Hard-sphere Equation Research Articles

A new pseudocubic perturbed hard-sphere equation of state

A new pseudocubic perturbed hard-sphere equation of state is proposed combining the hard-sphere expression of Carnahan and Starling and the empirical cubic form for apparent volume. The present equation of state is successfully applied to the saturated properties of pure substances including polar components. Given temperature and pressure, the proposed equation of state has at most three real roots for the volume, similar to the conventional cubic equations of state. All parameters in the present equation of state can be evaluated from the critical constants, normal boiling point, dipole moment and reduced temperature. The proposed equation of state strictly satisfies the critical point requirements.

Correlation of VLE of CO 2NH 3H 2O using the perturbed hard-sphere, equation of state

The use of the perturbed hard-sphere equation of state with a new mixing rule to correlate VLE of the CO 2NH 3H 2O system has been examined. Assuming a series reaction between CO 2 and NH 3 in aqueous solution, the VLE behaviour of the ternary system can be correlated with good agreement.

Viscoelastic properties of aqueous concentrated polystyrene latex dispersions containing grafted poly(ethylene oxide) chains

The viscoelastic properties of aqueous concentrated polystyrene latex dispersions containing grafted poly(ethylene oxide) (PEO) chains were investigated using oscillatory and steady-state shear stress-shear rate measurements. The relative viscosity-effective volume fraction results were fitted to the Dougherty-Krieger equation for hard spheres by adjusting the value for the adsorbed layer thickness Δ. The latter was found to decrease with an increase in volume fraction of the dispersion and near to close-packing considerable compression of the chains occurred. From the oscillatory measurements, the complex modules G ∗ , storage modules G′, and loss modulus G″ were obtained as a function of frequency at various latex volume fractions. The results showed that the dispersion changes from being more viscous ( G″ > G′) to more elastic ( G′ > G″) over a narrow range of volume fraction φ of the dispersion, i.e., when φ is increased from 0.465 to 0.5. Within the range of increase of φ, it is likely that the chains undergo some compression and interpenetration of the peripheries. When φ increased significantly above 0.5, the system becomes predominantly elastic and significant interpenetration and compression of the chains occur. Indeed when φ is further increased to 0.585 and 0.62, the moduli increase by several orders of magnitude and so does the dynamic viscosity. Under these conditions, the latex behaves as an elastic “gel.”

Prediction of binary and ternary critical properties using deiters' equation, hard sphere and hard convex body equations of state

The use of recently developed equations of state for critical phenomena is examined. Deiters' equation is incapable of accurately predicting the gas-liquid critical properties of mixtures of dissimilar size molecules whereas a hard convex body equation does not yield significantly better results than a simple hard sphere equation. In many cases a ‘hard sphere + attractive term’ equation can predict the critical properties of Type II mixtures to within experimental error. This conclusion is valid for a wide range of component molecules. However, an adjustable parameter is usually required to optimise agreement. Type II liquid-liquid properties can also be accurately calculated but the same precision can not be obtained for Type III phenomena. Some calculations for CHF 3 + alkane mixtures are reported which exhibited Type III behaviour. It appears that the most significant improvement in a priori prediction of critical phenomena lies in developing better combining rules rather than ‘improved’ equations of state. Work on ternary critical phenomena has hitherto been largely limited to simple gas-liquid properties which are continuous with respect to composition. In this work the critical properties of CF 4 + CHF 3 + CH 4 over a wide range of temperature and pressure are calculated. The results illustrate the potential of theory to classify the phase behaviour of ternary mixtures.

Application of hard convex body and hard sphere equations of state to the critical properties of binary mixtures

The critical properties of hydrocarbon mixtures, perfluorocarbon + hydrocarbon, perfluoromethylcyclohexane + siloxane, acetone + hydrocarbon and polydimethyl siloxane mixtures have been calculated from an equation of state for hard convex bodies and from Guggenheim's equation of state for hard spheres. In general, the results of both equations agree well with experimental data. It appears, however, that taking shape factors into account (by using the hard convex body equation) does not lead to a significant improvement in the agreement between theory and experiment for the critical properties.

A modified van der Waals one-fluid approximation based on a simple perturbed hard-sphere equation of state

A modification of van der Waals one-fluid approximation is proposed. This is based on a simple perturbed equation of state which consists of the van der Waals type attractive term and the Carnahan-Starling (for pure fluids) type or the Mansoori-Carnahan-Starling-Leland (for mixtures) type repulsive term. The new mixing rules have been tested with computer simulation results for Lennard-Jones fluids. The results, especially in highly dense regions, are satisfactory.

A rotational equation of state for normal paraffins

The hard convex body and hard sphere equations of state are used to derive a partition function that considers the effect of density on ratational motion in addition to translation. The inclusion of the rotational contribution in the perturbation theory gives rise to a more general principle of corresponding states for thermodynamic properties of polyatomic fluids. The argon equation of state developed by Twu—Lee—Starling is used to calculate the properties of a reference system in the perturbation expansion. In addition to adding the rotational pressure to the argon equation of state, the size of the reference system is also scaled up to that of real systems of interest to describe the properties of systems with either small or large molecules. The equation of state contains a structure parameter which acts as a third parameter in addition to molecular size and energy parameters. The size parameter can be calculated from the critical volume, the energy parameter from the critical temperature, and the structure parameter from the normal boiling-point temperature. Furthermore, the generalized equation of state for normal alkanes can be reduced to a function only of normal boiling-point temperature. The rotational equation of state represents accurately the vapor pressures, densities and enthalpy departures of the vapor, liquid and fluid states over the entire range of practical interest for the normal paraffins. For normal-paraffins, methane through n-eicosane, the average absolute deviation percent for all properties is less than 1% for 1541 points.

A perturbed hard-sphere equation of state for solutions containing an electrolyte

An equation of state was developed for mixtures containing polar components and a non-volatile electrolyte. The equation consists of hard-sphere, electrostatic and attractive contributions to the Helmholtz free energy. For mixtures without electrolytes the equation of state resembles a recent equation developed by Ludecke and Prausnitz. The equation of state was used to correlate mean ionic activity coefficients of various salts dissolved in water. The binary interaction parameter obtained from regressing data was shown to follow a trend with cation anion average hard-sphere diameter. Liquid liquid equilibria of aniline water salt systems and vapor liquid equilibria of methanol water sodium bromide were correlated with the equation of state.

Update

The present work aims to propose alternative corresponding states correlations for temperature-dependent parameters appeared in perturbed hard-sphere equation of state (PHS EOS). The microscopic scaling constants σ, the effective hard-sphere diameter, and ɛ, the non-bonded interaction energy between two spheres are employed to construct the correlations. The modified PHS EOS is applied to model the volumetric properties of pure and mixtures containing ionic liquids (ILs). For 3127 experimental density data points examined for the studied pure ILs, the total average absolute deviation (AAD) of the predicted densities from the literature data is found to be 0.52%. Also, 978 data points for binary mixtures is examined, the AAD of the calculated densities of mixtures from those reported in the literature is found to be 0.79%. Several thermodynamic coefficients have also been successfully determined by the use of the present PHS EOS. The effect of the solvent on the excess functions of the studied binary mixtures has also been investigated using the proposed model.

Perturbed hard sphere equations of state of real fluids—II. Residual parameter ap of non-polar liquids

Using the accurate Baker-Fisher-Watts potential, the dipole-dipole-dipole three-body potential, and the EXP approximation, the residual term a p of a perturbed hard sphere equation of state for argon has been calculated and compared with experimental data. Individual contributions to a p , as well as to the pressure, have been assessed and compared with simple analytical expressions currently available. It is shown that (1) the frequently used high temperature approximation alone yields, necessarily, a very poor equation of state and (2) that an accurate perturbed hard sphere equation of state must contain additional terms having their origin in non-additive three-body interactions and certain higher-order perturbation contributions.

RATIONAL CONSTRUCTION OF AN AUGMENTED HARD CORE EQUATION OF STATE FOR PURE COMPOUNDS AND STUDY OF ITS APPLICATION TO MIXTURES

The Carnahan-Starling hard-sphere equation of state perturbed by attractive forces expressed in a virial expansion has been studied in the vicinity of the critical point. The three van der Waals conditions for the critical isotherm at the critical point were used to determine the minimum number of terms required in the perturbation series. P-v-T data at other temperatures were used for the determination of the temperature dependence of the virial terms. The equation has been tested for mixtures using the mixing rules of Arai et at.

Application of a hard-sphere perturbation equation to calculate the heat capacity of argon in water over a wide temperature range

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTApplication of a hard-sphere perturbation equation to calculate the heat capacity of argon in water over a wide temperature rangeRoberto Fernandez-Prini and Maria L. JapasCite this: J. Phys. Chem. 1986, 90, 7, 1395–1396Publication Date (Print):March 1, 1986Publication History Published online1 May 2002Published inissue 1 March 1986https://doi.org/10.1021/j100398a036RIGHTS & PERMISSIONSArticle Views51Altmetric-Citations5LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (254 KB) Get e-Alerts

Group contributions to activity coefficients from the hard sphere expansion corresponding states theory

A new method has been developed for predicting liquid activity coefficients in ternary mixtures from group contributions. In this method, activity coefficients are obtained from the excess Gibbs free energy of mixing at constant temperature and pressure. In calculating this excess function, the constituent and mixture Gibbs free energies are each represented by an expansion about a pure reference fluid in powers of ratios of hard-sphere diameters and molecular attraction parameters. When the pure component differs from the reference by a single structural group, these ratios represent, respectively, the size contribution and the attraction contribution of this group to the thermodynamic property of the pure fluid. Contributions of intermolecular repulsion to the excess Gibbs free energy are calculated directly from hard-sphere equations of state for the mixture and pure components. The effect of polar contributions calculated by a Pade approximant is also examined. Results indicate that the method developed from the hard-sphere expansion corresponding-states theory is useful for predicting activity coefficients in ternary mixtures when unlike-pair interaction parameters are fitted to binary activity coefficient data. Furthermore, the method shows promise in providing a theoretical basis for applying group contributions to activity coefficients.

Nonlinear conductivity and entropy in the two-body Boltzmann gas

We find exact solutions of the two-particle Boltzmann equation for hard disks and hard spheres diffusing isothermally in an external field. The corresponding transport coefficient, one-particle current divided by field strength, decreases as the field increases. This nonlinear dependence of the current on the field and the corresponding nonlinear dependence of the distribution function on the current are compared to the predictions of “single-time” information theory. Our exact steady-state distribution function, from Boltzmann's equation, is quite different from the approximate information-theory analog. The approximate theory badly underestimates the nonlinear decrease of entropy with current. The exact two-particle solutions we find here should prove useful in testing and improving theories of steady-state and transient distribution functions far from equilibrium.

Short-wavelength collective modes and generalized hydrodynamic equations for hard-sphere particles

The extended hydrodynamic modes recently discussed by de Schepper, Cohen, and collaborators using an approximate hard-sphere kinetic equation are computed here with use of approximate hard-sphere generalized hydrodynamic equations. The theory presented here is completely analytic and reproduces the results of de Schepper et al. for dense hard-sphere fluids reasonably well. The connection to previous theories based on generalized hydrodynamics is discussed.

High-density properties of liquid-state theories: Physically intuitive meaning for the direct correlation functions.

We prove that the hypernetted-chain and soft-mean-spherical integral-equation theories, together with the variational perturbation theory (based on the Percus-Yevick approximation for the hard-sphere reference system) are all identical in the asymptotic high-density limit: They share the same Madelung energy that constitutes an exact lower bound to the true potential energy of the system. The Percus-Yevick equation for hard spheres is shown to diverge at total packing fraction equal to unity, with a direct correlation function c(r) given by the overlap volume of two spheres with separation r. We present the exact analytic solution of the mean-spherical approximation for any repulsive potential \ensuremath{\varphi}(r) satisfying \ensuremath{\varphi}\ifmmode \tilde{}\else \~{}\fi{}(k)\ensuremath{\ge}0 which is a Green's function for an operator of the Sturm-Liouville-type. The corresponding direct correlation function is given by the interaction between the particles when they are ``uniformly smeared'' inside a sphere. Our results are general, applicable to any number of dimensions and to mixtures. The physically intuitive meaning for the direct correlation functions is offered as a starting point for the statistical thermodynamic analysis of nonspherical hard, and/or charged, objects.

Erratum: Molecular dynamics calculations of the hard sphere equation of state

Erratum: Molecular dynamics calculations of the hard sphere equation of state

A note on the Boltzmann equation for hard spheres

A note on the Boltzmann equation for hard spheres

Analytical solutions for helium bubble and critical radius parameters using a hard sphere equation of state

Considerable theoretical and experimental work has verified the role of helium-stabilized bubbles as the precursor to void formation in fast neutron irradiated stainless steels. The concept of the critical bubble radius or critical helium number for bubble-to-void conversion has received particular attention. A hard sphere equation of state is used to compute these parameters for a variety of irradiation conditions and the results are compared with those computed using the ideal gas law. Simplified analytical solutions are developed which permit the calculation of the bubble radius and the critical bubble parameters without resorting to iterative techniques and yet retain the accuracy of the hard sphere equation of state. The use of these solutions is illustrated using a rate theory model of void swelling which has been calibrated using fast reactor swelling data.

On two-parameter equations of state and the limitations of a hard sphere Peng-Robinson equation

Simple two-parameter equations of state are exceptionally effective for calculations on systems of small, uncomplicated molecules. They are therefore extremely useful for vapour-liquid equilibrium calculations in cryogenic and light hydrocarbon process design. In a search for further improvement three two-parameter equations of state with a co-volume repulsion term and three with a hard sphere repulsion term have been investigated. Their characteristic constants at the critical point have been compared. The procedure for fitting the two parameters to empirical data in the subcritical region was analysed. A perturbed hard sphere equation with a Peng-Robinson attraction term was shown to be unsuitable for application over a wide range of p, T conditions. A similar equation with a Redlich-Kwong attraction term gives good service in the cryogenic range.