Conjugate forces and fluxes

Table 2.2 Conjugate Forces and Fluxes for Systems with Network-Constrained Components, i...

Show that products of the forces [i.e., the quantities (p q — p )], and the rates of reaction (i.e., the dYi/dt) which are present in Eq. 2.49 appear in the expression for the rate at which entropy is produced by the corresponding reactions. These quantities are therefore conjugate to one another just as are the conjugate forces and fluxes in Table 2.1. 

We begin here the study of thermodynamics in the proper sense of the word, by exploring a variety of physical situations in a system where one or more intensive variables are rendered nonuniform. So long as the variations in T, P, /x or other intensive quantities are small relative to their average values, one can still apply the machinery of equilibrium thermodynamics in a manner discussed later. It will be seen that the identification of conjugate forces and fluxes, the Onsager reciprocity conditions, and the rate of entropy production play a central role in the analysis provided later in the chapter. 

Next we will use Jv and JD to express the linear interdependence of conjugate forces and fluxes in irreversible thermodynamics (Eq. 3.29 ... 

It is asserted that if one determines pairs of conjugant forces and fluxes in the laboratory so that... 

Onsager s principle supplements these postulates and follows from the statistical theory of reversible fluctuations [5]. Onsager s principle states that when the forces and fluxes are chosen so that they are conjugate, the coupling coefficients are... 

If the forces and fluxes are properly selected, they are canonically conjugated in the sense of irreversible thermodynamics (Lf/ = L ) (Onsager relation). There are two distinctly different kinds of forces which may induce a flux. The first and more familiar is the electrical force. The second kind of forces has its origin in statistics it is not a force in the usual sense, but normally arises from concentration gradients and results in diffusive fluxes. In all our discussions we shall be concerned with an assembly of electrons, i.e. a Fermi system, for which the chemical potential (ji(T)) at r = 0 K is the Fermi energy. Then it is common practice to introduce the electrochemical potential, and to define an effective electric field E = — V(d> + pje), which is the field that is normally observed. 

Our starting point is a generic constmction for the extraction of work from a flow of heat (cf. Fig. 4). The auxiliary system—for example, a Brownian motor— performs work, W = Fx, against an external force F, where x is the corresponding variation of the thermodynamically conjugated variable. The system is at a temperature T and we introduce the corresponding thermodynamic force, Xi =F/T, and flux J = x (the dot referring to the time derivative). 

According to the second law, the dissipation function must be positive if not zero, which of course is to be expected here, since we are dealing with a spontaneously occurring passive process. The thermodynamic force A/x+, which contains both a concentration-dependent component and an electrical component, is the sole cause of the flow J+. In a system in which more than one process occurs, each process gives rise to a term in the dissipation function consisting of the product of an appropriate force and its conjugate flow. In the case of active transport of the cation, as found, for example, in certain epithelial tissues, the cation flux is coupled to a metabolic reaction. If we represent the flow or velocity of the reaction per unit area of membrane by Jr, the appropriate force driving the reaction is... 

Table 2.1 Selected Conjugate Forces, Fluxes, and Empirical...

What are the forces conjugate to the flux of heat, electric current, diffusion and chemical reaction if the dissipation function is used to define the forces ... 

The diffusional flux density (Eq. 3.35) is the difference between the mean velocities of solute and water. In mass flow (such as that described by Poiseuille s law Eq. 9.11), vs equals vw, so JD is then zero such flow is independent of All and depends only on AP. On the other hand, let us consider All across a membrane that greatly restricts the passage of some solute relative to the movement of water, i.e., a barrier that acts as a differential filter vs is then considerably less than vw, so JD has a nonzero value in response to its conjugate force, All. Thus Jo helps express the tendency of the solute relative to water to diffuse in response to a difference in osmotic pressure. 

A typical proposal of how the Onsager relations provides useful information about chemical reactions proceeds as follows [DeGroot (39)] The true kinetics of a chemical reaction are very often not known and the principle of detailed balancing cannot be readily applied. In such cases the symmetry of the matrix L can be used to obtain some useful information about the process. One proceeds by noting that the conjugant forces (x) and fluxes (j) in Eq. (436) are related by... 

Non-equilibrium thermodynamics was founded by Onsager. The theory was further elaborated by de Groot and Mazur and Prigogine. The theory is based on the hypothesis of local equilibrium a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this criterion. " Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the flux. - ... 

The non-equilibrium thermodynamic approach to models of epithehal transport begins with the identification of a system of relevant fluxes, J, and conjugate driving forces, A i [19]. With this system the energy dissipation of the model, 4>, may be expressed as a sum of the products of conjugate flows and forces... 

An even more potent concept comes from the Onsager reciprocal relations, which states that there is a coupling between conjugate force-flux pairs. For example, mass transfer of species by diffusion in an aqueous solution causes a change in concentration, which is accompanied by heat consumption or release due to the heat of dilution. This sets up a thermal gradient, which causes heat flow. The resulting link between heat flux and isothermal diffusion is the Dufour effect. Its conjugate is the Soret effect, which is the diffu-sional mass flux linked to heat flow. The Soret effect has been coupled with... 

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