Modified Van Der Waals Equation Research Articles

Anisotropic charged stellar models with modified Van der Waals EoS in f(Q) gravity

This paper is based on the study of compact stars in the context of electric fields and the nonmetricity effects of gravity. Due to this, we are motivated to build stellar models based on spherically symmetric space-time in f(Q) gravity. The space-time solution is obtained by Durgapal and Bannerji (Phys Rev D 27:328–331,1983) potential along with modified Van der Waals equation of state (EoS) p_r=eta rho ^2+ frac{beta rho }{gamma rho +1} by introducing a specific form of electric charge function q(r)=kr^3. In order to validate our charge model, we used observational data from the literature for celestial objects like Her X-1, 4U 1538-52, SAX J1808.4-3658, and SMC X-1. Furthermore, we have also retrieved the uncharged effects of gravity for the model SMC X-1 by taking k=0. Our present physical analysis shows that all the obtained features for the present solution are in excellent agreement with the viable model as far as observational data is concerned.

Physical implications of teleparallel gravity on stellar configurations with modified Van der Waals equation of state: a broader view

Physical implications of teleparallel gravity on stellar configurations with modified Van der Waals equation of state: a broader view

Self-gravitating anisotropic model in general relativity under modified Van der Waals equation of state: a stable configuration

The purpose of this paper consists in presenting models of compact stars described by a new class of exact solutions to the field equations, in the context of general relativity, for a fluid configuration which is locally anisotropic in the pressure. With current sensitivities, we considered a non-linear form of modified Van der Waals equation of state viz., p_{r}=alpha rho ^{2} +frac{beta rho }{1+gamma rho }, as well as a gravitational potential Z(x) as a generating function by exploiting an anisotropic source of matter which served as a basis for generating the confined compact stars. The exact solutions are formed by correlating an interior space-time geometry to an exterior Schwarzschild vacuum. Then, we analyze the physical viability of the model generated and compare it with observational data of some heavy pulsars coming from the Neutron Star Interior Composition Explorer. The model satisfies all the required pivotal physical and mathematical properties in the compact structures study, offering empirical evidence in support of the evolution of realistic stellar configurations. It is shown to be regular, viable, and stable under the influence generated by the parameters coming from the theory namely, alpha , beta , gamma , delta , everywhere within the astral fluid in the investigated high-density regime that supports the existence of realistic heavy pulsars such as PSR J0348+0432, PSR J0740+6620 and PSR J0030+0451.

Liquid Viscosity and Surface Tension of n-Hexane, n-Octane, n-Decane, and n-Hexadecane up to 573 K by Surface Light Scattering (SLS).

In the present study, the simultaneous and accurate determination of liquid viscosity and surface tension of the n-alkanes n-hexane (n-C6H14), n-octane (n-C8H18), n-decane (n-C10H22), and n-hexadecane (n-C16H34) by surface light scattering (SLS) in thermodynamic equilibrium is demonstrated. Measurements have been performed over a wide temperature range from 283.15 K up to 473.15 K for n-C6H14, 523.15 K for n-C8H18, and 573.15 K for n-C10H22 as well as n-C16H34. The liquid dynamic viscosity and surface tension data with average total measurement uncertainties (k = 2) of (2.0 and 1.7) % agree with the available literature and contribute to a new database at high temperatures. Over the entire temperature range, a Vogel-type equation for the dynamic viscosity and a modified van der Waals equation for the surface tension represent the measured data for the four n-alkanes within experimental uncertainties. By also considering our former SLS data for n-dodecane (n-C12H26) and n-octacosane (n-C28H58), empirical models for the liquid viscosity and surface tension of n-alkanes were developed as a function of temperature and carbon number covering values between 6 and 28. Agreement between these models and reference correlations for further selected n-alkanes which were not included in the development procedure was found.

Nonphysical Behavior in Several Statistical Mechanically Based Equations of State

In this work, we utilize bifurcation diagrams to study the role of mathematical artifacts in deteriorating the physical behavior in statistical mechanically based equations of state of pure fluids. We study the impact of common empirical approximations usually employed to overcome some of the mathematical and physical challenges such as the parametrization of mean field models or pair correlations functions at contact. The proposed diagrams elucidate how the reduced molar volume bifurcates with the variation of temperature at constant pressure. We generate bifurcation diagrams for the modified van der Waals equation of state (EOS) of Poole et al, SAFT-VR Mie, Soft-SAFT, CK-SAFT, and the original SAFT EOSs for spherical and nonspherical molecules. We find that the bifurcation diagram can serve as a useful schematic tool to reveal the unphysical PVT behavior, demonstrate the existence of physical and spurious two-phase separation regions, and illustrate how the number of molar volume roots vary with temperatures. Our method shows that the presence of unphysical branches can cause spurious two-phase separation regions and create erroneous behavior in the stability limit of vapor–liquid equilibrium. We demonstrate that the existence of customary and spurious phase envelopes is accompanied by S-shaped behavior in the volume-temperature bifurcation diagrams. The study reveals that none of the SAFT models is free from producing unphysical behavior. While the SAFT-VR Mie EOS exhibits solid–liquid-like behavior for nonspherical molecules, the CK-SAFT EOS shows liquid–liquid demixing behavior for spherical and nonspherical compounds. For the soft-SAFT EOS, three different two-phase separation regions are observed in addition to the common vapor–liquid phase separation region.

The modified van der Waals equation of state

PVT data of crystallizable materials (CM), minerals, alkali, alkali halides, metals, mineral oxides and hydroxides, rare gas, water, organic compounds and polymers, published in the literature are reanalyzed. It is shown that all these materials under pressure verify the modified van der Waals equation of state (mVW-EOS), discussed recently [J. Rault, Eur. Phys. J. E 40, 82 (2017)]. The characteristic parameters P*V* of this EOS depend only on the nature of the material and not on its state (liquid, glassy, solid of different structure) and whatever are its conductivity and magnetic properties (insulator, conductor, superconductor, paramagnetic, ferromagnetic). This EOS explains the following properties: (a) the fan structure of the isobars V(T), and of the tangents to the isotherms V(P); (b) the superposition principle of the isotherms V(P); (c) the αB rule: the constancy of the thermal pressure coefficient (dP/dT)V = αB, product of the thermal expansion coefficient α and the bulk modulus B; (d) its relation with the Slater conjecture: (dP/dT)V ~ dP/dTm in crystallized materials, Tm being the melting temperature. The characteristic pressure P* (T and V independent) is compared with the various pressures: (i) Pcoh = Ecoh/V, the cohesive energy density; (ii) PLm = Lm/ΔVm, Lm and ΔVm being the enthalpy and volume jumps at the melting, respectively; (iii) PD = ΔHa/DVa, ratio of the activation parameters of the autodiffusion coefficient; (iv) PX = X∕X′, X being the bulk modulus B, shear modulus G, elastic constants Cij, and the yield stress σy of the CM, X′ their pressure derivative (at ambient conditions). All the elastic constants B, G, Cij and the yield stress σy are linear functions of P at low pressure (P < P*) and extrapolate to zero at the same negative pressure − PX = −P*. (e) PBγ = B∕γ* ratio of the bulk modulus B and Gruneisen parameter γ* at zero pressure. (f) PΔVm is the pressure deduced from the linear relation between the volume jump ΔVm(P) at the transition (melting or crystalline transition) and the pressure. The universal relation P* = Pcoh = PLm = PD = PB = PG = PCij = Pσy = PΔVm is observed and discussed. In molecular compounds such as H2O, H2, and polymers with different intra- and intermolecular interactions, the compression involves two different processes, at low and high pressures, verifying the mVW-EOS with characteristic pressures P1* and P2*. The ratio of these pressures is about the ratio of the weak intermolecular and strong intramolecular bond energies. The generalized modified equation of state (gmVW-EOS) describes the two-step process of compression in materials having two (or several) types of bonds.

Liquid Viscosity and Surface Tension of n-Dodecane, n-Octacosane, Their Mixtures, and a Wax between 323 and 573 K by Surface Light Scattering

The present contribution provides experimental data for the liquid viscosity and surface tension of n-alkane based model systems at temperatures up to 573 K. The fundamental advantage of the used surface light scattering (SLS) method lies in its application in thermodynamic equilibrium without calibration in a contactless way. The investigated systems comprise the pure fluids n-dodecane (n-C12H26) and n-octacosane (n-C28H58), their binary mixture at a n-C12H26 mole fraction of about 0.3, and the commercially available hydrocarbon wax SX-70 representing a multicomponent mixture of n-alkanes with a broad chain length distribution. For the first time, it could be demonstrated that the SLS method can simultaneously access the liquid viscosity and surface tension of such medium- to long-chained n-alkane systems close to saturation conditions over a broad temperature range from 323 to 573 K. Typical measurement uncertainties of 2% based on a coverage factor k = 2, i.e., a level of confidence of more than 95%, were obtained. Over the entire temperature range, a simple polynomial equation for the dynamic viscosity and a modified van der Waals equation for the surface tension represent the measured data of the pure and binary systems well. The present investigations improve the data situation for hydrocarbon systems in the high-temperature range where no measurement results exist in literature.

Collapse of weakly nonspherical cavitation bubble

Possibility of realizing shock waves in a single cavitation bubble and the growth of bubble nonsphericity during its collapse in water, acetone, and tetradecane are studied. The radius of the bubble is 500 μm, the liquid pressure and temperature are in the ranges of 1–100 bar and 293–313 K, respectively. A relatively simple mathematical model is used in which the movement of the interphase boundary is governed by the Rayleigh-Plesset equation. The thermodynamic parameters of the vapor are assumed uniform, the state of the vapor being described by the modified Van der Waals equation. The calculations show that the shock waves inside a bubble in tetradecane are found to arise in all the conditions under consideration. Inside the bubble in acetone they do not appear at relatively low pressures while inside the bubble in water they never arise. Sphericity perturbations of the bubble grow to the highest degree in tetradecane (up to several thousand times) and to the smallest degree in acetone (up to several tens of times). In the case of water the perturbations increase up to a thousand times.

Modified Van der Waals equation and law of corresponding states

It is well known that the Van der Waals equation is a modification of the ideal gas law, yet it can be used to describe both gas and liquid, and some important messages can be obtained from this state equation. However, the Van der Waals equation is not a precise state equation, and it does not give a good description of the law of corresponding states. In this paper, we expand the Van der Waals equation into its Taylor’s series form, and then modify the fourth order expansion by changing the constant Virial coefficients into their analogous ones. Via this way, a more precise result about the law of corresponding states has been obtained, and the law of corresponding states can then be expressed as: in terms of the reduced variables, all fluids should obey the same equation with the analogous Virial coefficients. In addition, the system of 3He with quantum effects has also been taken into consideration with our modified Van der Waals equation, and it is found that, for a normal system without quantum effect, the modification on ideal gas law from the Van der Waals equation is more significant than the real case, however, for a system with quantum effect, this modification is less significant than the real case, thus a factor is introduced in this paper to weaken or strengthen the modification of the Van der Waals equation, respectively.

Construction of intermediate regions for a generalized van der Waals gas

An approach to constructing two-phase state regions for real gases, the state of which is described by the modified van der Waals equation in the form proposed by R.L. Fogel’son and E.R. Likhachev, has been discussed.

Numerical investigation of strong compression of vapor inside spherical cavitation bubbles

Numerical investigation of possibility of realizing shock waves in cavitation bubbles during their collapse in water, acetone, and tetradecane is performed. The radius of the bubble is 500 pm, the liquid pressure and temperature are in the ranges of 1-100 bar and 293-313 K, respectively. A numerical technique is used in which the movement of the interphase boundary is governed by the Rayleigh-Plesset equation. The thermodynamic parameters of the vapor are assumed uniform, the state of the vapor being described by the modified Van der Waals equation. The shock waves inside a bubble in tetradecane are found to arise in all the conditions under consideration. By contrast, inside the bubble in acetone they do not appear at relatively low pressures, while inside the bubble in water they never arise. At equal initial data the shock wave is formed much closer to the interface in the case of tetradecane.

Modeling and Simulations on CO&lt;sub&gt;2&lt;/sub&gt; Storage Using Graphene

In this paper, we adopt boththe Lennard-Jones potential and the mean field theory to determine themolecular interactions between carbon dioxide and the double layered graphenes.In addition, we employ a modified van der Waals equation which takes into accountthe multi-scale effect in the absorption regime todeduce the gravimetric uptakeof carbon dioxide between graphene sheets. We show that the full absorptionoccurs at rather low external pressure at low temperatures while this happensat roughly 0.2bar at room temperature. The current methodology has the merit ofrapid computational times and producing deductive results in comparison to theusual MD simulations.For graphene sheets of a separation of 10 Å, the maximumgravimetric uptake could reach 13.3 wt.%.

A universal modified van der Waals equation of state. Part I: Polymer and mineral glass formers.

PVT data of glass formers (minerals and polymers) published in the literature are re-analyzed. All the polymer glass formers (PS, PVAc, PVME, PMMA, POMS, PBMA, PVC, PE, PP, PMPS, PMTS, PPG) present two main properties which have never been noted: a) the isobars P(V) have a fan structure characterized by the two parameters T* and V*; b) the isotherms verify the principle of temperature-pressure superposition for P < P*. From these properties we show that the Equation Of State (EOS) can be put on a modified van der Waals form (VW-EOS), (V - V*) = (V0 - V*)P*/(P + P*). The characteristic pressure P* and the covolume V* are T and P independent. In polymer glass formers P* and V* have same values in the α (melt) and β (glass) domains. The characteristic temperatures T* deduced from the Fan Structure of the Isobar (FSIb) above and below T(g) are different. The characteristic temperature T*(α) of the melt state is found near the Vogel temperature T0 for linear polymers and more than 100 K below T0 for atactic polymers (with pendent groups). This difference in atactic polymers (and in some low molecular weight compounds) is explained by the importance of the β motions due to the pendent groups. The independence of T0 on P is discussed. A modified VFT equation (analogous to the compensation law and Meyer-Neldel rule) giving the relaxation time τ of the α motions as a function of P and T is proposed. The fan structure of the isotherm logτ versus P is explained. It is shown that organic non-polymeric liquids (C6H12, C6H14, DHIQ, OTP, Glycerol, Salol, PDE, DGEBA), mineral glass (SiO2, Se, GeSe4, GeSe2, GeO2, As2O3 and two metallic glasses (LaCe and CaAl alloys) verify this VW-EOS with similar accuracy. The relation P* = B 0/γ B among the characteristic pressure P*, the zero-pressure modulus B0 and the Slatter-Grüneisen anharmonicity parameter γ(B0 deduced from the VW-EOS, is observed in all the glass formers.

Anisotropic spheres with Van der Waals-type equation of state

We study static spherically symmetric space-time to describe relativistic compact objects with anisotropic matter distribution and derive two classes of exact models to the Einstein–Maxwell system with a modified Van der Waals equation of state. We motivate a Van der Waals-type equation of state to physically signify a high-density domain of quark matter, and the generated exact solutions are shown to contain several classes of exact models reported previously that correspond to various physical scenarios. Geometrical analysis shows that the physical quantities are well behaved so that these models may be used to describe anisotropic charged compact spheres.

Van der Waals Equation of State Revisited: Importance of the Dispersion Correction

One of the most basic equations of state describing nonideal gases and liquids is the van der Waals equation of state, and as a consequence, it is generally taught in most first year undergraduate chemistry courses. In this work, we show that the constants a and b in the van der Waals equation of state are linearly proportional to the polarizability volume of the molecules in a gas or liquid. Using this information, a new thermodynamic one-parameter equation of state is derived that contains experimentally measurable variables and physics constants only. This is the first equation of state apart from the Ideal Gas Law that contains experimentally measurable variables and physics constants only, and as such, it may be a very useful and practical equation for the description of dilute gases and liquids. The modified van der Waals equation of state describes pV as the sum of repulsive and attractive intermolecular interaction energies that are represented by an exponential repulsion function between the electron clouds of the molecules and a London dispersion component, respectively. The newly derived equation of state is tested against experimental data for several gas and liquid examples, and the agreement is satisfactory. The description of the equation of state as a one-parameter function also has implications on other thermodynamic functions, such as critical parameters, virial coefficients, and isothermal compressibilities. Using our modified van der Waals equation of state, we show that all of these properties are a function of the molecular polarizability volume. Correlations of experimental data confirm the derived proportionalities.

Transmission electron microscopy study of Ar‐filled bubbles in steel

Ar-filled bubbles in an oxide-dispersion-strengthened (ODS) alloy were characterized by analytical transmission electron microscopy. Energy dispersive X-ray analysis combined with nanoscale imaging was employed for quantitative analysis of Ar gas locked in a bubble 6 nm in size. As the Ar bubbles are formed at the ODS particles - matrix interface, the known Ti(2)Y(2)O(7) composition and size of ODS particles served as a reference in quantification of the Ar signal and for the calculation of the bubble size and the Ar density inside. The Ar pressure was also calculated by modified Van der Waals equation.

Lagrangian particle model for multiphase flows

A Lagrangian particle model for multiphase multicomponent fluid flow, based on smoothed particle hydrodynamics (SPH), was developed and used to simulate the flow of an emulsion consisting of bubbles of a non-wetting liquid surrounded by a wetting liquid. In SPH simulations, fluids are represented by sets of particles that are used as discretization points to solve the Navier–Stokes fluid dynamics equations. In the multiphase multicomponent SPH model, a modified van der Waals equation of state is used to close the system of flow equations. The combination of the momentum conservation equation with the van der Waals equation of state results in a particle equation of motion in which the total force acting on each particle consists of many-body repulsive and viscous forces, two-body (particle–particle) attractive forces, and body forces such as gravitational forces. Similar to molecular dynamics, for a given fluid component the combination of repulsive and attractive forces causes phase separation. The surface tension at liquid–liquid interfaces is imposed through component dependent attractive forces. The wetting behavior of the fluids is controlled by phase dependent attractive interactions between the fluid particles and stationary particles that represent the solid phase. The dynamics of fluids away from the interface is governed by purely hydrodynamic forces. Comparison with analytical solutions for static conditions and relatively simple flows demonstrates the accuracy of the SPH model.

On the physical state of molecules in microporous solids

Molecules occupying the intracrystalline volume of molecular sieves are known to be in strong interaction with their environment, i.e., “solvated” by the zeolite framework, and in a state that is neither gaseous, liquid, or solid. It is shown that the experimental effective densities of simple sorbed substrates, at saturation of the zeolite intracrystalline volume, may exceed their densities in the liquid and solid states, which can be explained in terms of confinement effects. A modified van der Waals equation is also proposed to describe the physical state of molecules sorbed in microporous solids below saturation coverage. It implies that these molecules could be considered as being in a pseudo-supercritical state at a temperature lower than their critical temperature. Here we attempt to explain how confinement effects can affect the physical state of molecules sorbed in the intracrystalline volume of microporous solids and how it may affect, eventually, their reactivity in microporous catalysts.

Liquid-liquid phase transition for an attractive isotropic potential with wide repulsive range

We investigate how the phase diagram of a repulsive soft-core attractive potential, with a liquid-liquid phase transition in addition to the standard gas-liquid phase transition, changes by varying the parameters of the potential. We extend our previous work on short soft-core ranges to the case of large soft-core ranges, by using an integral equation approach in the hypernetted-chain approximation. We show, using a modified van der Waals equation we recently introduced, that if there is a balance between the attractive and repulsive part of the potential this potential has two fluid-fluid critical points well separated in temperature and in density. This implies that for the repulsive (attractive) energy U(R)(U(A)) and the repulsive (attractive) range w(R)(w(A)) the relation U(R)/U(A) proportional to w(R)/w(A) holds for short soft-core ranges, while U(R)/U(A) proportional to 3w(R)/w(A) holds for large soft-core ranges.

Liquid-liquid phase transitions for soft-core attractive potentials.

Using event-driven molecular dynamics simulations, we study a three-dimensional one-component system of spherical particles interacting via a discontinuous potential combining a repulsive square soft core and an attractive square well. In the case of a narrow attractive well, it has been shown that this potential has two metastable gas-liquid critical points. Here we systematically investigate how the changes of the parameters of this potential affect the phase diagram of the system. We find a broad range of potential parameters for which the system has both a gas-liquid critical point C1 and a liquid-liquid critical point C2. For the liquid-gas critical point we find that the derivatives of the critical temperature and pressure, with respect to the parameters of the potential, have the same signs: they are positive for increasing width of the attractive well and negative for increasing width and repulsive energy of the soft core. This result resembles the behavior of the liquid-gas critical point for standard liquids. In contrast, for the liquid-liquid critical point the critical pressure decreases as the critical temperature increases. As a consequence, the liquid-liquid critical point exists at positive pressures only in a finite range of parameters. We present a modified van der Waals equation which qualitatively reproduces the behavior of both critical points within some range of parameters, and gives us insight on the mechanisms ruling the dependence of the two critical points on the potential's parameters. The soft-core potential studied here resembles model potentials used for colloids, proteins, and potentials that have been related to liquid metals, raising an interesting possibility that a liquid-liquid phase transition may be present in some systems where it has not yet been observed.