Robinson equation

As explained in section 3.6.1, many modifications have been proposed for the Debye-Hiickel relationship for estimating the mean ionic activity coefficient 7 of an electrolyte in solution and the Davies equation (equation 3.35) was identified as one of the most reliable for concentrations up to about 0.2 molar. More complex modifications of the Debye-Huckel equation (Robinson and Stokes, 1970) can greatly extend the range of 7 estimation, and the Bromley (1973) equation appears to be effective up to about 6 molar. The difficulty with all these extended equations, however, is the need for a large number of interacting parameters to be taken into account for which reliable data are not always available. 

Hydrocarbon mixtures are most often modeled by the equations of state of Soave, Peng Robinson, or Lee and Kesler. 

Zhu S-B, Lee J, Robinson G W and Lin S H 1988 A microscopic form of the extended Kramers equation. A simple friction model for cis-trans isomerization reactions Chem. Phys. Lett. 148 164-8... 

The values were calculated from the modified Debye-Hiickel equation utilizing the modifications proposed by Robinson and by Guggenheim and Bates ... 

Penetrating stains Penetration resistance Penetration theory Penetrometers Peng-Robinson equation DL-Pemcillamine Penicillamine [52-67-5]... 

In each of these expressions, ie, the Soave-Redhch-Kwong, 9gj j (eq. 34), Peng-Robinson, 9pj (eq. 35), and Harmens, 9 (eq. 36), parameter 9, different for each equation, depends on temperature. Numerical values for b and 9(7) are deterrnined for a given substance by subjecting the equation of state to the critical derivative constraints of equation 20 and by requiring the equation to reproduce values of the vapor—Hquid saturation pressure,... 

Corresponding states have been used in other equations. For example, the Peng-Robinson equation is a modified RedHch-Kwong equation formulated to better correlate vapor—Hquid equiHbrium (VLE) vapor pressure data. This equation, however, is not useful in reduced form because it is specifically designed to calculate accurate pressure data. Reduced equations generally presuppose knowledge of the pressure. 

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. 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

The methanol(l)/acetone(2) system serves as a specific example in conjunction with the Peng/Robinson equation of state. At a base temperature To of 323.15 K (50°C), both XT E data (Van Ness and Abbott, Jnt. DATA Ser, Ser A, Sel. Data Mixtures, 1978, p. 67 [1978]) and excess enthalpy data (Morris, et al., J. Chem. Eng. Data, 20, pp. 403-405 [1975]) are available. From the former. 

Several investigators have attempted to modify the basic Deutsch equation so that it would more nearly describe precipitator performance. Cooperman ( A NewTheoiy of Precipitator Efficiency, Pap. 69-4, APCA meeting, New York, 1969) introduced correction fac tors for diffusional forces arising from variations in particle concentration along the precipitator length and also perpendiciilar to the collecting surface. Robinson [Atmos. Environ. 1(31 193 (1967)] derived an equation for collec tion efficiency in which two erosion or reentrainment terms are introduced. 

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... 

In some earlier work the shift reaction was assumed always at equilibrium. Fiigacities were calculated with the SRK and Peng-Robinson equations of state, and correlations were made of the equilibrium constants. 

The existing equations of state (i.e., Benedict-Webb-Rubin (BWR), Soave-Redlich-Kwang, and Peng-Robinson) have some practical limitations. The equations of state developed by the University of Illinois... 

Cyclizations such as that which underlies the Knorr synthesis have also been successfully used in Robinson annelation sequences [72] (equation 12)... 

In this equation it is the cycle index sum Z(F) that we do not know. The cycle index sums Z(T) and Z( C) can be computed, though the method is not immediately apparent and is a story in itself. Thus equation (8.1) does, in theory, give a means for computing Z(F), but not a very practical one. The equation is the wrong way round, and what is needed is an expression for Z(F) in terms of Z( C ) and Z( T ). In what was an important breakthrough in this kind of enumeration, R. W. Robinson showed that equation (8.1) could be inverted to give an equation of the form... 

The fugacity coefficient can be calculated from other equations of state such as the van der Waals, Redlick-Kwong, Peng-Robinson, and Soave,d but the calculation is complicated, since these equations are cubic in volume, and therefore they cannot be solved explicitly for Vm, as is needed to apply equation (6.12). Klotz and Rosenburg4 have shown a way to get around this problem by eliminating p from equation (6.12) and integrating over volume, but the process is not easy. For the van der Waals equation, they end up with the relationship... 

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 A3.5 Comparison of the experimental r (dashed lines) with the r values calculated from the Peng-Robinson equation of state (solid lines). Values for the acentric factor are (a) = —0.218 (the value for HU), (b) = 0.011 (the value for CH4),...

NH3 (a = 0.250), and H20 (u> = 0.344). Thus, results for a wide range of acentric factors are compared. In Figure A3.5, we make the same comparisons with the Peng-Robinson equation. 

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. 

The value of v is important both in equation 7 and for accurate calculation of concentrations in other equations. For simplicity and accuracy, the Peng-Robinson equation of state has been used to calculate v for the model O). This equation expresses the P-V-T relationship as follows ... 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

Tropinone is another classic compound in the history of total synthesis. The celebrated plans of Willstatter and Robinson are shown in Schemes 4.20 and 4.21 and Figure 4.63 shows a synthesis map for different ways this compoimd has been made. The synthesis tree for the three-component Robinson plan is shown in Figure 4.64. Calcium carbonate and hydrochloric acid are added as inputs to complete the balanced chemical equation since these are involved in a neutralization reaction. It is clear from the results summarized in Table 4.30 that the Robinson plan is the clear front-rimner because the synthesis is achieved in a single step even though the reaction yield is modest. Any further improvements to this method would be directed to improving this parameter. 

Combining equations (7.4-24)-(7.4-26) gives a system of non-linear equations that can be solved using iterative techniques. Savings in equipment costs as compared to initial guesses are approximately 30 %. The real savings will be lower because the optimal choices for equipment units are usually not available on the chemical equipment market. The standard sizes greater but nearest to the optimal sizes will be selected. The total cost for the standard equipment is very close to the minimum found. Robinson and Loonkar (1972) extended their procedure for multiproduct batch plants. 

Volumetric equations of state (EoS) are employed for the calculation offluid phase equilibrium and thermo-physical properties required in the design of processes involving non-ideal fluid mixtures in the oil, gas and chemical industries. Mathematically, a volumetric EoS expresses the relationship among pressure, volume, temperature, and composition for a fluid mixture. The next equation gives the Peng-Robinson equation of state, which is perhaps the most widely used EoS in industrial practice (Peng and Robinson, 1976). 

Leu and Robinson (1992) reported data for this binary system. The data were obtained at temperatures of 0.0, 50.0, 100.0, 125.0, 133.0 and 150.0 °C. At each temperature the vapor and liquid phase mole fractions of isobutane were measured at different pressures. The data at 133.0 and 150.0 are given in Tables 14.9 and 14.10 respectively. The reader should test if the Peng-Robinson and the Trebble-Bishnoi equations of state are capable of describing the observed phase behaviour. First, each isothermal data set should be examined separately. 

Robinson and the Trebble-Bishnoi equations of state are capable of describing the observed phase behavior. First, each isothermal data set should be examined separately. 

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