Thermodynamic calculation of supercritical-fluid equilibria: New mixing rules for equations of state

Simple cubic equations of state with conventional mixing rules have played an important role in the calculation of phase equilibria and other thermodynamic properties of non-polar fluid mixtures. In the application of supercritical fluids to separation processes, volumetric as well as phase equilibrium properties are very important for rational process design.

Heyen (1980) proposed a cubic equation of state which shows better accuracy in the calculation of volumetric properties, compared to the Peng-Robinson equation of state. In order to apply his equation to polar mixtures, Heyen recently proposed a density-independent mixing rule, but this does not obey the universally-observed quadratic mixing rule of the second virial coefficient in the low-density limit.

This paper proposes a new density-dependent mixing rule for the Heyen equation of state. The Heyen equation of state with our new mixing rule appears to calculate the phase equilibria and the volumetric properties of CO₂-containing non-polar as well as polar mixtures with good accuracy.     

A new density-dependent mixing rule for the SRK equation of state

A new procedure for obtaining density-dependent mixing rules is applied to the Soave-Redlich-Kwong equation of state. The result is a one-parameter local-composition mixing rule which adequately represents the nonidealities possible in dense fluid mixtures but approaches the classical mixing rule at low densities. A three-parameter version of the mixing rule is also presented which allows for the local-composition effect in the low density limit. The expressions are tested with the Soave-Redlich-Kwong equation of state. Results for vapor-liquid and gas-liquid systems are discussed.     

Modelling the Vapor–Liquid Equilibrium of Polymer Solutions Using a Cubic Equation of State

This work presents the vapor–liquid equilibrium calculations in an isothermal flash, applied to polymer solutions, using the Peng–Robinson cubic equation of state modified by Stryjek–Vera, and the mixing rule introduced by Wong–Sandler. This rule allows combining the rigid lattice thermodynamic model of Flory–Huggins to the Peng–Robinson–Stryjek–Vera equation of state. As the Gibbs free energy must be minimum in the equilibrium state, a stochastic optimization method, the simulated annealing algorithm, was used to find out the extreme of this thermodynamic potential.     

Electron delocalization and aromaticity in low-lying excited states of archetypal organic compounds

Aromaticity is a property usually linked to the ground state of stable molecules. Although it is well-known that certain excited states are unquestionably aromatic, the aromaticity of excited states remains rather unexplored. To move one step forward in the comprehension of aromaticity in excited states, in this work we analyze the electron delocalization and aromaticity of a series of low-lying excited states of cyclobutadiene, benzene, and cyclooctatetraene with different multiplicities at the CASSCF level by means of electron delocalization measures. While our results are in agreement with Baird's rule for the aromaticity of the lowest-lying triplet excited state in annulenes having 4nπ-electrons, they do not support Soncini and Fowler's generalization of Baird's rule pointing out that the lowest-lying quintet state of benzene and septet state of cyclooctatetraene are not aromatic.     

An equation of state for hydrogen fluoride

Using a similar approach as Lencka and Anderko [AIChE J. 39 (1993) 533], we developed an equation of state for hydrogen fluoride (HF), which can correlate the vapor pressure, the saturated liquid and vapor densities of it from the triple point to critical point with good accuracy. We used an equilibrium model to account for hydrogen bonding that assumes the formation of dimer, hexamer, and octamer species as suggested by Schotte [Ind. Eng. Chem. Process Des. Dev. 19 (1980) 432]. The physical and chemical parameters are obtained directly from the regression of pure component properties by applying the critical constraints to the equation of state for hydrogen fluoride. This equation of state together with the Wong–Sandler mixing rule as well as the van der Waals one-fluid mixing rule are used to correlate the phase equilibria of binary hydrogen fluoride mixtures with HCl, HCFC-124, HFC-134a, HFC-152a, HCFC-22, and HFC-32. For these systems, new equation of state with the Wong–Sandler mixing rule gives good results.     

Extension of Huron-Vidal-type Mixing Rule to Three-parameter Harmens-Knapp Cubic Equation of State

In this paper was extended the HV-type mixing rules to Harmens-Knapp cubic equation state (HK CEOS). The new HV-type mixing rule with HK CEOS was tested for Vapor-liquid equilibrium(VLE) of different polar and nonpolar systems. The tested results are in good agreement with existing experimental data within a wide range of temperatures and pressures. In comparison with the VDW mixing rule, the new mixing rule gives much better predictions for the VLE of nonpolar and polar systems.     

Open-shell spherical aromaticity: the 2N2 + 2N + 1 (with S = N + ½) rule

4N Baird's rule represented the extension of Hückel's 4N + 2 rule to triplet state systems. In this work we extend the 2(N + 1)(2) Hirsch rule for spherical aromatic species to open-shell spherical compounds and we provide evidence that those spherical species having a same-spin half-filled last energy level with the rest of the levels being fully-filled, i.e., those having 2N(2) + 2N + 1 electrons and S = N + ?, are aromatic.     

Modeling nonlinear optics of nanosystems with sum-over-states model

Three-stage strategies (ladder rule, few state model (FSM), and parallelization) were proposed to improve the computational efficiency of the sum-over-states (SOS) model in nonlinear optics (NLO) modeling. Ladder rule decomposes NLO coefficients of the nth state into the (n-1)th term and the contribution from the (n-1)th to the nth state without loss of rigor in theory. FSM singles out the states with substantial contribution to NLO. Those strategies are universal to all (including revised and simplified) SOS models. The computing cost reduces roughly to C/(n(i-1)) (C is a constant and i is the rank (order) of the NLO coefficients).     

New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures

Huron, M.-J. and Vidal, J., 1979. New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilibria, 3: 255-271.Good correlations of vapour-liquid equilibria can be achieved by applying the same two-parameter cubic equation of state to both phases. The results primarily depend on the method used for calculating parameters and, for mixtures, on the mixing rule. True parameters are the covolume b and the energy parameter a/b. For this latter one, deviations from a linear weighting rule are closely connected to the excess free energy at infinite pressure. Thus any mixing rule gives a model for the excess free energy, or any accepted models for this property can be used as mixing rules.From the above, an empirical polynomial mixing rule is used for data smoothing and evaluation, while for practical work a local composition model is used. The mixing rule thus obtained can be reduced to the classical quadratic rule for some easily predicted values of the interaction energies. For highly polar systems, it includes three adjustable parameters. Using literature data, the new mixing rule is applied, in the low and high pressure range, to binary mixtures with one or two polar compounds, giving good data correlation and sometimes avoiding false liquid-liquid immiscibility.     

Phase transition induced by a shock wave in hard-sphere and hard-disk systems

Dynamic phase transition induced by a shock wave in hard-sphere and hard-disk systems is studied on the basis of the system of Euler equations with caloric and thermal equations of state. First, Rankine-Hugoniot conditions are analyzed. The quantitative classification of Hugoniot types in terms of the thermodynamic quantities of the unperturbed state (the state before a shock wave) and the shock strength is made. Especially Hugoniot in typical two possible cases (P-1 and P-2) of the phase transition is analyzed in detail. In the case P-1 the phase transition occurs between a metastable liquid state and a stable solid state, and in the case P-2 the phase transition occurs through coexistence states, when the shock strength changes. Second, the admissibility of the two cases is discussed from a viewpoint of the recent mathematical theory of shock waves, and a rule with the use of the maximum entropy production rate is proposed as the rule for selecting the most probable one among the possible cases, that is, the most suitable constitutive equation that predicts the most probable shock wave. According to the rule, the constitutive equation in the case P-2 is the most promising one in the dynamic phase transition. It is emphasized that hard-sphere and hard-disk systems are suitable reference systems for studying shock wave phenomena including the shock-induced phase transition in more realistic condensed matters.     

A Description of Vibrational Modes in Hexaphyrins: Understanding the Aromaticity Reversal in the Lowest Triplet State

Aromaticity reversal in the lowest triplet state, or Baird's rule, has been postulated for the past few decades. Despite numerous theoretical works on aromaticity reversal, experimental study is still at a rudimentary stage. Herein, we investigate the aromaticity reversal in the lowest excited triplet state using a comparable set of [26]‐ and [28]hexaphyrins by femtosecond time‐resolved infrared (IR) spectroscopy. Compared to the relatively simple IR spectra of [26]bis(rhodium) hexaphyrin ( R26H ), those of [28]bis(rhodium) hexaphyrin ( R28H ) show complex IR spectra the region for the stretching modes of conjugated rings. Whereas time‐resolved IR spectra of R26H in the excited triplet state are dominated by excited state IR absorption peaks, while those of R28H largely show ground state IR bleaching peaks, reflecting the aromaticity reversal in the lowest triplet state. These contrasting IR spectral features serve as new experimental aromaticity indices for Baird's rule.     

Violation of Hund's rule in the lowest excited singlet-triplet pairs of dicyclohepta[cd,gh]pentalene and dicyclopenta[ef,kl]heptalene

Using the Pariser-Parr-Pople type MO method, the energy ordering of the lowest excited singlet-triplet pairs of the title molecules is examined by taking into account ample configuration interactions. In dicyclohepta[cd,gh]pentalene, it is shown that the lowest excited singlet state lies below the corresponding triplet state by about 6 kcal/mol. This violation of Hund's rule is ascribed entirely to the correlation effects brought about by spin polarization terms. Such a violation of Hund's rule is expected to occur in the lowest excited states of dicyclopenta[ef,kl]heptalene.     

Interpretation of Hund's multiplicity rule for the carbon atom

Hund's multiplicity rule is investigated for the carbon atom using quantum Monte Carlo methods. Our calculations give an accurate account of electronic correlation and obey the virial theorem to high accuracy. This allows us to obtain accurate values for each of the energy terms and therefore to give a convincing explanation of the mechanism by which Hund's rule operates in carbon. We find that the energy gain in the triplet with respect to the singlet state is due to the greater electron-nucleus attraction in the higher spin state, in accordance with Hartree-Fock calculations and studies including correlation. The method used here can easily be extended to heavier atoms.     

Transition-metal-doped aluminum hydrides as building blocks for supramolecular assemblies

Density functional theory calculations were carried out to characterize a series of transition-metal-doped aluminum hydrides, forming TMAl(n)H(2n) and TMAl(n)H(2n+1) (TM = Sc, Ti, V; n = 3,4), in either charged or neutral form. A new electron-counting rule for these clusters was formulated as PSEN (paired skeleton electron number) = 4n, which can characterize both closed-shell and open-shell clusters. On the basis of this electron-counting rule, the superatomic clusters such as TiAl(4)H(9) and TiAl(3)H(6) were identified and can be used to assemble supramolecular structures. Electronic structure analysis showed that three-centered TM-H-Al bonds largely contributed to the structural stability. Also, the spin state of a wide range of clusters in their ground state can be predicted by the electron-counting rule.     

Comparison of different mixing rules for prediction of density and residual internal energy of binary and ternary Lennard–Jones mixtures

Mixing rules are necessary when equations of state for pure fluids are used to calculate various thermodynamic properties of fluid mixtures. The well-known van der Waals one-fluid (vdW1) mixing rules are proved to be good ones and widely used in different equations of state. But vdW1 mixing rules are valid only when molecular size differences of components in a mixture are not very large. The vdW1 type density-dependent mixing rule proposed by Chen et al. [1] is superior for the prediction of pressure and vapor–liquid equilibria when components in the mixture have very different sizes. The extension of the mixing rule to chain-like molecules and heterosegment molecules was also made with good results. In this paper, the comparison of different mixing rules are carried out further for the prediction of the density and the residual internal energy for binary and ternary Lennard–Jones (LJ) mixtures with different molecular sizes and different molecular interaction energy parameters. The results show that the significant improvement for the prediction of densities is achieved with the new mixing rule [1], and that the modification of the mixing rule for the interaction energy parameter is also necessary for better prediction of the residual internal energy.     

Connection between the manne—Åberg theorem and a sum rule derived in the framework of the green's-function formalism

A sum rule for ionization potentials, similar to the Manne-Åberg theorem, is derived in the framework of a many-body Green's-function formalism. This sum rule is shown to be valid under mainly two conditions: (i) the constant term and the affinity poles of the self-energy part have to be neglected; (ii) the final state wavefunction has to be separable in a free-electron and an (N ? 1)-electron part. The latter assumption is discussed in connection with the sudden approximation which is not used for the derivation of the new sum rule.     

Ethenedithione (S=C=C=S): Does It Obey Hund's Rule?

Significantly higher in energy (24 kJ mol⁻¹) than the triplet ground state (³Σ_g⁻) is the ¹Δ_g state of ethenedithione (S=C=C=S), in agreement with Hund's rule. This result was obtained from high-level ab initio calculations. Thus, ethenedithione cannot, as had been proposed, be considered as the first example for the violation of Hund's rule in an equilibrium structure.     

Reactivity and selectivity control by reactants and products. A general relationship between the selectivity and the position of the transition state.

Although the reactivity-selectivity principle and the frontier molecular orbital theory lead to opposed reactivity-selectivity relationships, they lead to an identical transition state position-selectivity relationship. The transition state position is generally reactant- and product-dependent. The RSP which neglects reactant effects and the FMO which neglects product effects can be considered as the two limits of the general rule.     

Theoretical studies on the hydrogen atom transfer reaction

The hydrogen atom transfer reaction between substituted methanes (substituents; H, F, CH₃, OH, and CN) and methyl radicals was studied by 4-31G (UHF) calculations using the MINDO/3 geometries. The transition state structures and energy barriers were determined, and variations of the transition state and of the reactivity due to the change of substituent were analyzed based on the potential energy surface characteristics. It was concluded that the reaction is of the S_H2 type with a backside attack, and transition state variations are controlled by the vector sum of the component parallel to (Hammond rule) and one perpendicular to the reaction coordinate (anti-Hammond rule). It was also concluded that the most important factor influencing the reactivity is bond dissociation energy effect directly related to the spin transfer of the radical species, and the polar effect need not be overemphasized.     

Theoretical Studies on Critical Properties of Binary System with Supercritical Carbon Dioxide

A new expression of mixing rule is suggested according to the Mayson‘s mixing rule in this paper,which adopts the Redlich-Kwong cubic equation of state and the modifed Chueh-Prausnitz method to calculate the experiment critical points of six binary mixtures CO2 toluene,CO2 cyclohexane,CO2 n-butanal,CO2 i-butanal,CO2 methanol,CO2 ethanol.The coefficients of interaction parameter in the expression of mixing rule were optimized from experimental data.The calculated results of critical temperature and critical pressure meet the experiment data well.The maximum relative errore of temperature and pressure between the calculation results and experiment data are 1.493% and 5.2236% respectively,indicating that the proposed expression of mixing rule is reasonable.This may provide a fundamental method for studying and predicting the properties of supercritical fluids.