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Particle distribution in a microporous material: theoretical concept.

Particle distribution and exchange equilibria in a microporous host material, built up of equivalent particle sites, which are grouped in larger subsets, are described. Simplified descriptions evolve from the exact formulae in the thermodynamic limit. We find that, for example, a single zeolite A nanocrystal consisting of about 1000 pseudo-unit-cells fixes a lower limit for the use of the approximate formula describing particle distribution. A rational selectivity coefficient, which is approximately constant over the whole exchange range, only results if a single zeolite crystal consists of one million pseudo-unit-cells or more, or if a sufficiently large number of smaller crystals is considered. On the basis of the statistical particle distribution model, a closed, simple formula for the ion-exchange isotherm is then derived, which is valid for systems involving a variable number of coupled-exchange reactions. Its similarity to the Langmuir isotherm is discussed. The theory on ion-exchange equilibria is used to derive formulae for the change of free-energy, enthalpy, and entropy occurring in coupled ion-exchange reactions. The findings, though applicable to virtually any particle exchanging system with the structural properties described above, are applied to zeolite A, since this material can be treated as a nearly ideal model. The results derived can straightforwardly be used to evaluate experimental data quantitatively, since the common inequivalence of the host sites can be taken into account.