Onsager reciprocal relations, equation

In addition, for chemically identical species A and A, the Onsager reciprocal relations. Equations 4-14 and 4-15, yield [64] ... 

What has been done so far is to take experimental laws and express them in the form of phenomenological equations, i.e., Eqs. (6.300) and (6.301). Just as the phenomenological equations describing the equilibrium properties of material systems constitute the subject matter of equilibrium thermodynamics, the above phenomenological equations describing the flow properties fall within the purview of nonequilibrium thermodynamics. In this latter subject, the Onsager reciprocity relation occupies a fundamental place (see Section 4.5.7). 

Equations (13.44) are the celebrated Onsager reciprocity relations [L. Onsager. Reciprocal relations in irreversible processes. Phys. Rev. 37, 405-26 38, 2265-79 (1931)] that relate off-diagonal Fj 7 and Ft Jj coupling and serve as a cornerstone of modem transport theory. They are derived here without reference to statistical mechanical or fluctuation assumptions. 

Onsager s reciprocal relations state that, provided a proper choice is made for the flows and forces, the matrix of phenomenological coefficients is symmetrical. These relations are proved to be an implication of the property of microscopic reversibility , which is the symmetry of all mechanical equations of motion of individual particles with respect to time t. The Onsager reciprocal relations are the results of the global gauge symmetries of the Lagrangian, which is related to the entropy of the system considered. This means that the results in general are valid for an arbitrary process. 

Equations (3.331) and (3.332) indicate that the first derivatives of the potentials represent linear phenomenological equations, while the second derivatives are the Onsager reciprocal relations. 

These equations obey the Onsager reciprocal relations, which state that the phenomenological coefficient matrix is symmetric. The coefficients Lqq and Lu arc associated with the thermal conductivity k and the mutual diffusivity >, respectively. In contrast, the cross coefficients Llq and Lql define the coupling phenomena, namely the thermal diffusion (Soret effect) and the heat flow due to the diffusion of substance / (Dufour effect). 

This equation is independent of the type of phenomenological relations between fluxes and forces. In contrast, linear phenomenological equations and the Onsager reciprocal relations yield... 

Linear phenomenological equations obey the Onsager reciprocal relations. For the nonlinear region, from the symmetry of the Jacobian of forces versus flows, we have... 

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... 

Equations 3.30 and 3.31 allow for the possibility that each of the flux densities can depend on the differences in both chemical potentials, Afxw and Afis. Four phenomenological coefficients are used in these two equations (Table 3-2). However, by the Onsager reciprocity relation, Lws equals Lsw. Thus, three different coefficients (Lww> Lws> and Lss) are needed to describe the relationship of these two flux densities to the two driving forces. Contrast this with Equation 3.7 [Jj = ufj(-dfLj/dx)] in which a flux density depends on but one force accordingly, only two coefficients are then needed to describe Jw and Js. If the solute were a salt dissociable into two ions, we would have three flux equations (foriM i+, and / ) and three forces (Afxw> Afx+> and Afx ) ... 

Equation 2.3.22 expresses the Onsager reciprocal relations (ORR) discussed briefly in Section 3.3. For ideal gases, this symmetry relation can be obtained from the kinetic theory of gases (HCB, 1964 Muckenfuss, 1973). 

Equation 3.3.7 expresses the Onsager reciprocal relations (ORR), named after Lars Onsager who first established the principles of irreversible thermodynamics (Onsager, 1931). The ORR have been the subject of many journal papers receiving support as well as criticism, the latter from, in particular, Coleman and Truesdell (1960) and Truesdell (1969). We shall assume the validity of the ORR in the development that follows. 

Differentiating equation (3) with respect to unrestrained forces Xt and making use of Onsager reciprocal relations viz., Ly = Ly we get... 

Equation (11.5.15) forms the basis of the Onsager reciprocal relations. These symmetry relations can be expressed in matrix form as... 

Again the same convention as described earlier is used, and flow into the cell is considered to be in the positive direction. The osmotic pressure due to the permeant solute is denoted by which is defined as = RT(Cf - Att has units of dyne/cm. In these equations we speak of differences in concentrations in bulk phases since the partition coefficient which relates the concentration in the membrane phase to that of the bulk phase is incorporated in the permeability coefficients. The subscripts i and s refer to impermeant and permeant solute respectively. Lpj is the cross-coefficient for the volume flow arising from differences in the osmotic pressure of the permeant solute, Aw, when there is no difference of either hydrostatic or osmotic pressure produced by impermeant solutes ( Attj = 0). L p is the relative diffusional solute mobility per unit hydrostatic (or impermeant solute) pressure difference when Asr = 0. Although is always positive, and L p are both negative and have the same units as L. If the Onsager reciprocal relation holds, then is the diffusional flow and is a measure of the relative... 

When the fluxes are taken to be and q, — (9kr/2m )j , as suggested by the equation of change for entropy, then the numerical coefficients in the cross terms are the same, in agreement with the Onsager reciprocal relations. This provides one check on the evaluation of the flux expressions for the Hookean dumbbell model. 

An examination of the derived flux equations [18] leads to the following conclusions (i) Onsager reciprocity relation (ORR) is obeyed, (ii) All coefficients are scalar, (iii) Higher powers of single force do not occur, (iv) Space derivatives of forces occur in the transport equations, (v) In none of the X, Xj terms, the tensorial order of X and Xj term is the same, (vi) All the X,X terms have the same tensorial order as the fluxes, (vii) Non-linearity arises on account of gradient of barycentric velocity. 

This equation shows that the phenomenological coefficients are related to the reaction rates at equilibrium. The principle of detailed balance or microscopic reversibility is incorporated into /rB = 7 )3 = rf3,eq, and hence the Onsager reciprocal relations are valid. 

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