Force conjugate

Let us now examine these electronic-nuclear coupling effects in more detail. The moderating exchange of electrons between the molecule and its hypothetical electron reservoir determines the effects of the electronic-nuclear coupling in the open molecular systems. Let us assume the initial electronic and geometric equilibria in such an initially open system p° = p.rej and F° = 0. The LeChatelier stability criteria of these two (decoupled) facets of the molecular structure requires that the conjugate forces A/jl(AN) or AFS(AQS) created by the primary electronic (AN> 0) or nuclear AQs > 0 displacements,... 

CONFORMATIONS Conjugated double bonds, DELOCALIZATION Conjugate force,... 

The common connection to underlying metric eigenmodes [cf. Eqs. (13.32), (13.34)] implies a deep-seated geometrical relationship between the conjugate forces and flows. To find this relationship, we can use the inverse of (13.34) [cf. the U-based (11.91b)]... 

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

Extensive Quantity Flux Conjugate Force Empirical Force—Flux Law ... 

In general, the fluxes may be expected to be a function of all the driving forces acting in the system, F) for instance, the heat flux Jq could be a function of other forces in addition to its conjugate force Fq that is,... 

Starting with Eq. 2.31 instead of Eq. 2.3 and repeating the procedure that led to Eq. 2.15, the conjugate force for the diffusion of component i in a network-constrained crystal takes the new form... 

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. 

On the other hand, irreversible thermodynamics has provided us with the insight that entropy generation is related to process flow rates like those of volume, V, mass in moles, h, chemical conversion, vl h, and heat, Q, and their so-called conjugated forces A(P/T), -A(p/T), A/T, and A(l/T). Although irreversible thermodynamics does not specify the relationship between these forces X and their conjugated flow rates /, it leaves no doubt about the... 

Phenomenological systems show that in relatively slow processes, the conjugate flow J is largely determined by frictional forces, and is linearly related to the conjugate force X... 

These equations are called the phenomenological equations, which are capable of describing multiflow systems and the induced effects of the nonconjugate forces on a flow. Generally, any force Xt can produce any flow./, when the cross coefficients are nonzero. Equation (3.175) assumes that the induced flows are also a linear function of non-conjugated forces. For example, ionic diffusion in an aqueous solution may be related to concentration, temperature, and the imposed electromotive force. 

This principle as originally stated by Curie in 1908, is quantities whose tensorial characters differ by an odd number of ranks cannot interact (couple) in an isotropic medium. Consider a flow J, with tensorial rank m. The value of m is zero for a scalar, it is unity for a vector, and it is two for a dyadic. If a conjugate force A) also has a tensorial rank m, than the coefficient Ltj is a scalar, and is consistent with the isotropic character of the system. The coefficients Lij are determined by the isotropic medium they need not vanish, and hence the flow J, and the force A) can interact or couple. If a force A) has a tensorial rank different from m by an even integer k, then Ltj has a tensor at rank k. In this case, Lfj Xj is a tensor product. Since a tensor coefficient Lt] of even rank is also consistent with the isotropic character of the... 

After identifying the conjugate forces and flows, for small forces of AT and Afi, the heat flow and mass flows may be represented by the following linear phenomenological equations... 

With the identified conjugate forces and flows, the linear phenomenological equations are... 

When a chemical reaction is the only process in a system, the entropy production is the products of flows Jr and conjugate forces X... 

Nonisothermal reaction-diffusion systems represent open, nonequilibrium systems with thermodynamic forces of temperature gradient, chemical potential gradient, and affinity. The dissipation function or the rate of entropy production can be used to identify the conjugate forces and flows to establish linear phenomenological equations. For a multicomponent fluid system under mechanical equilibrium with n species and A r number of chemical reactions, the dissipation function 1 is... 

Equation above shows the three contributions to the rate of entropy production due to heat flow, mass flow, and the chemical reaction, respectively, and excludes the viscous and electrical effects. As the membrane is assumed to be an isotropic medium, there will be no coupling between the vectorial heat and mass flows and scalar chemical reaction, according to the Curie-Prigogine principle. Under these conditions, entropy production equation identifies the conjugate forces and flows, and linear relations for coupled heat and mass flows become... 

Expanding these two flows as functions of their conjugate forces in a Taylor series about some reference steady state, and assuming all other forces as constant, yields the finite differences from the first-order terms... 

For nonequilibrium systems far from global equilibrium, the second law does not impose the sign of entropy variation due to the terms djS and d S, as illustrated in Figure 12.2. Therefore, there is no universal Lyapunov function. For a multicomponent fluid system with n components, entropy production in terms of conjugate forces Xu flows Jj, and / number of chemical reactions is... 

A thermodynamic flow system may be fully described in n-dimensions of flow and w-dimensions of conjugate force. According to Tellegen s theorem, we have... 

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. 

Now all spontaneous fluxes Jm of a given sign naturally bring about a change of opposite sign in their associated, conjugate forces [i.e., in 6Xp in this case see also Exercise 6.4.1]. This, in combination with the result just cited, means that Jm cannot be sustained the system ultimately returns to the initial, quiescent, stationary state. We thus deal here with an extension of Le Chdtelier s principle to steady-state... 

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