MULTICOMPONENT DIFFUSION: check the Nernst equation

The diffusion coefficient of a salt in water is given by the harmonic mean of the diffusion coefficients of the components of the salt, weighted by their molal concentration:
                                        (1),
which results, for a binary salt, in the Nernst equation:
                         (2)

For example, with DCa = 0.79*10-9 m2/s and DCl = 2.03*10-9 m2/s, the diffusion coefficient for CaCl2 is DCaCl2 = 1.34*10-9 m2/s. Thus, to maintain overall charge balance, diffusion of Ca2+ is accelerated, while Cl- is impeded somewhat.

PHREEQC input file mcd_ex1.phr calculates 3 years diffusion of 0.1 mM CaCl2 in a sediment with 0.1 µM KBr in the pores. A tracer with Dw = 1.34*10-9 m2/s is added to show that the PHREEQC calculations are in agreement with the Nernst equation.
In the figure, the lines for Ca2+, Cl-/2, and the tracer overlap.

  • The tracer Ca_t has zero charge. Will it diffuse differently if it is given a charge of +2? (Yes! (shown by the model points of 1.34e-9(z=+2); uncomment the plotting of Ca_tz in the input file)).
    The charged tracer is also dragged along by Cl-. Vinograd and McBain (1941) verified this effect experimentally with HCl / BaCl2 solutions. They were also the first to formulate the zero-charge flux condition that is used in PHREEQC's multicomponent diffusion module.

  • Will CaCl2 diffuse differently in a 100 mM KBr solution? (Yes! (try it out, change the concentrations in SOLUTION 1-20)).
    Note that the Ca2+ concentration in the sediment near the interface is higher than in the boundary solution, because diffusion is enhanced by the low activity-coefficients in the concentrated KBr solution. Also note that the concentrations of Cl-/2 and Ca2+ now diverge.

  • What happens if the Cl-concentration in the boundary solution is zero, resulting in a charge imbalance of +100%?
    Simple: the zero-charge flux forbids diffusion of Ca2+, while the uncharged tracer Ca_t continues to diffuse at the same rate. (By increased diffusion of OH- compared to H+ (plot pH on the second Y-axis), and also by counterdiffusion from the KBr solution, a tiny flux of Ca2+ does occur.) Diffusion also stops if Cl- is given a negligible diffusion coefficient, Dw = 1e-30 m2/s, or if Cl- is replaced by an immobile SURFACE. (try it out!)

    Reference
    Vinograd, J.R. and McBain, J.W., 1941. Diffusion of electrolytes and of the ions in their mixtures. J. Am. Chem. Soc. 63, 2008-2015.

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