This simulation plots the equilibrium isotherm for ion exchange of Cs⁺ in solution with Na⁺ on crystalline silicotitanate (CST), an inorganic ion exchanger used to remove radioactive cesium from nuclear waste. The solid has two types of exchange sites, and each site has its own selectivity coefficient K_C. The solid black curve is the overall isotherm; it is the sum of the contributions of the two sites (dashed curves).

Use the K_C1 and K_C2 sliders to set the selectivity coefficient of each site (log scale from 0.01 to 1000). The third slider sets how the exchange capacity is split between the sites: the left end is only site 1, the right end is only site 2, and the fraction of capacity in site 2 is shown beside the slider.

The grey dashed line is the isotherm for K_C = 1 (no preference for either ion). Click or drag on the plot to move the red point; hold the mouse over the point to show the liquid and solid equivalent fractions there. The initial slope of the isotherm (the Henry's law constant) is shown on the plot. Uncheck "show site contributions" to show only the overall isotherm.

Cesium in solution exchanges with sodium in the solid (both ions have a +1 charge):

\[ \mathrm{Cs^+} + \overline{\mathrm{Na^+}} \rightleftharpoons \overline{\mathrm{Cs^+}} + \mathrm{Na^+} \]

where the overbar denotes the solid phase. The selectivity coefficient of a site, written with equivalent fractions, is

\[ K_C = \frac{Q_{Cs}\,C_{Na}}{Q_{Na}\,C_{Cs}} = \frac{\bar{X}_{Cs}\,(1-X_{Cs})}{(1-\bar{X}_{Cs})\,X_{Cs}} \]

The equivalent fractions in the liquid and solid are

\[ X_{Cs} = \frac{C_{Cs}}{C_0}, \qquad \bar{X}_{Cs} = \frac{Q_{Cs}}{Q_0} \]

For a single type of site with an ideal solid phase, solving for the solid fraction gives the isotherm

\[ \bar{X}_{Cs} = \frac{K_C\,X_{Cs}}{1+(K_C-1)\,X_{Cs}} \]

which is concave for K_C > 1, a straight line for K_C = 1, and convex for K_C < 1. When two types of sites exchange independently, the isotherm is the capacity-weighted sum of the two single-site isotherms (eq 21 in Zheng et al.):

\[ \bar{X}_{Cs} = \bar{X}_{01}\,\frac{K_{C1}\,X_{Cs}}{1+(K_{C1}-1)\,X_{Cs}} + \bar{X}_{02}\,\frac{K_{C2}\,X_{Cs}}{1+(K_{C2}-1)\,X_{Cs}} \] \[ \bar{X}_{01} + \bar{X}_{02} = 1 \]

Each dashed curve in the plot is one term of this sum. At low cesium concentration the isotherm reduces to Henry's law, with an initial slope equal to the capacity-weighted average selectivity:

\[ \bar{X}_{Cs} \approx \left(\bar{X}_{01}K_{C1} + \bar{X}_{02}K_{C2}\right) X_{Cs} \]

When one site has K_C > 1 and the other has K_C < 1, the isotherm can have an inflection: the selective site fills first, and the other site loads only at high liquid-phase cesium fractions. For CST, the measured K_C for Cs⁺/Na⁺ is about 10⁴, so cesium is strongly removed even from waste that is about 5 M in sodium, where X_Cs is around 10⁻⁴.

Reference: Z. Zheng, C. V. Philip, R. G. Anthony, J. L. Krumhansl, D. E. Trudell, and J. E. Miller, "Ion exchange of group I metals by hydrous crystalline silicotitanates," Ind. Eng. Chem. Res. 35 (1996) 4246–4256.

Symbols:

• \(C_{Cs}, C_{Na}\) = liquid-phase concentrations (equiv/L)
• \(C_0\) = total liquid-phase concentration, \(C_{Cs}+C_{Na}\) (equiv/L)
• \(Q_{Cs}, Q_{Na}\) = solid-phase concentrations (equiv/kg)
• \(Q_0\) = total exchange capacity of the solid (equiv/kg)
• \(X_{Cs}\) = equivalent fraction of Cs⁺ in the liquid
• \(\bar{X}_{Cs}\) = equivalent fraction of Cs⁺ in the solid
• \(\bar{X}_{01}, \bar{X}_{02}\) = fraction of exchange capacity in site 1 and site 2
• \(K_C, K_{C1}, K_{C2}\) = selectivity coefficient (overall, site 1, site 2)

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.