Temperature Dependence Beyond Arrhenius

Plot $\ln k$ against $1/T$ for most simple gas-phase reactions and you get a beautifully straight line — the Arrhenius law at work. But push the temperature range wide enough, especially for combustion reactions, radical recombinations, or enzyme-catalyzed steps, and that line bends. This curvature is not experimental noise. It is real physical chemistry, telling you that the activation energy and pre-exponential factor are themselves functions of temperature. Understanding why requires looking past the simple two-parameter Arrhenius picture toward the collision and statistical mechanics that generate it. This article explains where non-Arrhenius curvature comes from. You will see how a modified power-law pre-factor arises naturally from collision theory. You will learn how to define and measure a temperature-dependent activation energy. Finally, you will see two worked numerical examples showing how to fit and interpret curved Arrhenius data, setting the stage for transition-state theory.

Key takeaways

The Arrhenius equation $k = A e^{-E_a/RT}$ assumes $A$ and $E_a$ are constants; real data over wide $T$ ranges often curve on an Arrhenius plot.
Collision theory and statistical mechanics predict a modified form $k = A' T^n e^{-E_a/RT}$ , with $n$ typically between 0 and 1 for simple reactions, negative for some radical recombinations.
Because of this, the "activation energy" becomes temperature-dependent: $E_a(T) = RT^2 \, d(\ln k)/dT$ , not a fixed number.
Non-Arrhenius curvature is common in combustion chemistry, low-temperature enzyme kinetics, and barrierless radical–radical reactions.
Fitting curved data to a single Arrhenius line over a narrow range still gives a useful local $E_a$ , but extrapolating far outside that range is unreliable.
Key idea: Curvature in an Arrhenius plot reflects a temperature-dependent activation energy and pre-exponential factor, both traceable to how molecular energy distributions and steric/entropic factors evolve with $T$ .

The core insight

Arrhenius parameters are not universal constants of a reaction— $A$ and $E_a$ are effective, temperature-averaged quantities extracted from a rate constant that fundamentally depends on the full energy-dependent reactivity of colliding molecules, so any theory (collision theory, transition-state theory) that resolves that energy dependence more finely predicts a rate law of the form $k(T) = A' T^n e^{-E_a(T)/RT}$ , whose curvature becomes visible once the temperature range is wide enough.

The physical picture

The simple Arrhenius equation packages an enormous amount of molecular detail into two numbers. It says that a reaction rate constant grows exponentially with temperature because only molecules with energy above some threshold $E_a$ can react, and the fraction of such molecules follows a Boltzmann-like factor $e^{-E_a/RT}$ . Everything else — steric requirements, the density of collisions, entropy changes in forming the transition state — gets absorbed into the prefactor $A$ . This works remarkably well over modest temperature ranges (say, a factor of 2 or 3 in absolute temperature), which is why the Arrhenius law dominates introductory treatments. But collision theory itself, derived from the kinetic theory of gases, predicts that the frequency of collisions between molecules scales with the mean relative speed, which goes as $T^{1/2}$ . So even a barebones hard-sphere collision model gives a rate constant that looks like

k = A' T^{1/2} e^{-E_a/RT}

not a pure exponential. Transition-state theory (which you will meet next) predicts an even richer temperature dependence, because the partition functions of the activated complex relative to the reactants change with $T$ in ways that are not simply exponential. The upshot: real rate constants often obey a modified Arrhenius (sometimes called Kooij or extended Arrhenius) form,

k(T) = A T^n e^{-E_a/RT}

where $n$ is an empirical or theoretically motivated exponent, frequently between $-1$ and $4$ depending on the reaction type. When you force such data onto a plot of $\ln k$ vs. $1/T$ , expecting a straight line, you see gentle curvature — concave down if $n>0$ dominates at high $T$ , or more complex shapes if $E_a$ itself varies with $T$ (as happens near phase transitions, in enzyme kinetics with conformational changes, or in radical reactions with pre-reactive complexes).

Figure

Arrhenius plot: straight line vs. curved (non-Arrhenius) data

A second useful picture is how the local slope of that curved plot — the effective activation energy — itself changes with temperature. Differentiating $\ln k$ with respect to $1/T$ at each point gives a temperature-dependent $E_a(T)$ , which can rise, fall, or even go negative (as seen in some barrierless radical recombination reactions, where rate constants decrease with increasing temperature).

Figure

Effective activation energy as a function of temperature

The core relations

k(T) = A\, T^{\,n}\, e^{-E_a/RT}

where: $A$ is a temperature-independent prefactor, $n$ is an empirical or theory-derived exponent (often $0.5$ – $1$ from collision theory, up to a few units in transition-state theory), $E_a$ is a reference activation energy near a chosen temperature.

E_a(T) \equiv RT^2 \frac{d\ln k}{dT}

where: this is the general, model-free definition of activation energy, reducing to a constant only in the strict Arrhenius limit.

E_a(T) = E_{a,0} + nRT

where: for the modified power-law form above, differentiating shows the effective activation energy grows (or shrinks) linearly with $T$ depending on the sign of $n$ .

\ln k = \ln A + n\ln T - \frac{E_a}{RT}

where: this is the linearized form used for a three-parameter nonlinear (or multi-linear-regression) fit to wide-range kinetic data.

Derivation AFrom the modified Arrhenius form to a temperature-dependent activation energy

Goal. Show that if a rate constant obeys the extended (modified) Arrhenius law $k(T) = A T^n e^{-E_a/RT}$ with $A$ , $n$ , $E_a$ all constants, the effective activation energy defined operationally from the slope of an Arrhenius plot is not constant, but depends linearly on $T$ .

Step 1. Start from the general definition of activation energy. The operational, model-independent definition of activation energy (the one an experimentalist actually measures from the local slope of an Arrhenius plot) is

E_a(T) = RT^2 \frac{d\ln k}{dT}

This definition reduces exactly to the familiar constant $E_a$ whenever $\ln k$ is a linear function of $1/T$ , but it remains well-defined even when the plot curves.

Step 2. Take the logarithm of the modified Arrhenius expression. Starting from $k = A T^n e^{-E_a/RT}$ , take the natural log:

\ln k = \ln A + n \ln T - \frac{E_a}{RT}

Here $A$ , $n$ , and $E_a$ are treated as true constants (this is the assumption of the "extended Arrhenius" model — a single power-law correction, not a fully general temperature dependence).

Step 3. Differentiate with respect to $T$ . Differentiate term by term:

\frac{d\ln k}{dT} = \frac{n}{T} + \frac{E_a}{RT^2}

The first term comes from the power-law prefactor $T^n$ ; the second comes from differentiating $-E_a/RT$ with $E_a$ constant.

Step 4. Substitute into the operational definition. Multiply through by $RT^2$ :

E_a^{\text{eff}}(T) = RT^2\left(\frac{n}{T} + \frac{E_a}{RT^2}\right)

E_a^{\text{eff}}(T) = nRT + E_a

Step 5. Interpret the result. This shows that even though the model activation energy $E_a$ appearing inside the exponential is a fixed constant, the effective, measured activation energy extracted from the local Arrhenius slope grows linearly with $T$ whenever $n \neq 0$ . For $n > 0$ (as collision theory predicts, $n = 1/2$ ), $E_a^{\text{eff}}$ increases modestly with temperature. For $n < 0$ (seen in some barrierless radical–radical recombination reactions), $E_a^{\text{eff}}$ decreases with $T$ and can even become negative, corresponding to a rate constant that falls as temperature rises.

\boxed{E_a^{\text{eff}}(T) = E_a + nRT}

This single relation is the bridge between the empirical modified-Arrhenius fit and the curvature seen on an Arrhenius plot: a nonzero $n$ is both the cause of the curvature and the reason the "activation energy" itself must be quoted at a specific reference temperature.

Worked examples

Example 1 — Foundational: fitting a three-parameter modified Arrhenius law. Suppose a gas-phase bimolecular reaction has been measured at four temperatures over a wide range, and a fit to $k(T) = A T^n e^{-E_a/RT}$ gives the parameters $A = 1.2\times10^{9}\ \text{dm}^3\,\text{mol}^{-1}\,\text{s}^{-1}\,\text{K}^{-n}$ , $n = 0.50$ (the value expected from simple hard-sphere collision theory), and $E_a = 45.0\ \text{kJ mol}^{-1}$ . Question: What is the effective activation energy at 300 K and at 1200 K, and by how much does it change? Using the boxed result,

E_a^{\text{eff}}(300\text{ K}) = 45{,}000\ \text{J mol}^{-1} + (0.50)(8.314\ \text{J mol}^{-1}\text{K}^{-1})(300\ \text{K})

E_a^{\text{eff}}(300\text{ K}) = 45{,}000 + 1247.1 = 46{,}247\ \text{J mol}^{-1} = 46.2\ \text{kJ mol}^{-1}

E_a^{\text{eff}}(1200\text{ K}) = 45{,}000 + (0.50)(8.314)(1200) = 45{,}000 + 4988.4 = 49{,}988\ \text{J mol}^{-1} = 50.0\ \text{kJ mol}^{-1}

So over a 900 K range, the effective activation energy rises by about $3.7\ \text{kJ mol}^{-1}$ — a modest but measurable curvature, roughly an 8% increase. This is exactly the kind of gentle bending you would see in the Arrhenius plot of a typical gas-phase reaction studied by shock-tube or flow methods across combustion-relevant temperatures. A rough check: since $RT$ at 300 K is only 2.49 kJ/mol, and $n = 0.5$ , the correction $nRT$ is on the order of a few kJ/mol, consistent with the small computed shift.

Example 2 — Applied: extracting $n$ and $E_a$ from curved rate data (harder, exam-level). A radical–radical recombination reaction is studied over a wide temperature range and shows the following rate constants (units $\text{dm}^3\,\text{mol}^{-1}\,\text{s}^{-1}$ ):

$T$ / K	$k$
300	$3.2\times10^{10}$
600	$2.0\times10^{10}$
1000	$1.35\times10^{10}$

Notice immediately that $k$ decreases as $T$ increases — impossible to fit with a standard Arrhenius law with positive $E_a$ , but a classic signature of a barrierless radical recombination where $n$ is negative and $E_a \approx 0$ . Assume the simplified pure power law $k = A T^n$ (i.e., $E_a = 0$ , appropriate for a genuinely barrierless process). Take logs:

\ln k = \ln A + n \ln T

Using the two extreme points to solve for $n$ :

n = \frac{\ln k_2 - \ln k_1}{\ln T_2 - \ln T_1} = \frac{\ln(1.35\times10^{10}) - \ln(3.2\times10^{10})}{\ln(1000) - \ln(300)}

Compute the numerator:

\ln(1.35\times10^{10}) = 23.325, \qquad \ln(3.2\times10^{10}) = 24.188

\text{numerator} = 23.325 - 24.188 = -0.863

Compute the denominator:

\ln(1000) = 6.908, \qquad \ln(300) = 5.704

\text{denominator} = 6.908 - 5.704 = 1.204

n = \frac{-0.863}{1.204} = -0.717

So $n \approx -0.72$ , close to the value often quoted for methyl radical recombination ( $\text{CH}_3 + \text{CH}_3 \rightarrow \text{C}_2\text{H}_6$ ), where $n$ is typically in the range $-0.6$ to $-1$ because increasing temperature reduces the effective capture cross-section of the loosely bound radical pair (an entropic/dynamical effect, not an energetic barrier). Check against the middle point: predict $A$ from the $T=300$ K data,

A = \frac{k_1}{T_1^{\,n}} = \frac{3.2\times10^{10}}{300^{-0.717}}

300^{-0.717} = e^{-0.717 \times \ln 300} = e^{-0.717 \times 5.704} = e^{-4.090} = 0.01671

A = \frac{3.2\times10^{10}}{0.01671} = 1.915\times10^{12}

Now predict $k$ at 600 K:

k_{\text{pred}} = A\, T^n = 1.915\times10^{12} \times 600^{-0.717}

600^{-0.717} = e^{-0.717\times \ln 600} = e^{-0.717 \times 6.397} = e^{-4.587} = 0.01019

k_{\text{pred}} = 1.915\times10^{12} \times 0.01019 = 1.95\times10^{10}

This is within 3% of the measured $2.0\times10^{10}\ \text{dm}^3\,\text{mol}^{-1}\,\text{s}^{-1}$ , confirming the negative power-law fit is self-consistent and that no exponential (Boltzmann-barrier) term is needed for this barrierless radical reaction.

Limiting cases & common traps

Situation	Behavior	Note
Narrow $T$ range (factor of ~2)	Arrhenius plot looks straight	$n$ -term contribution is small compared to scatter; simple Arrhenius fit is fine locally
Wide $T$ range (combustion, shock tube)	Visible curvature	Modified Arrhenius $k = AT^ne^{-E_a/RT}$ needed; $n$ from collision/TST theory
Barrierless radical–radical recombination	$k$ decreases with $T$	$E_a \approx 0$ , $n$ strongly negative (entropic control, not energetic)
Enzyme kinetics near denaturation	Sharp downturn at high $T$	Curvature reflects protein unfolding competing with catalysis, not a single elementary step
Diffusion-limited reactions in solution	Weak $T$ dependence	$E_a$ small and itself $T$ -dependent through solvent viscosity $\eta(T)$

The chemist behind it

Svante Arrhenius proposed the exponential temperature law in 1889, but he was explicit that it was an empirical fit inspired by van't Hoff's earlier thermodynamic analogy, not a claim that $A$ and $E_a$ were exactly constant. Later refinements came from collision theory (Trautz and Lewis, early twentieth century), which supplied the $T^{1/2}$ prefactor from kinetic gas theory, and from Henry Eyring's transition-state theory (1935), which showed how partition-function ratios naturally generate richer, non-constant temperature dependence — the theoretical ancestor of the modified Arrhenius law used throughout modern combustion and atmospheric chemistry databases (e.g., NASA/JPL and NIST kinetics compilations, which routinely tabulate $A$ , $n$ , and $E_a$ together for exactly this reason).

Origins · 1 minSvante Arrhenius and how this was discovered

Svante Arrhenius

1859–1927 · Swedish physical chemist who proposed the exponential rate–temperature law in 1889, building on van't Hoff's thermodynamic reasoning; later refinements by Trautz, Lewis, and Eyring revealed the temperature-dependent prefactor and activation energy this article explores.

Ask the tutor: Want to see how transition-state theory derives the $T^n$ prefactor directly from partition functions, and how that connects to the sign of $n$ found here?

Does non-Arrhenius behavior mean the Arrhenius equation is wrong?

No. The simple Arrhenius equation is an excellent local approximation over modest temperature windows, and it remains the standard starting point for reporting kinetics. Non-Arrhenius curvature simply reflects that $A$ and $E_a$ are not truly universal constants but effective, temperature-averaged parameters. Over a narrow enough range they behave as if constant; the curvature only becomes visible, and important, when data span a wide temperature range, as in combustion or atmospheric chemistry.

How is the modified Arrhenius exponent n determined experimentally?

Typically by fitting rate-constant data over a wide temperature range to the three-parameter form $k = AT^n e^{-E_a/RT}$ using nonlinear regression, or by using $\ln(k/T^n)$ vs. $1/T$ trial plots for several assumed $n$ values and choosing the one that linearizes the data best. Shock-tube and flash-photolysis experiments, which reach very high temperatures, are especially useful because they extend the range far beyond what simple Arrhenius fits can tolerate.

Can Ea become negative, and what does that mean?

Yes. A negative effective activation energy means the rate constant decreases as temperature increases, which happens in some barrierless reactions, especially radical–radical recombinations and certain ion-molecule reactions. Physically, raising the temperature reduces the effective capture cross-section or the lifetime of a loosely bound collision complex, so fewer collisions lead to reaction even though more collisions occur overall. It reflects entropic and dynamical effects, not a violation of any thermodynamic law.

How does this connect to transition-state theory?

Transition-state theory predicts $k(T) = (k_BT/h) K^{\ddagger}(T)$ , and because the equilibrium constant $K^{\ddagger}$ for forming the activated complex depends on temperature through partition functions, the resulting rate law naturally contains a power-law prefactor and a temperature-dependent effective activation energy, exactly the phenomenology described here. In that sense, the modified Arrhenius form is the empirical shadow cast by the more detailed statistical mechanics of transition-state theory, which is the natural next topic to study.