Expression For Internal Energy Research Articles

Investigation of thermodynamic properties of weakly interacting Fermi gas in weakly magnetic field by using the N-E-V distribution and pseudopotential method

Based on the fact that most of low temperature experiments of quantum systems are explored in an external field on condition that the particle numbers, volumes and energies of systems may be changed, the thermodynamic properties of weakly interacting Fermi systems in weak magnetic field are studied by using the statistical distribution of the completely open system with variable particle number, volume, and energy (N-E-V distribution). Firstly, the analytical expressions of internal energy and heat capacity, which are in the Fermi integral form, are obtained in the general case, and the analytical expressions and numerical results of energy and heat capacity are given under the extreme condition of supper-low temperature. The calculation results by the N-E-V distribution (with particle number density being variable) are compared with those by a pseudopotential method (with particle number density being unchanged). It can be found that the deviations of the internal energy and heat capacity calculated by the two different methods are very small, and the N-E-V distribution method can partially compensate for the error caused by the pseudo potential method. The most interesting point of the results obtained by the N-E-V distribution method is that there is a phase transition temperature in the weakly interacting Fermi system in weak magnetic field under the low temperature condition. The phase transition temperature is just in the range where occur the Fermi systems, Bose-Einstein condensation (BEC), Bardeen-Cooper-Schrieffer (BCS) phase transition, and BEC-BCS crossover, and does not vary with strength nor characteristic (attraction or repulsion) of the weak interaction, but it decreases with the strengthening of the external magnetic. When there is no external magnetic, the phase transition temperature is highest (more than 0.184 times Fermi temperature).

Internal Energy of the Non-Spinning Black Holes

The present paper derives an expression for internal energy of the Non-spinning black holes using first law of black hole thermodynamics and calculates their values of different test Non-spinning holes existing in X-ray binaries (XRBs) and Active Galactic Nuclei (AGN).

Redox Behavior of Nanoparticules: Nonextensive Thermodynamics Approach

We show here that the concepts of nonextensive thermodynamics (NET), described previously (J. Phys. Chem. B, 2004, 108, 18980), can be used to express the variations of a redox couple according to the size of the reduced solid species by a power law. The Gibbs−Thompson relation then appears as a particular case of this formulation. The NET adopts the same conceptual basis as classical thermodynamics but uses an extensity, χ, in the expression of internal energy. This extensity is an Euler’s function of the particle mass with a homogeneity degree, m, which can be other than one; m is the thermodynamic dimension of the system. The consequences are that the functions of state are no longer extensive and that the magnitudes such as the chemical potential or the pressure are no longer intensive extents. This approach can be used to describe complex systems (for example, nanoparticles, porous substrates, and dispersed or interpenetrated phases). We checked the validity of the relations established by considering experimental data concerning the Ag/Ag+ system. The silver clusters which grow on AgBr and AgBr in the presence of gelatin indeed have the properties of nonextensive phases of which we specify the characteristics.

Magnetic Properties of Layered Heisenberg Ferrimagnets

The method of spin-wave theory is used to study the magnetic properties of layered Heisenberg ferrimagnets. The internal energy and specific heat per site are calculated for different interlayer coupling strengths and temperatures. In the low temperature region θ (= K B T/6J) « θ 1 (= (2 + δ)/3), a characteristic temperature θ o = δ is given (δ is the ratio of ferrimagnetic interplanar coupling to planar coupling), the asymptotic expressions of internal energy anti specific heat are obtained for θ » θ o and θ « θ o , respectively. It is shown that the interlayer coupling has an important effect on the magnetic properties of ferrimagnets.

A new averaging scheme for the riemann problem in pure water

The numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms from the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations arc formulated in terms of total enthalpy, treatment of a pressure time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservation law. The present research establishes that successful upwind-biased shock capturing schemes can be based upon the pseudo-conservative system. A new averaging scheme for solving the associated Riemann problem is developed. The method is applied to numerical simulations of shock wave propagation in pure water.

A Weak Formulation of Roe's Approximate Riemann Solver Applied to the St. Venant Equations

The numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms from the Euler equations, then upwind-biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations arc formulated in terms of total enthalpy, treatment of a pressure time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservation law. The present research establishes that successful upwind-biased shock capturing schemes can be based upon the pseudo-conservative system. A new averaging scheme for solving the associated Riemann problem is developed. The method is applied to numerical simulations of shock wave propagation in pure water.

On shock capturing for pure water with general equation of state

AbstractThe numerical investigation of shock phenomena in gas or liquid media where enthalpy is the preferred thermodynamic variable poses special problems. When an expression for internal energy is available, the usual procedure is to employ a splitting scheme to remove source terms for the Euler equations, then upwind‐biased shock capturing algorithms are built around the Riemann problem for the conservative system which remains. However, when the governing equations are formulated in terms of total enthalpy, treatment of a pressure–time derivative as a source term leads to a Riemann problem for a system where one equation is not a conservative law. The present research establishes that successful upwind‐biased shock capturing schemes can be based upon the pseudoconservative system. The method is applied to numerical simulations of shock wave propagation in pure water.

Thermodynamic Functions and Properties of MgO at High Compression and High Temperature

The complete thermodynamic functions for MgO are presented for the temperature range 300–2000 K and the pressure range 0–150 GPa, both as tables and as graphs. Careful attention is given to the temperature and pressure dependence of the coefficient of thermal expansion α and the isothermal bulk modulus KT, which are the major corrections to the thermodynamic functions in extreme conditions. Our equations efficiently use the fact that the product αKT varies only slightly with either temperature or with volume, although scrupulous care is taken to account for the actual numbers. The corrections are now possible due to recent measurements of the bulk modulus up to 1800 K. The Birch–Murnaghan equation of state is used to account for certain terms in the expressions for internal energy and enthalpy. The parameters used in the Birch–Murnaghan equation of state are given as functions of temperature. The parameters which presently limit the calculations of thermodynamic functions to even higher temperatures and pressures and limit the applications of this method to other minerals are: (1) the uncertainty of the thermal expansivity at temperatures above 1500 K, and (2) the uncertainty of the temperature dependence of (∂KT/∂T)P at temperatures above 800 K.

Thermodynamic functions for some particular forms of the expression for internal energy of an ideal, compressible medium

Thermodynamic functions for some particular forms of the expression for internal energy of an ideal, compressible medium