Application of Irreversible Thermodynamics to Mass Transport in Ionic Conductors

Abstract

Despite the almost universal acceptance of the non-equilibrium, or irreversible, thermodynamics (NET) of Onsager¹ and his followers,^2,3 most of the published research concerning ion transport relies on the Nernst-Planck equations.⁴ The Nernst-Planck equations ignore cross-effects due to both equilibrium and non-equilibrium interactions. Cross-effects cannot be ignored in complete treatments of ion transport, as others^5–9 have also noted. For an ion of concentration c_i, charge z_i, and velocity v_i, the one dimensional Nernst-Planck equation is

$$ {{c}_{i}}{{v}_{i}}=-{{D}_{i}}\left( \frac{\partial {{c}_{i}}}{\partial x}-{{c}_{i}}{{z}_{i}}\frac{F}{RT}E \right) $$

(1.1)

where D_i is a diffusion coefficient, F is the Faraday constant, R the gas constant, T the temperature, and E the electric field. The principal inaccuracy in Eq. (1.1) is the omission of terms which represent the effects upon c_iv_i of interactions among ions of different kinds. The NET equation corresponding to (1.1) is, for n solute components,

$$ {{c}_{i}}{{v}_{i}}={{c}_{i}}{{v}_{0}}-\underset{j}{\mathop{\Sigma }}\,{{D}_{ij}}\frac{\partial {{c}_{j}}}{\partial x}+\frac{\lambda {{t}_{i}}}{F{{Z}_{i}}}E,i,j=1,\cdots ,n, $$

(1.2)

where v₀ is the velocity of the solvent (usually vanishingly small), λ is the specific conductance (often denoted σ by solid state physicists), and t_i is the Hittorf transference number for ion i in the particular solution. Eq. (1.2) reduces to Eq. (1.1) for very dilute solutions.