More on Arrhenius plots

Axel Lorke’s Quick Study in the May 2021 issue of Physics Today (page 66 ) describes Svante Arrhenius’s illustrious career and provides important insight into Arrhenius’s quantitative description of thermally induced processes. Lorke describes the broad power of the famous Arrhenius relationship $a=A\exp (-E_{\mathrm{a}}/k_{\mathrm{B}}T)$, where $k_{\mathrm{B}}$ is the Boltzmann constant and $T$ is absolute temperature, to capture complex physics with temperature-dependent measurements aimed at the extraction of a single parameter, the activation energy $E_{\mathrm{a}}$. Presenting an example from thermodynamics and another from kinetics, he shows how $E_{\mathrm{a}}$ connects closely to independently determined quantities such as a semiconductor’s bandgap energy and the UV-induced gelation energy of proteins.

The Quick Study focuses on the slope of logarithmic plots of rates and other temperature-dependent quantities versus inverse temperature $1/T$. In kinetics, the prefactor $A$ of the exponential also provides important physical information. It may be obtained by extrapolating an Arrhenius line like that of figure 3 in Lorke’s Quick Study to yield an intercept at the $1/k_{\mathrm{B}}T=0$ axis. If the quantity measured is the frequency of a process, as is often the case in solid-state physics, the prefactor $A$ can be called the attempt frequency, with $A^{-1}$ being the limiting time required to surmount the activation barrier as the temperature approaches infinity. In textbook examples, for small $E_{\mathrm{a}}$, this approach yields plausible values for such frequencies. Further, if the $E_{\mathrm{a}}$ of such a process is modified only slightly, the intercept does not change.

Starting with reports by Frederick Hurn Constable ¹ in 1925 and by Wilfried Meyer and Hans Neldel ² in 1937, researchers have done a great number of experiments on sets of closely related materials and systems in which the prefactor of Arrhenius plots of a related set varies systematically with $E_{\mathrm{a}}$. While care must be taken to avoid artifacts, it has been clear for some time that the phenomenon is real. ³ For a wide-ranging variety of sets of related physical, geological, biological, and chemical phenomena, the logarithms of those intercepts vary linearly with $E_{\mathrm{a}}$. That also means the Arrhenius fit lines cross at an isokinetic temperature at which the rate is independent of $E_{\mathrm{a}}$. Those observations have various names: the isokinetic rule, the compensation law (because the increase in the prefactor partially compensates for the increase in $E_{\mathrm{a}}$), and the Meyer–Neldel rule.

The meaning and explanation of the Meyer–Neldel rule were long considered to be a mystery, but work by a number of groups in the final decades of the past millennium provided a clear theoretical framework for both kinetic and equilibrium systems. The key to activation is not the energy or enthalpy; it is the free-energy change, which includes an entropy term. When the activation barrier is large, the entropy change increases with $E_{\mathrm{a}}$, and that increases $A$.

In 2006 one of us (Yelon) coauthored a review of the state of the art in experiment and theory, ⁴ which have continued to evolve since. Systematic studies yield information concerning the characteristic energy of the collective excitations—phonons or local vibrations—that are aggregated to surmount the activation barrier. ⁵ In some cases, notably studies of electronic or ionic conductivity, important information concerning mechanisms can be obtained. Like the Arrhenius relation that spawned it, the Meyer–Neldel rule is an elegant way to gain insight into the fundamental interactions governing temperature-dependent processes.