Universal behavior of the enthalpy of vaporization: an empirical equation

Values of the enthalpy of vaporization from the critical to the triple point are correlated by an empirical equation. The equation contains parameters which characterize each substance: the critical and triple point temperatures and the enthalpy of vaporization at the triple point, and for all substances the same universal critical ratio. This work suggests that a wide class of fluids with the exception of quantal liquids shows an universal behavior along the coexistence curve.     

Scaled equations for the coexistence curve,the capillary constant and the surface tension of n-alkanes

A review of experimental data of several fluids shows that their coexistence curve follows a power law in reduced temperature at the approach of the critical point, with an universal exponent equal to 0.325, their capillary constant a power law with an universal exponent equal to 0.925 and their surface tension a power law with an universal exponent equal to 1.26. In the critical region, the concept of two-scale-factor universality was used to predict the density difference amplitude, the capillary constant amplitude, and the surface tension amplitude between near critical vapor and liquid phases. A comparison with amplitudes determined from experimental data is given. In order to extend this universality all along the liquid–gas coexistence curve from the triple point to the critical point for n-alkanes, a mean field approximation was used far away from T_C. We show that the density difference, the capillary constant and the surface tension can be calculated with a reasonable accuracy by generalized scaled equations adding only two empirical constants. A comparison between calculated and experimental data is presented.     

General correlation model for some physical properties of saturated pure fluids

In this work, we use a general expression to accurately correlate the liquid density, the vaporization enthalpy, the surface tension, and the isobaric heat capacity of a saturated liquid versus temperature along the whole coexistence curve. The general expression used is the same for the four thermodynamic properties, and uses both critical and triple point values as reference. As representative examples of the use of the model, results are given for a set of 22 pure substances. We find that this general expression correlates the data with smaller or similar overall deviations when compared with other published models whose number of coefficients are the same or greater.     

A predictive vapor-pressure equation

A simple equation is presented for predicting the temperature dependence of the vapor-pressure of a pure substance along the entire (liquid + vapor) coexistence curve, from the triple point to the critical point. The proposed equation is based on the use of a dimensionless temperature reduced by using critical and triple point values, and of the Clausius–Clapeyron equation as a zeroth-order approximation. The pressure and temperature at the triple point, the normal boiling temperature, and the pressure and temperature at the critical point are required as input data. The proposed equation is verified for 53 fluids by using NIST data. These data are reproduced with an overall average deviation of 0.55%.     

Metastable extension of the liquid-vapor phase equilibrium curve and surface tension

The method of molecular dynamics has been used to calculate the parameters of liquid-vapor phase equilibrium and the surface tension in a two-phase system of 4096 Lennard-Jones particles. Calculations have been made in a range from the triple point to near-critical temperature and also at temperatures below the triple point corresponding to the metastable equilibrium of a supercooled liquid and supersaturated vapor. To determine the surface tension, along with a mechanical approach a thermodynamic one has been used as well. The latter was based on calculation of the excess internal energy of an interfacial layer. It has been shown that in accuracy the thermodynamic approach is as good as the more sophisticated mechanical one. Low-temperature asymptotics of the phase-equilibrium curve and also of liquid and vapor spinodals have been considered in the Lennard-Jones and the van der Waals models. The behavior of the surface tension and the excess internal energy of an interfacial layer at T-->0 is discussed.     

Cloud-Point Curves and Coexistence Curves of Several Polydisperse Polystyrenes in Cyclohexane

Cloud-point curves, critical points, and coexistence curves with feed concentrations close to the critical concentration were measured in three systems involving cyclohexane + different polydisperse polystyrenes. The shape of the coexistence curves is analyzed by using a scaling expression. In two systems the critical exponent β possesses values somewhat larger than in actual binary systems (where β ≈ 1/3), whereas in the third system a somewhat smaller value is found. By using a three-parameter Gibbs free energy relation, cloud-point curves and coexistence curves are calculated from the critical point data and from the slope of the cloud-point curve at this point. To account for polydispersity, the method of continuous thermodynamics is applied. The cloud-point curves are well described, but the prediction of the coexistence curves is bad due to the mean-field character of the Gibbs free energy relation resulting in β = 1/2. Hence, the often used practice of fitting the parameters of a mean-field Gibbs free energy relation to the critical point and to some cloud points and then to calculate the coexistence data is to be considered with great care.     

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.     

Critical Properties of Argon

Abstract

Measurements of P-V-T properties of argon in the critical region are reported. Isothermal compressibilities have been calculated from the data for liquid and vapor along the coexistence curve and for the gas above the critical temperature at the critical density. Densities of the liquid and vapor along the coexistence curve were measured, and the critical temperature, pressure and density of argon were redetermined. These data are compared with those of other workers; in addition critical indices which represent the manner in which these properties vary as one approaches the critical point were determined and compared with the predictions of several theoretical estimates of these quantities.     

Nonasymptotic critical behavior of a ternary ionic system

Refractive indices n and salt concentrations ms of coexisting phases of the ternary system 1,4-dioxane + water + potassium chloride were measured along the liquid-liquid-solid coexistence curve near the liquid-liquid critical end point. Refractive index measurements were carried out in the range 0.689 x 10-3 < t = (T - Tc)/Tc < 0.118 while salt concentrations were determined for the temperature range 1.84 x 10-3 < t < 8.07 x 10-2. From these experimental results, compositions fD (mass fraction of dioxane on a salt-free basis) and densities rho of coexisting phases were obtained. The shape of the coexistence curve was analyzed using alternatively n, ms, fD, and rho as order parameters. In all cases, the obtained coexistence curve displays, asymptotically, Ising behavior. Outside the asymptotic critical domain, n, ms, and rho show significant deviations of the effective critical exponent from its Ising value, while the concentration variable fD requires no corrections to simple scaling. On the basis of the present results, we conclude that this system shows no indication of multicritical behavior.     

On approximate mathematical modeling of the vapor–liquid coexistence curves by the Van der Waals equation of state and non-classical value of the critical exponent

Van der Waals equation of state as well as power laws and critical exponent theories are prototypes to study the cubic shape, asymmetries and “flatness” of the vapor–liquid equilibrium curves near the critical point. In this work we study two similar methods to determine the phase curves in analytical form, which differ from each other by simplicity of mathematical calculation. We analyze temperature dependence of the coexistence curves asymptotically close to the vapor–liquid critical point. We explain the novelty of our method with respect to the standard thermodynamic limit discussed in the literature. Therefore we show that the shape of the coexistence curves can strongly influence the accepted value of the critical exponent. The results of theoretical studies have been compared with the ones obtained by experimental methods.     

Gelatin hydrogels cross-linked with bis(vinylsulfonyl)methane (BVSM): 1. The chemical networks

This paper deals with chemical gelation of gelatin in the presence of a cross-linker, bis(vinylsulfonyl)methane (BVSM), which is able to create covalent C-N bonds with amine groups. The investigation is performed at 40 degrees C, where no triple helices are present. Gelatin is in random coil conformation. The influence of various parameters (gelatin concentration, cross-linker concentration, and pH (number of reacting sites along the gelatin chain)) was examined. Gel formation was followed by rheological and thermodynamic measurements (microcalorimetry) versus time (kinetic measurements). Furthermore, the storage moduli were compared to the number of links formed in the course of gelation. The experiments show that, within the experimental range investigated, a fully homogeneous network is not reached; the chemical gels, even upon completion of the reactions, are still in the critical domain, near the threshold. A power law behavior was put in evidence for the shear modulus versus the distance to the gel point, expressed as the concentration of links per gelatin chain. The exponent (f = 3.4 +/- 0.3) is close to that expected for the vulcanization of long chains. The storage moduli can be superposed on a single curve where the abscissa is the product of the number of C-N links per unit volume and the gelatin concentration at an exponent equal to -0.76 +/- 0.03. This exponent suggests the role of entanglements for interchain cross-linking.     

Vapor-liquid and vapor-solid phase equilibria for united-atom benzene models near their triple points: the importance of quadrupolar interactions

Gibbs ensemble Monte Carlo simulations were used to calculate the vapor-liquid and vapor-solid coexistence curves for benzene using two simple united-atom models. An extension of the Gibbs ensemble method that makes use of an elongated box containing a slab of the condensed phase with a vapor phase along one axis was employed for the simulations of the vapor-solid equilibria and the vapor-liquid equilibria at very low reduced temperatures. Configurational-bias and aggregation-volume-bias Monte Carlo techniques were applied to improve the sampling of particle transfers between the two simulation boxes and between the vapor and condensed-phase regions of the elongated box. An isotropic united-atom representation with six Lennard-Jones sites at the positions of the carbon atoms was used for both force fields, but one model contained three additional out-of-plane partial charge sites to explicitly represent benzene's quadrupolar interactions. Both models were fitted to reproduce the critical temperature and density of benzene and yield a fair representation of the vapor-liquid coexistence curve. In contrast, differences between the models are very large for the vapor-solid coexistence curve. In particular, the lack of explicit quadrupolar interactions for the 6-site model greatly reduces the energetic differences between liquid and solid phases, and this model yields a triple point temperature that is about a factor of 2 too low. In contrast, the 9-site model predicts a triple point of benzene at T = 253 +/- 6 K and p = 2.3 +/- 0.8 kPa in satisfactory agreement with the experimental data (T = 278.7 K and p = 4.785 kPa).     

Universal Asymptotics of the Thermodynamic Characteristics of the Nucleation on Small Macroscopic Nuclei of Soluble Surfactants

Asymptotic behavior of thermodynamic characteristics of nucleation on small macroscopic nuclei of soluble surfactants at their complete dissolution in a nucleating droplet is studied. It is taken into account that, in the region of small sizes of nuclei and corresponding small sizes of critical nuclei of liquid phase, the chemical potential of condensate and the work of droplet formation are affected by the presence of dense surfactant adsorption monolayer on the droplet surface. It is shown that, as the limiting surface area per surfactant molecule in adsorption monolayer increases, the behavior of thermodynamic characteristics of nucleation in the region of small nucleus sizes is characterized by the transition from asymptotics at the adsorption of almost all substance comprising nucleus in a monolayer to the asymptotics at constant adsorption. The study performed is not limited by the selection of specific adsorption isotherms; therefore, the obtained asymptotic dependences of thermodynamic characteristics on the nucleus size can be considered as universal for the heterogeneous nucleation on the nuclei of soluble surfactants.     

The evolution of viscoelasticity near the gel point of end-linking poly(dimethylsiloxane)s

The linear viscoelasticity of polymers near the gel point can be described by two scaling laws. The material at the gel point has a power-law linear viscoelastic relaxation modulus, and the relaxation exponent has been found to vary with the composition of the precursor materials, i.e., it is not universal for gelation. A second scaling law describes the evolution of the linear viscoelastic properties through the gel point. The rate of change of the dynamic mechanical modulus/viscosity is observed to scale as a power-law function of frequency. This power-law function defines a dynamic critical exponent, and this has been found to be independent of precursor composition for end-linking poly(dimethylsiloxane) polymers and equal to κ = 0.21 ± 0.02. This exponent may be a universal measure of gelation. The technique of Time Resolved Mechanical Spectroscopy is used to observe the evolution of linear viscoelastic properties of crosslinking polymers in situ in the rheometer. A stretched exponential relaxation modulus describes the evolution of mechanical properties in the vicinity of the gel point very well. The exponents which characterize the divergence of the zero-shear viscosity and the equilibrium modulus are not universal, since they are related to the relaxation exponent and the dynamic critical exponent.     

Molecular simulation of the shear viscosity and the self-diffusion coefficient of mercury along the vapor-liquid coexistence curve

In earlier work [G. Raabe and R. J. Sadus, J. Chem. Phys. 119, 6691 (2003)] we reported that the combination of an accurate two-body ab initio potential with an empirically determined multibody contribution enables the prediction of the phase coexistence properties, the heats of vaporization, and the pair distribution functions of mercury with reasonable accuracy. In this work we present molecular dynamics simulation results for the shear viscosity and self-diffusion coefficient of mercury along the vapor-liquid coexistence curve using our empirical effective potential. The comparison with experiment and calculations based on a modified Enskog theory shows that our multibody contribution yields reliable predictions of the self-diffusion coefficient at all densities. Good results are also obtained for the shear viscosity of mercury at low to moderate densities. Increasing deviations between the simulation and experimental viscosity data at high densities suggest that not only a temperature-dependent but also a density-dependent multibody contribution is necessary to account for the effect of intermolecular interactions in liquid metals. An analysis of our simulation data near the critical point yields a critical exponent of beta = 0.39, which is identical to the value obtained from the analysis of the experimental saturation densities.     

Critical behavior of binary mixture of {x C6H5CN + (1 − x) CH3(CH2)12CH3}: Measurements of coexistence curves,turbidity, and heat capacity

(Liquid + liquid) coexistence curve, turbidity, and isobaric heat capacity per unit volume for the critical solution of {benzonitrile + n-tetradecane} have been measured. The critical exponents β, ν, γ, and α and system-dependent critical amplitudes B, ξ₀, χ₀, and A^±, corresponding to the difference of the general density variable of two coexisting phases Δρ, the correlation length ξ, the osmotic compressibility χ, and the isobaric heat capacity per unit volume C_pV^?1, have been deduced and were used to test some universal ratios. The behavior of the diameter of the coexistence curves showed good agreement with the complete scaling theory. The analysis of effective critical exponent β_eff, which was well described by the crossover model proposed by Anisimov and Sengers, and effective critical exponent α_eff indicated monotonic crossover phenomena from 3D-Ising behavior to mean-field one as the temperature departed from the critical point.     

Phase diagram of softly repulsive systems: the Gaussian and inverse-power-law potentials

We redraw, using state-of-the-art methods for free-energy calculations, the phase diagrams of two reference models for the liquid state: the Gaussian and inverse-power-law repulsive potentials. Notwithstanding the different behaviors of the two potentials for vanishing interparticle distances, their thermodynamic properties are similar in a range of densities and temperatures, being ruled by the competition between the body-centered-cubic (bcc) and face-centered-cubic (fcc) crystalline structures and the fluid phase. We confirm the existence of a reentrant bcc phase in the phase diagram of the Gaussian-core model, just above the triple point. We also trace the bcc-fcc coexistence line of the inverse-power-law model as a function of the power exponent n and relate the common features in the phase diagrams of such systems to the softness degree of the interaction.     

Parachors of liquid/vapor systems: A set of critical amplitudes

In light of the available experimental data and of our current understanding of liquid–vapor critical phenomena, we examine the values of the parachors and of the parachor exponent, which are commonly used to estimate surface tension from the density difference between coexisting liquid and vapor phases. This is a controversial issue, as values for the parachor exponent ranging from 3.5 to 4 have been proposed in the literature. The parachor exponent and parachors can be viewed as a critical exponent and critical amplitudes, respectively. The Ising value, equal to 3.88, should be observed for the exponent “close enough” to the liquid/vapor critical point, i.e., for “low enough” tensions and densities. However, a review of experimental data for several fluids suggests an effective value in the range of 3.6, in line with the effective values observed for the exponents that describe the vanishing of the density difference and capillary length with the distance to the critical temperature. In fact, the asymptotic Ising regime is not reached experimentally, as confirmed by an estimation of the parachors very near the critical point. Those (Ising) parachors can be inferred from other critical amplitudes corresponding to bulk properties by using two-scale factor universality. Their values exceed those deduced from off-critical tension and density data by more than 10%, corresponding to surface tension differences larger than 50%. We argue that effective parachors (i.e., corresponding to an exponent in the range of 3.6) can be utilized in combination with two-scale-factor universality for determining the critical behavior of fluid systems in an extended range around their liquid/vapor critical point.