Force/flux pairs

Inasmuch as conjugate force-flux pairs have been selected, Onsager s reciprocity conditions apply Loi - L10. 

It is stressed that these relations apply only if the phenomenological equations are based on conjugate force-flux pairs. If other types of pairs are employed the coefficients with i j are functionally related but not equal. 

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... 

COMMON TRANSPORT MODES (FORCE/FLUX PAIRS)... 

You are likely already familiar with many of the simple direct force/flux pair relationships that are used to describe mass, charge, and heat transport—they include Pick s first law (diffusion). Ohm s law (electrical conduction), Fourier s law (heat conduction), and Poiseuille s law (convection). These transport processes are summarized in Table 4.1 using molar flux quantities. As this table demonstrates. Pick s first law of diffusion is really nothing more than a simplification of Equation 4.7 for... 

Table 1.1 Conjugate pairs of variables in work terms for the fundamental equation for the internal energy U. Here/is force of elongation, Z is length in the direction of the force, <7 is surface tension, As is surface area, , is the electric potential of the phase containing species i, qi is the contribution of species i to the electric charge of a phase, E is electric field strength, p is the electric dipole moment of the system, B is magnetic field strength (magnetic flux density), and m is the magnetic moment of the system. The dots indicate scalar products of vectors.

How is a concentration gradient of protons transformed into ATP We have seen that electron transfer releases, and the proton-motive force conserves, more than enough free energy (about 200 lcJ) per mole of electron pairs to drive the formation of a mole of ATP, which requires about 50 kJ (see Box 13-1). Mitochondrial oxidative phosphorylation therefore poses no thermodynamic problem. But what is the chemical mechanism that couples proton flux with phosphorylation ... 

To illustrate some new principles, a somewhat different approach will be used relative to the methods introduced in the earlier sections. As in Eqs. (6.9.1), we select (JS,VT) and (J,V(f/e)) as the conjugate set of variables but will include the T-1 factors in the phenomenological coefficients. Three new points are introduced at this time (i) Since fluxes may occur in two orthogonal directions, the conjugate flux-force pairs now are (J, VXT), (J, VyT), (Jx,Vx(C/e)), (Jy,Vy( /e)). The appropriate geometry is depicted in Fig. 6.10.1. (ii) For later convenience we shall select as independent variables from the this particular set the quantities VXT, VyT, J, J, so... 

We can extract three pairs of forces and fluxes the symmetric traceless strain rate, (Vu), and pressure, (P ), the difference between the director angular velocity and the angular velocity of the background fluid, (1/2)Vx u- 2 and... 

We can identify four pairs of thermodynamic forces and fluxes, the symmetric traceless strain rate (Vu) and the symmetric traceless pressure tensor, the director angular velocity relative to the background, (l/2)Vxu-I2 and the torque density X, the streaming angular velocity relative to the background (l/2)Vxu- and the torque density and the trace of the strain rate V-u and difference between the trace of the pressure tensor and the equilibrium... 

Consider first the case where no chemical reactions take place and where no particle fluxes occur. Then = —T Js O.lnXhe absence of any particle flux Js = Jq/T. ThenEq. (6.3.1) reads 9 = -T Jq-VT = Jq-Y 1/T). According to our standard interpretation J q and V (1/7) are conjugate flux-force pairs, so that the heat flux 7 q is driven by the gradient of 1/7. For small departures from equilibrium one may assume a linear dependence which is homogeneous, so that no additive constant prevents the flux from vanishing simultaneously with the force V(1/7). In short, we write... 

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

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