The Ruthven isotherm

The Ruthven isotherm

The Ruthven adsorption isotherm is defined by the following equation:

ln(bc) = Ei(x)

where b is the equilibrium binding constant, c the concentration of the adsorbate in solution, x = q / q_s, q is the adsorbed quantity and q_s a scaling parameter. Ei is the exponential integral :

Ei(x) =

⌠
⌡

x

− ∞

e^t

The Ruthven equation is solution of the differential equation:

d lnq

d lnc

= exp(− q / q_s)

So, the slope of the isotherm in log-log coordinates decreases exponentially when q increases (for the Freundlich isotherm, the slope is constant)

Fitting the Ruthven isotherm

The computer program Ruthven.bas written in FreeBASIC fits the curve by nonlinear regression, using simulated annealing followed by Marquardt's method. The fitted parameters are q_s and b.

Reference

K. H. Chu, M. A. Hashim, A. Mudhoo, J. Debord, Beyond Freundlich and Langmuir: the Ruthven–virial equilibrium isotherm for aqueous‑solid adsorption systems. Chemical Papers, 2022. https://doi.org/10.1007/s11696-022-02576-4