Corresponding states principle three-parameter

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). 

For nonassociating molecules, the methods of Tsonopoulos [14] and Hayden and O Connell [15] yield pretty good results for the estimation of the second virial coefficient. The first method works with the three-parameter corresponding-states principle (Section 2.4.4) the second method takes into account several molecular effects. In case of vapor phase association (Section 13.2), the model of Hayden-O Connell is extended by a chemical equilibrium term, whose parameters have to be fitted to experimental data. 

A modification of the corresponding-state principle by introducing a parameter related to the vapor pressure curve is reasonable, because experimental vapor pressure data as a function of temperature are easy to retrieve. Furthermore, the vapor-liquid equilibrium is a very sensitive indicator for deviations from the simple corresponding-state principle. The value Tr = 0.7 was chosen because this temperature is not far away from the normal boiling point for most substances. Additionally, the reduced vapor pressure at Tr = 0.7 of the simple fluids has the value = 0.1 (log = —1). As a consequence, the acentric factor of simple fluids is 0 and the three-parameter correlation simplifies to the two-parameter correlation. 

The three-parameter corresponding-state principle is applicable to many substances only for strongly polar or associating substances large deviations between the theoretical and experimental values can occur. 

For the description of thermophysical properties, estimation methods based on the three parameter corresponding states principle mostly employ the critical temperature (Tc), the critical pressure (Pc), and the acentric factor (co) as characteristic... 

As discussed in Section 2.5.4, the simple two-parameter corresponding states principle indicates that a generalized equation of state for all substances can be created using only two specific parameters, for example, T, and P. The success of this approach is restricted to simple, spherical molecules like Ar, Kr, Xe, or CH4, where vapor pressure and compressibility factor can be reasonably described. For other molecules, the simple two-parameter corresponding states principle leads to significant errors. A large improvement has been achieved with the introduction of a third parameter which describes the vapor pressure curve (extended three-parameter principle of corresponding states). The most common parameter of this kind is the so-called acentric factor, which is defined as... 

All theoretical equations of state suggest a corresponding state behavior of PVT properties that requires three scaling parameters P, V, and T ). These define the corresponding state and are used to scale/reduce the PVT toward P =P/P, V = V/V, f = T/T. In his Ph.D. thesis of 1873, van der Waals proposed the first EoS formulated in terms of corresponding state (reduced) variables. The relation can be written in terms of PVT or in terms of reduced variables PVf, indicating expected observance of the corresponding state principle ... 

An equation of state has been developed by Kosinski and Anderko (2001) for the representation of the phase behavior of high-temperature and supercritical aqueous systems containing salts. They improved the EOS by Anderko and Pitzer (1993a) to enhance the predictive capability of the EOS using the three-parameter corresponding-states principle. The model was successfully apphed to the H2O + NaCl solutions up to 573 K, and correctly predicts the p VTx properties of H2O + KCl solution up to 773 K. This EOS also considerably extended the validity range. The EOS is also applicable to water + nonelectrolyte solutions such as water + methane and water + n-decane systems. 

The second approach is to extend the simple two-parameter corresponding-states principle at its molecular origin. This is accomplished by making the intermolecular potential parameters functions of the additional characterization parameters /I, and the thermodynamic state, for example, the density p and temperature T. This can be justified theoretically on the basis of results obtained by performing angle averaging on a non-spherical model potential and by apparent three-body effects in the intermolecular pair potential. The net result of this substitution is a corresponding-states model that has the same mathematical form as the simple two-parameter model, but the definitions of the dimensionless volume and temperature are more complex. In particular the... 

Reflect a three-parameter corresponding states principle, i.e. predict successfully vapor pressures for nonpolar/weakly polar compounds where Jp, and values are available ... 

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 SRK and PR equations follow the principle of corresponding states in the three-parameter form only the commonly available critical properties T, p, and are required to apply the equation to a substance. The simple vdW mixing rules work well with these equations. Hence they are widely used for the calculation of vapor-liquid equilibrium in mixtures. 

The PR eos has been modified by Stryjek and Vera to extend to polar substances that do not follow the three-parameter principle of corresponding states. The modified eos is fitted to the vapor pressure of polar substances with additional substance-specific parameters. The PRSV equation has been described in Equation (4.163) et seq. The free-energy-matched mixture eos parameters are given in Equations (4.436) and (4.438) the fugacity coefficients are given in Equation (4.439). PRSV eos using the UNIEAC activity coefficient predicts the vie data for both ethanol/water mixtures at 423-623°K and acetone/water mixtures at 373-523°K from low to high pressure. 

If all parameter sets calculated in accordance with the procedure just described are evaluated, it turns out that, due to the principle of corresponding states, they form a net. All reactions characterized by a three-dimensional parameter combination positioned under this net may be performed in this vessel because, in the case of a failure scenario, they will produce mass flows vsdiich are smaller than the maximum allowed mass flow. All reactions with a parameter combination positioned above the net must not be performed in this plant except if the protecting safety concept has been adequately modified. A three-dimensional plot of the columns which carry the net for a set pressure of 6 bars is shown in Figure 7-7 for the example discussed here. 

A principle of corresponding states based on three parameters can be enunciated for substances to which the perturbation approach applies all substances that have the same value of the perturbation factor have the same reduced equations of state. For example, the reduced second virial coefficient can be written in terms of the acentric factor ... 

For molecular gases the extended principle of corresponding states holds only in a limited form, for the reasons discussed above. In particular, it holds only in the temperature range in which the repulsive wall of an effective potential is dominant, and so fails at low temperatures k T/e < 1) this eliminates the need for one of the three additional parameters introduced by the extended principle for the noble gases. However, the extension to higher temperatures is successful. The second virial coefficients can also be included, provided that corrections for specific contributions of nonspherical components of the potential are made. The overall accuracy of the correlation for molecular gases is estimated to be somewhat less than for the noble gases, but viscosity, diffusion and thermal diffusion are all included. 

Following the principle of corresponding states, the equation of state should be able to provide reasonably good results for nonpolar substances with two, and still better, with three adjustable parameters. Indeed, the typical cubic EoS contain two or three parameters. For very accurate results, however, non-cubic equations of state with a large number of parameters are used, such as the Benedict-Webb-Rubin one. 

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