Equations of state for CaSiO3 phases based on the Helmholtz free energy

Phase transitions are introduced using the van der Waals gas as an example. The equations of state are derived from the Helmholtz free energy of the ideal gas. The behavior of this model is analyzed, and an instability leads to a liquid-gas phase transition. The Maxwell construction for the pressure at which a phase transition occurs is derived. The effect of phase transition on the Gibbs free energy and Helmholtz free energy is shown. Latent heat is defined, and the Clausius–Clapeyron equation is derived. Gibbs' phase rule is derived and illustrated.

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VAPOR–LIQUID EQUILIBRIUM MODELING FOR MIXTURES OF HFC-32 + ISOBUTANE AND HFC-32 + HFO-1234ze(E)

International Journal of Air-Conditioning and Refrigeration ◽

10.1142/s2010132511000478 ◽

2011 ◽

Vol 19 (02) ◽

pp. 93-97 ◽

Cited By ~ 10

Author(s):

RYO AKASAKA

Keyword(s):

Free Energy ◽

Helmholtz Free Energy ◽

Equations Of State ◽

Dew Point ◽

Working Fluid ◽

Liquid Equilibrium ◽

Vapor Liquid Equilibrium ◽

Equilibrium Modeling ◽

Independent Variables ◽

Vapor Liquid

Vapor–liquid equilibrium (VLE) have been successfully modeled for the binary mixtures of difluoromethane (HFC-32) + isobutane and difluoromethane + trans-1,3,3,3-tetrafluoropropene (HFO-1234ze(E)). These mixtures are considered as possible replacements for conventional refrigerants far from negligible global warming potential (GWP). A multifluid approach explicit in the Helmholtz free energy forms the basis of the model. The independent variables are the temperature, density, and composition. Accurate published equations of state for pure HFC-32, isobutane, and HFO-1234ze(E) are incorporated to calculate the Helmholtz free energy of each component. Typical uncertainties of bubble- and dew-point pressures calculated using the model are within 2%. Although adjustable parameters of the model are determined only from experimental VLE data, it is highly probable that the model reasonably predicts other thermodynamic properties such as enthalpy and heat capacities. Therefore, the model allows practical design and simulation of refrigeration systems using the mixtures as a working fluid.

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A canonical ensemble derivation of the McMillan-Mayer solution theory

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19832888 ◽

1983 ◽

Vol 48 (10) ◽

pp. 2888-2892 ◽

Cited By ~ 2

Author(s):

Vilém Kodýtek

Keyword(s):

Free Energy ◽

Partition Function ◽

Energy Function ◽

Special Function ◽

Helmholtz Free Energy ◽

Canonical Ensemble ◽

Free Energy Function ◽

Solution Theory ◽

Pure Solvent ◽

The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

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Free Energy of Metals from Quasi-Harmonic Models of Thermal Disorder

Metals ◽

10.3390/met11020195 ◽

2021 ◽

Vol 11 (2) ◽

pp. 195

Author(s):

Pavel A. Korzhavyi ◽

Jing Zhang

Keyword(s):

Free Energy ◽

First Principles ◽

Density Functional ◽

Helmholtz Free Energy ◽

Complex Materials ◽

Predictive Capability ◽

Temperature Dependent ◽

Disordered Alloys ◽

Thermal Disorder ◽

Structure Calculations

A simple modeling method to extend first-principles electronic structure calculations to finite temperatures is presented. The method is applicable to crystalline solids exhibiting complex thermal disorder and employs quasi-harmonic models to represent the vibrational and magnetic free energy contributions. The main outcome is the Helmholtz free energy, calculated as a function of volume and temperature, from which the other related thermophysical properties (such as temperature-dependent lattice and elastic constants) can be derived. Our test calculations for Fe, Ni, Ti, and W metals in the paramagnetic state at temperatures of up to 1600 K show that the predictive capability of the quasi-harmonic modeling approach is mainly limited by the electron density functional approximation used and, in the second place, by the neglect of higher-order anharmonic effects. The developed methodology is equally applicable to disordered alloys and ordered compounds and can therefore be useful in modeling realistically complex materials.

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Thermodynamics Energy for Both Supervised and Unsupervised Learning Neural Nets at a Constant Temperature

International Journal of Neural Systems ◽

10.1142/s0129065799000162 ◽

1999 ◽

Vol 09 (03) ◽

pp. 175-186 ◽

Cited By ~ 6

Author(s):

HAROLD SZU

Keyword(s):

Free Energy ◽

Constant Temperature ◽

Helmholtz Free Energy ◽

Principal Component ◽

Neural Nets ◽

Mixing Condition ◽

Smart Cameras ◽

Minimization Principle ◽

Output Entropy ◽

Artificial Neural Network Ann

Unified Lyaponov function is given for the first time to prove the learning methodologies convergence of artificial neural network (ANN), both supervised and unsupervised, from the viewpoint of the minimization of the Helmholtz free energy at the constant temperature. Early in 1982, Hopfield has proven the supervised learning by the energy minimization principle. Recently in 1996, Bell & Sejnowski has algorithmically demonstrated. Independent Component Analyses (ICA) generalizing the Principal Component Analyses (PCA) that the continuing reduction of early vision redundancy happens towards the "sparse edge maps" by maximization of the ANN output entropy. We explore the combination of both as Lyaponov function of which the proven convergence gives both learning methodologies. The unification is possible because of the thermodynamics Helmholtz free energy at a constant temperature. The blind de-mixing condition for more than two objects using two sensor measurement. We design two smart cameras with short term working memory to do better image de-mixing of more than two objects. We consider channel communication application that we can efficiently mix four images using matrices [AO] and [Al] to send through two channels.

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Anharmonic Self-Consistent Theory of Crystals. II. The 2d and 3d Quartic Crystal Models

Zeitschrift für Naturforschung A ◽

10.1515/zna-1994-0602 ◽

1994 ◽

Vol 49 (6) ◽

pp. 663-670

Author(s):

S. Sh. Soulayman ◽

C. Ch. Marti ◽

Ch. Ch. Guilpin

Keyword(s):

Helmholtz Free Energy ◽

Equations Of State ◽

Three Dimensional ◽

Inert Gases ◽

Thermoelastic Properties ◽

Consistent Theory ◽

Jones Potential ◽

Lennard Jones ◽

2D And 3D ◽

Strong Anharmonicity

Abstract In this paper we apply the method developed in part I for describing the crystalline state of two and three dimensional inert gases. For strong anharmonicity of fourth order, the equations of state of these gases are obtained. This way we calculate the thermoelastic properties of two and three dimensional argon, krypton and xenon using the Lennard-Jones potential. The corrections to the Helmholtz free energy and thermodynamic properties due to quantum effects are considered. The results are compared with the available experimental data.

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Thermodynamics of Bidisperse Ferrofluids in Zero External Magnetic Field: Theory and Simulations

Solid State Phenomena ◽

10.4028/www.scientific.net/ssp.233-234.331 ◽

2015 ◽

Vol 233-234 ◽

pp. 331-334

Author(s):

Anna Yu. Solovyova ◽

Ekaterina A. Elfimova

Keyword(s):

Magnetic Field ◽

Field Theory ◽

Free Energy ◽

External Magnetic Field ◽

Volume Concentration ◽

Helmholtz Free Energy ◽

Hard Spheres ◽

Particle Volume ◽

Simulation Data ◽

Theoretical Predictions

The thermodynamic properties of a ferrofluid modeled by a bidisperse system of dipolar hard spheres in the absence of external magnetic field are investigated using theory and simulations. The theory is based on the virial expansion of the Helmholtz free energy in terms of particle volume concentration. Comparison between the theoretical predictions and simulation data shows a great agreement of the results.

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Hamiltonian Thermodynamics

Nelineinaya Dinamika ◽

10.20537/nd200403 ◽

2020 ◽

Vol 16 (4) ◽

pp. 557-580

Author(s):

S.A. Rashkovskiy ◽

Keyword(s):

Free Energy ◽

Hamiltonian Systems ◽

Thermal Energy ◽

Helmholtz Free Energy ◽

Random Processes ◽

Carnot Cycle ◽

Thermodynamic Process ◽

Thermodynamic Cycles ◽

Thermodynamic State ◽

Laws Of Thermodynamics

It is believed that thermodynamic laws are associated with random processes occurring in the system and, therefore, deterministic mechanical systems cannot be described within the framework of the thermodynamic approach. In this paper, we show that thermodynamics (or, more precisely, a thermodynamically-like description) can be constructed even for deterministic Hamiltonian systems, for example, systems with only one degree of freedom. We show that for such systems it is possible to introduce analogs of thermal energy, temperature, entropy, Helmholtz free energy, etc., which are related to each other by the usual thermodynamic relations. For the Hamiltonian systems considered, the first and second laws of thermodynamics are rigorously derived, which have the same form as in ordinary (molecular) thermodynamics. It is shown that for Hamiltonian systems it is possible to introduce the concepts of a thermodynamic state, a thermodynamic process, and thermodynamic cycles, in particular, the Carnot cycle, which are described by the same relations as their usual thermodynamic analogs.

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