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Is the Eyring Plot Misleading? A Case for Arrhenius Analysis of Activation Parameters

This paper challenges the prevailing view that the Eyring plot is more fundamental than the Arrhenius plot by demonstrating that harmonic models yield exact linearity in Arrhenius representations, that the Eyring formulation's quantum appearance stems from normalization conventions rather than physics, and that an alternative Arrhenius-based framework offers more physically relevant and interpretable activation parameters.

Original authors: Titus S. van Erp

Published 2026-07-13

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Titus S. van Erp

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out how fast a chemical reaction happens as the temperature changes. Scientists have been using two different "maps" to navigate this terrain for over a century. One map is the Arrhenius plot, which has been the standard since high school chemistry. The other is the Eyring plot, which many physical chemistry textbooks claim is the "superior" and more "fundamental" version because it looks fancier and comes from complex theories about quantum mechanics.

In this paper, author Titus van Erp argues that the Eyring map is actually misleading. He suggests that while the Eyring equation itself isn't wrong, the way we use it to draw straight lines and extract data is based on a hidden trick that breaks the rules of physics.

The "Dimension Mismatch" Mystery

To understand the problem, imagine you are comparing two groups of people.

Group A (The Reactants): These are people standing in a room. They have a certain number of ways they can move around (degrees of freedom).
Group B (The Transition State): This is the "saddle point" where the reaction happens. In the Eyring view, this group is defined as a flat wall or a surface that the people must cross.

Here is the catch: Group A is a 3D room full of people, but Group B is just a 2D wall. You can't directly compare the "probability" of being in a room to the "probability" of being on a wall because they have different dimensions. It's like trying to compare the weight of a cloud to the weight of a shadow.

To make the math work, the Eyring equation sneaks in a constant (the Planck constant, $h$ ) to fix the units. The paper argues that this isn't a deep quantum secret; it's just a bookkeeping trick to make the numbers match. Because of this trick, the "Activation Entropy" ( $\Delta S^\ddagger$ ) and "Activation Enthalpy" ( $\Delta H^\ddagger$ ) that scientists pull out of an Eyring plot are actually contaminated by this dimensional mismatch.

The "Straight Line" Illusion

The biggest misconception the paper tackles is the idea that an Eyring plot (graphing $\ln(k/T)$ vs. $1/T$ ) should always be a straight line.

The Old View: Textbooks say, "If you plot this, you get a straight line. The slope gives you the energy, and the intercept gives you the entropy. It's perfect."
The Paper's Reality Check: The author ran simulations using a simple, solvable model (a particle moving on a bumpy surface). In these simulations, the Eyring plot was not a straight line. It was slightly curved.

Why? Because the "Activation Enthalpy" ( $\Delta H^\ddagger$ ) in the Eyring method actually changes with temperature. It has a term that is linear with $T$ (specifically, it includes a $-k_B T$ term). The paper shows that the assumption that these values are constant is thermodynamically inconsistent.

The reason experimental data often looks like a straight line on an Eyring plot is simply that the curve is so gentle that our measuring tools aren't precise enough to see the bend. However, the Arrhenius plot (graphing $\ln(k)$ vs. $1/T$ ) remains linear in these same simulations because it doesn't suffer from the same dimensional mismatch.

The "Fake" Compensation

There is a famous phenomenon in chemistry called "enthalpy-entropy compensation," where if the energy barrier goes up, the entropy seems to go up too, canceling each other out. Many scientists think this is a real physical effect.

The paper argues that in the Eyring framework, this is largely an artifact. It's a mathematical illusion caused by trying to fit a slightly curved line with a straight ruler. When you force a straight line onto a curve, the slope and intercept will naturally correlate, making it look like the physics is compensating, when really it's just a fitting error.

A Better Way: The "Reference Frequency" Fix

So, if the Eyring plot is flawed, what should we do? The author proposes a clever fix that keeps the best parts of the Eyring idea but removes the dimensional mismatch.

Imagine we take that "flat wall" (the transition state) and give it a tiny, imaginary spring attached to it. We pretend the wall has a little bit of wiggle room, just like the reactants do. We call this a "reference frequency" ( $\nu_{ref}$ ).

By adding this wiggle, the transition state and the reactant state now have the same number of dimensions.
Now, when we calculate the free energy, the units match perfectly without needing a sneaky constant to fix them.
In this new framework, the activation enthalpy ( $\Delta H^\ddagger_{ref}$ ) becomes constant (just like the Arrhenius energy $E_a$ ), and the activation entropy becomes a clean, meaningful number.

The author suggests that instead of blindly trusting the Eyring plot, we should stick to the Arrhenius plot. If we want to talk about entropy and enthalpy, we should calculate them from the Arrhenius data using the new, corrected formulas. This way, we get values that are physically real and don't change just because of how we drew the graph.

The Bottom Line

The paper doesn't say the Eyring equation is "wrong" in a mathematical sense; it says the interpretation of the Eyring plot is flawed.

The Claim: The Eyring plot is not inherently more fundamental than the Arrhenius plot. In fact, for simple models, the Arrhenius plot is the one that stays linear.
The Evidence: The author proves this using classical statistical mechanics and exact solutions to a harmonic model (a simplified simulation of a reaction).
The Takeaway: The "quantum" feel of the Eyring equation is just a result of how we count units, not a deep physical truth. The "straight line" we see in Eyring plots is often an illusion of limited experimental precision. To get the true physical picture, we should rely on the Arrhenius representation and use a corrected definition of activation parameters that respects the dimensions of the system.

In short: Don't be fooled by the fancy quantum-looking constants. The old-school Arrhenius plot might actually be the more honest map of the reaction landscape.

Problem Statement
The paper addresses a prevalent misconception in physical chemistry literature and education: the belief that the Eyring representation (plotting $\ln(k/T)$ versus $1/T$ ) is more fundamental and physically rigorous than the Arrhenius representation (plotting $\ln(k)$ versus $1/T$ ). This perception is often motivated by the Eyring equation's derivation from statistical mechanics and its interpretation of the intercept as an activation entropy ( $\Delta S^\ddagger$ ), whereas the Arrhenius prefactor is frequently dismissed as purely phenomenological. The author argues that this view relies on flawed assumptions regarding the temperature independence of activation enthalpy ( $\Delta H^\ddagger$ ) and entropy, and obscures a fundamental dimensional mismatch between the reactant state and the transition state in the standard Eyring formulation.

Methodology
The author employs a theoretical analysis comparing three formulations of transition state theory (TST): the Arrhenius equation, the Eyring equation, and Chandler's reactive flux expression. The analysis proceeds through the following steps:

Formal Equivalence: The paper demonstrates that Chandler's reactive flux expression and the Eyring equation are mathematically equivalent when the microscopic definition of the activation free energy ( $\Delta G^\ddagger$ ) is fully resolved, specifically regarding the normalization of partition functions.
Harmonic Approximation Model: To test the linearity of Arrhenius versus Eyring plots, the author analyzes a two-dimensional potential energy surface using classical harmonic transition state theory (HTST). This allows for the analytical computation of rate constants and the decomposition of activation parameters.
Dimensionality Analysis: The paper critically examines the definition of $\Delta G^\ddagger$ in the Eyring framework, highlighting that it compares a finite probability (reactant state) with a probability density (transition state hypersurface). This comparison requires a dimensional correction factor (conventionally the Planck constant $h$ ), which introduces an artificial temperature dependence.
Reformulation: The author proposes an alternative convention for defining the transition state. By introducing a hypothetical harmonic curvature along the reaction coordinate with a reference frequency ( $\nu_{ref}$ ), the dimensionality of the transition state is matched to that of the reactant state.

Key Contributions and Results

Equivalence of Formulations: The study confirms that the Arrhenius, Eyring, and Chandler expressions describe the same underlying physics. The apparent differences arise from how the activation free energy is defined and how temperature dependence is distributed between the prefactor and the exponential term.
Failure of Eyring Linearity: Using the harmonic model, the paper shows that while the Arrhenius plot ( $\ln k$ vs. $1/T$ ) yields exact linearity with a constant activation energy ( $E_a$ ), the standard Eyring plot ( $\ln(k/T)$ vs. $1/T$ ) is inherently non-linear. The standard decomposition $\Delta G^\ddagger = \Delta H^\ddagger - T\Delta S^\ddagger$ leads to an activation enthalpy $\Delta H^\ddagger$ that contains a term linear in temperature ( $\Delta H^\ddagger = V^\ddagger - k_B T$ ), violating the assumption of temperature independence required for a linear Eyring plot.
Dimensionality Mismatch: The non-linearity and the resulting "enthalpy-entropy compensation" artifacts are traced to the dimensional mismatch in the standard Eyring definition. The transition state is defined as a hypersurface (removing one degree of freedom), while the reactant is a volume. The conventional use of $h$ to normalize this ratio introduces a temperature dependence that is not physically intrinsic to the barrier height but is an artifact of the convention.
Proposed Alternative Convention: The paper introduces a modified activation free energy, $\Delta G^\ddagger_{ref}$ $Δ G_{r e f}^{‡}$ , based on a reference frequency $\nu_{ref}$ $ν_{r e f}$ . This formulation restores dimensional equality between reactant and transition states. In this framework:
- The activation enthalpy $\Delta H^\ddagger_{ref}$ becomes strictly equal to the Arrhenius activation energy $E_a$ (in the harmonic limit) and is independent of the reference frequency.
- The activation entropy $\Delta S^\ddagger_{ref}$ is directly related to the Arrhenius prefactor $A$ via $\Delta S^\ddagger_{ref} = k_B \ln(A/\nu_{ref})$ .
- This approach yields an Arrhenius-type linear plot that is theoretically justified, avoiding the spurious temperature dependencies found in the standard Eyring analysis.

Significance and Claims
The paper argues that the preference for Eyring plots over Arrhenius plots is not theoretically grounded. The author contends that:

Arrhenius is More Fundamental for Analysis: Contrary to the view that Arrhenius is merely phenomenological, the paper shows that Arrhenius linearity is the exact result for harmonic systems, whereas Eyring linearity is an approximation that fails even in simple solvable models.
Misinterpretation of Quantum Mechanics: The presence of the Planck constant in the Eyring prefactor is often misinterpreted as a signature of quantum mechanics. The author clarifies that $h$ arises from the normalization convention of the partition function and cancels out in the final rate expression; thus, the Eyring equation can be derived purely from classical statistical mechanics.
Educational and Practical Implications: The author suggests that students and researchers should be taught that Arrhenius and Eyring plots are mathematically equivalent but that extracting activation parameters from an Arrhenius fit is physically more robust. The standard Eyring parameters ( $\Delta H^\ddagger, \Delta S^\ddagger$ ) often suffer from artificial temperature dependence and correlation artifacts (enthalpy-entropy compensation) due to the dimensional mismatch.
Recommendation: The paper concludes that activation parameters should be extracted from Arrhenius representations. If an entropy of activation is desired, it should be calculated from the Arrhenius prefactor using the proposed reference-frequency convention ( $\Delta S^\ddagger_{ref}$ ), which provides a transparent, dimensionally consistent definition that avoids the pitfalls of the traditional Eyring plot.

The "Dimension Mismatch" Mystery

The "Straight Line" Illusion

The "Fake" Compensation

A Better Way: The "Reference Frequency" Fix

The Bottom Line

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