Empirical Equation Of State Research Articles

Stresses due to phase changes in subduction zones and an empirical equation of state for the mantle

Summary. An empirical equation of state, assuming mineralogical equilibrium, is developed for the top 700 km of the mantle. Assuming a uniform viscosity, this equation of state is used to show that the stresses due to the changes in phase induced in a descending lithospheric plate in a subduction zone are an order of magnitude larger than those due to the negative buoyancy of the slab in the asthenosphere. The stresses predicted are well within the power law creep region for likely mantle materials and so the effective viscosity will vary within the slab. Consequently the stresses will be smaller than those of 7.0 × 108N/m2 obtained here using uniform viscosities. These stresses are relatively compressional near the sides of the slab and tensional in the centre.

An equation of state for dense nitrogen

An empirical equation of state for nitrogen at high pressure and density is considered. It is shown that for nitrogen at densities greater than 0.6 g/cm3, by using available data [1–3] on static compression of gaseous nitrogen and shock compression of liquid nitrogen, it is possible to construct a Mie-Gruneisen type equation of state which gives a pressuredensity relationship close to experiment along the shock adiabat of liquid nitrogen and agrees with the calculations of other authors for temperature values beyond the shock-wave front [2–3]. Heat capacity, entropy, and Gruneisen coefficient values beyond the shock-wave front in liquid nitrogen are calculated.

Thermodynamic properties of liquid polyethylene

AbstractThermodynamic properties of molten polyethylene are calculated for the temperature range 100° to 300°C and for pressures up to 3000 atmospheres. The calculations are based on limited experimental data augmented by a corresponding‐states correlation and an empirical equation of state. Specific volume, internal energy, enthalpy, entropy, and solubility parameter are presented as functions of temperature and pressure. The results of this study may be useful for engineering design in the high‐pressure polyethylene process.

Note on shock compression of solids

In the framework of a simple compression wave and a weak shock wave, an isentrope of the Tait type and a Hugoniot adiabat based on the linear velocity relation are shown to verify each other consistently. Thus, the two empirical equations of state are given a semianalytical treatment on favorable terms.

Treatment of the Steric Model of Nematic Liquid Crystals Based on the Empirical Equation of State

A treatment of the steric model of nematic liquid crystals based on the empirical equation of state is presented. The steric model of the nematic state assumes that the orientationally dependent molecular repulsive barrier potential is responsible for the isotropic-nematic phase transition. An examination of previous treatments of this model identifies one of the sources of computational difficulty as the need to calculate the form of the equation of state. A method is found which avoids this problem by use of an ansatz. The order parameter is found to be a function of a reduced temperature which depends on parameters which may be found by measuring the equation of state of the isotropic phase outside the transition region. The influence of short-range orientational order may be estimated in a more straightforward manner than possible in previous treatments. Theory and experiment are found to agree on two important aspects of this model.

An empirical equation of state for gases

An empirical equation of state for gases

Method of constructing empirical equations of state

A method is described for approximation of the equation of state p=p(ρ, T) in the form of a double polynomial by the method of least squares.

Equation of state of the alpha and epsilon phases of iron

An empirical equation of state that is thermodynamically consistent is found for the alpha (body-centered-cubic) and epsilon (hexagonal-close-packed) phases of iron. It is fit to all available types of experimental data-calorimetic, thermal expansion, acoustic, static compression, and shock data. All thermodynamic identities are satisfied, since expressions for all observables are derived from a thermodynamic potential. No specific theoretical assumptions are made, but the Helmholtz potential is formulated in terms of a number of adjustable functions of single variables by applying some general considerations of lattice vibrations, conduction electrons, and ferromagnetic exchange interaction. The effective Gruneisen parameter depends on temperature as well as volume. Ultrasonic and shock data for the alpha phase are found to be consistent with X-ray determinations of compression based on the sodium chloride standard. The Slater and Dugdale-MacDonald relations are tested against the fit at low pressure and are found to be in error. At high pressure not enough different types of data exist to establish a unique fit without theoretical assumptions.

Compressibilities and virial coefficients for methane, ethylene, and their mixtures

AbstractA Burnett type of apparatus was built for 12,000 lb./sq.in. pressure and temperatures of 25° to 100°C. Isothermal expansions were run on methane, ethylene, and four binary mixtures at 25°, 50°, and 75°C., and compressibility factors were derived from the pressure ratio observations.From the experimental data, second and third virial coefficients were derived by two techniques, that is, the slope‐intercept and the multiple regressions curve‐fit of the data by the density‐series virial equation. A comparison of these virial coefficients with those from the different empirical equations of state indicates areas of needed improvements in these equations. Second virial cross coefficients were also obtained.

Empirical Equations of State for the Earth's Lower Mantle and Core

Summary A quadratic representation of the incompressibility k in terms of the pressure p for the materials of the Earth inside the pressure range 0.4 < p < 3.2 × 1012 dyn/cm2 was empirically derived some years ago on the basis of seismic and other evidence. The numerical coefficients in the representation have been revised in the present paper, taking account of recent observational evidence. The revised quadratic law applies to the whole Earth below 1000 km. The new evidence includes the recently revised estimate of the Earth's moment of inertia, revised seismic evidence on the lower core, and data from free Earth oscillations. The quadratic model is compared with linear models derived for the lower mantle and core separately. Attention is drawn to errors made when any of these models is applied at pressures less than 0.4 × 1012 dyn/cm2. It is pointed out that unqualified application of Birch's finite-strain formulae leads to some results for the Earth's deeper interior that are somewhat discordant with the empirical evidence here used.

Adsorption in micropores

1. Two methods of precalculating the parameters of adsorption equilibrium within a broad range of pressures and temperatures according to the information contained in one adsorption isotherm have been proposed on the basis of an analysis of the experimental data on adsorption on microporous adsorbents. 2. The first method, applicable in the region of comparatively large degrees of filling, is based upon the virtual linearity of the adsorption isosteres and the exponential dependence of the limiting adsorption in micropores upon the temperature, taken with the opposite sign. 3. The second method, applicable in the region of not very significant degrees of filling, is based upon the assumption that the adsorption isosteres are practically linear all the way up to the so-called quasicritical temperature. A method is indicated for approximate estimation of the pressure and temperature for the quasicritical points on the basis of the empirical equations of state of the real gas. 4. Both methods are confirmed experimentally with quite satisfactory accuracy. They also provide the possibility of calculating the differential heats and entropies of adsorption. In addition, they permit a calculation of the adsorption of gas at temperatures above the critical temperatures according to the adsorption isotherm of the vapor (for example, at the boiling point).

Compressibility of elastomers with crystalline fillers and microvoid inhomogeneities related to various empirical equations of state for liquids and solids

AbstractModified classical compressibility techniques are described which measure both the adiabatic and isothermal bulk moduli of polymers. These differ from familiar methods principally in the utilization of sensitive electronic deformation sensors (differential transformers) to assess volume changes, the autographic recording of pressure and volume changes during an essentially quasi‐static pressurization cycle. The uniaxial and volumetric methods both characterize the compressibility behavior of polymers in the relatively low pressure region (0–100 atm.). Compressibility data for several well‐known liquids and solids are reported. The nonlinear nature of the PV behavior of liquids are shown to be adequately characterized by a simple equation of state which is applicable to rubbery polymerics as well. Equations of state for polymers and filled polymers containing dispersed void inhomogeneities are discussed. Relationships based upon the infinitesimal theory of the deformations of an externally pressurized hollow sphere are compared with more complex equations of state which consider finite deformations at the center of a hollow sphere. It is concluded that the experimental data for the compressibility of voided solids containing less than 0.5% of initial void volume are best represented by the simpler equations of state derived from infinitesimal theory.

Fugacity coefficients for hydrogen gas between O degrees and 1000 degrees C, for pressures to 3000 atm

An empirical equation of state is derived for hydrogen gas which permits computations of fugacity coefficients with reasonable confidence over a wide range of temperatures and pressures. Comparisons are given between fugacity coefficients from the tables and from graphical integration of P-V-T data, as well as comparisons with some previously published results. The tabulated values faithfully reproduce the trends shown by experimental data.

Preparation of Steam Tables with the Aid of a Digital Computer

This paper describes the use of a digital computer in deriving an empirical equation of state for steam, based on experimental data on its pressure-volume-temperature properties. A table of specific volume at round values of pressure and temperature, derived from this equation, is given. This table is preliminary and further work is going on to extend and improve it.

Critical phenomena in gases - I

The exact measurements of the isotherms of gases have proved extremely valuable in the determination of interatomic forces. For this purpose it has been found necessary to express the pv values of a gas as a finite power series in the density or in the pressure, and the coefficients so obtained have been compared with theoretical expressions in terms of interatomic fields. Many accounts of the method have been given and it is not necessary to give further details here (cf. Lennard-Jones 1931). While these methods are valid for gases at low densities where binary encounters are predominant, they fail for gases at high densities such as obtain in the neighbourhood of the critical point. Michels and his collaborators (Michels and others 1937) have recently studied the isotherms of gases at pressures as high as 3000 atm., and they find that the usual method of representing isotherms as simple functions of density or pressure ceases to be useful. The equation of state of van der Waals was astonishingly successful in accounting for the critical phenomena of gases and the form of the isotherms for temperatures below the critical temperature. Other empirical equations of state, for example that of Dieterici, were even more successful in reproducing the observed relations between the critical pressure, volume and temperature, and their very success has often obscured the fact that they were not logical theories of critical phenomena in gases, based as they were on arguments which were valid only for gases of low concentration. Thus the van der Waals equation, valuable as it has been and useful as it still is, implies that the internal energy of a vapour and its liquid phase is proportional only to the first power of the density, and this cannot be true for gases or vapours at densities comparable with those of liquids. The problem still remains of explaining why gases exhibit critical properties and of correlating the observed values of the critical temperature with the forces which atoms or molecules exert on each other.

An Empirical Equation of State

The law of Ramsay and Young, $P=\ensuremath{\psi}T\ensuremath{-}\ensuremath{\Phi}$, is shown to hold for diethyl ether both as a gas and as a highly compressed liquid. A single expression for $\ensuremath{\psi}$, $\ensuremath{\psi}=(\frac{R}{V}){e}^{\frac{a}{\mathrm{Vb}}}$ is shown to reproduce the experimental data for both conditions, with the same values of the constants $a$ and $b$. Similar relations exist for carbon dioxide. Hildebrand's expression for $\ensuremath{\Phi}$ for ether is shown to require alteration in order to fit the accurate values for $\ensuremath{\Phi}$ here obtained. His value of about 79 cc for ${V}_{0}$, the molal volume at the absolute zero of temperature and zero external pressure is verified with a more precise calculation of 79.46 cc. It is suggested that the close fit of $\ensuremath{\Phi}$ to a simple formula at high pressures and the deviations at lower pressures may be due to the existence of a more pronouncedly microcrystalline structure (Andrade Stewart) in the liquid at the higher pressures. The combination of expression for $\ensuremath{\Psi}$ and $\ensuremath{\Phi}$ is shown to produce an equation of state which is accurate for the dilute gas and the highly compressed liquid, but which fails for intermediate pressures.

XXXIV. On a proposed empirical equation of state for fluids

XXXIV. <i>On a proposed empirical equation of state for fluids</i>