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We show that the most prominent of the work theorems, the Jarzynski equality and the Crooks relation, can be applied to the momentum transfer at the air–sea interface using a hierarchy of local models. In the more idealized models, with and without a Coriolis force, the variability is provided from Gaussian white noise which modifies the shear between the atmosphere and the ocean. The dynamics is Gaussian, and the Jarzynski equality and Crooks relation can be obtained analytically solving stochastic differential equations. The more involved model consists of interacting atmospheric and oceanic boundary layers, where only the dependence on the vertical direction is resolved, the turbulence is modeled through standard turbulent models and the stochasticity comes from a randomized drag coefficient. It is integrated numerically and can give rise to a non-Gaussian dynamics. Also in this case the Jarzynski equality allows for calculating a dynamic beta βD of the turbulent fluctuations (the equivalent of the thermodynamic beta β=(kBT)-1 in thermal fluctuations). The Crooks relation gives the βD as a function of the magnitude of the work fluctuations. It is well defined (constant) in the Gaussian models and can show a slight variation in the more involved models. This demonstrates that recent concepts of stochastic thermodynamics used to study micro-systems subject to thermal fluctuations can further the understanding of geophysical fluid dynamics with turbulent fluctuations.

This equation of state was developed from PVT compressibility data on methane and ethane. The highly-constrained form originates on a given liquid-vapor coexistence boundary (described by equations for the vapor pressures and the orthobaric densities). It then requires only five least-squares coefficients, and ensures a qualitatively correct behavior of the P(ρ, T) surface and of its derivatives, especially about the critical point. This nonanalytic equation yields a maximum in the specific heats C(υ)(ρ, T) a t the critical point.

An updated thermodynamic description of the stable Cr-carbides was obtained by applying the same models as used for the elements within the 3rd generation of thermodynamic databases. The Einstein m ...

The peculiar chemical properties of nanoparticles are quite different from their bulk counterparts, the differences are related to their surface thermodynamic quantities, mainly depending on their particle sizes. In this paper, the relations between solid-gas surface thermodynamic quantities and particle size at constant temperature and constant volume were derived. And the theory of decomposition thermodynamics at constant temperature and constant volume was proposed and the relations of equilibrium constants and thermodynamic properties of decomposition reactions with particle size were derived respectively. Applying the theory, the method of obtaining the surface thermodynamic quantities by thermal decomposition was proposed. Experimentally, the thermal decomposition of nano-Cu 2 (OH) 2 CO 3 with different particle sizes was taken as a research system, the standard equilibrium constants, the thermodynamic properties and the heat capacity difference for decomposition of nano-Cu 2 (OH) 2 CO 3 with different sizes were obtained by calculating the partial pressure and the fugacity coefficient of each gas component by RK equation at different temperatures. Further, the solid-gas surface thermodynamic properties, surface tension and surface heat capacity were obtained. And the size dependence of surface thermodynamic quantities and decomposition thermodynamic quantities of nano-Cu 2 (OH) 2 CO 3 are consistent with the theoretical relations above.

Thermodynamic assessments for the Fe-Si-Cr and Fe-Si-Mg ternary systems were conducted based on the critically evaluated and optimized thermodynamic and phase diagram data in the literature. The Gibbs energy of the liquid phase was described using the modified quasi-chemical model in pair approximation. The obtained thermodynamic descriptions of the Fe-Si-Cr and Fe-Si-Mg systems can be used to calculate any sections of the phase diagrams and thermodynamic properties of these two systems with high accuracy from room temperature to above the melting temperature.

Development of thermodynamic property databases and thermodynamic modeling algorithms require thermodynamic functions of substances presented in a functional form. In this paper we consider substances in the gaseous state only. The most known methods for approximating dependences of the thermodynamic functions on temperature are overviewed. An algorithm is proposed to fit the heat capacity with polynomials splitting the temperature range into intervals where the interval number and boundaries are optimized with respect to a given maximum approximation error. This algorithm is used in the IVTANTHERMO project and the corresponding thermodynamic modeling code.

A thermodynamic analysis of the phase relations in the Na Sb Sn system was undertaken by applying the CALPHAD method and first principle calculations. The CALPHAD method was used for the optimization of the thermodynamic parameters of the binary and ternary phases, as well as for the calculation and prediction of the phase relations in the systems. The electronic structure calculations were performed using density functional theory to get a measure of formation enthalpies of binary and ternary compounds. New thermodynamic assessments are provided for the Na Sb and Na Sn systems. The formation of new compounds, NaSbSn, Na 3 Sb 3 Sn, and NaSb 2 Sn 3 , in the Na Sb Sn system was proposed.

Does non-transitivity in information theory have an analog in thermodynamics? A non-transitive game, “Swap”, is used as a toy thermodynamic model to explore concepts such as temperature, heat flow, equilibrium and entropy. These concepts, found to be inadequate for non-transitive thermodynamic, need to be generalized. Two kinds of temperatures, statistical and kinetic, are distinguished. Statistical temperature is a parameter in statistical distributions. Kinetic temperature is proportional to the expected kinetic energy based on its distribution. Identical for Maxwell-Boltzmann statistics, these temperatures differ in non-Maxwellian statistics when a force is present. Fourier’s law of conduction and entropy should be expressed using statistical temperature, not kinetic temperature. Kinetic temperature is always scalar but statistical temperature and statistical entropy in non-transitive systems have circulation, thereby allowing continuous and circular heat flow. Entropy is relative to underlying statistics, in analogy to the Kullback-Leibler divergence in information theory. The H-theorem, limited by assumptions of homogeneity and indistinguishability, only covers statistically homogeneous systems. The theorem does not cover non-transitive, statistically heterogeneous systems combining different distributions such as Maxwell-Boltzmann, biased half-Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein. The second law can be preserved if generalized by expressing it in terms of statistical temperature and statistical entropy.

The dynamic and thermodynamic properties of nanoclusters are studied in two different environments: the canonical and microcanonical ensembles. A comparison is made to thermodynamic properties of the bulk. It is shown that consistent and reproducible results on nanoclusters can only be obtained in the canonical ensemble. Nanoclusters in the microcanonical ensemble are trapped systems, and inconsistencies will be found if thermodynamic formalism is applied. An analytical model is given for the energy dependence of the phase space volume of nanoclusters, which allows the prediction of both dynamical and thermodynamical properties.

The thermodynamic properties of cesium oxides were calculated by combining ab initio calculations at 0 K and a quasi-harmonic statistical thermodynamic model to determine the temperature dependency of the thermodynamic properties. In a second approach, the CALPHAD method was used to derive a model describing the Gibbs energy for all the cesium oxide compounds and the liquid phase of the cesium–oxygen system. For this approach, available experimental data in the literature was reviewed and it was concluded that only experimental thermodynamic data for Cs2O are reliable. All these data together with the thermodynamic data calculated by combining ab initio and the statistical model were used to assess the Gibbs energy of all the phases of the cesium–oxygen system. A consistent thermodynamic model was obtained. The variation of the relative stability of the different oxides is discussed using structural and bond data for the oxides investigated by ab initio calculations. This work suggests that the melting point for Cs2O2 reported in the literature (863 K) is probably overestimated and should be re-measured.

We review the concept of nonequilibrium thermodynamic entropy and observables and internal variables as state variables, introduced recently by us, and provide a simple first principle derivation of additive statistical entropy, applicable to all nonequilibrium states by treating thermodynamics as an experimental science. We establish their numerical equivalence in several cases, which includes the most important case when the thermodynamic entropy is a state function. We discuss various interesting aspects of the two entropies and show that the number of microstates in the Boltzmann entropy includes all possible microstates of non-zero probabilities even if the system is trapped in a disjoint component of the microstate space. We show that negative thermodynamic entropy can appear from nonnegative statistical entropy.

Tin based alloys with addition of indium and copper are well discussed from the point of phase diagrams, but there are still properties which are not measured or calculated. Therefore, in this paper some calculations were performed in order to obtain the thermodynamic properties by prediction methods. The integral and partial molar excess Gibbs energies, as well as activities of tin in temperature range 873 - 1673 K were obtained for four sections In:Cu=1:1, 1:4, 4:1 and 9:1 in the Cu-In-Sn system.

An alternative definition of hardness (called the thermodynamic hardness) within the grand canonical ensemble formalism is proposed in terms of the partial derivative of the electronic chemical potential with respect to the thermodynamic chemical potential of the reservoir, keeping the temperature and the external potential constant. This temperature dependent definition may be interpreted as a measure of the propensity of a system to go through a charge transfer process when it interacts with other species, and thus it keeps the philosophy of the original definition. When the derivative is expressed in terms of the three-state ensemble model, in the regime of low temperatures and up to temperatures of chemical interest, one finds that for zero fractional charge, the thermodynamic hardness is proportional to T-1(I-A), where I is the first ionization potential, A is the electron affinity, and T is the temperature. However, the thermodynamic hardness is nearly zero when the fractional charge is different from zero. Thus, through the present definition, one avoids the presence of the Dirac delta function. We show that the chemical hardness defined in this way provides meaningful and discernible information about the hardness properties of a chemical species exhibiting integer or a fractional average number of electrons, and this analysis allowed us to establish a link between the maximum possible value of the hardness here defined, with the minimum softness principle, showing that both principles are related to minimum fractional charge and maximum stability conditions.

Thermodynamic systems can be defined as composed by many identical interacting subsystems. Here it is shown how the dynamics of relaxation toward equilibrium of a thermodynamic system is closely related to the symmetry group of the Hamiltonian of the subsystems of which it is composed. The transitions between states driven by the interactions between identical subsystems correspond to elements of the root system associated to the symmetry group of their Hamiltonian. This imposes constraints on the relaxation dynamics of the complete thermodynamic system, which allow formulating its evolution toward equilibrium as a system of linear differential equations in which the variables are the thermodynamic forces of the system. The trajectory of a thermodynamic system in the space of thermodynamic forces corresponds to the negative gradient of a potential function, which depends on the symmetry group of the Hamiltonian of the individual interacting subsystems.

The second law of thermodynamics is an example of the fundamental laws that govern our universe and is relevant to every branch of science exploring the physical world. This reference summarizes knowledge and concepts about the second law of thermodynamics and entropy. A verbatim explanation of chemical thermodynamics is presented by the author, making this text easy to understand for chemistry students, researchers, non-experts, and educators.

The basic philosophy of statistical representations of microscopic systems stems from the works of Boltzmann and Maxwell. Statistical mechanics is the bridge between the macroscopic properties of matter and the microscopic states of the system. The concepts of phase space and distribution functions provide the scaffold upon which the fundamental structure of statistical mechanics is built. The Liouville Theorem, the Poincare Recurrence Theorem, and the Ergodic Hypothesis validate the mathematical nature of distribution functions and the use of ensembles to calculate the thermodynamic properties of the microscopic systems. The underlying principles of statistical mechanics are introduced through an analysis of casino games that serve as macroscopic analogues to microscopic processes. Three ensembles and their corresponding partition functions are introduced as follows: the microcanonical ensemble for systems of constant number of particles (N), volume (V), and energy (E); the canonical ensemble for systems of constant N, V, and temperature (T); and the grand canonical ensemble for the description of systems of constant chemical potential (μ), V, and T. The equations that relate the partition functions to thermodynamic variables are derived.

Perhaps one of the most interesting and fruitful applications of ergodic optimization theory occurs in the study of freezing phenomena in equilibrium statistical mechanics. In this concluding chapter, we provide a first glimpse of such a rich interaction among theories, by scrutinizing with basic techniques the convergence of equilibrium states to a particular maximizing probability on certain examples.

Based on Virial Equation of State (V-EOS), a thermodynamic model was developed in this study. The applicability and performance of proposed modified-Virial EOS (mV-EOS) were assessed for calculation of thermodynamic properties of hydrogen and then extended to the hydrogen containing mixtures. For comparison purpose, a number of cubic equations of state were used. Four statistical parameters were defined as goodness criteria which can be regarded as Objective Functions (OF) introduced to optimization method i.e. Black Hole Method (BH). According to obtained results, the model was able to predict the studied thermodynamic properties with desirable accuracy.

Thermodynamic modeling of the Sn−Sr binary system and the Mg−Sn−Sr ternary system is carried out by means of the CALPHAD approach in combination with available experimental data in the literature and first-principles calculations in the present work. The finite temperature thermodynamic properties of the compounds in the Sn−Sr system and the MgSnSr ternary compound are predicted using the quasi-harmonic phonon calculations and the Debye model with inputs from first-principles calculations. The associate solution model is used to describe the short-range ordering behavior in the liquid phase. Combined with the previously modeled Mg−Sn and Mg−Sr binary systems in the literature, the thermodynamic modeling of the Mg−Sn−Sr system is performed.

All phase diagram and thermodynamic data of the MgO–MnO and MgO–MnO–MnO2 systems were critically evaluated and thermodynamically optimized to obtain one set of thermodynamic model parameters. The MgMn2O4–Mn3O4 spinel solutions were modeled with two sublattices Compound Energy Formalism by considering the cation distribution between tetrahedral and octahedral sites. The slag phase and monoxide solid solution were also described using the Modified Quasichemical Model and Bragg–Williams random mixing model, respectively. The optimized thermodynamic model parameters can be used with a Gibbs energy minimization routine to calculate the phase diagram and thermodynamic properties of the system at oxygen partial pressures between metal saturation and 1 atm.

We investigate the electronic, dynamic and thermodynamic properties of α-MnO2 using first-principles calculations based on density functional theory (DFT) with the GGA+U method. The results of electronic structures show that α-MnO2is a semiconductor with a direct band gap of 1.4 eV at Γ point. The results of dynamic properties indicate that the structure of α-MnO2 is dynamically unstable at ground-state. Several important thermodynamic quantities, such as entropy, enthalpy and Gibbs free energy, et al each as a function of temperature were presented.

We propose a new statistical theory for classical and quantum small systems. It is a generalized scheme of the Boltzmann–Gibbs statistical theory by extending the Boltzmann–Gibbs statistical factor from infinite systems to finite systems based on the microcanonical ensemble distribution function and keeping this factor in all thermodynamic processes. We reconstruct the statistical theory for finite systems by obtaining the expression of the average particle number and the thermodynamic quantities such as entropy and specific heat, in the finite systems. We also explore the discontinuous phase transitions in the interacting classical nanoscale gases without the thermodynamic limit.

The thermodynamic description of the Cu–Fe–Ni system has been updated considering the newly available experimental data, as well as compatibility of the present modeling with those used for the Cu and Fe systems. All of the experimental data available in the literature have been critically reviewed, and the inconsistent information has been excluded. The thermodynamic parameters have been evaluated in order to properly describe the thermodynamic properties of the liquid phase and miscibility gap in the solid state. A significant improvement in comparison with the previous thermodynamic descriptions has been achieved. Additionally, for the ordered L1 2 phase the four-sublattice model has been adopted to predict the ternary phase equilibria involving this phase. A set of thermodynamic parameters for the phases is given.

All known phases of the chromium-iron-sulfur ternary system are taken into consideration within a thermodynamic analysis of phase equilibria and thermodynamic data at 1 bar total pressure over the entire composition range between 298.15 K (25 °C) and temperatures greater than the liquidus. The modeling is based on recent evaluations of the Fe-S and Cr-S binary subsystems. The extended modified quasi-chemical model is applied for the liquid chromium-iron-sulfur phase. The Gibbs energy of solid solution phases as high-temperature chromium-iron pyrrhotite and the thiospinel Daubreelite is described by sublattice models within the framework of the compound energy formalism. Analysis of complex phase relations is now possible on the basis of a consistent thermodynamic description of the system. The thermodynamic modeling presented can be combined with models of other metal-sulfur systems to develop a thermodynamic multicomponent/multiphase database.

The type of physicochemical model dictates the choice of a minimized thermodynamic potential that provides the most adequate description of specific features of a natural or technological process. In the developed approach, the Gibbs energy was selected as the basis potential, and other thermodynamic potentials were considered as functions of G(T, P) and thermodynamic parameters with inclusion of independent factors of state. The presented thermodynamic functions of individual substances can be used in minimization of all basic thermodynamic potentials regardless of the conditions of existence of modeled systems.

The development of reliable methods for estimating microcanonical averages constitutes an important issue in statistical mechanics. One possibility consists of calculating a given microcanonical quantity by means of typical relations in the grand-canonical ensemble. But given that distinct ensembles are equivalent only at the thermodynamic limit, a natural question is if finite size effects would prevent such a procedure. In this work we investigate thoroughly this query in different systems yielding first- and second-order phase transitions. Our study is carried out from the direct comparison with the thermodynamic relation ( ∂ s ∂ e ) , where the entropy s is obtained from the density of states and e is the energy per site. A systematic analysis for finite sizes is undertaken. We find that, although results become inequivalent for extremely low system sizes, the equivalence holds true for rather small L ’s. Therefore direct, simple (when compared with other well established approaches) and very precise microcanonical quantities can be obtained from the proposed method.

Based on the property of extensivity (mathematically, homogeneity of first degree), we derive in a mathematically consistent manner the explicit expressions of the chemical potential and the Clausius entropy $S$ for the case of monoatomic ideal gases in open systems within phenomenological thermodynamics. Neither information theoretic nor quantum mechanical statistical concepts are invoked in this derivation. Considering a specific expression of the constant term of $S$, the derived entropy coincides with the Sackur-Tetrode entropy in the thermodynamic limit. We demonstrate however, that the former limit is not contained in the classical thermodynamic relations, implying that the usual resolutions of Gibbs paradox do not succeed in bridging the gap between the thermodynamics and statistical mechanics. We finally consider the volume of the phase space as an entropic measure, albeit, without invoking the thermodynamic limit to investigate its relation to the thermodynamic equation of state and observables.

Thermodynamic assessments of the La-B and La-Bi systems were carried out based on the available experimental data including thermodynamic properties and phase equilibria by means of the CALPHAD method. The enthalpies of formation at 0 K for the LaB4, LaB6, La2Bi, La5Bi3, La4Bi3 and LaBi were computed via first-principles calculations. These enthalpies of formation were used “key experimental data” in the CALPHAD approach in order to obtain the thermodynamic parameters with sound physical meaning. The liquid phase has been modeled with both the substitutional solution model and associate model, and two sets of self-consistent thermodynamic parameters for the La-B and La-Bi systems were finally obtained. The calculated phase diagram and thermodynamic properties agree reasonably with the literature experimental data and the present first-principles calculations.

The chapter describes the handling of the mathematical relations previously found within the framework of the canonical ensemble through the partition function in order to assimilate them to the expressions of classical thermodynamic functions. Concerning now the introduction of the nonmechanical functions such as the entropy and the temperature into the realm of statistical thermodynamics, the strategy consists in comparing the expressions concerning the mechanical quantities obtained (thanks to the theory of the canonical ensemble) and those stemming from classical thermodynamics. Therefore, the statistical expressions of internal energy, entropy, pressure, and chemical potential are obtained. Some of these functions are calculated with the aid of the characteristic function of the canonical function which spontaneously introduces itself into the calculations.

In this paper, we analyze the different lift of the dynamics of irreversible thermodynamic systems from the Legendre submanifold associated with the thermodynamic properties of the physical system to the full Thermodynamic Phase Space. Firstly, we define a set of Hamiltonian functions which generate a class of equivalent lifts, that it contact vector fields which are equal on some distinguished Legendre submanifold. Secondly we show that how one may construct an equivalent contact vector field which renders attractive of the inverse image of zero by the Hamiltonian function.

The basic principles and concepts of thermodynamics and statistical physics used in the following chapters are presented in Chapter 2 . The mathematical formalism relevant to describe thermodynamic systems is considered in detail. The microcanonical, canonical, isothermal–isobaric, and grand canonical ensembles that are a base of equilibrium statistical physics are described. The criteria of macroscopicity necessary to estimate a size of three-, two-, and one-dimensional systems admissible to application of the thermodynamics and statistical physics methods are derived. The equivalence of the thermodynamic quantities of multicomponent alloys obtained by using the microcanonical, canonical, and isothermal–isobaric ensembles is shown. The possibility to use the Helmholtz free energy instead of the Gibbs free energy and vice versa, due to the insignificant difference between them, to consider condensed matter (solids and liquids) is demonstrated. The separated degrees of freedom in the consideration of semiconductor alloys are described.

In this paper, we present a novel thermodynamic framework for graphs that can be used to analyze time evolving networks, relating the thermodynamics variables to macroscopic changes in network topology, and linking major structural transition to phase changes in the thermodynamic picture. We start from a recent quantum-mechanical characterization of the structure of a network relating the graph Laplacian to a density operator and resulting in a characterization of the network's entropy. Then we adopt a Schrodinger picture of the dynamics of the network, resulting in an estimation of a hidden time-varying Hamiltonian from the data, from which we derive a measure of Energy exchange. From these variables, using the thermodynamic identity, we obtain temperature under the assumption of constant volume of the system. Evaluation of real-world data shows that the thermodynamic variables thus extracted are effective in detecting critical events occurring during network evolution.

A generalized methodology for statistical analogues of derivatives of the fundamental equation of state in the microcanonical and canonical ensembles is suggested. Those analogues include rigorous measuring prescriptions for any thermodynamic property in molecular NVE and NVT simulation. In particular, optimal data sets for exhaustive thermodynamic data correlation become rigorously accessible.

Partition function and thermodynamic parameters of the system open with respect to energy E, volume V and number of particles N, whereas temperature T, chemical potential μ and pressure P were kept constant, were considered by the procedure proposed by Gibbs. The mean values of variable energy, volume and number of particles were obtained by universal expression for such values. The entropy was determined by its general statistical definition. The thermodynamic potential and other thermodynamic quantities were also turned out naturally.

A gas relaxing into equilibrium is often taken to be a process in which a system moves from an “improbable” to a “probable” state. Given that the thermodynamic entropy increases during such a process, it is natural to conjecture that the thermodynamic entropy is a measure of the probability of a macrostate. For nonideal classical gases, however, I claim that there is no clear sense in which the thermodynamic entropy of a macrostate measures its probability. We must therefore reject the idea that (in classical mechanics) thermodynamic entropy and probability are connected in a deep and general way.

Based on the CALPHAD approach, thermodynamic properties and phase diagram of the Am–Pu system are assessed with input from results of ab initio electronic-structure calculations within the framework of density-functional theory. Despite the limited availability of experimental data on this binary system, thermodynamic assessment has been performed and two phase diagrams are proposed that make distinction at high temperature between the relative stability of the bcc phase and the liquid state. Our investigation provides guidance and motivates further experimental studies.

With invoking analytical formulae in number theory and numerical calculations, we calculate the number of microstates in microcanonical ensemble for free particles in a single harmonic trap which in whole space defines a thermodynamic system but not a spatially homogeneous one. Once the number of excitation quanta m is larger than the square of the particle number N2 as m⪰O(N2) when N≫1, the number of microcanonical microstates for an ideal, harmonically trapped Bose or Fermi gas gradually converge to the Boltzmann microcanonical microstates for the classical particles with a proper consideration of the indistinguishability.

The experimental phase diagram data of the C–Cr–Ta ternary system available in the literature were critically reviewed. A thermodynamic modeling of the ternary system was then conducted by taking into account the literature data including the isothermal sections and invariant reactions. A set of self-consistent thermodynamic parameters for the Gibbs energies of individual phases in the C–Cr–Ta system was obtained by using the CALPHAD approach. Comprehensive comparisons between the calculated and measured phase diagrams show that all the reliable experimental information is satisfactorily accounted for by the present thermodynamic description.

The thermodynamic constraints to eliminate artificial phase relations were introduced with the Cu–Nd system as an example. The enthalpies of formation of the compounds Cu6Nd, Cu5Nd, Cu2Nd and αCuNd are calculated using density functional theory. Taking into account all the experimental data and the first-principles calculated enthalpies of formation of these compounds, the thermodynamic optimization of the Cu–Nd system was performed under the proposed thermodynamic constraints. It is demonstrated that the thermodynamic constraints are critical to obtain a set of thermodynamic parameters for the Cu–Nd system, which can avoid the appearance of all the artificial phase relations.

This paper purpose is to re-calculate thermodynamic or chemical–physical properties according to thermodynamic temperature T, where measured temperature values were obtained according to the ITS-90, T90 (or the reverse). The main aims of the calculations are: to remove the discontinuity of the first derivative at 273.16 K that is intrinsic in the ITS-90 definition; to use a single cubic spline to model the database of the available experimental data over the whole temperature range (2.37 to 1235) K, while maximising the medium-range smoothness of the correction function; to compute corrections using only local polynomial coefficients.

The crystal and electronic structure of TiH2 was calculated by the first-principle plane-wave density function theory pseudopotential method. Through the quasi-harmonic Debye model, the thermodynamic properties of TiH2 was successfully obtained in the whole temperature range from 0 to 1800 K and pressure range from 0 to 140 GPa. The thermodynamic properties including the relationship of elastic bulk modulus and pressure, heat capacities and temperature, thermal expansion and temperature and pressure. The calculated results indicated the lattice parameters are consistent with the experimental data, and when the temperature is constant, the relative volumes and heat capacities decreases with increasing pressure, Debye temperature and elastic bulk increase with increasing pressure, and temperature effect is smaller than pressure under high temperature and pressure.