Resolving a Mathematical Inconsistency in the Ho and McKay Adsorption Equation

Abstract Ho and McKay’s pseudo-second order equation is widely used for fitting adsorption data, often outperforming Lagergren’s model. However, the Ho and McKay equation contains a mathematical inconsistency that predicts infinite adsorption for q > qe. This issue should be resolved. This problem is illustrated graphically using the mathematical concept of ‘flow along the line’. This analysis reveals that the problem with the Ho and McKay model is its half-stable fixed point. Other variants of the Ho and McKay equation similarly predict infinite adsorption for q > qe. In contrast, the Lagergren equation does not show this anomaly – it has a stable fixed point. An analysis of more than 250 papers in the scientific literature that cite the Ho and McKay equation confirms that real adsorption experiments do not follow the predictions of the Ho and McKay equation for q > qe; they do not show ‘runaway’ behavior. That is, the issue raised in this work is purely mathematical/theoretical; it is about following recommended practices for making precise models. A simple solution to the inconsistency in the Ho and McKay equation is to only define it for q ≤ qe. Other possible fixes for this equation are also discussed.