Onsager coefficient

The matrix of Fick diffusion coefficients in the molar average reference velocity frame for the system acetone(l)-benzene(2)-carbon tetrachloride(3) at a temperature of 25°C and composition = 0.70, X2 = 0.15, X3 = 0.15 has been obtained from the experimental data of Cullinan and Toor (1965) as 

DATA The Hessian matrix of the Gibbs free energy [G] may be calculated with the nonrandom two liquid (NRTL) model. The NRTL parameters are 

SOLUTION The Hessian [G] is calculated from Eq. D.3.4 and the NRTL model equations in Table D.8. The result is 

Note that [G]/7 r is dimensionless and symmetric also, the cross-coefficients G12 and G21 are a large fraction of the main coefficients G and G22. Multiplying each element of the above matrix by RT gives 

Note that the ORR are not satisfied precisely pointing to experimental inaccuracies in the measured data.  

To calculate the constants in Onsager s formula for the dependence on concentration of ionic conductance in an aqueous solution of a sjnmnetrical electroljle at 25 C. 

We shall use the following notation in this and related problems c molar concentration, 

In a solution of a single sjnnmetrical dectroljdie Onsager s formula can be written 

For water at 25 °C we have (see problem 78) a = 1.172 (1/mole) and (Dorsey, Properties of ordinary water-substance , Reinhold, 1940) 

For an unsjTnmetrical electroljte, composed of v cations of charge 2+e and v anions of change —z e, the formulae are more complicated but involve the same constants 6 and a. At an ionic strength I = + v ) srz c the formulae are 

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Linear response theory is an example of a microscopic approach to the foundations of non-equilibrium thennodynamics. It requires knowledge of tire Hamiltonian for the underlying microscopic description. In principle, it produces explicit fomuilae for the relaxation parameters that make up the Onsager coefficients. In reality, these expressions are extremely difficult to evaluate and approximation methods are necessary. Nevertheless, they provide a deeper insight into the physics. 

Here L is the Onsager coefficient and the minus sign (-) indicates that the concentration flow occurs from regions of high p to low p in order that the system irreversibly flows towards the equilibrium state of a... 

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

It should be kept in mind that all transport processes in electrolytes and electrodes have to be described in general by irreversible thermodynamics. The equations given above hold only in the case that asymmetric Onsager coefficients are negligible and the fluxes of different species are independent of each other. This should not be confused with chemical diffusion processes in which the interaction is caused by the formation of internal electric fields. Enhancements of the diffusion of ions in electrode materials by a factor of up to 70000 were observed in the case of LiiSb [15]. 

Here A(r — r ) is the Onsager coefficient that specifies the transport properties of the considered system at a certain timescale and lengthscale, and which is nonlocal in general. The local chemical potential difference p(r) can be found in a standard way as a functional derivative of the coarse-grained free energy functional F [<()] ... 

Finally, the noise term in Eq. (53) should satisfy the appropriate fluctuation-dissipation relation [1], In this way, all information about specific properties of the system enters into the dynamic equation (53) via the free-energy functional and Onsager coefficient. 

The Onsager coefficient for a simple binary mixture is usually written as... 

For simplicity, the Onsager coefficient is assumed to be a constant. The velocity field should satisfy the Navier-Stokes (NS) equation, which in the general case has the following form [158,159] ... 

The Onsager coefficient is given by the fluctuation-dissipation theorem ... 

The positivity of the entropy production, dS/dt = J Xi + J2X2 > 0, which is a quadratic form in the thermodynamic forces, implies for the Onsager coefficients... 

The Onsager coefficients (L),y are here evaluated in terms of the real symmetric matrix... 

These constitute a set of linear relationships between the potential differences pi — p q, which drive the Y) toward equilibrium and their corresponding rates, dYi/dt. In terms of the Onsager coefficients, they have the form... 

Comparison of Eq. 2.50 with Eqs. 2.49 and 2.51 shows that Lij = Lji and therefore demonstrates the role of microscopic reversibility in the symmetry of the Onsager coefficients. More demonstrations of the Onsager principle are described in Lifshitz and Pitaerskii [6] and in Yourgrau et al. [8]. 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

The chemical potential gradients and Onsager coefficients in Eq. 6.7 can be converted to concentration gradients and interdiffusivities (Table 3.1). Each chemical potential in Eq. 6.7 is a function of the local concentration ... 

The formal description of thermodiffusion in the critical region has been discussed in detail by Luettmer-Strathmann [79], The diffusion coefficient of a critical mixture in the long wavelength limit contains a mobility factor, the Onsager coefficient a = ab + Aa, and a thermodynamic contribution, the static structure factor S(0) [7, 79] ... 

Debye-Hiickel-Onsager theory — Table. Debye-Huckel-Onsager coefficients of 1-1-electrolytes at 298 K... 

Net flux for highly reversible reactions is proportional to reverse flux Near equilibrium (for AG <thermodynamic driving force J = —AAG, where X is called the Onsager coefficient [154, 155], When the near-equilibrium approximation AG RT holds, the flux ratio J+/J is approximately equal to 1. In this case Equation (3.12) is approximated ... 

Therefore for highly reversible systems, the net flux is proportional to the reverse flux times the thermodynamic driving force the Onsager coefficient is equal to J-/RT. 

Coefficients Ly are also called Onsager coefficients because they have the same properties as coefficients Ly. The flux form of writing the function P or djS/dt is identical to the force form (2.6) and may appear sometimes to be preferable for mathematical analysis. 

Note that in the preceding example, Ai2 / A21. However, if the system under consideration is close to the equilibrium, the nondiagonal Onsager reciprocity coefficients Ay transform to the conventional Onsager coefficients Ly, which appear to be index symmetrical, too. 

We must emphasize that in this example, again, Ai2 / A21. However, it is also easy to demonstrate that Li2 = L2i on approaching the equilibrium of the stepwise transformations when the true Onsager coefficients Ly relate to true thermodynamic forces (the affinities of the stepwise processes). 

What can be said about the values of the Onsager reciprocal coeffi dents allowing for an interrelation of heat conductance and step wise chemical transformations What is the difference between the coefficients of classic and modified Onsager coefficients ... 

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