11.3 Conservation of charge [species/charge transport top]

In this section, we sum the Nernst-Planck equations for all species i, weighted by z_iF , to obtain the charge conservation equations. In so doing, we naturally obtain theconductivity σ or themolar conductivity Λ.

11.3.1 Charge conservation equation

The Nernst-Planck equation (Equation 11.10) shows that the change in species concentration is given by a divergence of species flux densities. Since the valence of species i is given by z_i, and the charge of a mole of species i is given by z_iF , each species conservation equation also describes the charge transport owing to species i. Summing over all species, we obtain

(11.16)

The charge fluxes from this equation are shown in Figure 11.2.

If the fluid velocity is small as compared to the electrophoretic motion of the ions, Equation 11.16 can be rearranged (Exercise 11.4) to give

(11.17)

and, if the diffusivities of all species are the same and equal to D, Equation 11.18 becomes

Here, σ [C∕V m s] is the conductivity and is defined as σ = ∑_ic_iz_iFμ_EP,i, and the charge density ρ_E is defined as ρ_E = ∑_ic_iz_iF . The electrical conductivity thus naturally comes from the charge conservation equation, and is directly related to μ_EP,i. Physically, this is consistent with the notion that charge is conducted in an electrolyte solution owing to ion motion—the higher the electrophoretic mobility of the ionic components, the higher the conductivity of the solution.

11.3.2 Diffusivity, electrophoretic mobility, and molar conductivity

Molar conductivity Λ [m² C∕V s mol] is definedsuch that

The molar conductivity is convenient if the species transport equation is to be solved simultaneous with the charge conservation equation. The molar conductivity is proportional to the electrophoretic mobility, as we might expect, since ohmic conductivity stems from the ability of charged ions to move in response to an electric field:

Note that the molar conductivity is always positive, since z and μ_EP always have the same sign and their product is always positive.

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