Constant-volume Specific Heat Research Articles

An equation of state for liquid iron and implications for the Earth's core

An equation of state is presented for liquid iron based on published ultrasonic, thermal expansion, and enthalpy data at 1 bar and on pulse‐heating and shock wave compression and sound speed data up to 10 Mbar. The equation of state parameters, centered at 1 bar and 1811 K (the normal melting point of iron), are density, ρ0 = 7019 kg/m3, isentropic bulk modulus, KS0 = 109.7 GPa, and the first‐and second‐pressure derivatives of KS, K′S0 = 4.66 and K″S0 = −0.043 GPa−1. A parameterization of the Grüneisen parameter γ as a function of density ρ and specific internal energy E is γ = γ0 + γ′(ρ/ρ0)n(E ‐ E0) where γ0 = 1.735, γ′ = −0.130 kg/MJ, n = −1.87, and E0 is the internal energy of the liquid at 1 bar and 1811 K. The model gives the temperature dependence of γ at constant volume as (∂γ/∂T)v|1bar,1811K = −8.4 × 10−5 K−1. The constant volume specific heat of liquid Fe at core conditions is 4.0–4.5 R. The model gives excellent agreement with measured temperatures of Fe under shock compression. Comparison with a preliminary reference Earth model indicates that the light component of the core does not significantly affect the magnitude of the isentropic bulk modulus of liquid Fe but does decrease its pressure derivative by ∼10%. Pure liquid Fe is 3–6% more dense than the inner core, supporting the presence of several percent of light elements in the inner core.

Adiabetic temperature changes in a one-component fluid near the liquid-vapor critical point

The theory of the fast adiabatic equilibration that results from the diverging thermal-expansion coefficient is extended to take into account the finite thermal conductivity of the container walls. As the critical point is approached and the fluid thermal impedance drops below that of the walls, a crossover takes place to a new regime in which the equilibration time varies as the square of the constant-volume specific heat. We also study the adiabatic changes that are generated by an internal heat pulse and by pressure modulation. The latter technique should make light-scattering experiments feasible very close to the critical point by avoiding the strong density stratification that sets in at thermal equilibrium. In the concluding portion of the paper, we propose experimental tests of the theory by interferometry and by sonic attenuation and dispersion.

Thermal expansion of mantle and core materials at very high pressures

The thermal expansivities (α) of MgO and high‐pressure phases of CaO, CaMgSi2O6, and Fe at ultrahigh pressure are obtained by comparing existing shock compression and temperature measurements to 300 K compression curves constructed from ultrasonic elasticity and static compression data. For MgO, α can be represented by: α = ρoγoCV(ρo/ρ)0.5±0.5/KT where γ is the Grüneisen parameter, CV is the constant volume specific heat, KT is the isothermal bulk modulus, and ρ is the density. Using this expression, the thermal expansivity of MgO is 28‐32×10−6K−1 at the pressure of the top of the lower mantle and 10‐16×10−6K−1 at its base (at 2000 K). New data for α of ε‐Fe, together with an inner core temperature of 6750 K, constrain the density of the inner core to be 5±2% less than the density of ε‐Fe, implying the inner core contains a light element.

Inelastic neutron and light scattering studies on liquid sulphur dioxide

The constant-volume specific heat and adiabatic sound velocity in liquid sulphur dioxide are studied by means of high resolution inelastic neutron and light scattering experiments. The generalized densities of states Z(E) are obtained from the inelastic neutron scattering spectra and show a marked temperature dependence, evidenced as quasicrystalline behaviour at temperatures just above melting, which develops a substantial intensity at zero and low frequencies as the temperature is raised. The hypersonic velocity derived from the light scattering study is shown to be consistent with previous ultrasound and thermal neutron data.

Lattice dynamics, thermodynamic and elastic properties of CdGeAs2

An extended Keating’s model with two-bond-stretching and one-angle-bending force constants within the rigid-ion approximation is applied to the investigation of the lattice dynamics of CdGeAs2. Five model parameters of the theory are determined by a least-square fitting to the infrared reflectivity measurements. The phonon spectrum along two high-symmetry directions and the one-phonon density of states are calculated. The theoretical values of the elastic constants, the constant-volume specific heat and the Debye temperature show a reasonable agreement with the experimental data reported by other authors.

Active control of nonlinear pressure oscillations in combustion chambers

A theoretical analysis of active control of nonlinear acoustic instabilities in combustion chambers is developed in this article. The formulation starts with a generalized wave equation that describes the dynamic behavior of second-order nonlinear oscillations with distributed feedback actions. Control inputs are provided by the burning of the injected seconday fuel in the chamber, with its instantaneous mass flow rate modulated by a proportionalplus-integral (PI) controller located between the pressure sensor and the fuel injection mechanism. Various nonlinear stability characteristics, including the existence and stability of limit cycles, are studied analytically using the method of time averaging. In addition, an optimization procedure is developed for selecting controller gains. Nomenclature Anij = parameters associated with nonlinear acoustic coupling a = speed of sound in mixture Bnij = parameters associated with nonlinear acoustic coupling b = spatial distribution of burning of control fuel, Eq. (5b), Cv - constant-volume specific heat for two-phase mixture c = amplification factor of sensor output Dni = linear parameters, Eq. (11) Eni = linear parameters, Eq. (11) e - error signal between desired response and actual measurement

On the lattice dynamics of AgGaS2

The vibrational spectrum and one-phonon density of states of a chalcopyrite crystal AgGaS2 are calculated in an extended Keating’s model with two-bond-stretching and one-bond-bending force constants. Three charges of ions and three force constants are determined by a least-square fitting to experimental frequencies of long-wave phonons taken from Raman-scattering experiments. The calculated constant-volume specific heat, Debye temperature and elastic constants, of AgGaS2 are in agreement with the experimental data of other authors.

Molecular dynamics simulation of interhalogen compounds. 1. Liquid chlorine trifluoride (ClF3): structure and thermodynamics

The results of a molecular dynamics (MD) simulation study of liquid chlorine trifluoride (ClF3) at 217, 260, and 287 K are reported. The cubic simulation cell consists of 108 ClF3 molecules assumed to be interacting via site–site Lennard–Jones 12–6 pair potential. The parameters for F–F and Cl–Cl interaction are the same as used for the simulation of F2, and Cl2, respectively, and those for the Cl–F cross interaction are calculated using Lorentz–Berthelot rules. These results are then used to calculate various radial distribution functions characteristic of the liquid structure. Thermodynamic properties, namely, configurational energy, constant volume specific heat, and internal pressure are also reported. The time-dependent properties, mean square force and torque, self diffusion coefficient, and the quantum corrections to the free energy, were also obtained. The dimer configuration drawn based on the observed contact distances was found to be in good agreement with the results of matrix isolation infrared and laser Raman spectroscopic studies. Keywords: MD simulation, interhalogens, liquid structure, thermodynamic properties.

Molecular dynamics simulation of ice at constant volume

The melting of ice, as modeled by the RSL2 potential, is investigated in a series of molecular dynamics simulations at constant volume. According to the behaviors of the total energy and structure factor with temperature, melting of the model system occurs between 260 and 330 K. Estimates of the constant volume specific heats are made which indicate that the molecules ought to be treated as rigid bodies. The necessity of running these simulations for many picoseconds, rather than only two or three, is illustrated.

Liquid–glass transition: A molecular dynamics study of the sodium system

Glass formation process of the sodium system is studied by means of molecular dynamics simulation with constant volume. An equilibrium liquid with 864 atoms is cooled with the rate of 8.00×1013 K s−1 to form the glassy state. A second peak splitting of pair-distribution function (PDF) disappears at high temperatures in the glassy state region and appears clearly with decreasing temperature. The disappearance is attributed to the thermal vibration effect of atoms in the glass. The harmonic vibration model is valid in the glassy state. The Wendt–Abraham parameter, the ratio of the first minimum to the first maximum in the PDF, is 0.08 at the glass transition temperature. The value is equal to the one for the soft-core system, which is lower than the value 0.14 for the Lennard-Jones and the Gaussian-core system. The constant-volume specific heat is found to be 3 Rg in the glassy state, where Rg is the gas constant. The difference of the specific heat between the supercooled liquid and the glassy state is Rg.

The three isentropic exponents of dry steam

The isentropic change of an ideal gas is described by the well known relations pvk=const, Tv(k−1) =const and p(1−k)Tk=const, where the exponent k is defined as the ratio of the constant pressure to the constant volume specific heat, k=cp/cv. The same relations are also used for real gases if small or differential isentropic changes are considered. A better examination of the differential isentropic change shows that for p, v, T systems, there are three different isentropic exponents corresponding to each pair formed out of the variables p, v, T. These three exponents noted kT,p, kT,v, kp,v after the corresponding pair of variables used, are interconnected by one relation, and accordingly only two out of the three are independent. The analysis of the present paper leads to the conclusion that calculations encountered in real gas isentropic change, with the exponent k taken as the ratio of the local specific heat values may lead to incorrect results. The numerical values of these exponents for steam, have been calculated and are presented. It can be seen that the deviations obtained, in comparison to the use of conventional k=cp/cv values are considerable.

Molecular dynamics at constant pressure and temperature

Methods are discussed for generating by molecular dynamics isobaric-isoenthalpic, NPH, isochoric-isothermal, NVT, and isobaric-isothermal, NPT, ensembles. Andersen's constant-pressure method is reformulated so that the ensemble rather than the scaled system is directly calculated. Four constant-temperature schemes were considered. Two involve the addition of a stochastic collision term to the molecular trajectories. The Andersen method and a stochastic dynamics approach were examined. The latter employed a velocity damping term in addition to the random force. Two other methods employed uniform velocity scaling applied to all molecules. The NPT algorithm induces a transition to the dilute phase for a Lennard-Jones fluid in the spinodal region (p* = 0.5, T* = 1.28) of the phase diagram. The thermodynamic equivalence of the ensembles is demonstrated by long calculations of the chemical potential of Lennard-Jones states by the particle insertion method. The internal energy, pressure, constant volume and pressure specific heats, adiabatic compressibilities, pair radial distribution functions and self-diffusion coefficients are also evaluated. Only for second-order thermodynamic quantities is there evidence of an ensemble dependence.

Consequences of the renormalization group for the thermodynamics of fluids near the critical point

We have obtained expressions including corrections to scaling terms for a number of thermodynamic properties of fluids near the critical point by specializing the appropriate derivatives of the logarithm of the grand partition function to trajectories of experimental interest. Our justification for applying Wegner's general predictions, for the functional form of the free energy, to this thermodynamic potential is that it is the potential for a fluid most closely analogous to the Helmholtz free energy for the Ising model. It is found that the average of the coexisting densities, the so-called rectilinear diameter, is a nonlinear function of the temperature with a temperature derivative which diverges like the constant volume specific heat, at the critical point, with a system-dependent coefficient. The second-temperature derivative of the chemical potential, along either the coexistence curve or the critical isochore, is found to be nondivergent at the critical point and its value is the same above and below ${T}_{c}$. The corrections due to the irrelevant scaling fields are found to be as important as those due to higher-order terms in the expansion of the scaling fields around the critical point. Using the parametric representation of the linear model, we have been able to obtain expressions for the elements of the matrix relating relevant scaling fields and physical variables in linear form, in terms of experimentally measurable quantities.

Interpretation of the unusual behavior of H2O and D2O at low temperatures: Tests of a percolation model

The unusual low-temperature behavior of liquid water is interpreted using a simple model based upon connectivity concepts from correlated-site percolation theory. Emphasis is placed on examining the physical implications of the continuous hydrogen-bonded network (or ’’gel’’) formed by water molecules. Each water molecule A is assigned to one of five species based on the number of ’’intact bonds’’ (the number of other molecules whose interaction energy with A is stronger than some cutoff VHB). It is demonstrated that in the present model the spatial positions of the various species are not randomly distributed but rather are correlated. In particular, it is seen that the infinite hydrogen-bonded network contains tiny ’’patches’’ of four-bonded molecules. Well-defined predictions based upon the putative presence of these tiny patches are developed. In particular, we predict the detailed dependence upon (a) temperature, (b) dilution with the isotope D2O, (c) hydrostatic pressure greater than atmospheric, and (d) ’’patch-breaking impurities’’—for four separate response functions, (i) the isothermal compressibility KT, (ii) the constant-pressure specific heat CP, (iii) the constant-volume specific heat CV, and (iv) the thermal expansivity αP, as well as for dynamic properties such as (v) the transport coefficients self-diffusivity Ds and shear viscosity η, (vi) the characteristic rotational relaxation time τch, and (vii) the Angell singularity temperature Ts. The experimentally observed dependence of these seven quantities upon the four parameters (a)–(d) is found in all cases to agree with the predicted behavior. The paradoxical behavior associated with the absence of a glass transition in pure liquid water is also resolved. Finally, we propose certain experiments and computer simulations—some of which are underway—designed to put the proposed percolation model to better tests than presently possible using available information.

Conformational effects on isobaric and constant volume specific heats in polymers

An equation was derived for the isobaric specific heat change ΔC p at T g for polymers by use of the configurational partition function normalized per unit volume for the polymer liquid. Eliminating the contributions of the conformational and free volume specific heat changes, the remaining contribution to ΔC p , relating to the thermal wave-length and the excess specific heat change, was considered thermodynamically. In order to investigate the causes of the abrupt change in C p at T g as observed in many polymers, for PE, PET, nylon-6 and PP, the conformational specific heat C c was calculated in the temperature range from 50 K to melting temperature. The characteristic temperature dependence of C c with a peak temperature of 100 K for PE, PET and nylon-6, 250 K near T g (263 K) for i-PP and about 320 K for s-PP was obtained. At T g of PE, PET and nylon-6, no peculiar change in C c was found. The abrupt change in C p at T g may be attributed almost entirely to the change in the free volume specific heat C f .

A pair interaction potential for rubidium calculated from thermodynamic and neutron diffraction data

Abstract A new thermodynamically self-consistent approximation relating the structure factor to the pair interaction potential is applied to the calculation of an effective pair potential for liquid rubidium by use of the structure factor measurements of Wingfield and Enderby (1972). Explicit treatment of the density dependence of the potential is avoided by using data taken at constant density and two temperatures, and constraining the approximation to reproduce the correct constant volume specific heat. It is further required that the temperature dependence of the resulting potential be minimal. The approximate potential is shown to reproduce the experimental structure data when used in a molecular dynamics computer calculation.

Specific heat of bone.

The specific heat of dry bone, as well as decalcified bone, obtained from bovine femur samples are measured as a function of temperature in the range 200 to 390K, using a differential-scanning-calorimetry technique. Special sample pans for volatile materials were used to provide a uniform thermal environment and to eliminate errors due to the evaporation of the moisture contained in the bone samples; the rate of heating was 10 K/min. From measurements of the constant pressure values, and using the Nernst-Lindermann equation, the constant-volume specific heat of both the collage and hydroxyapatite components are evaluated in the given temperature range.

Loci of maxima and minima of thermodynamic functions of a simple fluid

Some elementary properties of loci of extrema of thermodynamic functions are established and discussed in connection with maxima and minima of the constant volume specific heat, CV, the isothermal compressibility, χT, and the constant pressure specific heat, CP, along isotherms, and the extremum properties of the isobaric coefficient of expansion, αP, along isobars. Experimental results for fluid argon are used to construct the required loci of extrema. Van der Waals' equation is applied to obtain loci of extrema for χT, CP, and αP, for the speed of sound W, and for related inflexion loci.

A Study of Fluid Argon via the Constant Volume Specific Heat, the Isothermal Compressibility, and the Speed of Sound and their Extremum Properties along Isotherms

The properties of fluid argon are investigated via the maxima and minima along isotherms of selected thermodynamic functions, the isothermal compressibility, χT, the constant volume specific heat, CV, and the speed of sound, W. Calculations are based on an equation of state due to Gosman, McCarty, and Hust and on speed of sound data compiled by Thoen, Vangeel, and Van Dael. The calculation of CV in the dense liquid region, from the equation of state and from the speed of sound, is discussed in detail. Also, the linear dependence of W on the density in the liquid region is reconciled with the behaviour of W at temperatures above critical to obtain a unified description of the variation of W along isotherms.

Effects of an attractive-δ-potential impurity center on Bose-Einstein condensation

The exact density of states for one particle in the field of an attractive $\ensuremath{\delta}$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The Bose-Einstein phase is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature ${T}_{c}$. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, ${E}_{b}$. The transition temperature and the discontinuity of the specific heat at ${T}_{c}$ are also obtained as a function of ${E}_{b}$.