Beyond Freundlich and Langmuir: the Ruthven–virial equilibrium isotherm for aqueous-solid adsorption systems

Abstract

It is the purpose of this paper to investigate the data fitting attributes of the exponential integral isotherm, which was first derived by Ruthven in 2004 to describe adsorption at gas–solid interfaces. The exponential integral isotherm, a variant of the virial isotherm, incorporates the two fitting parameters of the Langmuir isotherm and has an expandable mathematical structure. It has lain dormant for nearly 20 years. This fact may seem surprising considering the versatility of the isotherm, as will be shown in this work. A slightly modified form of the exponential integral isotherm, called the Ruthven–virial isotherm, is used in this work to interpret previously published aqueous-phase isotherm data. It is observed that the Ruthven–virial isotherm is capable of accurately tracking experimental adsorption isotherms with and without an apparent plateau. The Freundlich isotherm is good at describing data without a plateau, while the Langmuir isotherm excels at fitting data with a plateau. The Ruthven–virial isotherm combines the individual strengths of the Freundlich and Langmuir isotherms into a single model. Furthermore, it is shown that the data fitting performance of the two-parameter Ruthven–virial isotherm is comparable to that of the three-parameter Sips isotherm. The Ruthven–virial isotherm is a hidden gem and a practical alternative to the Freundlich and Langmuir equations in the correlation of hyperbolic adsorption isotherms. We hope that the findings presented here will inspire the research community to evaluate the Ruthven–virial isotherm in the modeling of contaminant adsorption at aqueous-solid interfaces.

Graphical abstract

Appendix: derivation of Ruthven’s exponential integral isotherm

We start with Eq. (3) in Table 1 of Ruthven (2004), reproduced here as Eq. (A1), where b is the Langmuir equilibrium constant and Ei(x) is the exponential integral with x = q/q_m, which is defined by Eq. (A2).

$$\ln \left( {bp} \right) = Ei\left( x \right)$$

(A1)

$$Ei\left( x \right) = \int_{ - \infty }^{x} {\frac{\exp \left( t \right)}{t}} {\text{d}}t$$

(A2)

The series expansion of the exponential integral is given by Eq. (A3), where γ is the Euler–Mascheroni constant (0.5772156649…).

$$\ln \left( {bp} \right) = \gamma + \ln \left( x \right) + x + \frac{{x^{2} }}{4} + \frac{{x^{3} }}{18} + \cdot \cdot \cdot$$

(A3)

So, we obtain Eq. (A4) from Eq. (A3). In Eq. (6), the b_L value is in fact b/exp(γ) ≈ b/1.781.

$$bp = \exp \left( \gamma \right)x \, \exp \left( {x + \frac{{x^{2} }}{4} + \frac{{x^{3} }}{18} + \cdot \cdot \cdot } \right)$$

(A4)