Thermodynamics linear irreversible

In Sections 3.4, 3.5 and 3.6 we have derived expressions for the entropy production associated with spontaneous evolutions in systems which are not in equilibrium states. Inspection of these expressions as given in (3.19), (3.29), (3.33) and (3.49) shows us that the most general case of entropy production may be written in the form 

The summation index a in (3.78) runs through all irreversible processes in the system, and the superscript (i) in indicates that only the irreversible part of the change of entropy is included. 

From (3.79) we see that the forces are differences of intensive variables. This also applies to chemical reactions since according to (3.50) the affinity is defined as a balance of chemical potentials weighted by the stoichiometric coefficients. On the other hand, the fluxes in (3.79) are time derivatives of extensive variables. 

Clearly, the system is in complete equilibrium only if all of the forces vanish, F = 0, a = 1,2,. .. which implies that also all of the fluxes vanish, = 0, a = 1,2,. .. and vice versa. This conclusion suggests considering the forces F as causing the fluxes as their effects or vice versa, that is, to write 

Two important points should be emphasized in context with (3.80). First, it is usually observed in nonequilibrium systems that a force F not only gives rise to its conjugated flux I., but also to other fluxes I with B + a. This is expressed in (3.80) by writing the flux as a function of all forces F, F2 . .. The second 

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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... 

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 ... 

A more rigorous way to generalize Pick s law is to use phenomenological equations based on linear irreversible thermodynamics. In this treatment of an N-component system, the diffusive flux of component i is (De Groot and Mazur,... 

Solid state reactions occur mainly by diffusional transport. This transport and other kinetic processes in crystals are always regulated by crystal imperfections. Reaction partners in the crystal are its structure elements (SE) as defined in the list of symbols (see also [W. Schottky (1958)]). Structure elements do not exist outside the crystal lattice and are therefore not independent components of the crystal in a thermodynamic sense. In the framework of linear irreversible thermodynamics, the chemical (electrochemical) potential gradients of the independent components of a non-equilibrium (reacting) system are the driving forces for fluxes and reactions. However, the flux of one independent chemical component always consists of the fluxes of more than one SE in the crystal. In addition, local reactions between SE s may occur. 

Dissipation as an organizing factor can only occur under certain conditions, necessary prerequisites are energy input (open system), nonlinear dynamics and stabilization far away from thermal equilibrium i.e. beyond the regime of linear irreversible thermodynamics (vid. the requirements of Chapter I). 

In thermodynamic equilibrium, the electrochemical potential of a particle k (juk = Hk + zkeq>, juk = chemical potential,

electrical potential, zk = charge number of the particle, e = elementary charge) is constant. Gradients in jlk lead to a particle flux Jk and from linear irreversible thermodynamics [95] the fundamental transport... 

Two approaches will be considered one is to apply linear irreversible thermodynamics according to which the generalized rate... 

Formally, it will be even necessary to make corrections already in the starting flux equations. The detailed formulation of linear irreversible thermodynamics also includes coupling terms (cross terms) obeying the Onsager reciprocity relation. They take into account that the flux of a defect k may also depend on the gradient of the electrochemical potential of other defects. This concept has been worked out, in particular, for the case of the ambipolar transport of ions and electrons.230... 

In the framework of linear irreversible thermodynamics it is possible to handle responses to electrical and/or chemical driving forces, as could be seen from the previous sections, by what is called Nernst-Planck-... 

We start with the equations of linear irreversible thermodynamics... 

Dhar modelled the stretching of a polymer using the stochastic Rouse model, for which distributions of various definitions of the work can be obtained. Two mechanisms for the stretching were considered one where the force on the end of the polymer was constrained and the other where its end was constrained. Dhar commented that the variable selected for the work was only clearly identified as the entropy production in the latter case. In the former case they argue that the average work is non-zero for an adiabatic process, and therefore should not be considered as an entropy production, however we note that the expression is consistent with a product of flux and field as used in linear irreversible thermodynamics. 

Note that the above-mentioned phenomena are irreversible in nature and can be properly understood on the basis of the linear irreversible thermodynamics. The diffusion potential, Ed, arises if two solutions are in contact. This phenomenon is a result of the different mobility properties of the ion species and can theoretically be estimated if the individual ion electric conductivities and activities of the species are known. The diffusion potential should... 

Let us now summarize the underlying principle of the IPMC s actuation and sensing capabilities, which can be described by the standard Onsager formula using linear irreversible thermodynamics. When static conditions are imposed, a simple description of mechanoelectric effect is possible based upon two forms of transport ion transport (with a current density, /, normal to the material) and solvent transport (with a flux, Q, that we can assume is water flux). 

The experimental deformation characteristics are clearly consistent with the above predictions obtained by the above linear irreversible thermodynamics formulation, which are also consistent with Eqs. 2.5 and 2.6 in the steady state conditions, and have been used to estimate the value of the Onsager coefficient L to be of the order of 10 m A -s (Fig. 2.3). 

Macroscopic properties of nematic elastomers have been discussed [56, 60]. De Gennes focused on the static properties, emphasizing especially the importance of coupling terms associated with relative rotations between the network and the director field [60]. The electrohydrodynamics of nematic elastomers has been considered generalizing earlier work by the same authors [54,55] on the macroscopic properties of nematic sidechain polymers [56]. The static considerations of earlier work [60] were extended to incorporate electric effects in addition a systematic overview of all terms necessary for linear irreversible thermodynamics was given [56]. 

We also note that the vector or tensor responses (3.187), (3.189) depend only on the vector or tensor driving forces respectively. This fact is known in linear irreversible thermodynamics as the Curie principle [36, 80, 88, 89] (cf. discussion in [34, 38]). Present theory shows however, that this property follows from the isotropy of constitutive functions and from the representation theorems of such linear functions, see Appendix A.2, Eqs.(A.ll)-(A.13) and (A.57)-(A.59). But representation theorems for nonlinear isotropic constitutive functions [64, 65] show that the Curie principle is not valid generally. 

From a physical point of view, it seems that measurable quantities are mixture invariant (cf. end of Sect. 4.4). Such are the properties of mixture like y, T (see (4.94), (4.236), (4.240), (4.225)) but also the chemical potentials ga. Note that also heat flux is transformed as (4.118) (with functions (4.223)) and therefore heat flux is mixture invariant in a non-diffusing mixture (all = o) in accord with its measurability. But heat flux is mixture non-invariant in a diffusing mixture, consistently with our expectation of difficulties in surface exchange (of masses) of different constituents with different velocities together with heat. We note that all formulations of heat flux used in linear irreversible thermodynamics [1, 120] (cf. Rems. 11 in this chapter, 14 in Chap. 2) are contained (by arbitrariness of rjp) in expression (4.118) for heat flux in a diffusing mixture. 

These and all previous results of thermodynamic mixture which also fulfil Gibbs-Duhem equations (4.263) show the complete agreement with the classical thermodynamic of mixtures but moreover all these relations are valid much more generally. Namely, they are valid in this material model—linear fluid mixture—in all processes whether equilibrium or not. Linear irreversible thermodynamics [1-4], which studies the same model, postulates this agreement as the principle of local equilibrium. Here in rational thermodynamics, this property is proved in this special model and it cannot be expected to be valid in a more general model. We stress the difference in the cases when (4.184) is not valid—e.g. in a chemically reacting mixture out of equilibrium—the thermodynamic pressures P, Pa need not be the same as the measured pressure (as e.g. X =i Pa) therefore applications of these thermodynamic... 

Linearization of this equation in gives the phenomenological equation for reaction rate of linear irreversible thermodynamics [1, 3, 4, 130]. But there is a controversion here [159] 112 is contained in affinity (4.479) as well as in the first part of (4.489) which is considered as constant in such linearization, cf. below (4.494), see also [158]. 

We have studied a regular linear fluid mixture where most of the results for transport phenomena (4.137), (4.138), (4.165), (4.166) (viscosity, diffusion, heat conduction and cross effects) are not in a form useful in practice [76, 104, 180, 181]. In this section we transform them into a more convenient form which is also used in linear irreversible thermodynamics [1, 27, 28, 119, 120, 130, 182]. Onsager relations will be also noted and some applications, like Pick law and the electrical conductivity of electrolytes are discussed. 

This approach was furthered by Narasimhan and Peppas [65] to understand the dissolution of a glassy polymer in a solvent. The concentration field was divided into three regimes similar to the previous approach [61]. A linear irreversible thermodynamics argument was us to obtain the volumetric flux of the solvent as... 

Further conclusions can be drawn from the entropy production (3) applying the principles of linear irreversible thermodynamics Cl] - [3 3. By assuming a near equilibrium situation, an ideal gas state in the vapour and by taking into account that the pressure dependence on the chemical potential in the liquid phase is negligible, Eq.(3) can be recast into the following form ... 

Within the framework of linear irreversible thermodynamics (LIT) as introduced in Section 3.9 a negative interaction would be reflected by a negative coefficient Li2 = L21 < 0 In contrast to the scheme of LIT, however, the interaction incorporated in (5.10) is of a nonlinear type. Expansion of J. in terms of Ay. = y. - y. ... 

Uphill transport of molecules against increasing values of their concentration or potential can be discussed within the framework of linear irreversible thermodynamics (LIT) as presented in Section 3.9. If we have, for example, a 2-component system with linear phenomenological relations of the type of (4.81),... 

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