A Revised Pseudo-Second-Order Kinetic Model for Adsorption, Sensitive to Changes in Adsorbate and Adsorbent Concentrations

The development of new adsorbent materials for the removal of toxic contaminants from drinking water is crucial toward achieving the United Nations Sustainable Development Goal 6 (clean water and sanitation). The characterization of these materials includes fitting models of adsorption kinetics to experimental data, most commonly the pseudo-second-order (PSO) model. The PSO model, however, is not sensitive to parameters such as adsorbate and adsorbent concentrations (<i>C</i>₀ and <i>C</i>_s) and consequently is not able to predict changes in performance as a function of operating conditions. Furthermore, the experimental conditionality of the PSO rate constant, <i>k</i>₂, can lead to erroneous conclusions when comparing literature results. In this study, we analyze 103 kinetic experiments from 47 literature sources to develop a relatively simple modification of the PSO rate equation, yielding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>t</mml:mi></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>k</mml:mi><mml:mo>'</mml:mo><mml:msub><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>. Unlike the original PSO model, this revised rate equation (rPSO) provides the first-order and zero-order dependencies upon <i>C</i>₀ and <i>C</i>_s that we observe empirically. Our new model reduces the residual sum of squares by 66% when using a single rate constant to model multiple adsorption experiments with varying initial conditions. Furthermore, we demonstrate how the rPSO rate constant <i>k</i>' is more appropriate for comparing literature studies, highlighting faster kinetics in the adsorption of arsenic onto alumina versus iron oxides. This revised rate equation should find applications in engineering studies, especially since the rPSO rate constant <i>k</i>' does not show a counter-intuitive inverse relationship with increasing reaction rates when <i>C</i>₀ is increased, unlike the PSO rate constant <i>k</i>₂.