Entropy Of Ideal Gas Research Articles

Entropy approximations for simple fluids.

Liquid state entropy formulas based on configurational probability distributions are examined for Lennard-Jones fluids across a range temperatures and densities. These formulas are based on expansions of the entropy in a series of n-body distribution functions. We focus on two special cases. One, which we term the "perfect gas" series, starts with the entropy of an ideal gas; the other, which we term the "dense liquid" series, removes a many-body contribution from the ideal gas entropy and reallocates it among the subsequent n-body terms. We show that the perfect gas series is most accurate at low density, while the dense liquid series is most accurate at high density. We propose empirical interpolation methods that are capable of connecting the two series and giving consistent predictions in most situations.

Why Thermodynamic Entropy and Statistical Entropy are Two Different Physical Quantities

Background: Historically, the coincidence of classical thermodynamic entropy and statistical entropy is based on an assumption (the so-called Boltzmann-Planck relation). The study shows that the Boltzmann-Planck relation is not valid in general. Aims: The study demonstrates that thermodynamic entropy is a physical entity distinct from statistical entropy. Methods: Initially, the entropy of ideal gases is examined and demonstrated to be independent of the gas volume. The ideal gas's true entropy is then expressed correctly. Results: The Boltzmann-Planck entropy equation is demonstrated to be incapable of describing the thermodynamic entropy of ideal gases. Conclusion: In general, the Boltzmann-Planck entropy cannot be considered a statistical interpretation of the thermodynamic entropy. The two entropies represent two different physical quantities. The physical quantity that enters the second law of thermodynamics is the thermodynamic entropy, not the statistical entropy. 6: The study advocates for a fundamental rethinking of today's statistical understanding of thermodynamics.

Entropy Scaling of Molecular Dynamics in a Prototypical Anisotropic Model near the Glass Transition.

Dynamics and thermodynamics of molecular systems in the vicinity of the boundary between thermodynamically nonequilibrium glassy and metastable supercooled liquid states are still incompletely explored and their theoretical and simulation models are imperfect despite many previous efforts. Among them, the role of total system entropy, configurational entropy, and excess entropy in the temperature-pressure or temperature-density dependence of global molecular dynamics (MD) timescale relevant to the glass transition is an essential topic intensively studied for over half of a century. By exploiting a well-known simple ellipsoidal model recently successfully applied to simulate the supercooled liquid state and the glass transition, we gain a new insight into the different views on the relationship between entropy and relaxation dynamics of glass formers, showing the molecular grounds for the entropy scaling of global MD timescale. Our simulations in the anisotropic model of supercooled liquid, which involves only translational and rotational degrees of freedom, give evidence that the total system entropy is sufficient to scale global MD timescale. It complies with the scaling effect on relaxation dynamics exerted by the configurational entropy defined as the total entropy diminished by vibrational contributions, which was earlier discovered for measurement data collected near the glass transition. Moreover, we argue that such a scaling behavior is not possible to achieve by using the excess entropy that is in excess of the ideal gas entropy, which is contrary to the results earlier suggested within the framework of simple isotropic models of supercooled liquids. Thus, our findings also warn against an excessive reliance on isotropic models in theoretical interpretations of molecular phenomena, despite their simplicity and popularity, because they may reflect improperly various physicochemical properties of glass formers.

Vapor pressures and thermophysical properties of selected monoterpenoids II

In this work, a thermodynamic study of four enantiopure terpenoids, myrtenal, myrtenol, α-terpineol, and trans-sobrerol is presented. Vapor pressures of the selected terpenoids were determined using two static apparatus in a combined temperature range from 248 to 363 K. Heat capacities of the samples were determined by Tian-Calvet calorimetry in the temperature range from 261 to 353 K. Phase behaviour (melting temperature, fusion enthalpy, crystallization temperature, and potential polymorphic transformations) were studied using heat-flux calorimetry. Using a combination of quantum-chemical calculations and statistical thermodynamics, ideal-gas heat capacities and entropies were determined. Finally, consistency of the phase behaviour, vapor pressure, and heat capacity experimental data with the theoretically calculated data was assessed using the method of simultaneous correlation.

Partitioning Entropy with Action Mechanics: Predicting Chemical Reaction Rates and Gaseous Equilibria of Reactions of Hydrogen from Molecular Properties.

Our intention is to provide easy methods for estimating entropy and chemical potentials for gas phase reactions. Clausius’ virial theorem set a basis for relating kinetic energy in a body of independent material particles to its potential energy, pointing to their complementary role with respect to the second law of maximum entropy. Based on this partitioning of thermal energy as sensible heat and also as a latent heat or field potential energy, in action mechanics we express the entropy of ideal gases as a capacity factor for enthalpy plus the configurational work to sustain the relative translational, rotational, and vibrational action. This yields algorithms for estimating chemical reaction rates and positions of equilibrium. All properties of state including entropy, work potential as Helmholtz and Gibbs energies, and activated transition state reaction rates can be estimated, using easily accessible molecular properties, such as atomic weights, bond lengths, moments of inertia, and vibrational frequencies. We conclude that the large molecular size of many enzymes may catalyze reaction rates because of their large radial inertia as colloidal particles, maximising action states by impulsive collisions. Understanding how Clausius’ virial theorem justifies partitioning between thermal and statistical properties of entropy, yielding a more complete view of the second law’s evolutionary nature and the principle of maximum entropy. The ease of performing these operations is illustrated with three important chemical gas phase reactions: the reversible dissociation of hydrogen molecules, lysis of water to hydrogen and oxygen, and the reversible formation of ammonia from nitrogen and hydrogen. Employing the ergal also introduced by Clausius to define the reversible internal work overcoming molecular interactions plus the configurational work of change in Gibbs energy, often neglected; this may provide a practical guide for managing industrial processes and risk in climate change at the global scale. The concepts developed should also have value as novel methods for the instruction of senior students.

Nitrogen Dissolution in Liquid Ga and Fe: Comprehensive Ab Initio Analysis, Relevance for Crystallization of GaN.

The dissolution of molecular nitrogen in Ga and Fe was investigated by ab initio calculations and some complementary experiments. It was found that the N bonding inside these solvents is fundamentally different. For Ga, it is between Ga4s and Ga4p and N2p states whereas for Fe this is by N2p to Fe4s, Fe4p and Fe3d states. Accordingly, the energy of dissolution of N for arbitrarily chosen starting atomic configurations was 0.535 eV/mol and −0.299 eV/mol for Ga and Fe, respectively. For configurations optimized with molecular dynamics, the difference between the corresponding energy values, 1.107 eV/mol and 0.003 eV/mol, was similarly large. Full thermodynamic analysis of chemical potential was made employing entropy-derived terms in a Debye picture. The entropy-dependent terms were obtained via a normal conditions path to avoid singularity of ideal gas entropy at zero K. Nitrogen solubility as a function of temperature and N pressure was evaluated, being much higher for Fe than for Ga. For K and bar, the N concentration in Ga was at. fr. whereas for Fe, it was at. fr. in very good agreement with experimental data. It indicates that liquid Fe could be a prospective solvent for GaN crystallization from metallic solutions.

Scale-Invariant Model of Boltzmann Statistical Mechanics and Generalized Thermodynamics

Some implications of a scale-invariant model of Boltzmann statistical mechanics to the laws of generalized thermodynamics are investigated. Through definition of stochastic Planck and Boltzmann universal constants, dimension of Kelvin absolute temperature T (degrees kelvin) is identified as a length (meters) associated with Wien wavelength Tβ = λwβ of particle thermal oscillations. Hence, thermodynamic temperature and atomic mass of the field 𝔽β at scale β provide internal measures of (extension, duration) of background space 𝕊β+1 = 𝔽β needed to define external space and time coordinates and atomic-mass-unit of 𝔽β+1. Introduction of invariant internal thermodynamic spacetime and Boltzmann factor are in harmony with modern concepts of quantum gravity as deterministic dissipative dynamic system [73]. The connections between de Pretto number 8338 and Joule-Mayer mechanical equivalent of heat Jc = 4.169 kJ / kcal and universal gas constant Ro = 8338 J / kcal.m are identified leading to the modified mechanical equivalent of heat J = 2Jc = 8338 J / kcal . It is shown that with work defined as Helmholtz free energy W = F = U – TS, Helmholtz decomposition of total thermal energy into free heat U and latent heat pV results in modified form of the first law of thermodynamics Q = H = U- W = U + pV. Finally, by application of Boltzmann combinatorics method, entropy of ideal gas is expressed in terms of the number of Heisenberg-Kramers virtual oscillators as S = 4Nk in exact agreement with predicted entropy of black hole by Major and Setter [159].

Prediction of the ideal-gas thermodynamic properties for water

We report a suitable analytical representation of the ideal-gas Gibbs free energy for water, and also derive an efficient analytical representation of the ideal-gas entropy from the former expression. These two analytical representations only involve six molecular constants of water, whereas the conventional computation schemes require a great many experimental spectroscopy data or calorimetric data. The predicted results for the Gibbs free energies and entropy values of water are in excellent agreement with available experimental data. The present study provides a novel pathway of treating the anharmonic vibrations of the water molecule.

On the ambiguity of an ideal gas entropy concept

Based on a critical analysis of the existing characteristics of an ideal gas and the theory of thermodynamic potentials, the article considers its new model, which includes the presence of an ideal gas in addition to kinetic energy of potential (chemical) energy, in the framework of which the isothermal and adiabatic processes in it are studied both reversible and irreversible, in terms of changes in the entropy of the system in question, observed in case. In addition, a critical analysis was made of the process of introducing the concept of entropy by R. Clausius, as a result of which the main requirements for entropy were established, the changes of which are observed in isothermal and adiabatic quasistatic processes, in particular, it was revealed that if in isothermal processes involving one in a perfect gas, the entropy ST is uniquely characterized by the value , regardless of whether the process is reversible or not, then when the adiabatic processes occur, the only requirement made of them is the condition of mutual destruction adiabats in this Carnot cycle. As a result of this circumstance, in fact, in thermodynamics various “adiabatic” entropies are used, namely; const SA = const R ln V и C V ln T , and in addition, as established in this paper, CV, which leads, despite the mathematically perfect introduction of the concept of entropy for the Carnot cycle, to the impossibility of its unambiguous interpretation and, therefore, the determination of its physicochemical meaning even for perfect gas. A new concept is introduced in the work: “total” entropy of an ideal gas SS = R ln V + C V , satisfying the criteria of R. Clausius, on the basis of which it was established that this type of entropy multiplied by the absolute temperature characterizes a certain level of potential energy of the system, which can besuccessively converted to work in an isothermal reversible process, with the supply of an appropriate amount of heat, and in the adiabatic reversible process under consideration.

Methanol: association behaviour, third-law entropy analysis and determination of the enthalpy of formation

Chemical equilibrium constants for the methanol from CO/H2 reaction calculated from literature thermodynamic parameters are too high as compared to experimental results. Therefore, both the entropy value and the enthalpy of formation value of methanol were reviewed. A rigorous analysis of the association behaviour of methanol vapour confirms that dimers and cyclic tetramers play an important role, but physical non-ideal gas behaviour must also be taken into account. Accurate relationships for the association equilibrium constants and the physical contribution to the second virial coefficient were derived with the use of a combined fit of multiple experimental data sources including heat capacity, speed of sound, thermal conductivity, excess molar enthalpy of methanol and nitrogen and heat of vaporization. Additionally, high temperature second virial coefficients and measures for the consistent temperature dependencies of the entropy and enthalpy of formation were included in the parameter optimization to support the accuracy of the model. All experimental results and supporting data were taken or derived from the literature. The resulting virial equation of state was used to calculate new ideal-gas entropy and enthalpy of formation values of methanol, which now turn out to be consistent with values derived from experimental chemical equilibrium data. Furthermore, the new third-law entropy value turns out to be consistent with the current literature value. A new enthalpy of formation value is recommended and an improved chemical equilibrium relationship for the methanol from CO/H2 reaction is presented.

Modeling hydration-mediated ion-ion interactions in electrolytes through oscillating Yukawa potentials.

Classical Poisson-Boltzmann theory represents a mean-field description of the electric double layer in the presence of only Coulomb interactions. However, aqueous solvents hydrate ions, which gives rise to additional hydration-mediated ion-ion interactions. Experimental and computational studies suggest damped oscillations to be a characteristic feature of these hydration-mediated interactions. We have therefore incorporated oscillating Yukawa potentials into the mean-field description of the electric double layer. This is accomplished by allowing the decay length of the Yukawa potential to be complex valued. Ion specificity emerges from assigning individual strengths and phases to the Yukawa potential for anion-anion, anion-cation, and cation-cation pairs as well as for anions and cations interacting with an electrode or macroion. Excluded volume interactions between ions are approximated by replacing the ideal gas entropy by that of a lattice gas. We derive mean-field equations for the Coulomb and Yukawa potentials and use their solutions to compute the differential capacitance for an isolated planar electrode and the pressure that acts between two planar, like-charged macroion surfaces. Attractive interactions appear if the surface charge density of the macroions is sufficiently small.

Density functional analysis of atomic force microscopy in a dense fluid.

We present a density functional (DF) analysis for the entropic force in Atomic Force Microscopy (AFM) across the layers of a dense fluid. Previous theoretical analysis, based on the ideal gas entropy, was apparently supported by the similarity in the oscillatory decay for the force and density profile. We point out that such similarity is a generic DF result, which carries no information on the interface, since the decaying mode is characteristic of the bulk fluid correlation. The truly interfacial information, from the layering measured by AFM, comes in its amplitude and not in the decay mode. With our rigorous study of a simple hard sphere model, we find semiempirical clues to disentangle the role of the tip radius and to relate the amplitude of the molecular layering to the oscillatory force on the AFM tip.

First-principles calculation of ideal-gas thermodynamic properties of long-chain molecules by R1SM approach-Application to n-alkanes.

First-principles calculations, coupled with statistical thermodynamics, can provide ideal-gas thermodynamic properties but get complicated and less reliable with an increasing number of conformers. An approach designed for calculation of ideal-gas thermodynamic properties of long-chain molecules, R1SM, and its simplified version, sR1SM, is tested in this work by calculation of ideal-gas heat capacities and entropies for a homologous series of n-alkanes up to n-tetradecane. The R1SM approach incorporates the rigid rotor-harmonic oscillator approximation in combination with a correction for internal rotations of methyl tops using the one-dimensional hindered rotor scheme and the mixing model accounting for the population of conformers based on the Boltzmann distribution. The R1SM approach is applicable for compounds with up to hundreds of conformers, while the simplified sR1SM approach can be used for molecules with up to 105 conformers when coupled with rules for enumeration of stable conformers and estimation scheme for their energies. The obtained results for n-alkanes are compared with experimental values and previously employed computational schemes. As the conformational behavior and conformer energies are inherent parts of the proposed approaches, a thorough conformational study of n-alkanes is performed and compared with experiments and the Tasi rules for enumeration of n-alkane conformers. Finally, the standard uncertainty of the R1SM-calculated ideal-gas thermodynamic properties is estimated based on the error propagation from the used input quantities and approximations as well as on comparison to experimental values and amounts to less than 1% for both ideal-gas heat capacity and standard ideal-gas entropy.

Thermodynamic properties of 2-methylindole: experimental and computational results for gas-phase entropy and enthalpy of formation.

Measurements leading to the calculation of thermodynamic properties in the ideal-gas state for 2-methylindole (Chemical Abstracts registry number [95-20-5]) are reported. Experimental methods were adiabatic heat-capacity calorimetry, differential scanning calorimetry (d.s.c.), comparative ebulliometry, inclined-piston manometry, and oxygen bomb calorimetry. The critical temperature of 2-methylindole was determined experimentally with d.s.c. Molar thermodynamic functions for the condensed and ideal-gas states were derived from the experimental results. Statistical calculations were performed based on molecular geometry optimization and vibrational frequencies using B3LYP hybrid density functional theory with the def2-TZVPPD basis set. Excellent accord between computed and experimentally-derived ideal-gas entropies is shown. The enthalpy of formation for 2-methylindole in the gas phase was computed with an atomization-based protocol described recently, and excellent agreement with the experimental values is seen. The experimental literature for enthalpies of formation in the gas phase for 1- and 2-ring pyrrollic compounds is reviewed, and comparisons with computed values further support the findings here. All experimental results are compared with property values reported in the literature, where possible.

Information Entropy of Liquid Metals.

Correlations reduce the configurational entropies of liquids below their ideal gas limits. By means of first-principles molecular dynamics simulations, we obtain accurate pair correlation functions of liquid metals, then subtract the mutual information content of these correlations from the ideal gas entropies to predict the absolute entropies over a broad range of temperatures. We apply this method to liquid aluminum and copper and demonstrate good agreement with experimental measurements; then, we apply it to predict the entropy of a liquid aluminum-copper alloy. Corrections due to electronic entropy and many-body correlations are discussed.

Analysis of the Mean Field Free Energy Functional of Electrolyte Solution with Nonhomogenous Boundary Conditions and the Generalized PB/PNP Equations with Inhomogeneous Dielectric Permittivity

The energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is bounded, and people still use the similar energy form, PDE(s), but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for the electrolyte solution within a bounded domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term, and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical...

Thermodynamic properties of pyrrole, 1-methylpyrrole, 2,4-dimethylpyrrole, and 2,5-dimethylpyrrole: Experimental and computational results

New measurements of critical temperature, saturated liquid heat capacity from temperature T≈315K to T≈550K, and saturated liquid density from T=323K to T≈475K are reported for pyrrole (Chemical Abstracts registry number [109-97-7]), 1-methylpyrrole [96-54-8], 2,4-dimethylpyrrole [625-82-1], and 2,5-dimethylpyrrole [625-84-3]. New measurements of vapor pressure are reported for 2,4-dimethylpyrrole {338<(T/K)<477} and 2,5-dimethylpyrrole {341<(T/K)<479}, as well as the enthalpy of combustion determined with oxygen-bomb calorimetry for 2,4-dimethylpyrrole. These new measurements are combined with literature values to calculate thermodynamic properties in the ideal-gas state for extended ranges of temperature for all compounds: pyrrole {298<(T/K)<550}, 1-methylpyrrole {298<(T/K)<530}, 2,4-dimethylpyrrole {298<(T/K)<600}, and 2,5-dimethylpyrrole {298<(T/K)<560}. Molar thermodynamic functions (enthalpies, entropies, and Gibbs energies) for the liquid and ideal-gas states were derived from the experimental studies at selected temperatures. Statistical calculations were performed with optimized geometries, scaled vibrational frequencies, and methyl rotational potentials performed using B3LYP hybrid density functional theory with the def2-TZVPPD basis set. Methyl torsional barriers were evaluated at the DLPNO-CCSD(T)/def2-QZVP level of theory using the B3LYP/def2-TZVPPD geometries. Computed ideal-gas properties are shown to be in excellent accord with ideal-gas entropies derived from thermophysical property measurements, as well as with reported experimental heat capacities for the ideal-gas state. All experimental results are compared with property values available in the literature.

Analysis of Uncertainty in the Calculation of Ideal-Gas Thermodynamic Properties Using the One-Dimensional Hindered Rotor (1-DHR) Model

The uncertainty analysis of the calculation of ideal-gas thermodynamic properties using the one-dimensional hindered rotor (1-DHR), which is designed for quantum chemical treatment of internal rotations, is presented. The sensitivity of ideal-gas heat capacities and entropies to the input parameters of the 1-DHR model, that is, the reduced moments of inertia and barriers to internal rotation, is analyzed and evaluated. The calculations of energy barriers to internal rotations using the B3-LYP functional coupled with six basis sets and the empirical D3 correction accounting for dispersion interactions for a set of 60 molecular structures containing internal rotation motions were also performed, compared with available experimental values, and statistically evaluated to estimate the impact of uncertainty associated with the calculated barriers to internal rotation (15% on average, underestimated in two-thirds of cases) on ideal-gas thermodynamic properties.

Aneurysm of ascending aorta (aortic dissection)

We consider the effect of shear and normal viscous pressures on the non-equilibrium entropy of ideal gases in Couette flow. These results extend the previous ones (Bidar et al., Physica A 233 (1996) 163), where normal pressure effects were ignored. Furthermore, we analyze the non-equilibrium contributions to the chemical potential, which may be useful in the analysis of shear-induced effects on colligative properties and chemical equilibrium.

On a Scale-Invariant Model of Statistical Mechanics and the Laws of Thermodynamics

A scale-invariant model of statistical mechanics is applied to describe modified forms of zeroth, first, second, and third laws of classical thermodynamics. Following Helmholtz, the total thermal energy of the thermodynamic system is decomposed into free heat U and latent heat pV suggesting the modified form of the first law of thermodynamics Q = H = U + pV. Following Boltzmann, entropy of ideal gas is expressed in terms of the number of Heisenberg–Kramers virtual oscillators as S = 4 Nk. Through introduction of stochastic definition of Planck and Boltzmann constants, Kelvin absolute temperature scale T (degree K) is identified as a length scale T (m) that is related to de Broglie wavelength of particle thermal oscillations. It is argued that rather than relating to the surface area of its horizon suggested by Bekenstein (1973, “Black Holes and Entropy,” Phys. Rev. D, 7(8), pp. 2333–2346), entropy of black hole should be related to its total thermal energy, namely, its enthalpy leading to S = 4Nk in exact agreement with the prediction of Major and Setter (2001, “Gravitational Statistical Mechanics: A Model,” Classical Quantum Gravity, 18, pp. 5125–5142).