. Izrailev, and A. 1. Malyshev, Phys. Rev. Lett. 88, 154101 (2002).]...

The Lee/Kesler tables are adapted and published by permission from "A Generalized Thermodynamic Correlation Based on Three-Parameter Corresponding States," by Byungik Lee and Michael G. Kesler, AIChE J.,21,510-527 (1975). The numbers printed in italic type are liquid-phase properties. 

The midpoints of the 12 edges are the type-2 epikernels which are saddle-points connecting the minima. Thus, the three loops around the Tj intersection in Fig. 11 correspond to three equivalent paths from each vertex to its opposite vertex, and back, on the octahedron in Fig. 12. The geometrical parameters from CASSCF for each of the C2v minima is shown in the upper left of Fig. 12 for the both the singlet and triplet spin states. It should be clear that a very similar mechanism operates in both spin manifolds. 

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