A natural soil will always contain various exchangers which may have different selectivities for a given
cation. When the solute concentrations increase from zero, the most selective exchange sites fill up first,
the less selective sites follow later. As a result, the sorption isotherm becomes a parabola with a smaller
slope at higher concentrations. The effect can be modeled with the Rothmund-Kornfeld equation (see Appelo
and Postma, p. 258), but the exponents in this equation lack physical meaning.
Alternatively, the
active fraction model can be used, which is derived from the constant capacitance theory for surface
complexation (Appelo (1994): 763 kB pdf ). In the
active fraction model, the association constant of the exchange reaction is related to the equivalent
fraction occupied by the cation. For example for the exchange-half reaction of Na+,
Na+ + X- = NaX
the association constant becomes:
log K = log_k + a_f * (1 - βNa)
where log_k is the log of the equilibrium constant when all the sites are occupied by Na
+, a_f is the active fraction coefficient and β Na is the
equivalent fraction of Na+ in the exchanger. The Rothmund-Kornfeld and
active-fraction models for K+/Ca2+
exchange are compared in PHREEQC input file rk_af.phr. The output
shows that the 2 models can give the same results.
The active-fraction model is useful for modeling proton exchange on organic matter. It enables to calculate the fraction of base cations in the CEC of Dutch soils as a function of pH, shown in the figure (data from Van der Molen, cf. Appelo (1994): 763 kB pdf ). Note that protons occupy about 20% of the CEC at pH = 7, and 60% at pH = 5. The model line is obtained with PHREEQC input file pc_vdm.phr.
In a freshening aquifer, where calcite dissolves when Ca2+ is lost by cation exchange, the pH will be buffered by proton exchange, as in the Aquia aquifer.