This simulation shows the equilibrium isotherm for exchange of Cs+ for Na+ on a crystalline silicotitanate (CST) ion exchanger, which is used to remove radioactive cesium from nuclear waste. The isotherm plots the equivalent fraction of Cs+ in the solid versus the equivalent fraction of Cs+ in the liquid, for an ideal solid and constant selectivity coefficient KC.
Use the KC slider to change the selectivity coefficient on a log scale from 10−4 to 104. For KC > 1 the solid prefers Cs+ and the isotherm is favorable (concave down); for KC = 1 it is the diagonal; for KC < 1 the solid prefers Na+ and the isotherm is unfavorable (concave up). The reported value for Cs+/Na+ on CST is about 104.
Use the liquid fraction slider to move the point along the isotherm and read the equilibrium solid fraction. Check "show family of curves" to see isotherms for each decade of KC at once. Move a mouse over an isotherm to identify its KC value.
Ion exchange of Cs+ from the liquid with Na+ in the solid (both singly charged) is a stoichiometric process:
\[ \mathrm{Cs^+} + \overline{\mathrm{Na^+}} \rightleftharpoons \overline{\mathrm{Cs^+}} + \mathrm{Na^+} \]where the overbar denotes the solid phase. The selectivity coefficient (concentration-based equilibrium constant) is
\[ K_C = \frac{Q_{\mathrm{Cs}}\,C_{\mathrm{Na}}}{Q_{\mathrm{Na}}\,C_{\mathrm{Cs}}} \]For ion exchange the total equivalent concentration in each phase is fixed, so
\[ Q_{\mathrm{Cs}} + Q_{\mathrm{Na}} = Q_0, \qquad C_{\mathrm{Cs}} + C_{\mathrm{Na}} = C_0 \]Defining equivalent fractions in the liquid and solid,
\[ X = \frac{C_{\mathrm{Cs}}}{C_0}, \qquad \bar{X} = \frac{Q_{\mathrm{Cs}}}{Q_0} \]and substituting gives the binary ion exchange isotherm:
\[ \bar{X} = \frac{K_C\,X}{1 + (K_C - 1)\,X} \]Because each Cs+ that enters the solid displaces one Na+, the isotherm always passes through (0, 0) and (1, 1), unlike an adsorption isotherm. The shape is concave down (favorable) for KC > 1, the diagonal for KC = 1, and concave up (unfavorable) for KC < 1. The isotherm for KC is the mirror image of that for 1/KC about the line \( \bar{X} = 1 - X \).
At low loading the isotherm reduces to Henry's law:
\[ \bar{X} = K_C\,X \]and for KC >> 1 it takes the form of the Langmuir isotherm:
\[ \bar{X} = \frac{K_C\,X}{1 + K_C\,X} \]Zheng et al. measured KC ≈ 1.0 × 104 for Cs+/Na+ on CST (TAM-5) in 5.1 M NaNO3 with 0.6 M NaOH, and 1.8 × 104 in 5.7 M NaNO3. This very high selectivity lets CST remove trace Cs+ from wastes that are nearly saturated with Na+. The solid behaves ideally below its apparent Cs+ capacity (0.57 mequiv/g); at higher loadings step changes appear in the measured isotherms, which this ideal model does not describe.
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, 4246–4256 (1996).
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.