Analytical dependences between first order derivatives at the critical point of a pure substance

The finiteness and positive value of the slope of the saturation curve at the critical point according to its location in the region of a thermodynamic surface restricted by the curves of inversion and saturation are substantiated. It is shown that any two derivatives of the first order formed with respect to volume, internal energy, entropy, enthalpy, and the Helmholtz energy, one of which is taken at a fixed value of pressure and the other at a fixed value of temperature, are equal in the critical state. New sets of the critical conditions for a pure substance are obtained.     

Theoretical model based on the memory effect for the strange photoisomerization kinetics of diarylethene derivatives dispersed on polymer films

In the present paper the authors present a theoretical model to explain the kinetics involving the induction period observed by Irie et al. [Nature (London) 420, 759 (2002)] for photoisomerization of diarylethene derivatives dispersed on polymer films at a single molecular level. In the model we assume that both ground state and excited state free energy landscapes which result from the interaction between the photochromic molecule and the surrounding polymer are rugged and have several local minima along the pathway to the critical point at which isomerization actually occurs. We assume that after one photoexcitation a fraction of the photochromic molecule moves to a new local minimum and stays there, although the other fraction returns to the original local minimum. The former effect is referred to as the memory effect. After repeated photoexcitations the photochromic molecule moves gradually from one local minimum to another in the pathway to the isomerization point. It finally reaches the isomerization point, where isomerization occurs. Their model successfully reproduces the kinetics of photoisomerization of diarylethene derivatives dispersed on polymer films observed experimentally.     

Thermodynamic equations for the isobaric and isochoric heat capacities of pure substances at the critical point

It was proved on the assumption of a nonzero and finite slope of the elasticity curve at the critical point that the isochoric heat capacity at this point cannot be established on the basis of thermodynamics only. The critical conditions of a pure substance were derived from the differential equations of thermodynamics using a rigorous mathematical apparatus. Several indeterminate forms containing isochoric or isobaric heat capacities or their derivatives were evaluated.     

Efficient and reliable mixture critical points calculation by global optimization

Multicomponent systems may exhibit several critical points or no critical point at all. Local methods can find only one critical point for a given initial guess. Recently, several global methods have been proposed for finding all the solutions of the problem. In the present work, we propose a gradient-based calculation method using global optimization, with temperature and molar volume as primary variables, and with analytical partial derivatives calculated from a two-parameter cubic equation of state. The Tunneling global optimization method is used for finding all the global minima. The implementation is based on a unique feature of the Tunneling method, which is able to find efficiently and reliably multiple minima at the same level. Several mixtures from binaries to petroleum reservoir fluids are used to test the proposed method. Numerical experiments proved the efficiency and reliability of the Tunneling method for finding all mixture critical points.     

Solutions of the Klein–Gordon equation with the improved Tietz potential energy model

We present the relativistic rotation–vibrational energy equation of a diatomic molecule which moves under the improved Tietz potential energy model in higher spatial dimensions. The nonrelativistic limits of the bound state solutions of the Klein–Gordon equation are the bound state solutions of the Schrödinger equation with the same potential energy function. Numerical analysis results show that there exists a critical point around which the solution behaviors bifurcate into two extreme cases. Below the critical point, the behavior of the relativistic vibrational energies for the ground electronic state of carbon monoxide in higher dimensions keeps similar to that of the three-dimensional system, while this symmetry phenomenon breaks and the Klein–Gordon equation has no stability solution upon the critical point.     

Can the speed of sound be used for detecting critical states of fluid mixtures?

The phenomenology of sound speeds in fluid mixtures is examined near and across critical lines. Using literature data for binary and ternary mixtures, it is shown that the ultrasound speed along an isotherm-isopleth passes through a minimum value in the form of an angular (or V-shaped) point at critical states. The relation between critical and pseudo-critical coordinates is discussed. For nonazeotropic fixed-composition fluid mixtures, pseudo-critical temperatures and pressures are found to be lower than the corresponding critical temperatures and pressures. The analysis shows that unstable pseudo-critical states cannot be detected using acoustic methods. The thermodynamic link between sound speeds and isochoric heat capacities is formulated and discussed in terms of p-Vm-T derivatives capable of being calculated using cubic equations of state. Based on the Griffiths-Wheeler theory of critical phenomena, a new specific link between critical sound speeds and critical isochoric heat capacities is deduced in terms of the rate of change of critical pressures and critical temperatures along the p-T projection of the critical locus of binary fluid mixtures. It is shown that the latter link can be used to obtain estimates of critical isochoric heat capacities from the experimental determination of critical speeds of sound. The applicability domain of the new link does not include binary systems at compositions along the critical line for which the rate of change in pressure with temperature changes sign. The new equation is combined with thermodynamic data to provide approximate numerical estimates for the speed of sound in two mixtures of carbon dioxide and ethane at different temperatures along their critical isochores. A clear decrease in the sound speed is found at critical points. A similar behavior is suggested by available critical heat capacity data for several binary fluid mixtures. Using an acoustic technique, the critical temperature and pressure were determined for three different mixtures of methane and propane, and compared with literature data obtained using conventional methods. It is concluded that acoustic-based techniques are reliable to determine, for the most part, critical surfaces of fluid mixtures. The remaining few cases where the present analysis cannot be applied could be tested by the thermodynamic calculation of critical sound speeds using crossover equations of state in conjunction with experimentally determined critical isochoric heat capacities.     

Influence of molecular characteristics of nonionic cellulose ethers on their interaction with ionic surfactant investigated by conductometry

The interaction between nonionic derivatives of cellulose, hydroxypropylmethyl cellulose (HPMC) and methyl cellulose (MC), and ionic surfactant, sodium dodecylsulfate (SDS) were investigated by conductometric titration method, at 30°C. Obtained titration curves show two break points: critical aggregation concentration (cac) defined as the concentration of SDS at which interaction starts, and polymer saturation concentration (psp) as the concentration at which interaction finishes. Changes of characteristic concentration breaks were determined in dependence on concentration and molecular characteristics of cellulose derivatives (degree of substitution (DS) and molecular mass, i.e. intrinsic viscosity). It was shown that the first break point, cac, is independent of polymer concentration; while the second break point, psp, increases as polymer concentration increases, as described by a linear correlation. The slopes of linear relationship justify the DS on the intensity of the cellulose derivatives–SDS interaction. Changes in the intrinsic viscosity of cellulose derivatives do not exhibit influence on the interaction with SDS.     

Temperature dependence of the dynamic viscosity of liquid mercury

The dependence of the dynamic viscosity of mercury on temperature is calculated and expressed in terms of the cluster associate model, based on the Boltzmann distribution and with normalization at the melting point. The resulting refined equation for mercury viscosity adequately reflects this dependence over the range of the liquid state, including the critical point.     

Avoiding singularity problems associated with meta-GGA (generalized gradient approximation) exchange and correlation functionals containing the kinetic energy density

Convergence problems of meta-GGA (generalized gradient approximation) XC (exchange and correlation) functionals containing a self-interaction correction term are traced back to a singularity of the latter that occurs at critical points of the electron density. This is demonstrated for the bond critical point of equilibrium and stretched H2. A simple remedy is suggested that cures meta-XC functionals such as VSXC, TPSS, M05, M06, and their derivatives without extra cost.     

Predicting mixture phase equilibria and critical behavior using the SAFT-VRX approach

The SAFT-VRX equation of state combines the SAFT-VR equation with a crossover function that smoothly transforms the classical equation into a nonanalytical form close to the critical point. By a combinination of the accuracy of the SAFT-VR approach away from the critical region with the asymptotic scaling behavior seen at the critical point of real fluids, the SAFT-VRX equation can accurately describe the global fluid phase diagram. In previous work, we demonstrated that the SAFT-VRX equation very accurately describes the pvT and phase behavior of both nonassociating and associating pure fluids, with a minimum of fitting to experimental data. Here, we present a generalized SAFT-VRX equation of state for binary mixtures that is found to accurately predict the vapor-liquid equilibrium and pvT behavior of the systems studied. In particular, we examine binary mixtures of n-alkanes and carbon dioxide + n-alkanes. The SAFT-VRX equation accurately describes not only the gas-liquid critical locus for these systems but also the vapor-liquid equilibrium phase diagrams and thermal properties in single-phase regions.     

Density improvement of the SRK equation of state

The SRK equation of state has been modified by volume translation in order to improve its accuracy both in- and outside the critical region. A temperature dependent volume correction is proposed which can match the true critical point of the pure component, and provides accurate densities for polar and non-polar pure substances both near to and far from the critical point. It can also be easily extended to mixtures, and the calculation results show that it can shift the critical locus towards experimental values and gives good results for the liquid densities of mixtures.     

Experimental densities,vapor pressures,and critical point,and a fundamental equation of state for dimethyl ether

Densities, vapor pressures, and the critical point were measured for dimethyl ether, thus, filling several gaps in the thermodynamic data for this compound. Densities were measured with a computer-controlled high temperature, high-pressure vibrating-tube densimeter system in the sub- and supercritical states. The densities were measured at temperatures from 273 to 523 K and pressures up to 40 MPa (417 data points), for which densities between 62 and 745 kg/m³ were covered. The uncertainty (where the uncertainties can be considered as estimates of a combined expanded uncertainty with a coverage factor of 2) in density measurement was estimated to be no greater than 0.1% in the liquid and compressed supercritical states. Near the critical temperature and pressure, the uncertainty increases to 1%. Using a variable volume apparatus with a sapphire tube, vapor pressures and critical data were determined. Vapor pressures were measured between 264 and 194 kPa up to near the critical point with an uncertainty of 0.1 kPa. The critical point was determined visually with an uncertainty of 1% for the critical volume, 0.1 K for the critical temperature, and 5 kPa for the critical pressure. The new vapor pressures and compressed liquid densities were correlated with the simple TRIDEN model. The new data along with the available literature data were used to develop a first fundamental Helmholtz energy equation of state for dimethyl ether, valid from 131.65 to 525 K and for pressures up to 40 MPa. The uncertainty in the equation of state for density ranges from 0.1% in the liquid to 1% near the critical point. The uncertainty in calculated heat capacities is 2%, and the uncertainty in vapor pressure is 0.25% at temperatures above 200 K. Although the equation presented here is an interim equation, it represents the best currently available.     

Thermal epimerization of inositol 1,3-benzylidene acetals in the molten state

1,3-O-Benzylidene-2,4,5,6-tetra-O-substituted-myo-inositol derivatives obtained by the DIBAL-H reduction of the corresponding myo-inositol 1,3,5-orthobenzoate derivatives undergo epimerization at the acetal carbon on heating, in the molten state, just above their melting point. The same epimerization reaction does not proceed either in the crystalline state or in solution. DFT calculations suggest that the epimeric acetal obtained by this thermal process is relatively more stable than the starting acetal. Either of these acetals could not be obtained by the reaction of the corresponding inositol derived diol with benzaldehyde. These observations constitute a novel reaction solely in the molten state, which are rarely encountered in the literature. X-ray crystal structures of the epimeric acetals as well as their radical deoxygenation reaction are also reported.     

Hydrogen bonding in Schiff bases--NMR, structural and experimental charge density studies

A series of sixteen Schiff bases (derivatives of salicylaldehydes and aryl amines) was studied to reveal the influence of substituents and the length of the linker on the properties of the H-bonding formed. In theory, two groups of compounds, derivatives of 2-(2-hydroxybenzylidenoamine)phenol) and 2-hydroxy-N-(2-hydroxybenzylideno)benzylamine, can form different types of H-bonds using one or two hydroxyl groups present in the molecules. Two other groups of compounds, derivatives of 4-(2-hydroxybenzylidenoamine)phenol and N-(2-hydroxybenzyideno)benzylamine, can form only one type of H-bond. It was confirmed by (15)N and (13)C NMR experiments, that in all cases only traditional, H-bonded six-membered chelate rings were formed. The positions of the hydrogen atom in the rings depend on the substituent and phase. Generally, the OH H-bond form dominates in solution, with exception of the nitro derivatives, where the NH tautomer is present. In the solid state the tautomeric equilibrium is strongly shifted to the NH form. Only for the 5-Br derivative of one compound was the reverse relationship found. According to the results of experimental charge density investigations, two intramolecular H-bonds in the 5-methoxy derivative of 2-hydroxy-N-(2'-hydroxybenzylideno)benzylamine) differ significantly in terms of charge density properties. The intra- and intermolecular H-bonds formed by the deprotonated oxygen atom from 2-OH group are strong, with significant charge density concentration at the bond critical point and a straight, well-defined bond path, whereas the second intramolecular H-bond formed by the oxygen atom from the 2'-OH group is quite weak, with ca. five times smaller charge density concentration than in the previous case and a bent bond path. In terms of energy densities, the latter H-bond appears to be a non-bonding interaction, with total energy density being slightly positive. In terms of source contributions to the density at the H-bond critical point from the atoms involved, the intermolecular, linear H-bond is very strong and charge-assisted in the source function classification, the N(1)-H(1N)···O(1) H-bond is medium-strength, while the third H-bond is extremely weak.     

Crossover Peng-Robinson equation of state with introduction of a new expression for the crossover function

In this research, we use the original Peng-Robinson (PR) equation of state (EOS) for pure fluids and develop a crossover cubic equation of state which incorporates the scaling laws asymptotically close to the critical point and it is transformed into the original cubic equation of state far away from the critical point. The modified EOS is transformed to ideal gas EOS in the limit of zero density. A new formulation for the crossover function is introduced in this work. The new crossover function ensures more accurate change from the singular behavior of fluids inside the regular classical behavior outside the critical region. The crossover PR (CPR) EOS is applied to describe thermodynamic properties of pure fluids (normal alkanes from methane to n-hexane, carbon dioxide, hydrogen sulfide and R125). It is shown that over wide ranges of state, the CPR EOS yields the thermodynamic properties of fluids with much more accuracy than the original PR EOS. The CPR EOS is then used for mixtures by introducing mixing rules for the pure component parameters. Higher accuracy is observed in comparison with the classical PR EOS in the mixture critical region.     

The equation of state for an interface during the adsorption of organic substances on an electrode

The state of the interface between a metal and a solution of an electrolyte containing a neutral surfactant was investigated using a method alternative to the traditional thermodynamic approach. The method was based on the concept that there was a stability limit of a surfactant on an electrode, and the corresponding state could be described in terms of the catastrophe theory. The surface pressure was approximated by the Whitney polynomial in powers of the de Donder parameter (completeness of adsorption) with the coefficients depending on the chemical potential and polarization of the interface. The equation of state and the equation for the stability limit were obtained from the condition of zero first and second derivatives. These equations correctly described the results of electrocapillary measurements in the spirit of the law of corresponding states. The correlation between surface pressure maxima and critical stability potentials predicted by the theory was substantiated by the electrocapillary measurements data provided that the inflexions of surface pressure curves calculated from the electrocapillary data were related to the limiting stability at which the competing forces are balanced during the adsorption of surfactants. A simple equation for surface pressure was suggested in the form of a function of the state of thermodynamic parameters and completeness of adsorption. This function described the state of a surfactant at the interface. Equilibrium equations were derived for the state of a surfactant and the spinodal.     

A new cubic equation of state for simple fluids: pure and mixture

A two-parameter cubic equation of state is developed. Both parameters are taken temperature dependent. Methods are also suggested to calculate the attraction parameter and the co-volume parameter of this new equation of state. For calculating the thermodynamic properties of a pure compound, this equation of state requires the critical temperature, the critical pressure and the Pitzer’s acentric factor of the component. Using this equation of state, the vapor pressure of pure compounds, especially near the critical point, and the bubble point pressure of binary mixtures are calculated accurately. The saturated liquid density of pure compounds and binary mixtures are also calculated quite accurately. The average of absolute deviations of the predicted vapor pressure, vapor volume and saturated liquid density of pure compounds are 1.18, 1.77 and 2.42%, respectively. Comparisons with other cubic equations of state for predicting some thermodynamic properties including second virial coefficients and thermal properties are given. Moreover, the capability of this equation of state for predicting the molar heat capacity of gases at constant pressure and the sound velocity in gases are also illustrated.     

Physical properties of hexafluorobenzene

The critical properties of hexafluorobenzene were measured, and compared with the most reliable values reported in literature. The vapour pressure was determined from 226°C to the critical point and correlated by an equation of state.     

A crossover cubic equation of state near to and far from the critical region

In this research, we use the Patel–Teja (PT) cubic equation of state [N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473.] and develop a crossover cubic model near to and far from the critical region, which incorporates the scaling laws asymptotically close to the critical point and it transformed into original classical cubic equations of state far away from the critical point. This equation of state is used to calculate thermodynamic properties of pure systems (carbon dioxide, normal alkanes from methane to heptane). We show that, over a wide range of states, the equation of state yields the saturated vapour pressure data and the saturated density data with a much better accuracy than the original PT equation of state.