Redlich - Kwong equation

This equation is an improvement over the van der Waals equation in that it introduces a temperature dependency for the attraction parameter by dividing it by the term The equation also has a special quadratic term in the volume  

The parameters a and b are evaluated in terms of the critical constants by applying the critical point conditions (Equation 1.11). The results are 

An alternative method for determining the parameters is by the regression of experimental data with the objective of minimizing errors between the experimental and predicted properties. For the regression, either direct PVT data or properties that can be derived from the equation of state, such as VLE or enthalpy departure data, may be used. 

Later work on Redlich-Kwong-type equations (known as cubic equations because they are of the third degree in volume when expressed explicitly in the pressure) has shown that greater accuracy is achieved if the equation parameters are correlated as a function of temperature. The original Redlich-Kwong equation has therefore been dropped for the most part from practical applications in favor of the more recent cubic equations. 

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Compilation of binary experimental data reduced with the Wilson equation and, for high pressures, with a modified Redlich-Kwong equation. 

Given the estimate of the reactor effluent in Example 4.2 for fraction of methane in the purge of 0.4, calculate the.actual separation in the phase split assuming a temperature in the phase separator of 40°C. Phase equilibrium for this mixture can be represented by the Soave-Redlich-Kwong equation of state. Many computer programs are available commercially to carry out such calculations. 

TABLE 4.3 Vapor-Liquid Phase Split Using the Soave-Redlich-Kwong Equation of State... 

Soave, G. (1972), Equilibrium constants from a modified Redlich-Kwong equation of state . Chem. Eng. Sci., Vol. 27, p. 1197. 

Other pressure—volume—temperature (PVT) relationships may be found in the Hterature ie, Benedict, Webb, Rubin equations of state (4—7) the Benedict, Webb, Rubin, Starling equation of state (8) the Redlich equation of state (9) and the Redlich-Kwong equation of state (10). 

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]) ... 

Multiplication of the Redlich/Kwong equation (Eq. [4-220]) by V/RT leads to its expression in alternative form ... 

Since non-ideal gases do not obey the ideal gas law (i.e., PV = nRT), corrections for nonideality must be made using an equation of state such as the Van der Waals or Redlich-Kwong equations. This process involves complex analytical expressions. Another method for a nonideal gas situation is the use of the compressibility factor Z, where Z equals PV/nRT. Of the analytical methods available for calculation of Z, the most compact one is obtained from the Redlich-Kwong equation of state. The working equations are listed below ... 

For an isothermal change, the expression for P from the Redlich-Kwong equation can be substituted into the general formula for work done ... 

Reid (1976) and many other authors give pure propane a superheat temperature limit of 53 C at atmospheric pressure. The superheat temperature limit calculated from the Van der Waals equation is 38°C, whereas the value calculated from the Redlich-Kwong equation is S8°C. These values indicate that, though an exact equation among P, V, and 7 in the superheat liquid region is not known, the Redlich-Kwong equation of state is a reasonable alternative. 

Fig. 4. Vapor-phase solubility of naphthalene in ethylene. Data points from G. A. M. Diepen and F. E. C. Scheffer, J. Am. Chem. Soc. 70, 4085 (1948) vapor-phase fugacities from (---) Redlich-Kwong equation (-) Ideal gas law.

To obtain an analytic function / in Eq. (55), Chueh uses the Redlich-Kwong equation however, since the application is intended for liquids, the two constants in that equation were not evaluated (as is usually done) from critical data alone, but rather from a fit of the pure-component saturated-liquid volumes. The constants a and b in the equation of Redlich and Kwong are calculated from the relations... 

Calculated with universal constants in Redlich-Kwong equation. 

In their correlation, Chao and Seader use the original Redlich-Kwong equation of state for vapor-phase fugacities. For the liquid phase, they use the symmetric convention of normalization for y and partial molar volumes which are independent of composition, depending only on temperature. For the variation of y with temperature and composition, Chao and Seader use the equation of Scatchard and Hildebrand for a multicomponent solution ... 

To illustrate this thermodynamic consistency test, Figs. 15, 16, and 17 show plots of the appropriate functions needed to calculate Areas I, II, and III, respectively, for the nitrogen-carbon dioxide system at 0°C the data are taken from Muirbrook (M5). Fugacity coffiecients were calculated with the modified Redlich-Kwong equation (R4). 

Redlich-Kwong equation, 181 Rybczynski-Hadamard formula, 318, 332, 348... 

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... 

Figure A3.3 compares the experimental (corresponding states) results with the predictions from the van der Waals. modified Berthelot, Dieterici, and Redlich-Kwong equations of state.b The comparison is not so direct for the Soave and Peng-Robinson equations of state, since the reduced equation still includes to, the acentric factor. Figure A3.4 compares the corresponding states line, with the prediction from the Soave equation, using four different values of to. The acentric factors chosen are those for H (o> = —0.218), CH4 (to = 0.011),...

Figure A3.3 Comparison of the experimental r (dashed lines) with the c values calculated from the (a) van der Waals, (b) modified Berthelot, (c) Dieterici, and (d) Redlich-Kwong equations of state expressed in reduced form.

The Redlich-Kwong equation predicts the correct r reasonably well over almost the entire 77 and pr range shown in Figure A3.3. It appears that at pT > 5, significant deviations may occur, but overall, this equation seems to give the best fit of the equations compared. 

For both the Soave and Peng-Robinson equations, the fit is best for uj — 0. The Soave equation, which essentially reduces to the Redlich-Kwong equation when ui — 0, does a better job of predicting than does the Peng-Robinson equation. The acentric factors become important when phase changes occur, and it is likely that the Soave and Peng-Robinson equations would prove to be more useful when 77 < 1. 

For the monomers in the polymerization under consideration the fugacity coefficients were estimated by Redlich-Kwong equation of state and were found to be close to unity. The activity coefficients (8) for the monomers were estimated by Scatchard-Hildebrand s method (5) for the most volatile monomer there was a temperature dependence but none for the other monomer. These were later confirmed by applying the UNIFAC method (6). The saturation vapor pressures were calculated by Antoine coefficients (5). 

This equation is an extension of the more familiar Van der Waal s equation. The Redlich -Kwong equation is ... 

Soave (1972) modified the Redlich-Kwong equation to extend its usefulness to the critical region, and for use with liquids. 

For the vapour phase, the deviation of the specific enthalpy from the ideal state can be illustrated using the Redlich-Kwong equation, written in the form ... 

The basic features of equations of state are not complicated when they are expressed as PV/RT versus density. Figure 3 is a sample plot for methanol. These curves are characteristic of all fluids, and equations of state only differ in their ability to accurately predict these curves. The actual curves are relatively simple and they change only slightly from one material to another for this reason, simple equations of state such as the Redlich-Kwong equation have been about as successful as the BWR equation. The simplicity of the actual curves is often hidden because the data are not usually plotted as PV/RT versus density. More often the data are plotted as PV/RT versus pressure shown in Figure 4, or pressure versus volume shown in Figure 5. Both of these plots obscure the real simplicity shown in Figure 3. 

To illustrate the problem with mixtures, the Redlich-Kwong equation has been used to correlate literature data on three types of mixtures as follows. 

The present paper deals with one aspect of this problem the calculation of phase separation critical points in reacting mixtures. The model employed is the Soave-Redlich-Kwong equation of state (1 ), which is typical of several equations of state (2, 5) which have relatively recently come into wide use as phase equilibrium models for light gas mixtures, sometimes including water and the acid gases as components (4, . 5, 6). If the critical point contained in the equation of state (perhaps even for the mixture at reaction equilibrium) can be found directly, the result will aid in other equilibrium computations. 

For the Soave-Redlich Kwong equation, the fugacity derivatives are... 

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