The Redlich-Kwong equation of state

In this section the modified Redlich-Kwong equation will be described in detail. Of the many other equations of state which have been proposed, they either do not adequately represent available data or they contain too many adjustable parameters for the present purpose. 

Where the constant a is a measure of the cohesion between the molecules and is thus related to the attractive potential energy term of the Mie equation and b is a measure of the volume of the molecules, and is thus related to the repulsive potential term of the Mie equation. Although the van der Waals equation is useful over limited ranges of P and T, it fits experimental data poorly over extended ranges. 

Redlich-Kwong Equation. This equation, proposed by Redlich and Kwong (1949) has two adjustable parameters auid is convenient to use as a corresponding state equation. It is  

The constants a and b have the same physical significance as in the van der Waals equation. Redlich and Kwong (1949) showed that a and b parameters calculated by the corresponding state method resulted in good fits to experimental data for simple, non-polar molecules such as O2, CO and CH. For the corresponding state representation of a and b, Redlich and Kwong (1949) use  

In addition, equation (15) suggests that the a parameter should have a temperature dependence proportional to 1/T in the case of dipole moments (it can also be shown to hold for quadrupole moments). The Redlich-Kwong equation was modified by de Santis, et al. to take the above factors into account for H2O, CO2 and mixtures of them with non-polar molecules. The modified equation will be referred to as the MRK in this chapter. For H2O and C02r de Santis, et al. (1974) calculated the temperature dependence of a from experimental data up to 800 C. They write the expression for a as  

The early purpose of equations of state was the description of the volumetric behavior of fluids, especially in the vapor phase. The most popular and successful among them was that proposed by Redlich and Kwong (RK, 1949)  

Using the same approach as with the vdW EoS (Example 8.7), we arrive at the following expressions for a and b  

We explore the prediction of the volumetric behavior of a pure fluid with the RK EoS in the next Example. 

Introduction of these values into Eq.8.10.3 and solution of the resulting cubic equation gives  

TTie value of 1404.5 cm /mol at 420 K is very close to the experimental value with an error of -0.5%. 

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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). 

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 ... 

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. 

Dimensionless Constants for Saturated Liquids in the Redlich-Kwong Equation of State... 

To evaluate the first integral, it is necessary to substitute for or dP. A trial will show that it is simpler to substitute for dP. Thus, solving the Redlich-Kwong equation of state for P, we obtain... 

The fugacity coefficients are a function of pressure, temperature and the equilibrium mole fractions, so at given pressure and temperature eq. (2.4-20) can be solved for s and the equilibrium mole fractions can be calculated. Table 2.4-1 gives the calculated equilibrium composition of the reaction mixture at different pressures for an ideal gas mixture and in case the gas is described with the Redlich-Kwong equation of state. 

Figure 7.1-12. Energy stored in a gas pressurised vessel as function of pressure and the energy equivalent in mass of TNT according to a volume of 1 fit3 and a gas with an adiabatic coefficient c = 1.66 (Argon). Argon data are based on the Redlich-Kwong - Equation of State. These curves should only be used as a guide. Variation of temperature within a vessel must be considered [19],...

Fussell, D.D. and Yanosik, J.L. An Iterative Sequence for Phase Equilibria Calculations Incorporating the Redlich-Kwong Equation of State, Soc. Pet. Eng. J. (June 1978) 18, 173-182. 

Joffe J., Schroeder G.M., Zudkevitch D., "Vapor-liquid Equilibria with the Redlich-Kwong Equation of State", AIChF-J. 1970, 48,261-266. 

Mixing Rules for the Redlich-Kwonq Equation of State. The Redlich-Kwong equation of state, Equation 6, can be written in the following form ... 

These mixing rules, when joined with the Redlich-Kwong equation of state, will constitute the Redlich-Kwong equation of state for mixtures that is consistent with the statistical mechanical basis of the van der Waals mixing rules. 

The second term on the RHS of eq. 22 is the volume expansivity which can be calculated from a two-parameter, cubic equation of state such as the Redlich-Kwong EOS.(14) We chose the Redlich-Kwong EOS because the "a" term is independent of temperature, which simplifies the procedure of solving for the analytical solution. Using the method of implicit differentiation with the Redlich-Kwong equation of state,... 

The Chao-Seader and the Grayson-Streed methods are very similar in that they both use the same mathematical models for each phase. For the vapor, the Redlich-Kwong equation of state is used. This two-parameter generalized pressure-volume-temperature (P-V-T) expression is very convenient because only the critical constants of the mixture components are required for applications. For the liquid phase, both methods used the regular solution theory of Scatchard and Hildebrand (26) for the activity coefficient plus an empirical relationship for the reference liquid fugacity coefficient. Chao-Seader and Grayson-Streed derived different constants for these two liquid equations, however. 

These answers are close enough in view of the accuracy of the original data, d. Using the Redlich Kwong Equation of State... 

Estimate the molar volume of isopropyl alcohol vapor at 10 atm (1013 kPa) and 473 K (392°F) using the Redlich-Kwong equation of state. For isopropyl alcohol, use 508.2 K as the critical temperature Tc and 50 atm as the critical pressure Pc. The Redlich-Kwong equation is... 

Related Calculations. This two-constant equation of Redlich-Kwong is extensively used for engineering calculations and enjoys wide popularity. Many modifications of the Redlich-Kwong equations of state, such as those by Wilson, Barnes-King, Soave, and Peng-Robinson, have been made and are discussed in Reid et al. [1 ]. The constants for the equation of state may be obtained by least-squares fit of the equation to experimental P-V-T data. However, such data are often not available. When this is the case, estimate the constants on the basis of the critical properties, as shown in the example. 

Calculate the compressibility factor for the mixture. In a manner similar to that used in the previous problem, an expression for the fugacity coefficient in vapor mixtures can be derived from any equation of state applicable to such mixtures. If the Redlich-Kwong equation of state is used, the expression is... 

For an isothermal fluid flow described by the Redlich-Kwong equation of state, develop expressions in terms of the initial temperature and the initial and final volumes for the changes in internal energy, enthalpy, entropy, and the Gibbs free energy. 

Gillespie and Beattie [89] (see also [33]) were by for the most successful experimentally in establishing a firm basis for an analytical expression of the equilibrium constant in the range of industrial interest. The values in Tables 10 and 11 were calculated using their equation. A detailed description, with literature data and many tables, appears in [33]. A description of the equilibrium using the Redlich-Kwong equation of state is given in [90]. 

In this paper we present a new characterisation method for porous carbonaceous materials. It is based on a theoretical treatment of adsorption isotherms measured in wide temperature (303 to 383 K) and pressure ranges (0 to 10000 kPa) and for different adsorbates (N2, CH4, Ar, C3H8 and n-C4Hio). The theoretical treatment relies on the Integral Adsorption Equation concept. We developed a local adsorption isotherm model based on the extension of the Redlich-Kwong equation of state to surface phenomena and we improved it to take into account the multilayer formation. The pore size distribution fimction is assumed to be a bi-modal gaussian. By a minimisation procedure, it is possible to determine the bi-modal pore size distribution function witch can be used for purely characterisation purposes or to predict adsorption isotherms. 

Specialize the result for B to the Redlich/Kwong equation of state, express it in reduced form, and compare it numerically witli the generalized correlation for B for simple fluids, Eq. (3.61). Discuss what you find. 

The optimum binary interaction parameters are shown in Table 111. An example of the results is shown in Figure 1 for the PR-EOS applied to carbon dioxide/methyl oleate at 70 C. Comparing the results of those three simple equations of state, the Redlich-Kwong equation of state gave the poorest prediction. 

The two-constant, Peng-Robinson and Redlich-Kwong equations appear to be quite good according to Table 3.3. Figure 3.4 compares the van der Waals and the Redlich-Kwong equations of state with experimental data. 

Calculate the pressure of 176 g of CO2 in a 6250-cm tank at 298,15 K using the Redlich-Kwong equation of state. 

In these equations, T is the critical temperature (in absolute terms), p is the critical pressure, and T is the reduced temperature (the absolute temperature divided by the critical temperature). The a is particular to the Redlich-Kwong equation of state. 

The Redlich-Kwong equation of state was modified further by Soave to give the Redlich-Kwong-Soave equation of state (called RK-Soave in Aspen Plus), which is a common one in process simulators ... 

Find e specific volume of n-butane at 500 K and 18 atm using the Redlich-Kwong equation of state option in Aspen Plus. 

Find the specific volume of a mixture consisting of 630 kmol/h of carbon monoxide, 1130 kmol/h of water, 189 kmol/h of carbon dioxide, and 63 kmol/h of hydrogen at 1 atm and 500 K. The specific volume is the solution to the Redlich-Kwong equation of state, Eq. (2.8). 

Find the compressibility factor of ammonia gas at conditions from 50 to 250 atm and 4(X) K using the Redlich-Kwong equation of state in Excel. (Hint Before beginning your spreadsheet, think about how you can organize it so that you can copy formulas from cell to cell easily.)... 

Use the Redlich-Kwong equation of state to calculate the enthalpy departure of a mixture of acetone (1) and 1,3-butadiene (2) with mole fractions Y, = 0.3, Yj = 0-7 at 70°C and 200 kPa. Assume Kay s rules apply in calculating the equation of state mixture parameters. 

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