Linear nonequilibrium thermodynamic

The current density j is, according to linear nonequilibrium thermodynamics, proportional to the gradient of a chemical potential difference... 

Chaplan, S. R., and A. Essig, Bioenergetics and Linear Nonequilibrium Thermodynamics. The Steady State, Harvard University Press, Cambridge, Mass., 1983. 

The question of the efficiency of biological transport systems was examined extensively in the 1960s on the basis of linear nonequilibrium thermodynamics. I think it would be appropriate to give a brief account of the treatment here, especially since Professor Prigogine s early work was the source of most of our ideas at the time. The formal approach of... 

Errors in the description of nonequilibrium processes in the linear nonequilibrium thermodynamics (Glansdorff et al., 1971 Kondepudi et al., 2000 Prigogine, 1967 Zubarev, 1998) are caused primarily by the assumptions (unnecessary at MEIS application) on the linearity of motion equations. One of the main equations of this thermodynamics has the form... 

States away from global equilibrium are called the thermodynamic branch (Figure 2.2). Systems not far from global equilibrium may be extrapolated around equilibrium state. For systems near equilibrium, linear phenomenological equations may represent the transport and rate processes. The linear nonequilibrium thermodynamics theory determines the dissipation function or the rate of entropy production to describe such systems in the vicinity of equilibrium. This theory is particularly useful to describe coupled phenomena, and quantify the level of coupling in physical, chemical, and biological systems without detailed process mechanisms. 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The phenomenological coefficients are important in defining the coupled phenomena. For example, the coupled processes of heat and mass transport give rise to the Soret effect (which is the mass diffusion due to heat transfer), and the Dufour effect (which is the heat transport due to mass diffusion). We can identify the cross coefficients of the coupling between the mass diffusion (vectorial process) and chemical reaction (scalar process) in an anisotropic membrane wall. Therefore, the linear nonequilibrium thermodynamics theory provides a unifying approach to examining various processes usually studied under separate disciplines. 

The formulation of linear nonequilibrium thermodynamics is based on the combination of the first and second laws of thermodynamics with the balance equations including the entropy balance. These equations allow additional effects and processes to be taken into account. The linear nonequilibrium thermodynamics approach is widely recognized as a useful phenomenological theory that describes the coupled transport without the need for the examination of the detailed coupling mechanisms of complex processes. 

Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

Some options for achieving a thermodynamic optimum are to improve an existing design so the operation will be less irreversible and to distribute the irreversibilities uniformly over space and time. This approach relates the distribution of irreversibilities to the minimization of entropy production based on linear nonequilibrium thermodynamics. For a transport of single substance, the local rate of entropy production is... 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

Various formulations and methodologies have been suggested for describing combined heat and mass transfer problems, such as the integral transform technique, in the development of general solutions. In this chapter, cross phenomena or coupled heat and mass transfer are discussed using the linear nonequilibrium thermodynamics theory. 

The linear nonequilibrium thermodynamics approach can provide a quantified description of the fully coupled phenomena for systems in the vicinity of global equilibrium. 

The set of Eq. (10.85) is related to various classical studies of electrokinetic phenomena, since the equations describe the coupled processes and yield naturally a number of symmetry relationships, which have been observed experimentally. Therefore, they provide a practical application of the linear nonequilibrium thermodynamic approach. For example, we may consider studies with identical solutions at each surface of the membrane, so that An = Ans = 0. Then the system has only two degrees of freedom, and we have... 

Here, a molecule of the salt dissociates into V cations of charge z, and v2 anions of charge z2, and F is the Faraday constant. The set of Eq. (10.104) is useful for the treatment of a composite membrane consisting of compartments in series. The practical parameters above were derived long before the linear nonequilibrium thermodynamics formulations... 

Note (ELM) Emulsion liquid membranes (SLM) supported liquid membranes (LNET) linear nonequilibrium thermodynamics (TR) conventional transport equations (R) conventional rate equations. 

Note. (LNET) Linear nonequilibrium thermodynamics approach (R) conventional rate equations. 

According to the Onsager s relations, three coefficients are to be determined. They are the passive permeability to sodium ZNa, the metabolic reaction coefficient if there is no sodium transport Zr, and the cross-coefficient between the chemical reaction and the sodium flow ZNar. The linear nonequilibrium thermodynamics formulation for the active transport of sodium and the associated oxygen consumption in frog skin and toad urinary bladders are studied experimentally. Sodium flow JNa is taken as positive in the direction from the outer to the inner surface of the tissue. The term JT is the rate of suprabasal oxygen consumption assumed to be independent of the oxygen consumption associated with the metabolic functions. 

This chapter starts with a simplified analysis of biological processes using the basic tools of physics, chemistry, and thermodynamics. It provides a brief description of mitochondria and energy transduction in the mitochondrion. The study of proper pathways and multi-inflection points in bioenergetics are summarized. We also summarize the concept of thermodynamic buffering caused by soluble enzymes and some important processes of bioenergetics using the linear nonequilibrium thermodynamics formulation. 

A starting point in linear nonequilibrium thermodynamic formulations is the representative dissipation function... 

The linear nonequilibrium thermodynamics formulations start with the rate of entropy production... 

The forces can be controlled in various ways to find a proper pathway leading to quasi-linear force-flow relationships so that the theory of linear nonequilibrium thermodynamics can be applied. For a first-order reaction S -> P, doubling the concentrations of S and P will double the reaction rate for an ideal system, although the affinity remains the same, and a distinction must be made between thermodynamic and kinetic linearity. Proper pathways are associated with thermodynamic linearity. The rate of a process depends not only on the force but also on the reference state the flow of a solute across a membrane depends on its chemical potential and on its thermodynamic state on both sides of the membrane. 

The determination of flow control coefficients is difficult, and requires the independent variation of the activity of all the enzymes within the pathway. Based on linear nonequilibrium thermodynamics, the kinetics of enzyme reactions can be described by the linear functions of the change in Gibbs free energy. This yields a direct relation between the elasticity coefficients and the change in Gibbs free energy for the reactions in a simple two-step pathway. 

The control coefficients can be determined by the linear nonequilibrium thermodynamics formulation. Schuster and Westerhoff (1999) provide a simple example for the coupled processes of oxidative phosphorylation with slipping enzymes, for which a representative dissipative function is... 

Control coefficients related to slipping enzymes can be calculated by the linear nonequilibrium thermodynamics approach. The overall control coefficients in the modular approach describe the control exerted by the particular degrees of freedom of a module on the measurable variables at steady state. Using the degree of coupling q (qop = Z0p/ ZpZ0) and the phenomenological stoichiometry Z (Z = Zp/Z0), the reation flows of. /ro and. /rp in Eq. (11.47) become... 

Stucki (1980, 1984) applied the linear nonequilibrium thermodynamics theory to oxidative phosphorylation within the practical range of phosphate potentials. The nonvanishing cross-phenomenological coefficients Ly(i v /) reflect the coupling effect. This approach enables one to assess the oxidative phosphorylation with H+pumps as a process driven by respiration by assuming the steady-state transport of ions. A set of representative linear phenomenological relations are given by... 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

Linear nonequilibrium thermodynamics has some fundamental limitations (i) it does not incorporate mechanisms into its formulation, nor does it provide values for the phenomenological coefficients, and (ii) it is based on the local equilibrium hypothesis, and therefore it is confined to systems in the vicinity of equilibrium. Also, properties not needed or defined in equilibrium may influence the thermodynamic relations in nonequilibrium situations. For example, the density may depend on the shearing rate in addition to temperature and pressure. The local equilibrium hypothesis holds only for linear phenomenological relations, low frequencies, and long wavelengths, which makes the application of the linear nonequilibrium thermodynamics theory limited for chemical reactions. In the following sections, some of the attempts that have been made to overcome these limitations are summarized. 

The first two terms on the right are the dissipation (entropy production times the absolute temperature), same as in the linear nonequilibrium thermodynamics. The last term is the contribution of the internal variables. The Pv acts as an internal variable and as a rate variable. The evolution equation for Pv is... 

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

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