A modification of the martin equation of state for calculating vapour-liquid equilibria

A modification of the Martin (1979) equation of state suitable for vapour-liquid equilibrium calculations is presented. The temperature dependence of this modification is determined from the second virial coefficient correlation of Tsonopoulos (1974) and the pure-component vapour pressures. Excellent predictions of phase equilibria have been obtained for a number of binary systems, including a system containing hydrogen. For the systems tested, the equation of this study is more accurate than the Peng-Robinson (1976) equation of state.     

Comparison of the differential isosteric adsorption enthalpies and entropies calculated from chromatographic data

The differential isosteric enthalpies, -deltaH(ads), and entropies, -deltaS(ads), of adsorption were calculated taking the retention times of the peak maxima and the centres of gravity of peaks into account and compared with the results obtained from the adsorption second virial coefficients. A mathematical link between the -deltaH(ads) and -deltaS(ads) magnitudes and experimental data was derived through the Antoine-type equation which enables the -deltaH(ads) and -deltaS(ads) magnitudes to be found from adsorption second virial coefficients, B2S, calculated on the basis of chromatographically determined adsorption isotherm data. The virial coefficients were calculated employing the values of the Tóth and Unilan equation parameters. There are no significant differences to be found between the isosteric enthalpies obtained, whereas the values of the adsorption entropies were the highest for the centre of peak gravity data.     

Estimation of third virial coefficients at low reduced temperatures

We use the Clausius–Clapeyron equation to calculate third virial coefficients at low reduced temperatures. This procedure gives an alternative to predict third virial coefficients in a region where the third virial coefficient is difficult to measure. We compare the results of this method with published third virial coefficient data. Calculated third virial coefficients have average percentage deviations within 5% of the experimental values at reduced temperatures between 0.8 and 1.0.     

Calculation of the limiting activity coefficients of components in a solution from isothermal data on the equilibrium compositions of liquid and vapor phases

The method of calculating the limiting activity coefficients of components was developed and illustrated by a number of examples. Isothermal binary data on the compositions of the solution and vapor coexisting phases were fitted to the Porter, Margules, Van Laar, Wilson, Redlich-Kister, and NRTL interpolation equations. In this method, nonideality of a vapor phase is directly taken into account and the composition of the azeotrope in the system, if exists, may be calculated.     

A multisolute osmotic virial equation for solutions of interest in biology

The osmotic virial equation was used to predict osmolalities of solutions of interest in biology. The second osmotic virial coefficients, Bi, account for the interactions between identical solute molecules. For multisolute solutions, the second osmotic virial cross coefficient, Bij, describes the interaction between two different solutes. We propose to use as a mixing rule for the cross coefficient the arithmetic average of the second osmotic virial coefficients of the pure species, so that only binary solution measurements are required for multisolute solution predictions. Single-solute data were fit to obtain the osmotic virial coefficients of the pure species. Using those coefficients with the proposed mixing rule, predictions were made of ternary solution osmolality, without any fitting parameters. This method is shown to make reasonably accurate predictions for three very different ternary aqueous solutions: (i) glycerol + dimethyl sulfoxide + water, (ii) hemoglobin + an ideal, dilute solute + water, and (iii) bovine serum albumin + ovalbumin + water.     

A simple method for evaluation of parameters of the Bender equation of state from experimental data

A simple numerical method for evaluation of parameters (constants) of Bender equation of state for pure fluids is proposed. The minimisation of the objective function leads to a set of linear equations. The method employs experimental data on state behaviour (p–ρ–T) of fluid phases, vapour–liquid equilibrium data (saturated vapour pressures and orthobaric densities), second virial coefficients, and the coordinates of the gas–liquid critical point. Results of the tests using data for two fluids (methane and n-pentane) are presented.     

Rescaled density expansions and demixing in hard-sphere binary mixtures

The demixing transition of a binary fluid mixture of additive hard spheres is analyzed for different size asymmetries by starting from the exact low-density expansion of the pressure. Already within the second virial approximation the fluid separates into two phases of different composition with a lower consolute critical point. By successively incorporating the third, fourth, and fifth virial coefficients, the critical consolute point moves to higher values of the pressure and to lower values of the partial number fraction of the large spheres. When the exact low-density expansion of the pressure is rescaled to higher densities as in the Percus-Yevick theory, by adding more exact virial coefficients a different qualitative movement of the critical consolute point in the phase diagram is found. It is argued that the Percus-Yevick factor appearing in many empirical equations of state for the mixture has a deep influence on the location of the critical consolute point, so that the resulting phase diagram for a prescribed equation has to be taken with caution.     

Transformations of thermodynamic systems with the retention of projective invariants

Equations that determined the transformations of thermodynamic properties with the retention of projective invariants and had a solution for thermodynamic systems whose fundamental equation satisfied a certain phenomenological condition were obtained. (The geometric meaning of this condition is the possibility of projective bending of the surface described by the fundamental equation.) It was shown that, for a mixture of ideal gases, there were three sets of solutions, and, for a mixture of real gases in the region of convergence of the virial expansion (with an accuracy to at least the forth virial coefficient inclusive), there was one solution. A second solution appeared if the virial coefficients of intermolecular interactions were related by an additional equation.     

Equation of state for the phase coexistence region of insoluble monolayers under consideration of the entropy nonideality

The equation of state for the monolayer with a fluid (G, LE)/condensed (LC) phase transition derived earlier (Fainerman, V.B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145) in the framework of a quasichemical approach is generalized. A term is added that takes into account the entropy nonideality of mixing of the monomers and clusters of amphiphilic molecules. The results calculated from the proposed equations agree well with the experimental Pi-A isotherms obtained for various types of amphiphilic monolayers. The values of molecular areas of the amphiphilic molecules estimated from the fitting of experimental data to the proposed equation are quite similar to the real values. Another equation of state capable of describing the fluid state of insoluble monolayers and based on equations for the chemical potential of the solvent in the bulk phase and in the surface layer (Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2006, 110, 10436) is also generalized to be extended to the fluid/condensed phase transition region (A < A(c)), taking into account entropy nonideality for mixing solvent molecules, monomers, and clusters of amphiphilic molecules. The values calculated on this basis agree also well with the experimental data.     

Application of symplectic algorithms to QCT calculation:H+H_2 system

The basic theory of symplectic algorithm was introduced. A comparison between Runge-Kutta method and symplectic integration method was preformed in the simulation of the long time behavior of H + H2 system on BKMP potential energy surface. Our results reveal a dis-sipative behavior in the integral of ordinary differential equation by the fourth order Runge-Kutta method, which causes incorrect simulation results in QCT calculations. However, when the symplectic integration method is applied, the dissipative behavior is not found in the same system. When the initial state is the same, the energy deviation of fourth order symplectic integral method is almost one percent of that of fourth order Runge-Kutta method in a 60000-step simulation, and that of sixth order symplectic integral method is much less. These results show that the symplectic integral methods are always the better choice in the integral calculation of the long time behavior in maintaining energy conservation.     

Generalised virial equation of state for natural gas systems

In this work we present a generalised virial equation of state for natural gas systems under custody transfer conditions. The model is based on corresponding states expressions for the second and third virial coefficients with argon as the reference fluid. These functional forms involve 12 adjustable coefficients. For the extension to mixtures we propose a one-fluid mixture model with binary interaction parameters in the combining rules for the mixture critical temperature and density. We obtained overall average absolute deviations (AAD) of 0.04 and 0.08% in pure-fluid compression factors and speeds of sound; AADs of 0.07 and 0.19% in compression factors and speeds of sound, respectively, of binary mixtures and AADs of 0.047, and 0.13% in natural gas compression factors and speeds of sound, respectively. These results compare favourably with equivalent calculations with other generalised virial coefficient models.     

Vapour pressure and excess Gibbs energy of binary 1,2-dichloroethane + cyclohexanone,chloroform + cyclopentanone and chloroform + cyclohexanone mixtures at temperatures from 298.15 to 318.15 K

The vapour pressures of the binary systems 1,2-dichloroethane + cyclohexanone, chloroform + cyclopentanone and chloroform + cyclohexanone mixtures were measured at temperatures between 298.15 and 318.15 K. The vapour pressures vs. liquid phase composition data for three isotherms have been used to calculate the activity coefficients of the two components and the excess molar Gibbs energies, G^E, for these mixtures, using Barker's method. Redlich–Kister, Wilson, NRTL and UNIQUAC equations, taking into account the vapour phase imperfection in terms of the 2-nd virial coefficient, have represented the G^E values. No significant difference between G^E values obtained with these equations has been observed. Our data on vapour–liquid equilibria (VLE) and excess properties of the studied systems are examined in terms of the DISQUAC and modified UNIFAC (Dortmund) predictive group contributions models.     

A simple perturbed-hard-sphere equation of state applicable to subcritical and supercritical temperatures

A Carnahan—Starling—van der Waals equation of state is applied to eight fluids, including ammonia and water. The two constants are evaluated from supercritical-density data, subcritical saturated-liquid densities and vapor pressures; both constants decrease weakly as temperature rises. Good agreement with experiment is obtained over a wide range of conditions but, because of inherent weaknesses in perturbed-hard-sphere theory, agreement in the immediate critical region is poor and calculated second virial coefficients tend to be too positive. Nevertheless, this simple equation may be useful for engineering calculations.     

Application of a generalized van der Waals equation of state to several nonpolar mixtures

A generalized van der Waals equation of state, applied recently (Nguyen Van Nhu and Kohler, 1995) to the calculation of excess properties and phase equilibria for the mixture methane + ethane, is now extended to several nonpolar binary mixtures.Improved mixing rules for the van der Waals attractive term and for the correction term are proposed. With these mixing rules, the equation gives good agreement for vapour-liquid and liquid-liquid equilibria over a large temperature range for 29 binary mixtures. The agreement of mixture volumes and cross second virial coefficients is also satisfactory.     

Application of symplectic algorithms to QCT calculation: H + H2 system

The basic theory of symplectic algorithm was introduced. A comparison between Runge-Kutta method and symplectic integration method was preformed in the simulation of the long time behavior of H + H₂ system on BKMP potential energy surface. Our results reveal a dissipative behavior in the integral of ordinary differential equation by the fourth order Runge-Kutta method, which causes incorrect simulation results in QCT calculations. However, when the symplectic integration method is applied, the dissipative behavior is not found in the same system. When the initial state is the same, the energy deviation of fourth order symplectic integral method is almost one percent of that of fourth order Runge-Kutta method in a 60000-step simulation, and that of sixth order symplectic integral method is much less. These results show that the symplectic integral methods are always the better choice in the integral calculation of the long time behavior in maintaining energy conservation.     

Calculation of the second virial coefficients of alkali metals by modified Peng–Robinson equation

In this work, the Peng–Robinson (P–R) equation of state has been modified by proposing a new α function for calculating the second virial coefficients of alkali metals. The relationship between α^0.5 and (1???T _r ^0.5 ) is a nonlinear function. The correlation between the second virial coefficient and P–R equation was presented by expanding the P–R equation into its Taylor series form. For P–R equation, the linear correlation between parameters C₁ and C₂ of α function and acentric factors \( \omega \) of alkali metals was proposed. The new α function and its first, second and third derivatives are continuous. The average standard deviations of compressibility factor which calculated by modified P–R equation are less than 4.3%. The second virial coefficients of alkali metals were calculated over the temperature range 600–3000 K by using the modified P–R equation. Comparison with literature data, the new equation provides more reliable and accurate second virial coefficient predictions for alkali metals than the original P–R equation. It is useful to guide and improve calculation of the second virial coefficients of other metal vapors for design and operation of separation processes in vacuum metallurgy.     

Application of symplectic algorithms to QCT calculation: H + H2 system

The basic theory of symplectic algorithm was introduced. A comparison between Runge-Kutta method and symplectic integration method was preformed in the simulation of the long time behavior of H + H₂ system on BKMP potential energy surface. Our results reveal a dissipative behavior in the integral of ordinary differential equation by the fourth order Runge-Kutta method, which causes incorrect simulation results in QCT calculations. However, when the symplectic integration method is applied, the dissipative behavior is not found in the same system. When the initial state is the same, the energy deviation of fourth order symplectic integral method is almost one percent of that of fourth order Runge-Kutta method in a 60000-step simulation, and that of sixth order symplectic integral method is much less. These results show that the symplectic integral methods are always the better choice in the integral calculation of the long time behavior in maintaining energy conservation.     

Isothermal (vapour + liquid) equilibria and excess Gibbs free energies in some binary (cyclopentanone + chloroalkane) mixtures at temperatures from 298.15 K to 318.15 K

The vapour pressures of binary (cyclopentanone + 1-chlorobutane, +1,3-dichloropropane, and +1,4-dichlorobutane) mixtures, were measured at the temperatures of (298.15, 308.15, and 318.15) K. The vapour pressures vs. liquid phase composition data have been used to calculate the excess molar Gibbs free energies G^E of the investigated systems, using Barker’s method. Redlich–Kister, Wilson and NRTL equations, taking into account the vapor phase imperfection in terms of the second virial coefficient, have represented the G^E values. No significant difference between G^E values obtained with these equations has been observed.     

Interaction second virial coefficients at low temperatures for NeKr and NeO2 systems

Published results of vapour—liquid and vapour—solid equilibrium studies for neon—krypton and neon—oxygen mixtures have been analysed to give values for the interaction second virial coefficients. B₁₂. Comparisons are made with other results for the NeKr system and with the predictions of some proposed intermolecular potentials.     

An optimized explicit Runge-Kutta method with increased phase-lag order for the numerical solution of the Schrödinger equation and related problems

In this paper we present an optimized explicit Runge-Kutta method, which is based on a method of Fehlberg with six stages and fifth algebraic order and has improved characteristics of the phase-lag error. We measure the efficiency of the new method in comparison to other numerical methods, through the integration of the Schrödinger equation and three other initial value problems.