Fundamental Thermodynamic Relations Research Articles

Non-Boltzmann stationary distributions and nonequilibrium relations in active baths.

Most natural and engineered processes, such as biomolecular reactions, protein folding, and population dynamics, occur far from equilibrium and therefore cannot be treated within the framework of classical equilibrium thermodynamics. Here we experimentally study how some fundamental thermodynamic quantities and relations are affected by the presence of the nonequilibrium fluctuations associated with an active bath. We show in particular that, as the confinement of the particle increases, the stationary probability distribution of a Brownian particle confined within a harmonic potential becomes non-Boltzmann, featuring a transition from a Gaussian distribution to a heavy-tailed distribution. Because of this, nonequilibrium relations (e.g., the Jarzynski equality and Crooks fluctuation theorem) cannot be applied. We show that these relations can be restored by using the effective potential associated with the stationary probability distribution. We corroborate our experimental findings with theoretical arguments.

CFD simulations of turbulent nonreacting and reacting flows for rocket engine applications

A consistent real gas framework based on a preconditioning scheme is implemented into the CFD code TASCOM3D to simulate nonreacting and reacting turbulent flows with transcritical injection. All thermodynamic properties and partial derivatives are rigorously evaluated from the reduced residual Helmholtz energy by use of fundamental thermodynamic relations. The Soave–Redlich–Kwong equation of state with an optional volume translation is employed in combination with high-pressure models for the calculation of fluid transport properties. Three test cases are simulated to validate the implemented real gas framework using different turbulence modeling strategies. A good match between numerical and experimental data is obtained.

Thermodynamics of Noncommutative Quantum Kerr Black Holes

The thermodynamic formalism for rotating black holes, characterized by noncommutative and quantum corrections, is constructed. From a fundamental thermodynamic relation, the equations of state and thermodynamic response functions are explicitly given, and the effect of noncommutativity and quantum correction is discussed. It is shown that the well-known divergence exhibited in specific heat is not removed by any of these corrections. However, regions of thermodynamic stability are affected by noncommutativity, increasing the available states for which some thermodynamic stability conditions are satisfied.

Determination of solubility parameters from density measurements for non-polar hydrocarbons at temperatures from (298–433) K and pressures up to 137 MPa

The Hildebrand's solubility parameter, δ, gives a quantitative representation of the well-known “like dissolves like” aphorism for non-polar hydrocarbon liquids. This is of great commercial importance for certain polymers and hydrocarbon mixtures. However, the solubility parameters are usually available only at ambient conditions for a limited number of substances. Extrapolation of these parameters to high pressure and high temperature conditions is not attained by common empirical correlations or cubic equations of state. In this work a set of equations are derived from fundamental thermodynamic property relations to account for the pressure and temperature dependence of this cohesive energy parameter. The equations, which require a reference value for the solubility parameter –usually at ambient conditions– and volumetric data, are then applied to calculate the solubility parameters of four pure non-polar hydrocarbons, namely n-heptane, n-dodecane, benzene and toluene, at temperatures from (298–433) K and pressures up to 137 MPa. A comparison against literature data and predictions from the Perturbed Chain version of the Statistical Association Fluid Theory (PC-SAFT) and Peng-Robinson equations of state is also provided. The proposed method provides a novel approach for a quick, simple and accurate determination of the solubility parameters at elevated temperatures and pressures for non-polar hydrocarbons using density measurements.

MODELING OF THE SURFACE TENSION OF COLLOIDAL SUSPENSIONS

Surface tension is one of the fundamental properties of the colloids, which can be altered by concentration and size of colloidal particles. In the current work, modeling of the surface tension of suspension as it would be analyzed by maximum bubble pressure method has been performed. A new modified equation to correlate the surface tension with the bubble pressure is derived by applying fundamental thermodynamic relation considering the presence of particles in suspension and curvature of the interface between the particles and bubbles inside liquid. Moreover, the change of particles concentration in air–water interface due to capillary force is also considered. The predicted surface tension using the developed model has been verified by numerous experimental data with deviation less than 5% in most of cases. It was found that the calculated surface tension is altered by contact angle and particle radius as well as particle concentration. The obtained model may have potential application to predict the surface tension of colloidal suspension.

From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics

This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

Theoretical problems in solid electrocapillarity

The main problem in electrocapillarity of solid electrodes is the lack of clarity in determining the surface stress and basic equations. Within the framework of the Gibbs concept of geometrical dividing surface, the “surface stress” cannot be defined because methods of continuum mechanics can be applied to a physical surface layer (of finite thickness), but not to a mathematical surface. Gibbs never used the concept of surface stress, introducing only “surface tension” for a liquid electrode and “closely related quantity” for a solid electrode. Revisiting the derivation of the Gibbs adsorption equation, we prove its applicability to solid surfaces without the limiting requirement of constant state of strain, which was undeservedly interpreted by Eriksson as a shortcoming of the Gibbs theory caused to look for other approaches to surface stress problem. A critical analysis shows that the attempts (Shuttleworth, Eriksson, Couchman, Gokhstein, Weissmuller, etc.) to create a thermodynamic definition of the surface stress (as well as the formulation of fundamental thermodynamic equations and Maxwell relations operating with surface stresses) contain mathematical defects. It is shown that confusing interpretations of some Gibbs’ concepts encountered in the literature have led to “modifications” of the Lippmann equation based on the critical error in the Gibbs–Duhem relation due to the occurrence of an extensive variable, which is inadmissible. The famous Lippmann equation should not be modified, and it remains a unique electrocapillary relation applicable to liquid and solid electrodes.

Consistent thermostatistics forbids negative absolute temperatures

A considerable body of experimental and theoretical work claims the existence of negative absolute temperatures in spin systems and ultra-cold quantum gases. Here, we clarify that such findings can be attributed to the use of a popular yet inconsistent entropy definition, which violates fundamental thermodynamic relations and fails to produce sensible results for simple analytically tractable classical and quantum systems. Within a mathematically consistent thermodynamic formalism, based on an entropy concept originally derived by Gibbs, absolute temperature remains positive even for systems with bounded spectrum. We address spurious arguments against the Gibbs formalism and comment briefly on heat engines with efficiencies greater than one.

Black Hole Dynamic Potentials

In the following paper, certain black hole dynamic potentials have been developed definitively on the lines of classical thermodynamics. These potentials have been refined in view of the small differences in the equations of the laws of black hole dynamics as given by Bekenstein and those of thermodynamics. Nine fundamental black hole dynamical relations have been developed akin to the four fundamental thermodynamic relations of Maxwell. The specific heats C Ω,Φ and C J,Q have been defined. For a black hole, these quantities are negative. The κdA equation has been obtained as an application of these fundamental relations. Time reversible processes observing constancy of surface gravity are considered and an equation connecting the internal energy of the black hole E, the additional available energy defined as the first free energy function K, and the surface gravity κ, has been obtained. Finally as a further application of the fundamental relations, it has been proved for a homogeneous gravitational field in black hole space times or a de Sitter black hole that $C_{\Omega ,\Phi } -C_{J,Q} =\kappa \left[ {\left( {\frac{\partial J}{\partial \kappa }} \right)_{\Omega ,\Phi } \left( {\frac{\partial \Omega }{\partial \kappa }} \right)_{J,Q} +\left( {\frac{\partial Q}{\partial \kappa }} \right)_{\Omega ,\Phi } \left( {\frac{\partial \Phi }{\partial \kappa }} \right)_{J,Q} } \right]$ . This is dubbed as the homogeneous fluid approximation in context of the black holes.

Acoustic determination of the heavy oil thermophysical properties: Thermodynamic consistency revisited

We employed an Initial Value approach, so as to integrate the compressional wave propagation velocity data, to determine the thermo-physical properties of heavy oil samples. We present our results for several thermo-physical characteristics of the heavy oil, including isothermal compressibility, thermal expansivity, isobaric heat capacity, and the Joule–Thomson coefficient, for the temperatures between 283.15K and 373.15K and pressures up to 20MPa. The acoustically-determined heavy oil densities exhibited outstanding accuracy within 0.4% of the experimental counterparts. Owing to the nature of the integration methodology applied, derived from the fundamental thermodynamic relations, reasonably accurate results should be expected as for the average thermo-physical properties established for the heavy oil mixtures.

Efficiency bounds for nonequilibrium heat engines

We analyze the efficiency of thermal engines (either quantum or classical) working with a single heat reservoir like an atmosphere. The engine first gets an energy intake, which can be done in an arbitrary nonequilibrium way e.g. combustion of fuel. Then the engine performs the work and returns to the initial state. We distinguish two general classes of engines where the working body first equilibrates within itself and then performs the work (ergodic engine) or when it performs the work before equilibrating (non-ergodic engine). We show that in both cases the second law of thermodynamics limits their efficiency. For ergodic engines we find a rigorous upper bound for the efficiency, which is strictly smaller than the equivalent Carnot efficiency. I.e. the Carnot efficiency can be never achieved in single reservoir heat engines. For non-ergodic engines the efficiency can be higher and can exceed the equilibrium Carnot bound. By extending the fundamental thermodynamic relation to nonequilibrium processes, we find a rigorous thermodynamic bound for the efficiency of both ergodic and non-ergodic engines and show that it is given by the relative entropy of the nonequilibrium and initial equilibrium distributions. These results suggest a new general strategy for designing more efficient engines. We illustrate our ideas by using simple examples.

The Acoustic Determination of the Thermophysical Properties of n-Hexane + n-Hexadecane from 298.15 K to 373.15 K and Up to 400 MPa

The thermophysical properties for a mixture of n-hexane + n-hexadecane were determined via an initial value integration method using acoustic data. A distinct functional form was devised for the compressional wave propagation velocity in terms of pressure along each isotherm. Comparison with the experiment confirmed excellent accuracy for the acoustically determined first-order thermodynamic properties within a maximum deviation of 3.64% from the experimental counterparts and up to 400.7 MPa. Given the nature of the integration methodology applied, derived using fundamental thermodynamic relations, exact results should be expected for the thermophysical properties established, within the realm of the thermodynamic principles.

Thermal Decomposition Kinetics of Some Di‐ and Triorganotin(IV) Carboxylates Derived from para‐Nitrophenylethanoic Acid

AbstractThermogravimetric (TG) investigations of organotin(IV) carboxylates with the general formula RmSnL4−m (where R=CH3, C2H5, n‐C4H9, C6H5, cyclo‐C6H11, n‐C8H17, m=2, 3, and L=para‐nitrophenylethanoate anion) have been performed. Derivative thermogravimetry (DTG) and differential thermal analysis (DTA) techniques, Horowitz‐Metzger method and the fundamental thermodynamic relations are used to evaluate the thermokinetic parameters of each thermal degradation pattern. Results reveal that the thermal stability is functional to SnC and SnO bonds. In the case of R2SnL2, activation energy, reaction order and pre‐exponential factor associated with the bulk degradation processes increase as the alkane chain length increases. Hence, Oct2SnL2 is thermally more stable than Bu2SnL2, which in turn is more resistant to thermal dissociation than Et2SnL2. The same phenomenon is not observed for R3SnL compounds because their degradation is highly irregular. Furthermore, R2SnL2 has larger values of kinetic parameters than those of corresponding triorganotin(IV) para‐nitrophenylethanotes. Thermodynamic parameters of these compounds also reinforce the above facts.

First principles molecular dynamics simulations of diopside (CaMgSi 2O 6) liquid to high pressure

We use first principles molecular dynamics simulations based on density functional theory in the local density approximation to investigate CaMgSi 2O 6 liquid over the entire mantle pressure regime. We find that the liquid structure becomes much more densely packed with increasing pressure, with the mean Si–O coordination number increasing nearly linearly with volume from fourfold near ambient pressure to sixfold at the base of the mantle. Fivefold Si–O coordination environments are most abundant at intermediate compression. The properties of Mg and Si coordination environments are nearly identical to those in MgSiO 3 liquid, whereas Ca is more highly coordinated with larger mean Ca–O bond length as compared with Mg. The density increases smoothly with increasing pressure over the entire range studied. The Grüneisen parameter increases by a factor of three on twofold compression. The density contrast between diopside composition liquid and the isochemical crystalline assemblage is less than 2% at the core mantle boundary, less than that in the case of MgSiO 3. Thermodynamic properties are described in terms of a liquid-state fundamental thermodynamic relation.

급속 압축장치(RCM)의 압축 조건에 따른 최대 압력 변화에 관한 실험적 연구

RCM(Rapid Compression Machine)은 엔진 내부의 복잡한 연소현상을 연구하는 데 적합한 장치이다. 엔진의 연구를 위해서는 내부의 최대압력을 측정하는 것이 중요하다. 그 이유는 최대압력이 내부 연료의 연소특성에 큰 영향을 끼치기 때문이며, 연소실 내부의 최대온도가 최대 압력에 의해 추정되기 때문이다. 그러나 최대압력은 실린더 벽면과의 열전달로 인해 정량적으로 계산하기 어려우며 보통 실험적인 방법을 통해 측정하고 있다. 본 연구를 통해 RCM의 구동시간에 따른 연소실의 최대압력과의 관계를 실험적으로 확인하고, 실험결과를 바탕으로 최대압력과 구동시간에 관한 일반적인 공식을 도출한다.

Molecular Thermodynamic Modeling and Design of Microencapsulation Systems for Drug Delivery

A systematic design strategy is given for computer-aided design of microparticle drug-delivery systems produced by solvent evaporation. In particular, design of solvents, polymer material, and external phase composition are considered for the case when the active ingredient is known. The procedure is based on fundamental thermodynamic relations and group contributions to properties of pure species (solvent, active ingredient and polymer) and their mixtures. The method is intended for pharmaceuticals with complex molecular structures, for which limited experimental information is known. Case studies of solvent design are given.

Hamiltonian and Thermodynamic Modeling of Quantum Turbulence

The state variables in the novel model introduced in this paper are the fields playing this role in the classical Landau-Tisza model and additional fields of mass, entropy (or temperature), superfluid velocity, and gradient of the superfluid velocity, all depending on the position vector and another tree dimensional vector labeling the scale, describing the small-scale structure developed in 4He superfluid experiencing turbulent motion. The fluxes of mass, momentum, energy, and entropy in the position space as well as the fluxes of energy and entropy in scales, appear in the time evolution equations as explicit functions of the state variables and of their conjugates. The fundamental thermodynamic relation relating the fields to their conjugates is left in this paper undetermined. The GENERIC structure of the equations serves two purposes: (i) it guarantees that solutions to the governing equations, independently of the choice of the fundamental thermodynamic relation, agree with the observed compatibility with thermodynamics, and (ii) it is used as a guide in the construction of the novel model.

Microscopic diagonal entropy and its connection to basic thermodynamic relations

We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as S d = - ∑ n ρ nn ln ρ nn with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy S n = −Tr ρ ln ρ. However, in contrast to S n , the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that S d behaves consistently with expectations from thermodynamics.

Fault Tolerant Oxygen Control of a Diesel Engine Air System

This paper is devoted to the fault tolerant control problem of a Diesel engine air system having a jammed Exhaust Gas Recirculation (EGR) valve. The fault tolerant control is based on replaning the trajectory in order to track a new controlled variable which is the oxygen concentration in the intake manifold instead of the fresh air mass flow. The trajectory planning is based on an inverse model approach, utilizing the fundamental thermodynamic relations of the air system.

Are Long-Chain Alkanes Hydrophilic?

Although short n-alkane chains are classic examples of hydrophobic solutes, mounting evidence points to a hydrophilic crossover for the hydration free energies (DeltaG) of sufficiently long n-alkane chains. Experimental and simulation results for the hydration of n-alkanes from methane (C1) to docosane (C22) are combined with fundamental thermodynamic relations to elucidate intermolecular contributions to DeltaG. Theoretical bounds on the influence of solute conformation on DeltaG are inferred by considering the hydration of idealized linear (all-trans) and globular (spherical) model solutes. More detailed theoretical extrapolations of experimental and simulation results imply that the water-mediated free energy change associated with collapsing an all-trans C100 chain is on the order of -100 kJ/mol and thus that n-alkane chains of this length and longer may be hydrophilic (DeltaG < 0).