Equation Cubic

The average accuracy of the Lee and Kesler model is much better than that of all cubic equations for pressures higher than 40 bar, as well as those around the critical point. 

Figure A2.5.23. Reduced temperature = T/T versus reduced density p. = p/p for Ne, Ar, Kr, Xe, N2, O2, CO, and CH. The frill curve is the cubic equation (A2.5.26). Reproduced from [10], p 257 by pennission of the American Institute of Physics.

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

Special cases of equation 33 are obtained on specification of values or expressions for various parameters. Because of their generality, two-parameter cubic equations are the most popular. The following equations are modem examples (10—12) ... 

Cubic equations, although simple and able to provide semiquantitative descriptions of real fluid behavior, are not generally useful for accurate representation of volumetric data over wide ranges of T and P. For such appHcations, more comprehensive expressions with large numbers of adjustable parameters are needed. 7h.e simplest of these are the extended virial equations, exemplified by the eight-constant Benedict-Webb-Rubin (BWR) equation of state (13) ... 

Mixing mles for the parameters in an empirical equation of state, eg, a cubic equation, are necessarily empirical. With cubic equations, linear or quadratic expressions are normally used, and in equations 34—36, parameters b and 9 for mixtures are usually given by the following, where, as for the second virial coefficient, = 0-. 

AppHcation of equation 226 requires the availabiHty of a single equation of state suitable for both vapor and Hquid mixtures. Cubic equations of state are widely used for VLE calculations. 

A generalized cubic equation is the Redhch-Kwong equation (77) ... 

Vapor densities for pure compounds can also be predicted by cubic equations of state. For hydrocarbons, relatively accurate Redlich-Kwong-type equations such as the Soave and Peng-Robinson equations are often used. Both require only T, and (0 as inputs. For organic compounds, the Lee-Erbar-EdmisteF" equation (which requires the same input parameters) has been used with errors essentially equivalent to those determined for the Lydersen method. While analytical equations of state are not often used when only densities are required, values from equations of state are used as inputs to equation of state formulations for thermal and equilibrium properties. 

Cubic Equations A cubic equation, in one variable, has the form x + bx + CX + d = Q. Eveiy cubic equation having complex coefficients... 

Cubic Equations of State The simplest expressions that can (in... 

The modern development of cubic equations of state started in 1949 with publication of the Redlich/Kwong equation (Redhch and Kwong, Chem. Rev., 44, pp. 233-244 [1949]) ... 

The application of cubic equations of state to mixtures requires expression of the equation-of-state parameters as func tions of composition. No exact theory like that for the virial coefficients prescribes this composition dependence, and empirical mixing rules provide approximate relationships. The mixing rules that have found general favor for the Redhch/Kwong equation are ... 

The two values kp and k are usually not very different, and kp is not strongly composition dependent. Nevertheless, the quadratic dependence of Z — a/RT) on composition indicated by Eq. (4-305) is not exactly preserved. Since this quantity is not a true second virial coefficient, only a value predicted by a cubic equation of state, a strict quadratic dependence is not required. Moreover, the composition-dependent kp leads to better results than does use of a constant value. 

A variety of equations-of-state have been applied to supercritical fluids, ranging from simple cubic equations like the Peng-Robinson equation-of-state to the Statistical Associating Fluid Theoiy. All are able to model nonpolar systems fairly successfully, but most are increasingly chaUenged as the polarity of the components increases. The key is to calculate the solute-fluid molecular interaction parameter from the pure-component properties. Often the standard approach (i.e. corresponding states based on critical properties) is of limited accuracy due to the vastly different critical temperatures of the solutes (if known) and the solvents other properties of the solute... 

The unknown partial pressure at the external surface can be eliminated as P 5 = Pag — r/ki), which results in a cubic equation for / ... 

Rearranging Equation 13-36 into a cubic equation becomes g R -H 3fg2R2 - 4 d2R2 + 3f2gR2 - 2adR2 P -H - a 4 = 0... 

Equation 13-39 is a cubic equation in terms of the larger aspect ratio R2. It can be solved by a numerical method, using the Newton-Raphson method (Appendix D) with a suitable guess value for R2. Alternatively, a trigonometric solution may be used. The algorithm for computing R2 with the trigonometric solution is as follows ... 

Using the cubic equation presented earlier, Bartknecht [54] developed for vessels of different sizes for the same process system in closed or vented vessels, valid for flammable gases and combustible dusts ... 

Third-degree equations cubic equations), in the general case, have the form, after division by the coefficient of the highest-order term,... 

For instance, a quadratic expression for C (T) will require the solution of a cubic equation in T. ... 

If one sets p r2 this leads ultimately to the determination of p from the cubic equation... 

Equations of state that are cubic in volume are often employed, since they, at least qualitatively, reproduce the dependence of the compressibility factor on p and T. Four commonly used cubic equations of state are the van der Waals, Redlich-Kwong, Soave, and Peng-Robinson. All four can be expressed in a reduced form that eliminates the constants a and b. However, the reduced equations for the last two still include the acentric factor u> that is specific for the substance. In writing the reduced equations, coefficients can be combined to simplify the expression. For example, the reduced form of the Redlich-Kwong equation is... 

The four cubic equations of state are summarized in Table A3.1. 

Finally, these results clearly show yet again that attachment of hydrogen to the aromatic is not a fast step, thereby eliminating the A-l mechanism yet the data in mixed solvents followed the prediction of the Gross-Butler cubic equation, and once again this test turns out to be valueless. 

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