Van Der Waals Form Research Articles

Molecular thermodynamics of fluid mixtures at low and high densities

A molecular-thermodynamic framework is proposed to describe phase equilibria over a wide range of densities and compositions. The proposed framework is expressed through a model for the Helmholtz energy that incorporates arbitrary mixing rules at high densities, while at low densities the model reduces to the correct second-virial-coefficient limit. In effect, this framework provides density-dependent mixing rules. The proposed framework is illustrated with an equation of state of the Boublik—Mansoori—van der Waals form. Special attention is given to high-pressure vapor—liquid equilibria for systems containing water and hydrocarbons.

Non-linear characteristics of lipid monolayers in relation to the ability to self-organize

It is known that electrostatic interactions strongly influence the growth of condensed phase domains at the main phase transition of lipid membranes. It is demonstrated in this paper that the inclusion of electrostatic interactions into the pressure-area-temperature equation of state provides a much better fit to the observed equilibrium behaviour than the classical van der Waals form. Both of these equations of state predict a region of negative differential compressibility (NDC), which is not described on long time scales because of a competing first-order phase transition. It is shown in this paper that dynamic NDC can account for most of the features of the observed oscillatory behaviour of monolayers at the oil-water interface. Such behaviour may be related to self-organization of spatiotemporal structures in living organisms.

Interaction potential between rare-gas atoms and metal surfaces.

The interaction potential of rare-gas atoms with metal surfaces is calculated following the Zaremba-Kohn scheme. The attractive part of the potential is described by the usual van der Waals form. The change in the energy eigenvalues of the metal electron is calculated by using a pseudopotential formalism for the rare-gas-atom perturbation, the metal electrons being described by the atomic wave functions. It turns out that this model gives a qualitatively and semiquantitatively good description of the potential experienced by the metal s electrons at the surface. Calculations are performed not only for helium interactions with the noble metals, but also in the case of the other heavier rare gases.

Comparison of two hard-sphere reference systems for perturbation theories for mixtures

The Carnahan—Starling equation of state for hard spheres can be extended to mixtures using either a one-fluid theory, or the generalization of scaled-particle (or Percus—Yevick theory) proposed by Boublik and by Mansoori and coworkers. The two reference systems are combined with a perturbation term of the van der Waals form; they are then used to correlate the phase behavior of binary mixtures of nonpolar molecules differing significantly in molecular size. In each case, one adjustable binary parameter ( a 12) is used to correlate vapor—liquid equilibria over the entire composition range. Predicted Henry's constants and liquid densities for the saturated mixture are compared with experiment. The Boublik—Mansoori hard-sphere-mixture equation is superior to the Carnahan—Starling One-Fluid theory, expecially in the dilute region.

Phase equilibria for strongly nonideal mixtures from an equation of state with density-dependent mixing rules

Abstract An equation of state of the van der Waals form represents vapor—liquid and liquid—liquid equilibria in binary, particularly aqueous, mixtures. The Mansoori—Carnahan—Starling expression is used for the repulsive part of the equation of state while the attractive part uses a simple van der Waals form. For a mixture, the usual (density-independent) quadratic mixing rule is used for the leading attractive term but a density-dependent correction is added to allow for noncentral intermolecular forces of disimilar components at high densities. This procedure gives the necessary quadratic dependence of the second virial coefficient at low densities but includes also cubic terms for representation of phase equilibria at liquid-like densities. Good results are obtained for vapor—liquid and liquid—liquid equilibria in binary systems containing water, hydrocarbons, phenol, pyridine and methanol. However, extension to liquid—liquid equilibria in ternary systems is not successful because the equation of state is not able properly to represent phase equilibria of binary systems at conditions only slightly removed from binary liquid-phase instability; at these conditions, the equation of state erroneously predicts a two-liquid region. For further progress toward application of equations of state to ternary liquid—liquid equilibria, it will be necessary to introduce some fundamental modifications toward better representation of phase behavior in the critical region.

Theoretical calculation of Penning-ionization cross sections for collisions of He(2(1),3S) with sodium atoms.

Cross sections have been calculated for the Penning ionization of ground-state sodium atoms in collisions with metastable He(2 $^{1}$,3S) atoms for the range of center-of-mass energies 0.01 eV \ensuremath{\le}E\ensuremath{\le}1.0 eV. The cross sections were calculated with use of optical-model potentials. The real parts of the relevant potential-energy curves were determined by carrying out a configuration-interaction (CI) calculation on the NaHe molecule. The potentials were adjusted to yield the correct asymptotic van der Waals form. The imaginary parts of the potentials correspond to autoionization rates and were obtained from a Stieltjes moment analysis of a discrete (${L}^{2}$) representation of the ${e}^{\mathrm{\ensuremath{-}}}$+${\mathrm{NaHe}}^{+}$ continuum, as represented by diffuse molecular orbitals generated by the CI procedure. The calculated cross section for He(2 $^{3}S$) on Na at E=0.04 eV is in good agreement with stationary-afterglow measurements at room temperature. The singlet-to-triplet ratio of cross sections is in good agreement with that measured by beam techniques. Elastic differential cross sections for He(2 $^{1}S$) scattered from Na have also been calculated. Satisfactory agreement is obtained with recent crossed-beam measurements.

Supercritical‐Fluid Extraction Calculations for High‐Boiling Petroleum Fractions Using Propane. Application of Continuous Thermodynamics

AbstractSupercritical‐fluid extraction is a useful process for upgrading heavy multicomponent mixtures such as high‐boiling petroleum fractions. Design‐oriented calculations are reported for an extraction process using propane to extract intermediate components from a petroleum fraction with molecular weights in the range 150‐−500. The composition of the heavy‐hydrocarbon mixture is described by two continuous distribution functions of molecular weight, one for paraffinic and one for aromatic components. Process calculations are based on a recently developed thermodynamic framework for mixtures with very many components, coupled with an equation of state of the van der Waals form. Particular attention is given to multicomponent phase equilibria in the retrograde region where solvent recovery is favorable. Quantitative results are given for solvent capacity and selectivity as a function of operating conditions.

ChemInform Abstract: EMPIRICAL CORRECTIONS TO THE VAN DER WAALS PARTITION FUNCTION FOR DENSE FLUIDS

A simple equation of the van der Waals form is used to fit thermodynamic data for methane in the region 100 to 600 K and pressures to 600 bar. Only two temperature-dependent constants are used. In determining constants, particular attention is given to vapor-pressure data well below the critical temperature and to superheated volumetric data well above the critical temperature. With these constants, calculated second virial coefficients are in error as is the two-phase boundary in the critical region. Upon adding simple analytic correction functions to the calculated Helmholtz energy, agreement with experiment is very much improved. For the region 2 to 12 K below the critical temperature, the calculated critical exponent ..beta.. = 0.38, in good agreement with experiment (..beta.. = 0.36). These empirical corrections may suggest new techniques for establishing theoretical modifications to improve the van der Waals model.

Equations of state for strongly nonideal fluid mixtures: Application of local compositions toward density-dependent mixing rules

A local-composition, two-fluid model has been developed for equation-of-state calculations of fluid-phase equilibria for asymmetric mixtures; it is applicable to any equation of state of the van der Waals form. A modification of the quasichemical theory of Guggenheim is applied to mixtures at all fluid densities. Desirable boundary conditions are met at low densities, at high densities, and at high temperatures.

Empirical corrections to the van der Waals partition function for dense fluids

A simple equation of the van der Waals form is used to fit thermodynamic data for methane in the region 100 to 600 K and pressures to 600 bar. Only two temperature-dependent constants are used. In determining constants, particular attention is given to vapor-pressure data well below the critical temperature and to superheated volumetric data well above the critical temperature. With these constants, calculated second virial coefficients are in error as is the two-phase boundary in the critical region. Upon adding simple analytic correction functions to the calculated Helmholtz energy, agreement with experiment is very much improved. For the region 2 to 12 K below the critical temperature, the calculated critical exponent ..beta.. = 0.38, in good agreement with experiment (..beta.. = 0.36). These empirical corrections may suggest new techniques for establishing theoretical modifications to improve the van der Waals model.

Dynamics of freely suspended lyotropic films. I. An inelastic light scattering study of thermal surface fluctuations

We have studied the spectrum and intensity of light scattered by thermal surface displacement fluctuations on freely suspended lyotropic films. Films consisted of a liquid core and surface soap layers and were drawn from solution containing water, glycerol, NaCl, and the ionic surfactant hexadecyltrimethyl ammonium bromide (HTAB). Two modes were observed: a propagating undulation mode in which the film surfaces move together and a damped peristaltic mode having oppositely moving surface soap layers. Dispersion relations for these modes, obtained from the dependence of the scattered light intensity correlation function on film thickness h and wave vector k, confirm the macroscopic hydrodynamic description of film motion. In particular, the overdamped peristaltic mode is shown to involve Poiseuille flow of the fluid core with the flow velocity zero within 2 Å of the surfactant–solution interface, indicating no significant slip or rigid interfacial water layer. No evidence of dispersion in the effective viscosity of the fluid core h(k,w) over the range 0<k<106 cm−1, 0<w<103 sec−1 was found. The undulation mode was found to be underdamped, with frequency in the range of 400 kHz to 1 MHz, the largest observed to date for surface waves in fluids. The dynamic surface tension term s(k,w) for k∼106 cm−1 and w∼6×10 sec−1 was found to be the same as the static value within experimental error. Analysis of the total scattered intensity and of the peristaltic mode dynamics allows the determination of R(h), that part of the film pressure due to electrostatic repulsive and van der Waals attractive forces. The measured R(h) are well represented by the sum of a repulsive screened electrostatic interaction and an attractive van der Waals term. The screened electrostatic interaction is consistent with the concentration of NaCl used and the attractive part of R could be fitted equally well by the simple nonretarded van der Waals form for a uniform dielectric slab, or the Ninham–Parsegian form for a three layer hydrocarbon–water–hydrocarbon slab.

A density functional theory for inhomogeneous charged fluids

We formulate a density functional theory for the one particle densities and the thermodynamic properties of multicomponent, classical, inhomogeneous, charged fluids. We show that for slowly varying densities, the Helmholtz free energy can be expanded as a series of density gradients plus an explicit electrostatic contribution. The coefficients in this expansion are directly proportional to moments of the non-coulombic part of the Ornstein-Zernike direct correlation functions of a uniform mixture. An explicit formula for the stress tensor is derived. The theory is applied to the liquid-vapour interface of a molten alkali halide and some formally exact results for the ionic density profiles and surface tension are derived. Using the truncated gradient expansion, we develop a tractable approximation scheme for the surface properties. The surface tension is of the van der Waals form plus a contribution which involves the integral of the off-diagonal part of the Maxwell stress tensor through the interface. The...

Configurational Thermodynamic Properties of Amorphous Polymers and Polymer Melts. II. Theoretical Considerations

The three methacrylate polymers and melts of low and high density polyethylenes investigated in the preceding paper are discussed in terms of theory. Corresponding literature data on n-paraffins and hevea rubbers are also considered. The good agreement between experimental and predicted PVT relations obtained for the high polymers is similar to that found earlier in several instances. The extensive results available at present make possi- ble comparisons of the characteristic scaling parameters for different systems, with a variation of the characteristic temperatures by a factor of 2. A relationship between the characteristic compressibility factor or entropy per unit mass and the temperature scaling factor ensues, which results in a correlation between the characteristic segmental energy factor and the two-thirds power of the segmental'mass. Proceeding to the pressure dependence of the ther- modynamic functions, we find again good agreement between experimental and theoretical energies andentropies. The results once more illustrate the inadequacy of a van der Waals form for the configurational internal energy. The analysis of the liquid-glass transition line in the methacrylate systems yields the variation of the hole fraction 1 - y along the boundary for the glasses formed by a variable pressure history, and a satisfactory constancy for the glass formed under atmospheric pressure. From a combination of the theoretical and the experimental equation of state of the latter we derive the temperature and pressure dependence of y and compute an internal energy without further adjustments. Whereas the frozen fraction at Tg, as defined earlier, increases significantly with pressure in polystyrene, it remains nearly constant in the methacrylates. From the magnitude of the hole fractions (or free vol- umes) it follows that a significant extent of hole clustering should occur in high Tg systems. In a series of recent papers we have compared experi- mental results originating from our laboratory or the litera- ture with theoretical predictions. These have included polymers of styrene (PS) and o-methylstyrene (PoMS),' atactic and isotactic methyl methacrylate (PMMA),2J vinyl chloride (PVC),* and vinyl acetate (PVAC).4 The re- sults and a detailed examination of dilatometric data at at- mospheric pressure516 revealed the good agreement be- tween the experimental and theoretical5 equations of state. This enabled us to examine the liquid-glass transition region in terms of the equilibrium the~ry,~,~ and to predict the pressure dependence of the glass temperature. Finally we have explored the equation of state of the glass itself and the modifications required in the equilibrium theory and the temperature and pressure dependence of the or- dering parameter appearing in the theory.4.7 We wish to pursue these directions with the polymers in- vestigated in the preceding paper.s The theoretical descrip- tion of a particular liquid system proceeds in terms of char- acteristic volume (V*), temperature (T*), and pressure (P*) scaling parameter^,^ which are to reflect structural characteristics of the segment within, of course, the as- sumptions of the theory. The analysis of the recent liquid state results8 will enlarge the collection of such parameter values, besides providing the intrinsic information. Consid- ering possible correlations between these quantities,g the methacrylates will furnish further examples of relatively high and intermediate T, systems. The polyethylene melts, on the other hand, represent examples of low T,'s. The scale of T*'s will be further extended by the inclusion of n- alkanes. 10~ Finally we examine the liquid-glass transition zone, the glassy state of the methacrylate polymers, and the compar- ative behavior of the ordering parameter in the glass. I. Equation of State The relations to be employed are in reduced variables5

Second virial coefficient of adsorbed para-hydrogen

The second virial coefficient of two-dimensional systems of para-hydrogen are calculated under the assumption that the interaction between molecules is given by the Lennard-Jones 6–12 potential. An estimate of the critical temperature for a liquid-gas transition is obtained by fitting the calculated virial coefficient to a van der Waals form. The results of the analysis indicate that the increase of the specific heat of parahydrogen with decreasing temperature observed by Bretz and Chung is the behavior expected from a nonideal two-dimensional quantum gas. Further, the peaks in the specific heat indicate a liquid-gas transition.

Born-Green correlation functions for fluids with van der Waals forces

In recent work on the asymptotic form of correlation functions in `classical' fluids it has been proposed that the direct correlation function f(r) ~ -ϕ(r)/kT at large r, where ϕ(r) is the interatomic potential. It is pointed out here that for the Born-Green theory, and for ϕ(r) which has the van der Waals form -Ar-6 at large r this relation is modified to read f(r) ~ - (ϕ(r)/kT)(1/KT(2P-n0kT)) where n0 is the mean density, P the fluid pressure and KT the isothermal compressibility. The multiplying factor can be significantly different from unity and negative. It is emphasized that small-angle scattering of X rays or neutrons from liquid argon, for example, could in principle decide between the two results.

Negative Liquid Pressure at High Temperatures

IT must have been remarked in the discussions of the various forms of equation of state for vapour-liquid (cf. K. Onnes and Keesom, “Ency. der Math., or in Leyden Communications, xi., 1912, p. 727) that this equation should determine the range of possible negative pressures in liquid. If we could assume the van der Waals form of equation to hold over the wide range that is concerned, it would readily follow that negative pressure could subsist only at absolute temperatures below 27 3/2 of the critical point of the substance. For water the latter is 365° C.; thus in that substance internal tension could (theoretically) persist up to 538° absolute, which is 265° C. Such an order of magnitude appears at first sight surprisingly high, though really there is nothing to compare it with. By an oversight I have recently (Proc. Lond. Math. Soc., 1916, p. 191) quoted the critical point of water as 365° absolute, and so obtained the much lower limit 35° C.; and it was a reference to experiments by Prof. H. H. Dixon (Proc. R. Dublin Soc., 1914, p. 233), realising, for vegetable sap, tensions of the order of a hundred atmospheresat temperatures around 80° C., that has given rise to this correction.