Thermodynamics, irreversible

Displacements toward equilibrium are irreversible or, more descriptively, one way only. An elegant discipline describing these displacements is irreversible thermodynamics, sometimes called nonequilibrium thermodynamics. The four fundamental postulates of irreversible thermodynamics are (1)  

Entropy S increases continuously for irreversible processes in isolated systems. 

and other thermodynamic (equilibrium) functions can be defined and used even though equilibrium does not exist. 

Fluxes are proportional to gradients or so-called generalized forces. For example, diffusional flux is proportional to the concentration gradient dc/dx, not to (dc/dx)2. Electrical current is proportional to voltage, and so on. This linearity is expressed by 

Onsager s reciprocity relationships are valid. According to Onsager the coefficients are related by 

Considerable effort has been expended in the attempt to develop a general theory of reaction rates through some extension of thermodynamics or statistical mechanics. Since neither of these sciences can, by themselves, yield any information about rates of reactions, some additional assumptions or postulates must be introduced. An important method of treating systems that are not in equilibrium has acquired the title of irreversible thermodynamics. Irreversible thermodynamics can be applied to those systems that are not too far from equilibrium. The theory is based on the thermodynamic principle that in every irreversible process, that is, in every process proceeding at a finite rate, entropy is created. This principle is used together with the fact that the entropy of an isolated system is a maximum at equilibrium, and with the principle of microscopic reversibility. The additional assumption involved is that systems that are slightly removed from equilibrium may be described statistically in much the same way as systems in equilibrium. 

Central to the thermodynamic discussion of irreversible processes is the concept of entropy production. Consider the Clausius inequality, dS Q/T, which we can rearrange to the form 

Principle of microscopic reversibility at equilibrium, any molecular process occurs at the same rate as the reverse of that process. 

The quantity on the left is greater than or equal to zero, so we may write 

If we suppose that the system is in contact with a reservoir at T, and a quantity of heat dQ flows into the system, then a quantity, — 4Q, flows into the reservoir. If the quantity, — 4Q, is transferred reversibly to the reservoir, then the entropy change of the reservoir is dS s = —4Q/T, and we can write Eq. (33.13) as 

The multicomponent flux equations given in Eq. 7.1-2 are empirical generalizations of Pick s law that define a set of multicomponent diffusion coefficients. Because such definitions are initially intimidating, many have felt the urge to rationalize the origin of these equations and buttress this rationale with more fundamental principles. This emotional need is often met with derivations based on irreversible thermodynamics. 

Three basic postulates are involved in the derivation of Eq. 7.1-2 (Fitts, 1962). The first postulate states that thermodynamic variables such as entropy, chemical potential, and temperature can in fact be correctly defined in a differential volume of a system that is not at equilibrium. This is an excellent approximation, except for systems that are very far from equilibrium, such as explosions. In the simple derivation given here, we assume a system of constant density, temperature, and pressure, with no net flow or chemical reaction. More complete equations without these assumptions are derived elsewhere (e.g., Haase, 1969). 

The mass balance for each species in this type of system is given by 

In this continuity equation, we use the fact that at no net flow and constant density, n,-equals /, the flux relative to the volume or mass average velocity. We also imply that the concentration is expressed in mass per unit volume. The left-hand side of this equation represents solute accumulation, and the right-hand side represents the solute diffusing in minus that diffusing out. The energy equation is similar  

By parallel arguments, we can write a similar equation for entropy  

This is a situation, similar, as a plunger would move in the center of the capillary. Likewise, we could rearrange Eq. (19.11) to obtain 

Transport phenomena can be explained by the theory of irreversible thermodynamics [3]. First of all we recall the nature of energy. Energy has the physical dimension of kgm s . Consequently, the derivative of energy with respect to a length x, dUldx, has the physical dimension of a force F, namely kgms.  

This differential is nonzero, if the energy is changing in space. Pictorially, a force is acting on some properties or variables that make up the energy, such as entropy, volume, matter, in order to change these variables to boil down the force to zero. 

We exemplify the situation with matter. At constant temperature and pressure, as explained elsewhere, instead of ordinary energy we must use the free enthalpy G(T, p, n) instead of energy U S, V, n). Moreover, we use molar quantities, G(T, p). 

Now we subdivide the system into small subsystems and we assume that the molar free enthalpy is varying with position. Thus, in addition, the molar free enthalpy will become a function of position (jc, y, z) = r and in addition, if we allow a variation in time, also a function of time t. 

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Caiien H B 1960 Thermodynamics, an Introduction to the Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics (New York Wiiey)... 

Zwanzig R 1961 Memory effects in irreversible thermodynamics Phys. Rev. 124 983... 

Reverse osmosis models can be divided into three types irreversible thermodynamics models, such as Kedem-Katchalsky and Spiegler-Kedem models nonporous or homogeneous membrane models, such as the solution—diffusion (SD), solution—diffusion—imperfection, and extended solution—diffusion models and pore models, such as the finely porous, preferential sorption—capillary flow, and surface force—pore flow models. Charged RO membrane theories can be used to describe nanofiltration membranes, which are often negatively charged. Models such as Dorman exclusion and the... 

Kelvin showed the interdependence of these phenomena by thermodynamic analysis, assuming that the irreversible processes were independent of the reversible ones. This approach was later proved theoretically sound using Onsager s concepts of irreversible thermodynamics (9). 

Dialysis transport relations need not start with Eickian diffusion they may also be derived by integration of the basic transport equation (7) or from the phenomenological relationships of irreversible thermodynamics (8,9). 

Retention Rejection and Reflection Retention and rejection are used almost interchangeably. A third term, reflection, includes a measure of solute-solvent coupling, and is the term used in irreversible thermodynamic descriptions of membrane separations. It is important in only a few practical cases. Rejection is the term of trade in reverse osmosis (RO) and NF, and retention is usually used in UF and MF. 

Irreversible thermodynamics has also been used sometimes to explain reverse osmosis [14,15]. If it can be assumed that the thermodynamic forces responsible for reverse osmosis are sufficiently small, then a linear relationship will exist between the forces and the fluxes in the system, with the coefficients of proportionality then referred to as the phenomenological coefficients. These coefficients are generally notoriously difficult to obtain, although some progress has been made recently using approaches such as cell models [15]. 

The thermodynamic method has limitations. Since the method ignores the intermediate stages, it cannot be used to determine shock-wave parameters. Furthermore, a shock wave is an irreversible thermodynamic process this fact complicates matters if these energy losses are to be fully included in the analysis. Nevertheless, the thermodynamic approach is a very attractive way to obtain an estimate of explosion energy because it is very easy and can be applied to a wide range of explosions. Therefore, this method has been applied by practically every worker in the field. 

It should be noted that the simple Nernst equation cannot be used since the standard electrode potential is markedly temperature dependent. By means of irreversible thermodynamics equations have been computed to calculate these potentials and are in good agreement with experimentally determined results. 

It should be kept in mind that all transport processes in electrolytes and electrodes have to be described in general by irreversible thermodynamics. The equations given above hold only in the case that asymmetric Onsager coefficients are negligible and the fluxes of different species are independent of each other. This should not be confused with chemical diffusion processes in which the interaction is caused by the formation of internal electric fields. Enhancements of the diffusion of ions in electrode materials by a factor of up to 70000 were observed in the case of LiiSb [15]. 

The reason for the exponential increase in the electron transfer rate with increasing electrode potential at the ZnO/electrolyte interface must be further explored. A possible explanation is provided in a recent study on water photoelectrolysis which describes the mechanism of water oxidation to molecular oxygen as one of strong molecular interaction with nonisoenergetic electron transfer subject to irreversible thermodynamics.48 Under such conditions, the rate of electron transfer will depend on the thermodynamic force in the semiconductor/electrolyte interface to... 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

According to irreversible thermodynamics, the entropy production per unit volume S for an isothermal system can be written... 

Yao, Y. L., Irreversible Thermodynamics, Science Press, Beijing and Van Nostrand-Reinhold, New York, 1981. 

Since the ionic fluxes cannot be measured individually, it is preferable to introduce the salt flux, besides solvent flux and charge flux (current density). The driving forces would then be the gradients or differences of the chemical potentials in media with different salt concentrations and different pressures, multiplied by -1. These differences must be relatively small to remain within the framework of linear irreversible thermodynamics, so that... 

The transport of both solute and solvent can be described by an alternative approach that is based on the laws of irreversible thermodynamics. The fundamental concepts and equations for biological systems were described by Kedem and Katchalsky [6] and those for artificial membranes by Ginsburg and Katchal-sky [7], In this approach the transport process is defined in terms of three phenomenological coefficients, namely, the filtration coefficient LP, the reflection coefficient o, and the solute permeability coefficient to. 

SZ Song, JR Cardinal, SJ Wisniewski, SW Kim. Mechanisms of solute permeation through hydrogel films An irreversible thermodynamic approach. Abstracts of Papers Presented at the 126th National Meeting of the American Pharmaceutical Association, 1979. 

Another theoretical basis of the superheated liquid-film concept lies on the irreversible thermodynamics developed by Prigogine [43]. According to this theory, irreversible chemical processes would be described (Equation 13.17) by extending the equation of De Donder, provided that simultaneous reactions were coupled in a certain thermodynamic model, as follows ... 

Various theories, ranging from qualitative interpretations to those rooted in irreversible thermodynamics and geochemical kinetics, have been put forward to explain the step rule. A kinetic interpretation of the phenomenon, as proposed by Morse and Casey (1988), may provide the most insight. According to this interpretation, Ostwald s sequence results from the interplay of the differing reactivities of the various phases in the sequence, as represented by Ts and k+ in Equation 26.1, and the thermodynamic drive for their dissolution and precipitation of each phase, represented by the (1 — Q/K) term. 

Irreversible perturbation reactions, 14 617 Irreversible thermodynamics models, 21 638, 661... 

Yong RN, Samani HMV (1989) Analysis of two-dimensional solute transport in clay soils using irreversible thermodynamics. Proc CANCAM 23 54... 

K. G. Denbigh, The Thermodynamics of the Steady State, Methuen, London, 1951 H. B. CaUen, Thermodynamics and an Introduction to Thermostatics, 2nd ed., Wiley, New York, 1985, Chapter 14 B. C. Eu, Kinetic Theory and Irreversible Thermodynamics, Wiley, New York, 1992 D. Kondepudi and I. Prigogine, Modem Thermoodynamics, Wiley, New York, 1990 Y. Demitel and S. I. Sandler, J. Phys. Chem. B 108, 31-43 (2004). 

LS.4. I. Prigogine, Etude Thermodynamique des Phenomenes Irreversibles, (Thermodynamic Study of Irreversible Phenomena) (These d agregation de I Enseignement superieur, Universite Libre de Bruxelles), Dunod, Paris et Desoer, Liege. 

TNC.6. 1. Prigogine and R. Balescu, Cychc Processes in Irreversible Thermodynamics, in Proceedings International Symposium on Transport Processes in Statistical Mechanics, Brussels, 1958, Interscience Publishers, New York, 1958, pp. 343-345. 

MSN.92. I. Prigogine, M. Theodosopoulou, and A. Grecos, On the derivation of linear irreversible thermodynamics for classical fluids, Proc. Natl. Acad. Sci. USA, 75, 1632-1636 (1978). 

The power (work by the system per unit time) is thus W = —Fx = —JiXiT. The work is performed under the influence of a heat flux Q leaving the hot reservoir at temperature Ti. The cold reservoir is at temperature T2 (where T > T2). The corresponding thermodynamic force is X2 = I/T2 — 1/Ti, and the flux is J2 = Q. The temperature difference Ti —T2 = AT is assumed to be small compared to T2 T kT, so one can also write X2 = AT/T. Linear irreversible thermodynamics is based on the assumption of local equilibrium with the following linear relationship between the fluxes and forces ... 

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