Formulation Of Nonequilibrium Thermodynamics Research Articles

Comments on the Onsager Reciprocal Relations in the Frictional Formalism of Non-equilibrium Thermodynamics.

Comments on the Onsager Reciprocal Relations in the Frictional Formalism of Non-equilibrium Thermodynamics.

Material transport in hyperfiltration

Transport equations for hyperfiltration, based on the frictional formalism of non-equilibrium thermodynamics, are derived for a gel-type membrane. Taking the concentration distribution in the membrane into account and allowing for possible non-equilibrium partition effects at the membrane boundaries, the flow equations are integrated and equations for solute rejection in dilute solutions are derived. It is shown that the main parameters determining solute rejection are the solute partition coefficient on the high-pressure side of the membrane and a frictional factor, involving the solute and solvent friction coefficients. Partition effects on the low-pressure side are unimportant for processes at high flow rates.

Viscous effects in the frictional formalism of non-equilibrium thermodynamics

A hydrodynamic interpretation of the frictional formalism of non-equilibrium thermodynamics indicates that viscous effects are inherent in the formalism, provided the component velocities are properly averaged. Under these circumstances transport processes in porous media can be treated without reference to the structure of the medium. The treatment can be applied to open pore structures as well as to dense gels, and takes into account both diffusive transport and viscous flow phenomena.

Effect of furosemide on the thermodynamic parameters of frog skin.

The formalism of nonequilibrium thermodynamics (NET) was used to analyze the effect of diuretic furosemide on the transport mechanism of frog skin. Mucosal furosemide increased the active conductance of Na+ across the mucosal membrane of the cells, producing an increased transport of Na+ (short-circuit current). Furosemide also increased both the thermodynamic affinity of the metabolic reaction supplying energy to the sodium pump and the degree of coupling between transport and metabolism. It changed the phenomenological cross-coefficient of the NET description as well as the stoichiometric ratio, indicating that its effect cannot be explained simply on the basis of a change in Na+ conductance. The effect on the NET parameters was independent of the presence of either chloride or sulfate as the principal anion in the solutions, and was qualitatively similar to the effect produced by mucosal application of triphenylmethylphosphonium ion.

Dynamic equation of state of a gas containing vaporizing droplets

On the basis of the relaxational formalism of nonequilibrium thermodynamics, a differential equation is obtained relating the pressure and volume of a gas containing vaporizing droplets. The relaxation time is calculated.

Thermodynamic analysis of active sodium and potassium transport in the frog corneal epithelium.

The formalism of linear nonequilibrium thermodynamics for a three-flow system was applied to the isolated frog corneal epithelium to study the coupling between metabolism and the Na-K transport system across this layer. There is little or no net ion transport across the isolated frog corneal epithelium bathed in Na2SO4 Ringer. Addition of amphotericin B to the tear side solution increases apical membrane permeability, which results in a net Na transport (from tear to stroma) and a net K transport in the opposite direction. Corneas were mounted in a modified Ussing chamber that permitted the simultaneous measurements of electrical parameters and O2 consumption by means of Clark-type oxygen electrodes. The overall degree of coupling, q, of the Na-K transport system to metabolism was calculated from measuring the suprabasal O2 consumption rate at "static head" and "level flow" conditions and by a second independent technique. Measurements of electrical conductance used in conjunction with other previously measured parameters allowed the calculation of the affinity, A, of the metabolic reaction driving transport, all phenomenological coefficients, and the electromotive forces of sodium (ENa) and potassium transport (EK). Values of q determined by the two techniques agreed (q = 0.80 and 0.84, respectively). This indicates incomplete coupling and a variable stoichiometric relationship among O2 consumption rate, net Na transport, and net K transport. The value calculated for A was 70.5 kcal.mol-1, for ENa 142.5 mV, and for EK -34.9 mV.

Mesoscopic thermodynamics of nonequilibrium open systems. I. Negentropy consumption and residual entropy

In order to treat active phenomena of open systems, we present a mesoscopic-level formulation of nonequilibrium thermodynamics, which is adaptable to the intermediate level between microscopic description and macroscopic one. In this formulation, an open system is represented by a mesoscopic modelM=(S,T),s, σ, γ).S is a set of mesoscopically coarse-grained states, each of which has the residual entropy s(a) (a∈S).T is a set of transitionsτ from I(τ)∈S toF(τ)∈S, and each τ∈T occurs at the transition probability rateγ(τ), and with the consumptionσ(τ) of the negentropy stored within the environment. The dynamical quantityγ is related to the thermostatic oness andσ through a postulate,individual detailed balance. This scheme is applied to various examples such as chemical reactions, reaction-random walks and lasers. A difference-operator method for lattice systems is also developed.

The optimal efficiency and the economic degrees of coupling of oxidative phosphorylation.

A phenomenological theory considering the output characteristics of oxidative phosphorylation has been worked out by adopting the formalism of linear nonequilibrium thermodynamics. The linearity of oxidative phosphorylation in the range of the output forces of practical interest has been experimentally verified. the efficiency of oxidative phosphorylation is zero if either a load with a zero conductance (open-circuited situation) or a load with an infinite conductance (short-circuited situation) is attached to oxidative phosphorylation. In between these extreme conductances there exists a finite load conductance permitting oxidative phosphorylation to operate with optimal efficiency. The necessary and sufficient condition for optimal efficiency was found to be L33/L11 = square root 1 - q2 where L11 is the phenomenological conductance of phosphorylation, L33 the phenomenological conductance of the load and q the degree of coupling of oxidative phosphorylation driven by respiration. This condition was called conductance matching. Under the condition of conductance matching, four output functions of oxidative phosphorylation of practical interest were optimized. A maximal net rate of oxidative phosphorylation occurs at a degree of coupling qf = 0.78. A maximal output power of oxidative phosphorylation, i.e. net rate times established phosphate potential, resuls at qp = 0.91. The maximization of the function net rate times efficiency yielded an economic degree of coupling qfec = 0.95 for maximal ATP flow. Finally, maximization of the function output power times efficiency led to a degree of coupling qpec = 0.97. This last function simultaneously maximized net rate of ATP production, developed phosphate potential and efficiency and reflects therefore the most economic solution to the output problem under the condition of conductance matching. In isolated rat livers perfused in a metabolic resting state, the condition of conductance matching is fulfilled. In addition, the degree of coupling oxidative phosphorylation under these conditions corresponds to the economic degree of coupling qpec.

The use of linear nonequilibrium thermodynamics in the study of renal physiology

Classical formulations for the analysis of membrane transport processes, which ignored possible interactions between flows of diverse permeant species, often led to inconsistencies in the evaluation of permeability coefficients. For water flow induced by an osmotic pressure difference this difficulty was resolved by Staverman's introduction of the reflection coefficient sigma, a parameter which incorporates the interaction between solute and solvent in the course of their passage through a membrane. A comprehensive nonequilibrium thermodynamic (NET) formalism suitable for many biological systems was provided by Kedem and Katchalsky. For an n-flow system each flow is in general dependent on n forces; the assumption of Onsager reciprocity, however, reduces the number of independent phenomenological coefficients. Although NET is widely applied in the study of renal physiology, fundamental theoretical and practical problems remain. Basic considerations are the need to control or evaluate the influence of all coupled flows and to establish conditions fostering linear dependencies of flows on forces. When this is done a transport system may be characterized in terms of intrinsic membrane parameters, facilitating the systematic study of the effects of drugs, hormones, and various experimental perturbations.

Transients in toad skin: Short circuit current and ionic fluxes related to inner sodium substitution by monovalent cations

When the Na electrochemical potential difference across the skin (delta muNa) is altered by perturbing the transmembrane electrical potential difference or the external Na concentration, effects on transport and associated oxygen consumption can be described by the formalism of linear nonequilibrium thermodynamics (Vieira, Caplan & Essig, 1972, J. Gen. Physiol. 59:77; Danisi & Lacaz-Vieira, 1974, J. Gen. Physiol. 64:372; Procópio and Lacaz-Vieira, 1977, J. Membrane Biol. 35:219). We now show that with modifications of delta muNa by substitution of Li or choline for Na in the inner bathing solution, this formalism is no longer applicable. Inner Na by K substitution ((Na X K)i) causes profound alterations in short-circuit current (SCC), JinNa, K efflux (JeffK) and PD. SCC drops transiently after (Na X K)i in Cl and in SO4 media, increasing subsequently. In Cl medium, following the initial transient, there is a late decline in SCC toward a steady state. The rate of SCC decline in Cl medium is more pronounced than that observed in SO4 medium. (Na X K)i causes a transient increase in JinNa with a peak synchronous to the minimum in SCC, both in Cl and in SO4 media. This was interpreted as due to depolarization of the inner membrane. In SO4 medium, following the peak observed after (Na X K)i, JimNa drops, to increase again toward a steady state in which SCC and JinNa are not statistically different, resembling the control condition before (Na X K)i. In Cl medium, however, the JinNa steady state is approximately 100% higher than SCC. This difference is due to an important K efflux (JeffK), which builds up progressively after the substitution. The apparent K permeability [JeffK/(Ki)] is of comparable magnitude in Cl and in SO4 media before (Na X K)i and also in SO4 medium after (Na X K)i. However, in Cl medium, after (Na X K)i, the apparent K permeability increases one order of magnitude as compared to the control condition before the ionic substitution. In Cl medium, the high levels of JinNa and of Jeff(K) observed in the steady state after (Na X K)i were interpreted as being a consequence of cell swelling. SCC and PD follow very different temporal patterns after (Na X K)i which are characterized by transients in SCC and a simple fall in PD. Reasons for these differences are discussed.

Effective thermodynamic functions of a gas with solid particles

Using the relaxation formalism of nonequilibrium thermodynamics, a dynamic equation of state and hyperbolic thermal-conductivity equation are derived for a gas with solid particles.

Thermodynamic analysis of active sodium transport and oxidative metabolism in toad urinary bladder

Measurements of electrical current and oxygen consumption were carried out concurrently under voltage clamp conditions in 11 toad hemibladders. Inhibition of active transport with amiloride then permitted evaluation of the passive conductance and the rate of basal oxygen consumption Jbr, allowing the simultaneous determination of the rates of active sodium transport JaNa and suprabasal oxygen consumption Jsbr-JaNa and Jabr were linear functions of the electrical potential difference over a range of +/- 80 mV. This allowed the comprehensive application of a linear nonequilibrium thermodynamic formalism, leading to the evaluation of the affinity A (negative free energy) of the metabolic reaction driving transport, all phenomenological coefficients, and the degree of coupling q relating transport to metabolism. Values of A determined by two techniques were A1=56.0 +/- 5.8 and A2=58.2 +/- 6.5 kcal per mole. Values of q determined by two techniques agreed well and were less than 1, indicating incompleteness of coupling, and hence lack of fixed stoichiometry between Na transort and O2 consumption. The affinity and the electromotive force of sodium transport ENa are not closely correlated, reflecting the fact that ENa comprises both kinetic and energetic factors.

On transient relativistic thermodynamics and kinetic theory

In the conventional Onsager formulation of non-equilibrium thermodynamics the Fourier heat conduction law is parabolic, and permits arbitrarily large propagation velocities for temperature discontinuities. While embarassing in a non-relativistic theory, this is unacceptable in a relativistic one. This paper presents a detailed discussion of the thermodynamics and (in the dilute gas limit) kinetic theory of such transients. When proper account of the two length scales involved is taken into account, the theory becomes hyperbolic; propagation velocities for a dilute gas never exceed √3/5 c = 0.775 c , where c is the speed of light.

Electrical potential differences across salt transporting membranes

The electrical potential differences across membranes where active transport of ions occurs has been examined using the formalism of linear non-equilibrium thermodynamics, and can be represented as the arithmetic sum of a resistive term, a term directly dependent on metabolism (i.e. electrogenic) and terms appropriate for describing a diffusion potential. The Hittorf transport number for each ion in the latter terms is the ratio of the partial conductances of the membrane to that ion to the total membrane conductance, and the conductance to an ion consists of the arithmetic sum of conductance of active and passive pathways providing these are independent. The conductances of active transport mechanisms arise from variation of the rate of transport with the electrochemical potentials against which they operate. The electrogenic term arises from imbalance between anion and cation transport. If an ion is transported by an obligatorily electrically neutral exchange for some other ion such transport gives rise to no electrogenic effect. A membrane will transport salt most efficiently if there is no imbalance between anion and cation transport, when it will not be electrogenic, but modest deviations from this condition will not degrade the efficiency of active transport markedly.

The mechanism of water transport in membranes.

Biological membranes, lipid membranes and organic polymer membranes have some chemical similarities. All can transport water and it is likely that the molecular transport processes have some common features in the three types of system. Polymer membranes, being stable and strong, can be subjected to more varied and intensive study than can either lipid or biological membranes. Furthermore, the structures, dimensions and molecular organization of polymer membranes, which are very simple compared with their biological counterparts, can be characterized in some detail. It is already a reasonable objective to analyse transport phenomena in polymer membranes in terms of the molecular processes likely to occur in media of known structure. If the level of understanding in this area could be improved it might become possible to make some confident deductions about structure in lipid and biological membranes from observations on their transport properties. This review is intended to assess the current situation regarding the interpretation of flux data in structural terms and, perhaps, to encourage further developments. Although observations on transport processes are made by observing changes in the bulk phases in contact with the membrane, it is the processes within the membrane that are under study. The mathematical formulation of membrane transport must be designed to emphasize the role of the membrane as a phase in which irreversible processes are occurring. This requires the determination of diffusion coefficients, concentrations, activity coefficients and their profiles within the membrane. The concentration and activity of water in a membrane are studied through equilibrium absorption isotherms. These vary widely in form with the chemical structure and organization in polymer membranes. Almost nothing detailed is known about sorption by lipid membranes. Most flow processes are studied between pairs of aqueous solutions that contain solutes which may also be able to permeate the membrane. The possibility of several irreversible processes occurring simultaneously in the membrane can best be handled through the formalism of non-equilibrium thermodynamics. This formulation provides also a convenient method of introducing pressure and osmotic pressure as driving forces in addition to that of simple Fickian diffusion flow. A bridge between the phenomenological coefficients of thermodynamics and molecular processes can be built only on a model system. By regarding the membrane as intrinsically homogeneous and isotropic the frictional model can be applied in which steady flow is represented by a balance between thermodynamic driving forces and frictional retarding forces among the various flowing components and the membrane. A case of particular interest arises where the solute is an isotopically labelled species of water. The number of independent frictional coefficients in the two-component system is reduced from three to two. They can be determined from measurements of the tracer diffusion coefficient and the hydrodynamic or osmotic permeability of the membrane. This has been done for two sets of polymer membranes: highly hydrated hydrogels and moderately hydrated cellulose acetates. The membranes were prepared so as to be as nearly homogeneous as possible and it was found that the frictional coefficients observed were consistent with the homogeneous model theory. By applying a form of argument which has several times been used to diagnose the existence of pores in biological and lipid membranes, it was possible to deduce from the data on the synthetic membranes that they too transport water in, sometimes quite large, pores. This conclusion would be at variance with their known structures and shows that the argument is unsound when it is used as a criterion for the presence or absence of pores. It is clear also that the argument is valid only when applied to a homogeneous system and so would not be valid if applied to a system that were truly porous but also transported water through the matrix material supporting the pores.

Properties of One-Dimensional Diagenetic Models

It is not possible to treat both absolute and relative time in the diagenetic equation of Berner, and the equation cannot therefore be particularized to obtain an expression that isolates the diagenetic contribution to a sediment property. This problem can be solved by using a Lagrangian approach in which the depositional contribution to the sediment property becomes a boundary condition. A simplistic steady-state model is used to define the convective velocities of pore fluid and sediment particles in terms of porosity and the rate of accumulation of compacted sediment. The argument is extended to include values of sedimentation rate in nature and diffusion coefficients in sediment pore fluid, and the changing relative importance of convective and diffusive transport during compaction is considered. A continuity equation is developed for the concentration of a component i in multi-component pore fluid. If the equation is used in a reference frame moving at the convective velocity of the sediment particles, it appears to be possible to follow the course of chemical reaction in a compacting system, even though pore fluid and sediment particles are necessarily moving in opposite directions. The formalism of nonequilibrium thermodynamics will be called for in more complex diagenetic models that treat such additional phenomena as heat flow, stress distribution, and the semi-permeable membrane behavior of aggregates of minerals with high surface area and surface charge. The requirement of linearity between thermodynamic forces and fluxes is well obeyed in electrolyte diffusion, but poorly in chemical reactions. The nonequilibrium thermodynamic approach requires detailed knowledge of a considerable variety of parameters of a system, if it is to yield greater understanding than could be obtained from a set of continuity equations.

Role of surface integrals in the hamiltonian formulation of general relativity: Tullio Regge. Institute for Advanced Study, Princeton, New Jersey 08540 and Claudio Teitelboim. Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540

We construct a generally-covariant formulation of non-equilibrium thermodynamics in General Relativity. We find covariant entropic forces arising from gradients of the entropy density, and a corresponding non-conservation of the energy momentum tensor in terms of these forces. We also provide a Hamiltonian formulation of General Relativity in the context of non-equilibrium phenomena and write the Raychaudhuri equations for a congruence of geodesics. We find that a fluid satisfying the strong energy condition could avoid collapse for a positive and sufficiently large entropic-force contribution. We then study the forces arising from the internal production of “bulk” entropy of hydrodynamical matter, as well as from the entropy gradients in the boundary terms of the action, like those associated with black hole horizons. Finally, we apply the covariant formulation of non-equilibrium thermodynamics to the expanding universe and obtain the modified Friedmann equations, with an extra term corresponding to an entropic force satisfying the second law of thermodynamics.

Thermodynamics of Ion Exchange

AbstractThe local formulation of nonequilibrium thermodynamics is used to define the electrochemical potentials in the ion‐clay‐water system, thus enabling development of an improved nonempirical approach to cation‐exchange processes. The presence of the electrical field in the system is expressed by a proportional increase of the pressure in the double layer. The effect of the electrical field on the chemical potentials of the exchangeable cations is divided into the common electrostatic term and a term dependent on their polarizabilities. For systems containing a mixture of monovalent ions it can be assumed that the electrostatic effect of the electrical field is the same for each of the exchangeable cations. Using values of the polarizabilities and intrinsic volumes of the alkali metal cations, the free energies of exchange with sodium in montmorillonite were calculated. The calculated free energies were found to be in good agreement with experimental values.

Relative anion mobilities in molten AgNO3-AgCl by the measurement of density and the gravitational component of cell potential

Densities of molten AgNO3-AgCl mixtures have been measured from the liquidus temperature to 320°C. Emf's of gravitational cells: Ag(h1)/AgNO3,AgCl/Ag(h2) have been investigated up to 260°C. These measurements are related to the mobilities of the anion constituents through a non-equilibrium thermodynamic formalism. The experimental results are compared to the liquid-junction potential and calorimetric data which have been published for this system, and it is shown that the relative anion mobilities determined by the measurement of the gravitational component of cell potential are in better agreement than those obtained by direct Hittorf-type experiments. The mobility of the nitrate-ion constituents is found to be slightly greater than that of the chloride-ion constituents in the temperature range considered. These facts are discussed in relation to the formation of low mobility complex ions.

Nonequilibrium Thermodynamics in Chemical Kinetics

The description of chemical reaction rates, in terms of the language of nonequilibrium thermodynamics, is compared with the conventional language of chemical kinetics. It is concluded that the formalism of nonequilibrium thermodynamics possesses considerable disadvantages, even in the regime where the rate is a linear function of affinity; this conclusion also holds for cases where chemical rates are coupled to diffusion processes. Some of the difficulties involved are illustrated using the experimental data of Nebeker and Pings (28).