[Paper Review] A speculative study of non-linear Arrhenius plot by using fractional calculus

Nelson H. T. Lemes, Valentino A. Simpao+1 more|arXiv (Cornell University)|Mar 22, 2016

Fractional Differential Equations Solutions3 citations

TL;DR

This paper proposes a fractional calculus-based generalization of the Van't Hoff equation, introducing a non-integer order derivative to model non-linear Arrhenius behavior in chemical kinetics. The resulting Fractional Van't Hoff Equation (FVHE) fits experimental rate data with relative percentage errors below 3%, outperforming traditional Arrhenius and d-Arrhenius models, particularly for systems showing curvature in Arrhenius plots.

ABSTRACT

In this study, the Van't Hoff differential equation is taken under consideration by making use of fractional derivative tools. In this context, the nonlinear Arrhenius behaviour can be obtained and some experimental values of reaction rate as function of temperature were fitted, with the proposed model. The new model showed better performance to fit rate constant data for different kinetics process, when compared with Arrhenius law. In these case, the Van't Hoff differential equation with noniteger order found relative percentage error less that 3% within experimental error. The fractional order plays an important role in modeling temperature dependence of these kinetic processes. Thus it provides a new perspective in the handling of many problems (e.g., as solubility as function of temperature; temperature dependency of the viscosity and conductivity, etc).

Motivation & Objective

• To address deviations from linearity in Arrhenius plots observed in experimental data across various kinetic processes.
• To develop a generalized model of the Van't Hoff equation using fractional derivatives to capture non-exponential, non-linear temperature dependence of rate constants.
• To improve fitting accuracy over conventional Arrhenius and d-Arrhenius models, especially where curvature in ln(k) vs. 1/T plots indicates non-ideal behavior.
• To explore the physical significance of the fractional order parameter α in modeling anomalous diffusion and memory effects in reaction kinetics.
• To extend the applicability of the model beyond chemical kinetics to other temperature-dependent phenomena such as solubility, viscosity, and conductivity.

Proposed method

• Formalizing the Van't Hoff differential equation using Riemann-Liouville fractional derivatives of non-integer order α.
• Deriving an exact analytical solution of the Fractional Van’t Hoff Equation (FVHE) via operational methods and integral transforms.
• Implementing numerical methods, including the Frac PECE subroutine, to solve the FVHE for practical data fitting.
• Fitting experimental rate data from diverse systems (e.g., sugar inversion, bacterial growth, thermal decomposition) using the FVHE model.
• Comparing model performance against Arrhenius, d-Arrhenius, and quadratic models using relative percentage error as a metric.
• Adjusting the fractional order α to capture curvature in Arrhenius plots, treating α as a fitting parameter that reflects system-specific kinetic behavior.

Experimental results

Research questions

• RQ1 Can a fractional derivative formulation of the Van't Hoff equation explain non-linear Arrhenius behavior observed in experimental data?
• RQ2 How does the fractional order α in the generalized Van’t Hoff equation influence the curvature of Arrhenius plots?
• RQ3 Does the proposed FVHE model achieve better fitting accuracy than the classical Arrhenius and d-Arrhenius models for non-linear kinetic data?
• RQ4 Is there a physical or mechanistic interpretation for the fractional order α in terms of memory effects or anomalous diffusion in chemical systems?
• RQ5 Can the FVHE model be extended to other temperature-dependent processes such as solubility, viscosity, and conductivity?

Key findings

• The FVHE model achieved relative percentage errors below 3% when fitting experimental rate data, significantly outperforming the Arrhenius and d-Arrhenius models.
• For the inversion of cane sugar and other reactions, the FVHE model reduced relative errors to less than 0.3% in some cases, compared to over 10% for the Arrhenius model.
• The fractional order α was found to be a critical parameter in capturing curvature in Arrhenius plots, with optimal values ranging from 0.142 to 0.830 across different datasets.
• No observed correlation was found between the fractional order α and the d-parameter in the d-Arrhenius model, indicating distinct physical interpretations.
• The model successfully described non-linear behavior in diverse systems, including bacterial growth, thermal decomposition, and sugar inversion, suggesting broad applicability.
• The FVHE provides a physically grounded alternative to empirical models by embedding memory effects and non-local dynamics through fractional calculus.